Combinatorics of Lyndon words - Murdoch UniversityMany easily formulated problems are difficult to...
Transcript of Combinatorics of Lyndon words - Murdoch UniversityMany easily formulated problems are difficult to...
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Combinatorics of Lyndon words
Amy Glen
School of Chemical & Mathematical Sciences
Murdoch University, Perth, Australia
[email protected]://amyglen.wordpress.com
Mini-Conference“Combinatorics, representations, and structure of Lie type”
The University of Melbourne
February 16–17, 2012Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 1
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Background
Words
By a word, I mean a finite or infinite sequence of symbols (letters) taken from anon-empty finite set A (alphabet).
Examples:
001
(001)∞ = 001001001001001001001001001001 · · ·
1100111100011011101111001101110010111111101 · · ·
100102110122220102110021111102212222201112012 · · ·
1121212121212 · · ·
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Background
Words
By a word, I mean a finite or infinite sequence of symbols (letters) taken from anon-empty finite set A (alphabet).
Examples:
001
(001)∞ = 0.↑01001001001001001001001001001 . . .
1100111100011011101111001101110010111111101 · · ·
100102110122220102110021111102212222201112012 · · ·
1121212121212 · · ·
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 2
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Background
Words
By a word, I mean a finite or infinite sequence of symbols (letters) taken from anon-empty finite set A (alphabet).
Examples:
001
(001)∞ = 0.01001001001001001001001001001 . . . = (2/7)2
1100111100011011101111001101110010111111101 · · ·
100102110122220102110021111102212222201112012 · · ·
1121212121212 · · ·
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 2
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Background
Words
By a word, I mean a finite or infinite sequence of symbols (letters) taken from anon-empty finite set A (alphabet).
Examples:
001
(001)∞ = 0.01001001001001001001001001001 . . . = (2/7)2
1.↑100111100011011101111001101110010 . . .
100102110122220102110021111102212222201112012 · · ·
1121212121212 · · ·
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 2
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Background
Words
By a word, I mean a finite or infinite sequence of symbols (letters) taken from anon-empty finite set A (alphabet).
Examples:
001
(001)∞ = 0.01001001001001001001001001001 . . . = (2/7)2
1.100111100011011101111001101110010 . . . = ((1 +√
5)/2)2
100102110122220102110021111102212222201112012 · · ·
1121212121212 · · ·
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 2
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Background
Words
By a word, I mean a finite or infinite sequence of symbols (letters) taken from anon-empty finite set A (alphabet).
Examples:
001
(001)∞ = 0.01001001001001001001001001001 . . . = (2/7)2
1.100111100011011101111001101110010 . . . = ((1 +√
5)/2)2
10.↑0102110122220102110021111102212222201112012 . . .
1121212121212 · · ·
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Background
Words
By a word, I mean a finite or infinite sequence of symbols (letters) taken from anon-empty finite set A (alphabet).
Examples:
001
(001)∞ = 0.01001001001001001001001001001 . . . = (2/7)2
1.100111100011011101111001101110010 . . . = ((1 +√
5)/2)2
10.0102110122220102110021111102212222201112012 . . . = (π)3
1121212121212 · · ·
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Background
Words
By a word, I mean a finite or infinite sequence of symbols (letters) taken from anon-empty finite set A (alphabet).
Examples:
001
(001)∞ = 0.01001001001001001001001001001 . . . = (2/7)2
1.100111100011011101111001101110010 . . . = ((1 +√
5)/2)2
10.0102110122220102110021111102212222201112012 . . . = (π)3
[1; 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, . . .] =√
3
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Background
Words
By a word, I mean a finite or infinite sequence of symbols (letters) taken from anon-empty finite set A (alphabet).
Examples:
001
(001)∞ = 0.01001001001001001001001001001 . . . = (2/7)2
1.100111100011011101111001101110010 . . . = ((1 +√
5)/2)2
10.0102110122220102110021111102212222201112012 . . . = (π)3
[1; 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, . . .] =√
3
The study of combinatorial properties of words, known as combinatorics on
words (MSC: 68R15), has connections to many modern, as well as classical, fieldsof mathematics with applications in areas ranging from theoretical computerscience (from the algorithmic point of view) to molecular biology (DNAsequences).
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Background
Combinatorics on Words
In particular, connections with algebra are very deep.
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Background
Combinatorics on Words
In particular, connections with algebra are very deep.
Indeed, a natural environment of a finite word is a free monoid.
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 3
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Background
Combinatorics on Words
In particular, connections with algebra are very deep.
Indeed, a natural environment of a finite word is a free monoid.
Formally, under the operation of concatenation, the set A∗ of allfinite words over A is a free monoid with identity element ε (theempty word) and set of generators A.
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Background
Combinatorics on Words
In particular, connections with algebra are very deep.
Indeed, a natural environment of a finite word is a free monoid.
Formally, under the operation of concatenation, the set A∗ of allfinite words over A is a free monoid with identity element ε (theempty word) and set of generators A.
The set of non-empty finite words over A is the free semigroupA+ := A∗ \ {ε}.
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Background
Combinatorics on Words
In particular, connections with algebra are very deep.
Indeed, a natural environment of a finite word is a free monoid.
Formally, under the operation of concatenation, the set A∗ of allfinite words over A is a free monoid with identity element ε (theempty word) and set of generators A.
The set of non-empty finite words over A is the free semigroupA+ := A∗ \ {ε}.
It follows immediately that the mathematical research of wordsexploits two features: discreteness and non-commutativity.
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Background
Combinatorics on Words
In particular, connections with algebra are very deep.
Indeed, a natural environment of a finite word is a free monoid.
Formally, under the operation of concatenation, the set A∗ of allfinite words over A is a free monoid with identity element ε (theempty word) and set of generators A.
The set of non-empty finite words over A is the free semigroupA+ := A∗ \ {ε}.
It follows immediately that the mathematical research of wordsexploits two features: discreteness and non-commutativity.
The latter aspect is what makes this field a very challenging one.
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 3
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Background
Combinatorics on Words
In particular, connections with algebra are very deep.
Indeed, a natural environment of a finite word is a free monoid.
Formally, under the operation of concatenation, the set A∗ of allfinite words over A is a free monoid with identity element ε (theempty word) and set of generators A.
The set of non-empty finite words over A is the free semigroupA+ := A∗ \ {ε}.
It follows immediately that the mathematical research of wordsexploits two features: discreteness and non-commutativity.
The latter aspect is what makes this field a very challenging one.
Many easily formulated problems are difficult to solve . . .
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Background
Combinatorics on Words
In particular, connections with algebra are very deep.
Indeed, a natural environment of a finite word is a free monoid.
Formally, under the operation of concatenation, the set A∗ of allfinite words over A is a free monoid with identity element ε (theempty word) and set of generators A.
The set of non-empty finite words over A is the free semigroupA+ := A∗ \ {ε}.
It follows immediately that the mathematical research of wordsexploits two features: discreteness and non-commutativity.
The latter aspect is what makes this field a very challenging one.
Many easily formulated problems are difficult to solve . . . mainlybecause of the limited availability of mathematical tools to deal withnon-commutative structures compared to commutative ones.
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Background
Combinatorics on Words . . .
There have been important contributions on “words” dating as farback as the beginning of the last century.
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Background
Combinatorics on Words . . .
There have been important contributions on “words” dating as farback as the beginning of the last century.
Many of the earliest contributions were typically needed as tools toachieve some other goals in mathematics
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 4
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Background
Combinatorics on Words . . .
There have been important contributions on “words” dating as farback as the beginning of the last century.
Many of the earliest contributions were typically needed as tools toachieve some other goals in mathematics, with the exception ofcombinatorial group theory which studies combinatorial problems onwords as representing group elements.
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 4
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Background
Combinatorics on Words . . .
There have been important contributions on “words” dating as farback as the beginning of the last century.
Many of the earliest contributions were typically needed as tools toachieve some other goals in mathematics, with the exception ofcombinatorial group theory which studies combinatorial problems onwords as representing group elements.
Early 1900’s: First investigations by Axel Thue (repetitions in words)
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 4
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Background
Combinatorics on Words . . .
There have been important contributions on “words” dating as farback as the beginning of the last century.
Many of the earliest contributions were typically needed as tools toachieve some other goals in mathematics, with the exception ofcombinatorial group theory which studies combinatorial problems onwords as representing group elements.
Early 1900’s: First investigations by Axel Thue (repetitions in words)
1938: Marston Morse & Gustav Hedlund
Initiated the formal development of symbolic dynamics.
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 4
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Background
Combinatorics on Words . . .
There have been important contributions on “words” dating as farback as the beginning of the last century.
Many of the earliest contributions were typically needed as tools toachieve some other goals in mathematics, with the exception ofcombinatorial group theory which studies combinatorial problems onwords as representing group elements.
Early 1900’s: First investigations by Axel Thue (repetitions in words)
1938: Marston Morse & Gustav Hedlund
Initiated the formal development of symbolic dynamics.
This work marked the beginning of the formal study of words.
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 4
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Background
Combinatorics on Words . . .
There have been important contributions on “words” dating as farback as the beginning of the last century.
Many of the earliest contributions were typically needed as tools toachieve some other goals in mathematics, with the exception ofcombinatorial group theory which studies combinatorial problems onwords as representing group elements.
Early 1900’s: First investigations by Axel Thue (repetitions in words)
1938: Marston Morse & Gustav Hedlund
Initiated the formal development of symbolic dynamics.
This work marked the beginning of the formal study of words.
1960’s: Systematic study of words initiated by M.P. Schützenberger.
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Background
Combinatorics on Words . . .
There have been important contributions on “words” dating as farback as the beginning of the last century.
Many of the earliest contributions were typically needed as tools toachieve some other goals in mathematics, with the exception ofcombinatorial group theory which studies combinatorial problems onwords as representing group elements.
Early 1900’s: First investigations by Axel Thue (repetitions in words)
1938: Marston Morse & Gustav Hedlund
Initiated the formal development of symbolic dynamics.
This work marked the beginning of the formal study of words.
1960’s: Systematic study of words initiated by M.P. Schützenberger.
Recent developments in combinatorics on words have culminated inthe publication of three books by a collection of authors, under thepseudonym of M. Lothaire . . .
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Background
Combinatorics on Words . . .
Lothaire Books
1 “Combinatorics on words”, 1983 (reprinted 1997)
2 “Algebraic combinatorics on words”, 2002
3 “Applied combinatorics on words”, 2005
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Background
Combinatorics on Words . . .
Lothaire Books
1 “Combinatorics on words”, 1983 (reprinted 1997)
2 “Algebraic combinatorics on words”, 2002
3 “Applied combinatorics on words”, 2005
In the introduction to the first edition, Roger Lyndon stated
“This is the first book devoted to broad study of the
combinatorics of words, that is to say, of sequences of symbols
called letters. This subject is in fact very ancient and has
cropped up repeatedly in a wide variety of subjects.”
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Background
Combinatorics on Words . . .
Lothaire Books
1 “Combinatorics on words”, 1983 (reprinted 1997)
2 “Algebraic combinatorics on words”, 2002
3 “Applied combinatorics on words”, 2005
In the introduction to the first edition, Roger Lyndon stated
“This is the first book devoted to broad study of the
combinatorics of words, that is to say, of sequences of symbols
called letters. This subject is in fact very ancient and has
cropped up repeatedly in a wide variety of subjects.”
Combinatorics on words has now become a very active and challengingfield in its own right.
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Background
Combinatorics on Words . . .
Combinatorics on Words
Number Theory
algebra
Free Groups, SemigroupsMatrices
RepresentationsBurnside Problems
Discrete
Dynamical Systems
TopologyTheoretical Physics
Theoretical
Computer Science
AlgorithmicsAutomata Theory
ComputabilityCodes
Logic
Probability Theory
Biology
DNA sequencing, Patterns
Discrete Geometry
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Lyndon Words Definitions
Basic Definitions: Rotations
Given a word w = x1x2 · · · xn (with xi letters), the first rotation of w
is R(w) = x2 · · · xnx1.
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Lyndon Words Definitions
Basic Definitions: Rotations
Given a word w = x1x2 · · · xn (with xi letters), the first rotation of w
is R(w) = x2 · · · xnx1.
So R(w) shifts w by one position to the left, R2(w) shifts by twopositions to the left, and so on . . .
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Lyndon Words Definitions
Basic Definitions: Rotations
Given a word w = x1x2 · · · xn (with xi letters), the first rotation of w
is R(w) = x2 · · · xnx1.
So R(w) shifts w by one position to the left, R2(w) shifts by twopositions to the left, and so on . . .
By convention, w = R0(w), the 0-th (trivial) rotation of itself.
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Lyndon Words Definitions
Basic Definitions: Rotations
Given a word w = x1x2 · · · xn (with xi letters), the first rotation of w
is R(w) = x2 · · · xnx1.
So R(w) shifts w by one position to the left, R2(w) shifts by twopositions to the left, and so on . . .
By convention, w = R0(w), the 0-th (trivial) rotation of itself.
Example
Consider the word w = aabac. This word has five distinct rotations:
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Lyndon Words Definitions
Basic Definitions: Rotations
Given a word w = x1x2 · · · xn (with xi letters), the first rotation of w
is R(w) = x2 · · · xnx1.
So R(w) shifts w by one position to the left, R2(w) shifts by twopositions to the left, and so on . . .
By convention, w = R0(w), the 0-th (trivial) rotation of itself.
Example
Consider the word w = aabac. This word has five distinct rotations:
R0(w) = aabac
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Lyndon Words Definitions
Basic Definitions: Rotations
Given a word w = x1x2 · · · xn (with xi letters), the first rotation of w
is R(w) = x2 · · · xnx1.
So R(w) shifts w by one position to the left, R2(w) shifts by twopositions to the left, and so on . . .
By convention, w = R0(w), the 0-th (trivial) rotation of itself.
Example
Consider the word w = aabac. This word has five distinct rotations:
R0(w) = aabac
R1(w) = abaca
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Lyndon Words Definitions
Basic Definitions: Rotations
Given a word w = x1x2 · · · xn (with xi letters), the first rotation of w
is R(w) = x2 · · · xnx1.
So R(w) shifts w by one position to the left, R2(w) shifts by twopositions to the left, and so on . . .
By convention, w = R0(w), the 0-th (trivial) rotation of itself.
Example
Consider the word w = aabac. This word has five distinct rotations:
R0(w) = aabac
R1(w) = abaca
R2(w) = bacaa
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Lyndon Words Definitions
Basic Definitions: Rotations
Given a word w = x1x2 · · · xn (with xi letters), the first rotation of w
is R(w) = x2 · · · xnx1.
So R(w) shifts w by one position to the left, R2(w) shifts by twopositions to the left, and so on . . .
By convention, w = R0(w), the 0-th (trivial) rotation of itself.
Example
Consider the word w = aabac. This word has five distinct rotations:
R0(w) = aabac
R1(w) = abaca
R2(w) = bacaa
R3(w) = acaab
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Lyndon Words Definitions
Basic Definitions: Rotations
Given a word w = x1x2 · · · xn (with xi letters), the first rotation of w
is R(w) = x2 · · · xnx1.
So R(w) shifts w by one position to the left, R2(w) shifts by twopositions to the left, and so on . . .
By convention, w = R0(w), the 0-th (trivial) rotation of itself.
Example
Consider the word w = aabac. This word has five distinct rotations:
R0(w) = aabac
R1(w) = abaca
R2(w) = bacaa
R3(w) = acaab
R4(w) = caaba
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Lyndon Words Definitions
Basic Definitions: Rotations . . .
A word w of length |w| has at most |w| distinct rotations.
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Lyndon Words Definitions
Basic Definitions: Rotations . . .
A word w of length |w| has at most |w| distinct rotations.
More precisely, any word w can be uniquely expressed in the form
w = zp = zz · · · z︸ ︷︷ ︸
p times
where p ≥ 1 and z is a primitive word (i.e., not a positive integerpower of a shorter word).
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Lyndon Words Definitions
Basic Definitions: Rotations . . .
A word w of length |w| has at most |w| distinct rotations.
More precisely, any word w can be uniquely expressed in the form
w = zp = zz · · · z︸ ︷︷ ︸
p times
where p ≥ 1 and z is a primitive word (i.e., not a positive integerpower of a shorter word).
Expressing a given word w in this way, it is easy to see that w hasexactly |w|/p (= |z|) distinct rotations
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Lyndon Words Definitions
Basic Definitions: Rotations . . .
A word w of length |w| has at most |w| distinct rotations.
More precisely, any word w can be uniquely expressed in the form
w = zp = zz · · · z︸ ︷︷ ︸
p times
where p ≥ 1 and z is a primitive word (i.e., not a positive integerpower of a shorter word).
Expressing a given word w in this way, it is easy to see that w hasexactly |w|/p (= |z|) distinct rotations (the length of its primitiveroot z).
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Lyndon Words Definitions
Basic Definitions: Rotations . . .
A word w of length |w| has at most |w| distinct rotations.
More precisely, any word w can be uniquely expressed in the form
w = zp = zz · · · z︸ ︷︷ ︸
p times
where p ≥ 1 and z is a primitive word (i.e., not a positive integerpower of a shorter word).
Expressing a given word w in this way, it is easy to see that w hasexactly |w|/p (= |z|) distinct rotations (the length of its primitiveroot z).
In particular, we note that any primitive word w has exactly |w|distinct rotations.
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Lyndon Words Definitions
Basic Definitions: Rotations . . .
Examples
1 From the previous example, aabac is a primitive word of length 5 having 5distinct rotations.
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Lyndon Words Definitions
Basic Definitions: Rotations . . .
Examples
1 From the previous example, aabac is a primitive word of length 5 having 5distinct rotations.
2 The word v = abcabc = (abc)2 has |v|/2 = 3 distinct rotations:
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Lyndon Words Definitions
Basic Definitions: Rotations . . .
Examples
1 From the previous example, aabac is a primitive word of length 5 having 5distinct rotations.
2 The word v = abcabc = (abc)2 has |v|/2 = 3 distinct rotations:
R0(v) = abcabc
R1(v) = bcabca
R2(v) = cabcab
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Lyndon Words Definitions
Basic Definitions: Rotations . . .
Examples
1 From the previous example, aabac is a primitive word of length 5 having 5distinct rotations.
2 The word v = abcabc = (abc)2 has |v|/2 = 3 distinct rotations:
R0(v) = abcabc
R1(v) = bcabca
R2(v) = cabcab
The rotations of a word w are also known as its circular shifts or conjugates.
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Lyndon Words Definitions
Basic Definitions: Rotations . . .
Examples
1 From the previous example, aabac is a primitive word of length 5 having 5distinct rotations.
2 The word v = abcabc = (abc)2 has |v|/2 = 3 distinct rotations:
R0(v) = abcabc
R1(v) = bcabca
R2(v) = cabcab
The rotations of a word w are also known as its circular shifts or conjugates.
Two words x and y over A are said to be conjugate if there exist finitewords u and v such that x = uv and y = vu.
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Lyndon Words Definitions
Basic Definitions: Rotations . . .
Examples
1 From the previous example, aabac is a primitive word of length 5 having 5distinct rotations.
2 The word v = abcabc = (abc)2 has |v|/2 = 3 distinct rotations:
R0(v) = abcabc
R1(v) = bcabca
R2(v) = cabcab
The rotations of a word w are also known as its circular shifts or conjugates.
Two words x and y over A are said to be conjugate if there exist finitewords u and v such that x = uv and y = vu.
Equivalently, x and y are conjugate if and only if there exists a word z suchthat xz = zy
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Lyndon Words Definitions
Basic Definitions: Rotations . . .
Examples
1 From the previous example, aabac is a primitive word of length 5 having 5distinct rotations.
2 The word v = abcabc = (abc)2 has |v|/2 = 3 distinct rotations:
R0(v) = abcabc
R1(v) = bcabca
R2(v) = cabcab
The rotations of a word w are also known as its circular shifts or conjugates.
Two words x and y over A are said to be conjugate if there exist finitewords u and v such that x = uv and y = vu.
Equivalently, x and y are conjugate if and only if there exists a word z suchthat xz = zy (that is, y = z−1xz).
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Lyndon Words Definitions
Basic Definitions: Rotations . . .
Examples
1 From the previous example, aabac is a primitive word of length 5 having 5distinct rotations.
2 The word v = abcabc = (abc)2 has |v|/2 = 3 distinct rotations:
R0(v) = abcabc
R1(v) = bcabca
R2(v) = cabcab
The rotations of a word w are also known as its circular shifts or conjugates.
Two words x and y over A are said to be conjugate if there exist finitewords u and v such that x = uv and y = vu.
Equivalently, x and y are conjugate if and only if there exists a word z suchthat xz = zy (that is, y = z−1xz).
This is an equivalence relation on A∗ since x is conjugate to y if and only ify can be obtained by a cyclic permutation (rotation) of the letters of x .
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Lyndon Words Definitions
Basic Definitions: Necklaces
The set of all conjugates (rotations) of a given word is called its conjugacy class.
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Lyndon Words Definitions
Basic Definitions: Necklaces
The set of all conjugates (rotations) of a given word is called its conjugacy class.
The conjugacy class of a word can be represented by a necklace (also known as acircular word).
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Lyndon Words Definitions
Basic Definitions: Necklaces
The set of all conjugates (rotations) of a given word is called its conjugacy class.
The conjugacy class of a word can be represented by a necklace (also known as acircular word).
•
•
•
•
• •
a a
aa
b
b•
•
•
•
• •
a
aa
b
b
b
aabaab aabbab
Periodic Primitive
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Lyndon Words Definitions
Basic Definitions: Necklaces
The set of all conjugates (rotations) of a given word is called its conjugacy class.
The conjugacy class of a word can be represented by a necklace (also known as acircular word).
•
•
•
•
• •
a a
aa
b
b•
•
•
•
• •
a
aa
b
b
b
aabaab aabbab
Periodic Primitive
Note:
A necklace of length n over a k-letter alphabet can be thought of as n circularlyconnected beads of up to k different colours.
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Lyndon Words Definitions
Basic Definitions: Necklaces
The set of all conjugates (rotations) of a given word is called its conjugacy class.
The conjugacy class of a word can be represented by a necklace (also known as acircular word).
•
•
•
•
• •
a a
aa
b
b•
•
•
•
• •
a
aa
b
b
b
aabaab aabbab
Periodic Primitive
Note:
A necklace of length n over a k-letter alphabet can be thought of as n circularlyconnected beads of up to k different colours.
A necklace can also be classified as an orbit of the action of the cyclic group onwords of length n.
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Lyndon Words Definitions
Basic Definitions: Necklaces
The set of all conjugates (rotations) of a given word is called its conjugacy class.
The conjugacy class of a word can be represented by a necklace (also known as acircular word).
•
•
•
•
• •
a a
aa
b
b•
•
•
•
• •
a
aa
b
b
b
aabaab aabbab
Periodic Primitive
Note:
A necklace of length n over a k-letter alphabet can be thought of as n circularlyconnected beads of up to k different colours.
A necklace can also be classified as an orbit of the action of the cyclic group onwords of length n.
Each aperiodic (primitive) necklace can be uniquely represented by a so-calledLyndon word . . .
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Lyndon Words Definitions
Basic Definitions: Lyndon Words
Suppose the alphabet A is totally ordered by the relation ≺.
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Lyndon Words Definitions
Basic Definitions: Lyndon Words
Suppose the alphabet A is totally ordered by the relation ≺.
Then we can totally order the semigroup A+ by the lexicographical order �(which is the usual alphabetical order in a dictionary) induced by the totalorder ≺ on A.
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Lyndon Words Definitions
Basic Definitions: Lyndon Words
Suppose the alphabet A is totally ordered by the relation ≺.
Then we can totally order the semigroup A+ by the lexicographical order �(which is the usual alphabetical order in a dictionary) induced by the totalorder ≺ on A.
Definition
A finite word w ∈ A+ is a Lyndon word if w is strictly smaller in lexicographicalorder than all of its non-trivial rotations for the given total order ≺ on A.
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Lyndon Words Definitions
Basic Definitions: Lyndon Words
Suppose the alphabet A is totally ordered by the relation ≺.
Then we can totally order the semigroup A+ by the lexicographical order �(which is the usual alphabetical order in a dictionary) induced by the totalorder ≺ on A.
Definition
A finite word w ∈ A+ is a Lyndon word if w is strictly smaller in lexicographicalorder than all of its non-trivial rotations for the given total order ≺ on A.
Lyndon words are named after mathematician Roger Lyndon, who introducedthem in 1954 under the name of standard lexicographic sequences.
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Lyndon Words Definitions
Basic Definitions: Lyndon Words
Suppose the alphabet A is totally ordered by the relation ≺.
Then we can totally order the semigroup A+ by the lexicographical order �(which is the usual alphabetical order in a dictionary) induced by the totalorder ≺ on A.
Definition
A finite word w ∈ A+ is a Lyndon word if w is strictly smaller in lexicographicalorder than all of its non-trivial rotations for the given total order ≺ on A.
Lyndon words are named after mathematician Roger Lyndon, who introducedthem in 1954 under the name of standard lexicographic sequences.
Being the singularly lexicographically smallest rotation implies that a Lyndon wordis different from all of its non-trivial rotations, and is therefore primitive.
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Lyndon Words Definitions
Basic Definitions: Lyndon Words
Suppose the alphabet A is totally ordered by the relation ≺.
Then we can totally order the semigroup A+ by the lexicographical order �(which is the usual alphabetical order in a dictionary) induced by the totalorder ≺ on A.
Definition
A finite word w ∈ A+ is a Lyndon word if w is strictly smaller in lexicographicalorder than all of its non-trivial rotations for the given total order ≺ on A.
Lyndon words are named after mathematician Roger Lyndon, who introducedthem in 1954 under the name of standard lexicographic sequences.
Being the singularly lexicographically smallest rotation implies that a Lyndon wordis different from all of its non-trivial rotations, and is therefore primitive.
So a Lyndon word w is a primitive word that is the lexicographically smallest wordin its conjugacy class
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Lyndon Words Definitions
Basic Definitions: Lyndon Words
Suppose the alphabet A is totally ordered by the relation ≺.
Then we can totally order the semigroup A+ by the lexicographical order �(which is the usual alphabetical order in a dictionary) induced by the totalorder ≺ on A.
Definition
A finite word w ∈ A+ is a Lyndon word if w is strictly smaller in lexicographicalorder than all of its non-trivial rotations for the given total order ≺ on A.
Lyndon words are named after mathematician Roger Lyndon, who introducedthem in 1954 under the name of standard lexicographic sequences.
Being the singularly lexicographically smallest rotation implies that a Lyndon wordis different from all of its non-trivial rotations, and is therefore primitive.
So a Lyndon word w is a primitive word that is the lexicographically smallest wordin its conjugacy class, i.e., w ∈ A or w ≺ vu for all words u, v such that w = uv.
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Lyndon Words Definitions
Basic Definitions: Lyndon Words
Suppose the alphabet A is totally ordered by the relation ≺.
Then we can totally order the semigroup A+ by the lexicographical order �(which is the usual alphabetical order in a dictionary) induced by the totalorder ≺ on A.
Definition
A finite word w ∈ A+ is a Lyndon word if w is strictly smaller in lexicographicalorder than all of its non-trivial rotations for the given total order ≺ on A.
Lyndon words are named after mathematician Roger Lyndon, who introducedthem in 1954 under the name of standard lexicographic sequences.
Being the singularly lexicographically smallest rotation implies that a Lyndon wordis different from all of its non-trivial rotations, and is therefore primitive.
So a Lyndon word w is a primitive word that is the lexicographically smallest wordin its conjugacy class, i.e., w ∈ A or w ≺ vu for all words u, v such that w = uv.
It follows that w ∈ A+ is a Lyndon word iff w ∈ A or w ≺ v for all proper suffixesv of w.
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Lyndon Words Definitions
Examples
Example 1: w = aabac with alphabet A = {a, b, c}
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Lyndon Words Definitions
Examples
Example 1: w = aabac with alphabet A = {a, b, c}
w is a Lyndon word for the orders a ≺ b ≺ c and a ≺ c ≺ b
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Lyndon Words Definitions
Examples
Example 1: w = aabac with alphabet A = {a, b, c}
w is a Lyndon word for the orders a ≺ b ≺ c and a ≺ c ≺ b
(i.e., a Lyndon word for the orders on A with min(A) = a).
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Lyndon Words Definitions
Examples
Example 1: w = aabac with alphabet A = {a, b, c}
w is a Lyndon word for the orders a ≺ b ≺ c and a ≺ c ≺ b
(i.e., a Lyndon word for the orders on A with min(A) = a).
R2(w) = bacaa is a Lyndon word for the orders on A with min(A) = b.
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Lyndon Words Definitions
Examples
Example 1: w = aabac with alphabet A = {a, b, c}
w is a Lyndon word for the orders a ≺ b ≺ c and a ≺ c ≺ b
(i.e., a Lyndon word for the orders on A with min(A) = a).
R2(w) = bacaa is a Lyndon word for the orders on A with min(A) = b.
R4(w) = caaba is a Lyndon word for the orders on A with min(A) = c.
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Lyndon Words Definitions
Examples
Example 1: w = aabac with alphabet A = {a, b, c}w is a Lyndon word for the orders a ≺ b ≺ c and a ≺ c ≺ b
(i.e., a Lyndon word for the orders on A with min(A) = a).
R2(w) = bacaa is a Lyndon word for the orders on A with min(A) = b.
R4(w) = caaba is a Lyndon word for the orders on A with min(A) = c.
Example 2
For the 2-letter alphabet {a, b} with a ≺ b, the Lyndon words up to length fiveare as follows (sorted lexicographically for each length):
a, b, ab, aab, abb, aaab, aabb, abbb, aaaab, aaabb, aabab, aabbb, abbbb, . . .
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Lyndon Words Definitions
Examples
Example 1: w = aabac with alphabet A = {a, b, c}w is a Lyndon word for the orders a ≺ b ≺ c and a ≺ c ≺ b
(i.e., a Lyndon word for the orders on A with min(A) = a).
R2(w) = bacaa is a Lyndon word for the orders on A with min(A) = b.
R4(w) = caaba is a Lyndon word for the orders on A with min(A) = c.
Example 2
For the 2-letter alphabet {a, b} with a ≺ b, the Lyndon words up to length fiveare as follows (sorted lexicographically for each length):
a, b︸︷︷︸
2
, ab︸︷︷︸
1
, aab, abb︸ ︷︷ ︸
2
, aaab, aabb, abbb︸ ︷︷ ︸
3
, aaaab, aaabb, aabab, aabbb, ababb, abbbb︸ ︷︷ ︸
6
, . . .
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Lyndon Words Definitions
Examples
Example 1: w = aabac with alphabet A = {a, b, c}w is a Lyndon word for the orders a ≺ b ≺ c and a ≺ c ≺ b
(i.e., a Lyndon word for the orders on A with min(A) = a).
R2(w) = bacaa is a Lyndon word for the orders on A with min(A) = b.
R4(w) = caaba is a Lyndon word for the orders on A with min(A) = c.
Example 2
For the 2-letter alphabet {a, b} with a ≺ b, the Lyndon words up to length fiveare as follows (sorted lexicographically for each length):
a, b︸︷︷︸
2
, ab︸︷︷︸
1
, aab, abb︸ ︷︷ ︸
2
, aaab, aabb, abbb︸ ︷︷ ︸
3
, aaaab, aaabb, aabab, aabbb, ababb, abbbb︸ ︷︷ ︸
6
, . . .
How do we generate them?
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Lyndon Words Definitions
Examples
Example 1: w = aabac with alphabet A = {a, b, c}w is a Lyndon word for the orders a ≺ b ≺ c and a ≺ c ≺ b
(i.e., a Lyndon word for the orders on A with min(A) = a).
R2(w) = bacaa is a Lyndon word for the orders on A with min(A) = b.
R4(w) = caaba is a Lyndon word for the orders on A with min(A) = c.
Example 2
For the 2-letter alphabet {a, b} with a ≺ b, the Lyndon words up to length fiveare as follows (sorted lexicographically for each length):
a, b︸︷︷︸
2
, ab︸︷︷︸
1
, aab, abb︸ ︷︷ ︸
2
, aaab, aabb, abbb︸ ︷︷ ︸
3
, aaaab, aaabb, aabab, aabbb, ababb, abbbb︸ ︷︷ ︸
6
, . . .
How do we generate them? How do we count them?
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Lyndon Words Definitions
Examples
Example 1: w = aabac with alphabet A = {a, b, c}w is a Lyndon word for the orders a ≺ b ≺ c and a ≺ c ≺ b
(i.e., a Lyndon word for the orders on A with min(A) = a).
R2(w) = bacaa is a Lyndon word for the orders on A with min(A) = b.
R4(w) = caaba is a Lyndon word for the orders on A with min(A) = c.
Example 2
For the 2-letter alphabet {a, b} with a ≺ b, the Lyndon words up to length fiveare as follows (sorted lexicographically for each length):
a, b︸︷︷︸
2
, ab︸︷︷︸
1
, aab, abb︸ ︷︷ ︸
2
, aaab, aabb, abbb︸ ︷︷ ︸
3
, aaaab, aaabb, aabab, aabbb, ababb, abbbb︸ ︷︷ ︸
6
, . . .
How do we generate them? How do we count them?
These questions will be answered in a momentAmy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 12
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Lyndon Words Definitions
Examples
Example 1: w = aabac with alphabet A = {a, b, c}w is a Lyndon word for the orders a ≺ b ≺ c and a ≺ c ≺ b
(i.e., a Lyndon word for the orders on A with min(A) = a).
R2(w) = bacaa is a Lyndon word for the orders on A with min(A) = b.
R4(w) = caaba is a Lyndon word for the orders on A with min(A) = c.
Example 2
For the 2-letter alphabet {a, b} with a ≺ b, the Lyndon words up to length fiveare as follows (sorted lexicographically for each length):
a, b︸︷︷︸
2
, ab︸︷︷︸
1
, aab, abb︸ ︷︷ ︸
2
, aaab, aabb, abbb︸ ︷︷ ︸
3
, aaaab, aaabb, aabab, aabbb, ababb, abbbb︸ ︷︷ ︸
6
, . . .
How do we generate them? How do we count them?
These questions will be answered in a moment, but first . . .Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 12
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Lyndon Words Applications
Some Applications of Interest
There are many and varied applications of Lyndon words in algebraand combinatorics – far too many to mention here.
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Lyndon Words Applications
Some Applications of Interest
There are many and varied applications of Lyndon words in algebraand combinatorics – far too many to mention here.
For instance, Lyndon words have an application to the description offree Lie algebras in constructing bases.
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Lyndon Words Applications
Some Applications of Interest
There are many and varied applications of Lyndon words in algebraand combinatorics – far too many to mention here.
For instance, Lyndon words have an application to the description offree Lie algebras in constructing bases.
This was, in fact, Lyndon’s original motivation for introducing thesewords.
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Lyndon Words Applications
Some Applications of Interest
There are many and varied applications of Lyndon words in algebraand combinatorics – far too many to mention here.
For instance, Lyndon words have an application to the description offree Lie algebras in constructing bases.
This was, in fact, Lyndon’s original motivation for introducing thesewords.
Lyndon words may also be understood as a special case of Hall sets.
[C. Reutenauer, Free Lie Algebras, Oxford University Press, 1993]
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Lyndon Words Applications
Some Applications of Interest
There are many and varied applications of Lyndon words in algebraand combinatorics – far too many to mention here.
For instance, Lyndon words have an application to the description offree Lie algebras in constructing bases.
This was, in fact, Lyndon’s original motivation for introducing thesewords.
Lyndon words may also be understood as a special case of Hall sets.
[C. Reutenauer, Free Lie Algebras, Oxford University Press, 1993]
Lyndon words also have applications to semigroups, patternmatching, and representation theory of certain algebras (cf. recentwork of Ram et al.).
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Lyndon Words Applications
Some Applications of Interest . . .
All such applications make use of combinatorial properties of Lyndonwords, particularly factorisation theorems and lexicographical properties.
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Lyndon Words Applications
Some Applications of Interest . . .
All such applications make use of combinatorial properties of Lyndonwords, particularly factorisation theorems and lexicographical properties.
The rest of this talk will be concerned with some of the most importantcombinatorial properties of Lyndon words, which you will hopefully findinteresting and which may be useful in your own work.
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 14
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Lyndon Words Applications
Some Applications of Interest . . .
All such applications make use of combinatorial properties of Lyndonwords, particularly factorisation theorems and lexicographical properties.
The rest of this talk will be concerned with some of the most importantcombinatorial properties of Lyndon words, which you will hopefully findinteresting and which may be useful in your own work.
In particular, we’ll learn about the following in regards to Lyndon words:
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Lyndon Words Applications
Some Applications of Interest . . .
All such applications make use of combinatorial properties of Lyndonwords, particularly factorisation theorems and lexicographical properties.
The rest of this talk will be concerned with some of the most importantcombinatorial properties of Lyndon words, which you will hopefully findinteresting and which may be useful in your own work.
In particular, we’ll learn about the following in regards to Lyndon words:
Enumeration
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Lyndon Words Applications
Some Applications of Interest . . .
All such applications make use of combinatorial properties of Lyndonwords, particularly factorisation theorems and lexicographical properties.
The rest of this talk will be concerned with some of the most importantcombinatorial properties of Lyndon words, which you will hopefully findinteresting and which may be useful in your own work.
In particular, we’ll learn about the following in regards to Lyndon words:
Enumeration
Generation
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Lyndon Words Applications
Some Applications of Interest . . .
All such applications make use of combinatorial properties of Lyndonwords, particularly factorisation theorems and lexicographical properties.
The rest of this talk will be concerned with some of the most importantcombinatorial properties of Lyndon words, which you will hopefully findinteresting and which may be useful in your own work.
In particular, we’ll learn about the following in regards to Lyndon words:
Enumeration
Generation
Factorisation
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 14
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Lyndon Words Applications
Some Applications of Interest . . .
All such applications make use of combinatorial properties of Lyndonwords, particularly factorisation theorems and lexicographical properties.
The rest of this talk will be concerned with some of the most importantcombinatorial properties of Lyndon words, which you will hopefully findinteresting and which may be useful in your own work.
In particular, we’ll learn about the following in regards to Lyndon words:
Enumeration
Generation
Factorisation
I’ll return to some of the aforementioned applications along the way . . .
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Lyndon Words Enumeration
Counting Lyndon Words
From now on, we consider words over a totally ordered finite alphabet Aconsisting of at least two distinct letters, unless stated otherwise.
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Lyndon Words Enumeration
Counting Lyndon Words
From now on, we consider words over a totally ordered finite alphabet Aconsisting of at least two distinct letters, unless stated otherwise.
We know that the conjugacy class of any primitive word contains asingle Lyndon word.
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Lyndon Words Enumeration
Counting Lyndon Words
From now on, we consider words over a totally ordered finite alphabet Aconsisting of at least two distinct letters, unless stated otherwise.
We know that the conjugacy class of any primitive word contains asingle Lyndon word.
That is, each primitive word is a conjugate (rotation) of a uniqueLyndon word.
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Lyndon Words Enumeration
Counting Lyndon Words
From now on, we consider words over a totally ordered finite alphabet Aconsisting of at least two distinct letters, unless stated otherwise.
We know that the conjugacy class of any primitive word contains asingle Lyndon word.
That is, each primitive word is a conjugate (rotation) of a uniqueLyndon word.
So Lyndon words form representatives of conjugacy classes ofprimitive words.
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Lyndon Words Enumeration
Counting Lyndon Words
From now on, we consider words over a totally ordered finite alphabet Aconsisting of at least two distinct letters, unless stated otherwise.
We know that the conjugacy class of any primitive word contains asingle Lyndon word.
That is, each primitive word is a conjugate (rotation) of a uniqueLyndon word.
So Lyndon words form representatives of conjugacy classes ofprimitive words.
Hence the number of Lyndon words of length n over A is preciselythe number of aperiodic (primitive) necklaces of length n over A.
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Lyndon Words Enumeration
Counting Lyndon Words
From now on, we consider words over a totally ordered finite alphabet Aconsisting of at least two distinct letters, unless stated otherwise.
We know that the conjugacy class of any primitive word contains asingle Lyndon word.
That is, each primitive word is a conjugate (rotation) of a uniqueLyndon word.
So Lyndon words form representatives of conjugacy classes ofprimitive words.
Hence the number of Lyndon words of length n over A is preciselythe number of aperiodic (primitive) necklaces of length n over A.
How do we count these?
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Lyndon Words Enumeration
Counting Lyndon Words . . .
Suppose A has size |A| = k.
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Lyndon Words Enumeration
Counting Lyndon Words . . .
Suppose A has size |A| = k.
Let Nk(n) denote the number of primitive necklaces of length n on A.
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Lyndon Words Enumeration
Counting Lyndon Words . . .
Suppose A has size |A| = k.
Let Nk(n) denote the number of primitive necklaces of length n on A.
If |w| = n and w = vp = vv · · · v︸ ︷︷ ︸
p times
where v is a primitive word,
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Lyndon Words Enumeration
Counting Lyndon Words . . .
Suppose A has size |A| = k.
Let Nk(n) denote the number of primitive necklaces of length n on A.
If |w| = n and w = vp = vv · · · v︸ ︷︷ ︸
p times
where v is a primitive word, then
n = pd where d = |v|
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Lyndon Words Enumeration
Counting Lyndon Words . . .
Suppose A has size |A| = k.
Let Nk(n) denote the number of primitive necklaces of length n on A.
If |w| = n and w = vp = vv · · · v︸ ︷︷ ︸
p times
where v is a primitive word, then
n = pd where d = |v| and the number of distinct conjugates of w isexactly d.
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Lyndon Words Enumeration
Counting Lyndon Words . . .
Suppose A has size |A| = k.
Let Nk(n) denote the number of primitive necklaces of length n on A.
If |w| = n and w = vp = vv · · · v︸ ︷︷ ︸
p times
where v is a primitive word, then
n = pd where d = |v| and the number of distinct conjugates of w isexactly d.
Hence, the total number of different words of length n on A is
kn =∑
d|n
d · Nk(d)
where the sum is over all positive divisors of n.
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Lyndon Words Enumeration
Counting Lyndon Words . . .
Now, by the well-known Möbius inversion formula, we have
Nk(n) =1
n
∑
d|n
µ(d) · kn/d
where µ is the Möbius function defined on N+ as follows:
µ(n) =
1 if n = 1,
(−1)i if n = p1 · · · pi where the pi are distinct primes,
0 if n is divisible by a square.
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Lyndon Words Enumeration
Counting Lyndon Words . . .
Now, by the well-known Möbius inversion formula, we have
Nk(n) =1
n
∑
d|n
µ(d) · kn/d
where µ is the Möbius function defined on N+ as follows:
µ(n) =
1 if n = 1,
(−1)i if n = p1 · · · pi where the pi are distinct primes,
0 if n is divisible by a square.
This “necklace-counting formula” (often called Witt’s formula) was provedby E. Witt in 1937 in connection with the theorem on free Lie algebrasnow called the Poincaré–Birkhoff–Witt Theorem.
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Lyndon Words Enumeration
Counting Lyndon Words . . .
By Witt’s formula, the numbers of binary Lyndon words of each length, startingwith length 0 (empty word), form the integer sequence
1, 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, . . .
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Lyndon Words Enumeration
Counting Lyndon Words . . .
By Witt’s formula, the numbers of binary Lyndon words of each length, startingwith length 0 (empty word), form the integer sequence
1, 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, . . . [sequence A001037 in OEIS].
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Lyndon Words Enumeration
Counting Lyndon Words . . .
By Witt’s formula, the numbers of binary Lyndon words of each length, startingwith length 0 (empty word), form the integer sequence
1, 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, . . . [sequence A001037 in OEIS].
Note
There are as many Lyndon words of length n over an alphabet of size p
(a prime) as there are irreducible monic polynomials of degree n over a finitefield of characteristic p. [Gauss, 1900]
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Lyndon Words Enumeration
Counting Lyndon Words . . .
By Witt’s formula, the numbers of binary Lyndon words of each length, startingwith length 0 (empty word), form the integer sequence
1, 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, . . . [sequence A001037 in OEIS].
Note
There are as many Lyndon words of length n over an alphabet of size p
(a prime) as there are irreducible monic polynomials of degree n over a finitefield of characteristic p. [Gauss, 1900]
However, there is no known bijection between such irreducible polynomialsand Lyndon words.
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Lyndon Words Enumeration
Counting Lyndon Words . . .
By Witt’s formula, the numbers of binary Lyndon words of each length, startingwith length 0 (empty word), form the integer sequence
1, 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, . . . [sequence A001037 in OEIS].
Note
There are as many Lyndon words of length n over an alphabet of size p
(a prime) as there are irreducible monic polynomials of degree n over a finitefield of characteristic p. [Gauss, 1900]
However, there is no known bijection between such irreducible polynomialsand Lyndon words.
We’ve just seen how to count the number of Lyndon words of each length
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Lyndon Words Enumeration
Counting Lyndon Words . . .
By Witt’s formula, the numbers of binary Lyndon words of each length, startingwith length 0 (empty word), form the integer sequence
1, 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, . . . [sequence A001037 in OEIS].
Note
There are as many Lyndon words of length n over an alphabet of size p
(a prime) as there are irreducible monic polynomials of degree n over a finitefield of characteristic p. [Gauss, 1900]
However, there is no known bijection between such irreducible polynomialsand Lyndon words.
We’ve just seen how to count the number of Lyndon words of each length, buthow do we generate them?
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Lyndon Words Enumeration
Counting Lyndon Words . . .
By Witt’s formula, the numbers of binary Lyndon words of each length, startingwith length 0 (empty word), form the integer sequence
1, 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, . . . [sequence A001037 in OEIS].
Note
There are as many Lyndon words of length n over an alphabet of size p
(a prime) as there are irreducible monic polynomials of degree n over a finitefield of characteristic p. [Gauss, 1900]
However, there is no known bijection between such irreducible polynomialsand Lyndon words.
We’ve just seen how to count the number of Lyndon words of each length, buthow do we generate them?
Duval (1988) gave a beautiful, and clever, recursive process that generatesLyndon words of bounded length over a totally ordered finite alphabet.
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Lyndon Words Enumeration
Counting Lyndon Words . . .
By Witt’s formula, the numbers of binary Lyndon words of each length, startingwith length 0 (empty word), form the integer sequence
1, 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, . . . [sequence A001037 in OEIS].
Note
There are as many Lyndon words of length n over an alphabet of size p
(a prime) as there are irreducible monic polynomials of degree n over a finitefield of characteristic p. [Gauss, 1900]
However, there is no known bijection between such irreducible polynomialsand Lyndon words.
We’ve just seen how to count the number of Lyndon words of each length, buthow do we generate them?
Duval (1988) gave a beautiful, and clever, recursive process that generatesLyndon words of bounded length over a totally ordered finite alphabet.
At the core of Duval’s efficient algorithm is the following important factorisationtheorem . . .
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Lyndon Words Factorisation
Factorisation of Lyndon words
Theorem (Lyndon)
A word w ∈ A+ is a Lyndon word if and only if w ∈ A or there exists two Lyndonwords u and v such that w = uv and u ≺ v.
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Lyndon Words Factorisation
Factorisation of Lyndon words
Theorem (Lyndon)
A word w ∈ A+ is a Lyndon word if and only if w ∈ A or there exists two Lyndonwords u and v such that w = uv and u ≺ v.
In general, this factorisation is not unique since, for example, w = aabac has twosuch factorisations:
w = (a)(abac) and w = (aab)(ac).
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Lyndon Words Factorisation
Factorisation of Lyndon words
Theorem (Lyndon)
A word w ∈ A+ is a Lyndon word if and only if w ∈ A or there exists two Lyndonwords u and v such that w = uv and u ≺ v.
In general, this factorisation is not unique since, for example, w = aabac has twosuch factorisations:
w = (a)(abac) and w = (aab)(ac).
However, there is a unique factorisation of a given Lyndon word as a product uv
of two Lyndon words u, v with u ≺ v, called the standard factorisation.
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Lyndon Words Factorisation
Factorisation of Lyndon words
Theorem (Lyndon)
A word w ∈ A+ is a Lyndon word if and only if w ∈ A or there exists two Lyndonwords u and v such that w = uv and u ≺ v.
In general, this factorisation is not unique since, for example, w = aabac has twosuch factorisations:
w = (a)(abac) and w = (aab)(ac).
However, there is a unique factorisation of a given Lyndon word as a product uv
of two Lyndon words u, v with u ≺ v, called the standard factorisation.
Theorem (Chen-Fox-Lyndon 1958)
If w = uv is a Lyndon word with v its lexicographically smallest proper suffix,then u and v are also Lyndon words and u ≺ v.
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Lyndon Words Factorisation
Factorisation of Lyndon words
Theorem (Lyndon)
A word w ∈ A+ is a Lyndon word if and only if w ∈ A or there exists two Lyndonwords u and v such that w = uv and u ≺ v.
In general, this factorisation is not unique since, for example, w = aabac has twosuch factorisations:
w = (a)(abac) and w = (aab)(ac).
However, there is a unique factorisation of a given Lyndon word as a product uv
of two Lyndon words u, v with u ≺ v, called the standard factorisation.
Theorem (Chen-Fox-Lyndon 1958)
If w = uv is a Lyndon word with v its lexicographically smallest proper suffix,then u and v are also Lyndon words and u ≺ v.
So the standard factorisation of a Lyndon word w = uv is obtained by choosing v
to be the lexicographically least proper suffix of w
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Lyndon Words Factorisation
Factorisation of Lyndon words
Theorem (Lyndon)
A word w ∈ A+ is a Lyndon word if and only if w ∈ A or there exists two Lyndonwords u and v such that w = uv and u ≺ v.
In general, this factorisation is not unique since, for example, w = aabac has twosuch factorisations:
w = (a)(abac) and w = (aab)(ac).
However, there is a unique factorisation of a given Lyndon word as a product uv
of two Lyndon words u, v with u ≺ v, called the standard factorisation.
Theorem (Chen-Fox-Lyndon 1958)
If w = uv is a Lyndon word with v its lexicographically smallest proper suffix,then u and v are also Lyndon words and u ≺ v.
So the standard factorisation of a Lyndon word w = uv is obtained by choosing v
to be the lexicographically least proper suffix of w, which also happens to be thelongest proper suffix of w that is Lyndon.
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Lyndon Words Factorisation
Factorisation of Lyndon words
Theorem (Lyndon)
A word w ∈ A+ is a Lyndon word if and only if w ∈ A or there exists two Lyndonwords u and v such that w = uv and u ≺ v.
In general, this factorisation is not unique since, for example, w = aabac has twosuch factorisations:
w = (a)(abac) and w = (aab)(ac).
However, there is a unique factorisation of a given Lyndon word as a product uv
of two Lyndon words u, v with u ≺ v, called the standard factorisation.
Theorem (Chen-Fox-Lyndon 1958)
If w = uv is a Lyndon word with v its lexicographically smallest proper suffix,then u and v are also Lyndon words and u ≺ v.
So the standard factorisation of a Lyndon word w = uv is obtained by choosing v
to be the lexicographically least proper suffix of w, which also happens to be thelongest proper suffix of w that is Lyndon.
Example: w = aabac has standard factorisation w = (a)(abac).Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 19
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Lyndon Words Factorisation
An Application in Algebra
In algebraic settings, Lyndon words give rise to commutators usingstandard factorisation iteratively.
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Lyndon Words Factorisation
An Application in Algebra
In algebraic settings, Lyndon words give rise to commutators usingstandard factorisation iteratively.
For example, the Lyndon word aababb with standard factorisation(a)(ababb) gives rise to the commutator [a, [[a, b], [[a, b], b]]].
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Lyndon Words Factorisation
An Application in Algebra
In algebraic settings, Lyndon words give rise to commutators usingstandard factorisation iteratively.
For example, the Lyndon word aababb with standard factorisation(a)(ababb) gives rise to the commutator [a, [[a, b], [[a, b], b]]].
These commutators can be viewed either as elements of the freegroup with [x, y] = xyx−1y−1, or as elements of the free Lie algebrawith [x, y] = xy − yx .
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Lyndon Words Factorisation
An Application in Algebra
In algebraic settings, Lyndon words give rise to commutators usingstandard factorisation iteratively.
For example, the Lyndon word aababb with standard factorisation(a)(ababb) gives rise to the commutator [a, [[a, b], [[a, b], b]]].
These commutators can be viewed either as elements of the freegroup with [x, y] = xyx−1y−1, or as elements of the free Lie algebrawith [x, y] = xy − yx .
In either case, Lyndon words give rise to a basis of some algebra.
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Lyndon Words Factorisation
An Application in Algebra
In algebraic settings, Lyndon words give rise to commutators usingstandard factorisation iteratively.
For example, the Lyndon word aababb with standard factorisation(a)(ababb) gives rise to the commutator [a, [[a, b], [[a, b], b]]].
These commutators can be viewed either as elements of the freegroup with [x, y] = xyx−1y−1, or as elements of the free Lie algebrawith [x, y] = xy − yx .
In either case, Lyndon words give rise to a basis of some algebra.
For specific details, see [C. Reutenauer, Free Lie Algebras, 1993] and[M. Lothaire, Combinatorics on words, 1983].
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Lyndon Words Generation
Generation of Lyndon Words
Duval’s Algorithm: Generates the Lyndon words over A of length at most n
(n ≥ 2).
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Lyndon Words Generation
Generation of Lyndon Words
Duval’s Algorithm: Generates the Lyndon words over A of length at most n
(n ≥ 2).
If w is one of the words in the list of Lyndon words up to length n (not equal tomax(A)), then the next Lyndon word after w can be found by the following steps:
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Lyndon Words Generation
Generation of Lyndon Words
Duval’s Algorithm: Generates the Lyndon words over A of length at most n
(n ≥ 2).
If w is one of the words in the list of Lyndon words up to length n (not equal tomax(A)), then the next Lyndon word after w can be found by the following steps:
1 Repeat the letters from w to form a new word x of length exactly n, wherethe i-th letter of x is the same as the letter at position i (mod |w|) in w.
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Lyndon Words Generation
Generation of Lyndon Words
Duval’s Algorithm: Generates the Lyndon words over A of length at most n
(n ≥ 2).
If w is one of the words in the list of Lyndon words up to length n (not equal tomax(A)), then the next Lyndon word after w can be found by the following steps:
1 Repeat the letters from w to form a new word x of length exactly n, wherethe i-th letter of x is the same as the letter at position i (mod |w|) in w.
2 If the last letter of x is max(A) for the given order on A, remove it,producing a shorter word, and take this to be the new x .
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Lyndon Words Generation
Generation of Lyndon Words
Duval’s Algorithm: Generates the Lyndon words over A of length at most n
(n ≥ 2).
If w is one of the words in the list of Lyndon words up to length n (not equal tomax(A)), then the next Lyndon word after w can be found by the following steps:
1 Repeat the letters from w to form a new word x of length exactly n, wherethe i-th letter of x is the same as the letter at position i (mod |w|) in w.
2 If the last letter of x is max(A) for the given order on A, remove it,producing a shorter word, and take this to be the new x .
3 If the last letter of x is max(A), then repeat Step 2.
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Lyndon Words Generation
Generation of Lyndon Words
Duval’s Algorithm: Generates the Lyndon words over A of length at most n
(n ≥ 2).
If w is one of the words in the list of Lyndon words up to length n (not equal tomax(A)), then the next Lyndon word after w can be found by the following steps:
1 Repeat the letters from w to form a new word x of length exactly n, wherethe i-th letter of x is the same as the letter at position i (mod |w|) in w.
2 If the last letter of x is max(A) for the given order on A, remove it,producing a shorter word, and take this to be the new x .
3 If the last letter of x is max(A), then repeat Step 2. Otherwise, replace thelast letter of x by its successor in the sorted ordering of the alphabet A.
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Lyndon Words Generation
Generation of Lyndon Words
Duval’s Algorithm: Generates the Lyndon words over A of length at most n
(n ≥ 2).
If w is one of the words in the list of Lyndon words up to length n (not equal tomax(A)), then the next Lyndon word after w can be found by the following steps:
1 Repeat the letters from w to form a new word x of length exactly n, wherethe i-th letter of x is the same as the letter at position i (mod |w|) in w.
2 If the last letter of x is max(A) for the given order on A, remove it,producing a shorter word, and take this to be the new x .
3 If the last letter of x is max(A), then repeat Step 2. Otherwise, replace thelast letter of x by its successor in the sorted ordering of the alphabet A.
Note: Since, in general, the factorisation of a Lyndon word as a product uv oftwo Lyndon words u, v with u ≺ v is not unique, Duval’s algorithm may producethe same Lyndon word more than once.
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 22
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b.
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 22
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 22
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 22
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
b
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
b −→ bb
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
b −→ bb −→ b
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 22
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
b −→ bb −→ b −→
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 22
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
b −→ bb −→ b −→ ε
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 22
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
b −→ bb −→ b −→ ε
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 22
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
b −→ bb −→ b −→ ε
Updated List: L = {a, b, ab, . . .}
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
b −→ bb −→ b −→ ε
Updated List: L = {a, b, ab, . . .}Lyndon words of length at most 3:
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 22
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
b −→ bb −→ b −→ ε
Updated List: L = {a, b, ab, . . .}Lyndon words of length at most 3:
a
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
b −→ bb −→ b −→ ε
Updated List: L = {a, b, ab, . . .}Lyndon words of length at most 3:
a −→ aaa
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
b −→ bb −→ b −→ ε
Updated List: L = {a, b, ab, . . .}Lyndon words of length at most 3:
a −→ aaa −→ aab
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
b −→ bb −→ b −→ ε
Updated List: L = {a, b, ab, . . .}Lyndon words of length at most 3:
a −→ aaa −→ aab
ab
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
b −→ bb −→ b −→ ε
Updated List: L = {a, b, ab, . . .}Lyndon words of length at most 3:
a −→ aaa −→ aab
ab −→ aba
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
b −→ bb −→ b −→ ε
Updated List: L = {a, b, ab, . . .}Lyndon words of length at most 3:
a −→ aaa −→ aab
ab −→ aba −→ abb
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Lyndon Words Generation
Duval’s Algorithm in Action
Let’s use the algorithm to generate all the Lyndon words up to length 4 over thealphabet {a, b} with a ≺ b.
We begin with the Lyndon words of length 1: a and b. List: L = {a, b, . . .}Lyndon words of length at most 2:
a −→ aa −→ ab
b −→ bb −→ b −→ ε
Updated List: L = {a, b, ab, . . .}Lyndon words of length at most 3:
a −→ aaa −→ aab
ab −→ aba −→ abb
Updated List: L = {a, b, ab, aab, abb, . . .}
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
a
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
a −→ aaaa
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
a −→ aaaa −→ aaab
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
a −→ aaaa −→ aaab
ab
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
a −→ aaaa −→ aaab
ab −→ abab
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
a −→ aaaa −→ aaab
ab −→ abab −→ abb
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
a −→ aaaa −→ aaab
ab −→ abab −→ abb (repeat)
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
a −→ aaaa −→ aaab
ab −→ abab −→ abb (repeat)
aab
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
a −→ aaaa −→ aaab
ab −→ abab −→ abb (repeat)
aab −→ aaba
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
a −→ aaaa −→ aaab
ab −→ abab −→ abb (repeat)
aab −→ aaba −→ aabb
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
a −→ aaaa −→ aaab
ab −→ abab −→ abb (repeat)
aab −→ aaba −→ aabb
abb
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
a −→ aaaa −→ aaab
ab −→ abab −→ abb (repeat)
aab −→ aaba −→ aabb
abb −→ abba
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
a −→ aaaa −→ aaab
ab −→ abab −→ abb (repeat)
aab −→ aaba −→ aabb
abb −→ abba −→ abbb
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
a −→ aaaa −→ aaab
ab −→ abab −→ abb (repeat)
aab −→ aaba −→ aabb
abb −→ abba −→ abbb
Updated List: L = {a, b, ab, aab, abb, aaab, aabb, abbb, . . .}
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Duval’s Algorithm in Action . . .
Lyndon words of length at most 4:
a −→ aaaa −→ aaab
ab −→ abab −→ abb (repeat)
aab −→ aaba −→ aabb
abb −→ abba −→ abbb
Updated List: L = {a, b, ab, aab, abb, aaab, aabb, abbb, . . .}
And on it goes . . .
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 23
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Lyndon Words Generation
Remarks on Duval’s Algorithm
The worst-case time to generate a successor of a Lyndon word w byDuval’s procedure is O(n).
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Lyndon Words Generation
Remarks on Duval’s Algorithm
The worst-case time to generate a successor of a Lyndon word w byDuval’s procedure is O(n).
This can be improved to constant time if the generated words arestored in an array of length n and the construction of x from w isperformed by appending letters to w instead of making a new copyof w.
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Lyndon Words Generation
Remarks on Duval’s Algorithm
The worst-case time to generate a successor of a Lyndon word w byDuval’s procedure is O(n).
This can be improved to constant time if the generated words arestored in an array of length n and the construction of x from w isperformed by appending letters to w instead of making a new copyof w. [Berstel-Pocchiola 1994]
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Lyndon Words Generation
Remarks on Duval’s Algorithm
The worst-case time to generate a successor of a Lyndon word w byDuval’s procedure is O(n).
This can be improved to constant time if the generated words arestored in an array of length n and the construction of x from w isperformed by appending letters to w instead of making a new copyof w. [Berstel-Pocchiola 1994]
Duval (1983) also developed an algorithm for standard factorisationthat runs in linear time and constant space.
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Lyndon Words More on Factorisations
Back to Factorisations of Lyndon Words
A famous theorem concerning Lyndon words asserts that every word w can beuniquely factorised as a non-increasing product of Lyndon words . . .
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Lyndon Words More on Factorisations
Back to Factorisations of Lyndon Words
A famous theorem concerning Lyndon words asserts that every word w can beuniquely factorised as a non-increasing product of Lyndon words . . .
Theorem
Any word w ∈ A+ may be uniquely written as a non-increasing product ofLyndon words, i.e.,
w = ℓ1ℓ2 · · · ℓn
where the ℓi are Lyndon words such that ℓ1 � ℓ2 � · · · � ℓn
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Lyndon Words More on Factorisations
Back to Factorisations of Lyndon Words
A famous theorem concerning Lyndon words asserts that every word w can beuniquely factorised as a non-increasing product of Lyndon words . . .
Theorem
Any word w ∈ A+ may be uniquely written as a non-increasing product ofLyndon words, i.e.,
w = ℓ1ℓ2 · · · ℓn
where the ℓi are Lyndon words such that ℓ1 � ℓ2 � · · · � ℓn, called the Lyndonfactorisation.
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Lyndon Words More on Factorisations
Back to Factorisations of Lyndon Words
A famous theorem concerning Lyndon words asserts that every word w can beuniquely factorised as a non-increasing product of Lyndon words . . .
Theorem
Any word w ∈ A+ may be uniquely written as a non-increasing product ofLyndon words, i.e.,
w = ℓ1ℓ2 · · · ℓn
where the ℓi are Lyndon words such that ℓ1 � ℓ2 � · · · � ℓn, called the Lyndonfactorisation.
Example: abaacaab
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Lyndon Words More on Factorisations
Back to Factorisations of Lyndon Words
A famous theorem concerning Lyndon words asserts that every word w can beuniquely factorised as a non-increasing product of Lyndon words . . .
Theorem
Any word w ∈ A+ may be uniquely written as a non-increasing product ofLyndon words, i.e.,
w = ℓ1ℓ2 · · · ℓn
where the ℓi are Lyndon words such that ℓ1 � ℓ2 � · · · � ℓn, called the Lyndonfactorisation.
Example: abaacaab = (ab)(aac)(aab)
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Lyndon Words More on Factorisations
Back to Factorisations of Lyndon Words
A famous theorem concerning Lyndon words asserts that every word w can beuniquely factorised as a non-increasing product of Lyndon words . . .
Theorem
Any word w ∈ A+ may be uniquely written as a non-increasing product ofLyndon words, i.e.,
w = ℓ1ℓ2 · · · ℓn
where the ℓi are Lyndon words such that ℓ1 � ℓ2 � · · · � ℓn, called the Lyndonfactorisation.
Example: abaacaab = (ab)(aac)(aab)
The origin of the theorem is unclear
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Lyndon Words More on Factorisations
Back to Factorisations of Lyndon Words
A famous theorem concerning Lyndon words asserts that every word w can beuniquely factorised as a non-increasing product of Lyndon words . . .
Theorem
Any word w ∈ A+ may be uniquely written as a non-increasing product ofLyndon words, i.e.,
w = ℓ1ℓ2 · · · ℓn
where the ℓi are Lyndon words such that ℓ1 � ℓ2 � · · · � ℓn, called the Lyndonfactorisation.
Example: abaacaab = (ab)(aac)(aab)
The origin of the theorem is unclear – it is usually credited toChen-Fox-Lyndon (1958), whose paper does not explicitly contain the statement,but it can be recovered from their results.
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Lyndon Words More on Factorisations
Back to Factorisations of Lyndon Words
A famous theorem concerning Lyndon words asserts that every word w can beuniquely factorised as a non-increasing product of Lyndon words . . .
Theorem
Any word w ∈ A+ may be uniquely written as a non-increasing product ofLyndon words, i.e.,
w = ℓ1ℓ2 · · · ℓn
where the ℓi are Lyndon words such that ℓ1 � ℓ2 � · · · � ℓn, called the Lyndonfactorisation.
Example: abaacaab = (ab)(aac)(aab)
The origin of the theorem is unclear – it is usually credited toChen-Fox-Lyndon (1958), whose paper does not explicitly contain the statement,but it can be recovered from their results.
We note that the Lyndon factorisation of a word can be computed in linear time[Duval 1983].
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Lyndon Words More on Factorisations
Applications of Lyndon Factorisations
Used as part of a bijective variant of the Burrows-Wheeler transform fordata compression. [Burrow-Wheeler 1994, Gil-Scott 2009, Kufleitner 2009]
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Lyndon Words More on Factorisations
Applications of Lyndon Factorisations
Used as part of a bijective variant of the Burrows-Wheeler transform fordata compression. [Burrow-Wheeler 1994, Gil-Scott 2009, Kufleitner 2009]
Used in algorithms for digital geometry. [Brlek et al. 2009]
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Lyndon Words More on Factorisations
Applications of Lyndon Factorisations
Used as part of a bijective variant of the Burrows-Wheeler transform fordata compression. [Burrow-Wheeler 1994, Gil-Scott 2009, Kufleitner 2009]
Used in algorithms for digital geometry. [Brlek et al. 2009]
Connection to de Bruijn sequences: If one concatenates together, inlexicographical order, all the Lyndon words over an alphabet of size k thathave length dividing a given number n, then the result is a de Bruijnsequence (1975)
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Lyndon Words More on Factorisations
Applications of Lyndon Factorisations
Used as part of a bijective variant of the Burrows-Wheeler transform fordata compression. [Burrow-Wheeler 1994, Gil-Scott 2009, Kufleitner 2009]
Used in algorithms for digital geometry. [Brlek et al. 2009]
Connection to de Bruijn sequences: If one concatenates together, inlexicographical order, all the Lyndon words over an alphabet of size k thathave length dividing a given number n, then the result is a de Bruijnsequence (1975) – a necklace of length kn on an alphabet of size k in whicheach possible word of length n appears exactly once as one of its factors(contiguous subwords).
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Lyndon Words More on Factorisations
Applications of Lyndon Factorisations
Used as part of a bijective variant of the Burrows-Wheeler transform fordata compression. [Burrow-Wheeler 1994, Gil-Scott 2009, Kufleitner 2009]
Used in algorithms for digital geometry. [Brlek et al. 2009]
Connection to de Bruijn sequences: If one concatenates together, inlexicographical order, all the Lyndon words over an alphabet of size k thathave length dividing a given number n, then the result is a de Bruijnsequence (1975) – a necklace of length kn on an alphabet of size k in whicheach possible word of length n appears exactly once as one of its factors(contiguous subwords).
For example, the concatenation, in lexicographical order, of the Lyndonwords on two letters with lengths dividing 4 yields the de Bruijn wordB(2, 4):
a · aaab · aabb · ab · abbb · b
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Lyndon Words More on Factorisations
Applications of Lyndon Factorisations
Used as part of a bijective variant of the Burrows-Wheeler transform fordata compression. [Burrow-Wheeler 1994, Gil-Scott 2009, Kufleitner 2009]
Used in algorithms for digital geometry. [Brlek et al. 2009]
Connection to de Bruijn sequences: If one concatenates together, inlexicographical order, all the Lyndon words over an alphabet of size k thathave length dividing a given number n, then the result is a de Bruijnsequence (1975) – a necklace of length kn on an alphabet of size k in whicheach possible word of length n appears exactly once as one of its factors(contiguous subwords).
For example, the concatenation, in lexicographical order, of the Lyndonwords on two letters with lengths dividing 4 yields the de Bruijn wordB(2, 4):
a · aaab · aabb · ab · abbb · b
This fact was discovered by Fredericksen and Maiorana in 1978.
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Lyndon Words More on Factorisations
Applications of Lyndon Factorisations
Used as part of a bijective variant of the Burrows-Wheeler transform fordata compression. [Burrow-Wheeler 1994, Gil-Scott 2009, Kufleitner 2009]
Used in algorithms for digital geometry. [Brlek et al. 2009]
Connection to de Bruijn sequences: If one concatenates together, inlexicographical order, all the Lyndon words over an alphabet of size k thathave length dividing a given number n, then the result is a de Bruijnsequence (1975) – a necklace of length kn on an alphabet of size k in whicheach possible word of length n appears exactly once as one of its factors(contiguous subwords).
For example, the concatenation, in lexicographical order, of the Lyndonwords on two letters with lengths dividing 4 yields the de Bruijn wordB(2, 4):
a · aaab · aabb · ab · abbb · b
This fact was discovered by Fredericksen and Maiorana in 1978.
Interestingly, there is a relationship between Lyndon words, shift-registersequences, and de Bruijn words [Knuth 2005].
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Lyndon Words More on Factorisations
Some Applications of de Bruijn Sequences
A de Bruijn sequence can be used to shorten a brute-force attack on aPIN-like code lock that does not have an “enter” key and accepts the last n
digits entered.
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Lyndon Words More on Factorisations
Some Applications of de Bruijn Sequences
A de Bruijn sequence can be used to shorten a brute-force attack on aPIN-like code lock that does not have an “enter” key and accepts the last n
digits entered.
Example
Consider such a digital door lock with a 3-digit code.
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Lyndon Words More on Factorisations
Some Applications of de Bruijn Sequences
A de Bruijn sequence can be used to shorten a brute-force attack on aPIN-like code lock that does not have an “enter” key and accepts the last n
digits entered.
Example
Consider such a digital door lock with a 3-digit code.
All possible solution codes are contained exactly once in B(10, 3) — the de BruijnBruijn sequence with alphabet {0, 1, . . . , 9} that contains each word of length 3exactly once.
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Lyndon Words More on Factorisations
Some Applications of de Bruijn Sequences
A de Bruijn sequence can be used to shorten a brute-force attack on aPIN-like code lock that does not have an “enter” key and accepts the last n
digits entered.
Example
Consider such a digital door lock with a 3-digit code.
All possible solution codes are contained exactly once in B(10, 3) — the de BruijnBruijn sequence with alphabet {0, 1, . . . , 9} that contains each word of length 3exactly once.
B(10, 3) has length 103 = 1000
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Lyndon Words More on Factorisations
Some Applications of de Bruijn Sequences
A de Bruijn sequence can be used to shorten a brute-force attack on aPIN-like code lock that does not have an “enter” key and accepts the last n
digits entered.
Example
Consider such a digital door lock with a 3-digit code.
All possible solution codes are contained exactly once in B(10, 3) — the de BruijnBruijn sequence with alphabet {0, 1, . . . , 9} that contains each word of length 3exactly once.
B(10, 3) has length 103 = 1000, so the number of presses required to open thelock would be at most 1000 + 2 = 1002 (since the solutions are cyclic)
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Lyndon Words More on Factorisations
Some Applications of de Bruijn Sequences
A de Bruijn sequence can be used to shorten a brute-force attack on aPIN-like code lock that does not have an “enter” key and accepts the last n
digits entered.
Example
Consider such a digital door lock with a 3-digit code.
All possible solution codes are contained exactly once in B(10, 3) — the de BruijnBruijn sequence with alphabet {0, 1, . . . , 9} that contains each word of length 3exactly once.
B(10, 3) has length 103 = 1000, so the number of presses required to open thelock would be at most 1000 + 2 = 1002 (since the solutions are cyclic), comparedto up to 4000 presses if all possible codes were tried separately.
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Lyndon Words More on Factorisations
Some Applications of de Bruijn Sequences
A de Bruijn sequence can be used to shorten a brute-force attack on aPIN-like code lock that does not have an “enter” key and accepts the last n
digits entered.
Example
Consider such a digital door lock with a 3-digit code.
All possible solution codes are contained exactly once in B(10, 3) — the de BruijnBruijn sequence with alphabet {0, 1, . . . , 9} that contains each word of length 3exactly once.
B(10, 3) has length 103 = 1000, so the number of presses required to open thelock would be at most 1000 + 2 = 1002 (since the solutions are cyclic), comparedto up to 4000 presses if all possible codes were tried separately.
de Bruijn sequences are also of general use in neuroscience and psychologyexperiments, and even robotics . . .
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Lyndon Words More on Factorisations
Some Applications of de Bruijn Sequences
A de Bruijn sequence can be used to shorten a brute-force attack on aPIN-like code lock that does not have an “enter” key and accepts the last n
digits entered.
Example
Consider such a digital door lock with a 3-digit code.
All possible solution codes are contained exactly once in B(10, 3) — the de BruijnBruijn sequence with alphabet {0, 1, . . . , 9} that contains each word of length 3exactly once.
B(10, 3) has length 103 = 1000, so the number of presses required to open thelock would be at most 1000 + 2 = 1002 (since the solutions are cyclic), comparedto up to 4000 presses if all possible codes were tried separately.
de Bruijn sequences are also of general use in neuroscience and psychologyexperiments, and even robotics . . .
See Wikipedia for more information and other interesting applications.
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Lyndon Words More on Factorisations
Standard Factorisations
Recall that if w = uv is a Lyndon word with v the longest properLyndon suffix of w, then u is also a Lyndon word and u ≺ v.
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Lyndon Words More on Factorisations
Standard Factorisations
Recall that if w = uv is a Lyndon word with v the longest properLyndon suffix of w, then u is also a Lyndon word and u ≺ v.
This factorisation of w as an increasing product of two Lyndon wordsis called the standard or right-standard factorisation of w.
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Lyndon Words More on Factorisations
Standard Factorisations
Recall that if w = uv is a Lyndon word with v the longest properLyndon suffix of w, then u is also a Lyndon word and u ≺ v.
This factorisation of w as an increasing product of two Lyndon wordsis called the standard or right-standard factorisation of w.
An alternative left-standard factorisation is due to Shirshov (1962)and Viennot (1978).
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Lyndon Words More on Factorisations
Standard Factorisations
Recall that if w = uv is a Lyndon word with v the longest properLyndon suffix of w, then u is also a Lyndon word and u ≺ v.
This factorisation of w as an increasing product of two Lyndon wordsis called the standard or right-standard factorisation of w.
An alternative left-standard factorisation is due to Shirshov (1962)and Viennot (1978).
Theorem (Shirshov 1962, Viennot 1978)
If w = uv is a Lyndon word in A+ with u the longest proper Lyndon prefixof w, then v is also a Lyndon word and u ≺ v.
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Lyndon Words More on Factorisations
Standard Factorisations
Recall that if w = uv is a Lyndon word with v the longest properLyndon suffix of w, then u is also a Lyndon word and u ≺ v.
This factorisation of w as an increasing product of two Lyndon wordsis called the standard or right-standard factorisation of w.
An alternative left-standard factorisation is due to Shirshov (1962)and Viennot (1978).
Theorem (Shirshov 1962, Viennot 1978)
If w = uv is a Lyndon word in A+ with u the longest proper Lyndon prefixof w, then v is also a Lyndon word and u ≺ v.
Example
The left-standard and right-standard factorisations of aabaacab are:
(aabaac)(ab) and (aab)(aacab).
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Lyndon Words More on Factorisations
Standard Factorisations . . .
The left-standard and right-standard factorisations of a Lyndon wordsometimes coincide.
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Lyndon Words More on Factorisations
Standard Factorisations . . .
The left-standard and right-standard factorisations of a Lyndon wordsometimes coincide.
Example
The Lyndon word aabaabab has coincidental left-standard andright-standard factorisations: (aab)(aabab).
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Lyndon Words More on Factorisations
Standard Factorisations . . .
The left-standard and right-standard factorisations of a Lyndon wordsometimes coincide.
Example
The Lyndon word aabaabab has coincidental left-standard andright-standard factorisations: (aab)(aabab).
Let’s take a closer look at some “coincidental Lyndon words” over {a, b}with a ≺ b . . .
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Lyndon Words More on Factorisations
Standard Factorisations . . .
The left-standard and right-standard factorisations of a Lyndon wordsometimes coincide.
Example
The Lyndon word aabaabab has coincidental left-standard andright-standard factorisations: (aab)(aabab).
Let’s take a closer look at some “coincidental Lyndon words” over {a, b}with a ≺ b . . .
a, b, ab, ab, abb, aaab, abbb, aaaab, aabab, ababb, abbbb, . . .
Do you notice any structural property that holds for all these words?
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Lyndon Words More on Factorisations
Standard Factorisations . . .
The left-standard and right-standard factorisations of a Lyndon wordsometimes coincide.
Example
The Lyndon word aabaabab has coincidental left-standard andright-standard factorisations: (aab)(aabab).
Let’s take a closer look at some “coincidental Lyndon words” over {a, b}with a ≺ b . . .
a, b, ab, ab, abb, aaab, abbb, aaaab, aabab, ababb, abbbb, . . .
How about now?
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Lyndon Words More on Factorisations
Standard Factorisations . . .
The left-standard and right-standard factorisations of a Lyndon wordsometimes coincide.
Example
The Lyndon word aabaabab has coincidental left-standard andright-standard factorisations: (aab)(aabab).
Let’s take a closer look at some “coincidental Lyndon words” over {a, b}with a ≺ b . . .
a, b, ab, ab, abb, aaab, abbb, aaaab, aabab, ababb, abbbb, . . .
All such words take the form aub where u is a palindrome.
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Lyndon Words More on Factorisations
Standard Factorisations . . .
The left-standard and right-standard factorisations of a Lyndon wordsometimes coincide.
Example
The Lyndon word aabaabab has coincidental left-standard andright-standard factorisations: (aab)(aabab).
Let’s take a closer look at some “coincidental Lyndon words” over {a, b}with a ≺ b . . .
a, b, ab, ab, abb, aaab, abbb, aaaab, aabab, ababb, abbbb, . . .
All such words take the form aub where u is a palindrome.
But not just any old palindrome . . .
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
For example: (glen)+ =
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
For example: (glen)+ = glenelg
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
For example: (glen)+ = glenelg
(race)+ =
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
For example: (glen)+ = glenelg
(race)+ = racecar
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
For example: (glen)+ = glenelg
(race)+ = racecar
We define Pal(ε) = ε
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
For example: (glen)+ = glenelg
(race)+ = racecar
We define Pal(ε) = ε, and for any word w and letter x ,
Pal(wx) = (Pal(w)x)+.
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
For example: (glen)+ = glenelg
(race)+ = racecar
We define Pal(ε) = ε, and for any word w and letter x ,
Pal(wx) = (Pal(w)x)+.
For example:Pal(abab) =
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
For example: (glen)+ = glenelg
(race)+ = racecar
We define Pal(ε) = ε, and for any word w and letter x ,
Pal(wx) = (Pal(w)x)+.
For example:Pal(abab) = a
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
For example: (glen)+ = glenelg
(race)+ = racecar
We define Pal(ε) = ε, and for any word w and letter x ,
Pal(wx) = (Pal(w)x)+.
For example:Pal(abab) = ab
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
For example: (glen)+ = glenelg
(race)+ = racecar
We define Pal(ε) = ε, and for any word w and letter x ,
Pal(wx) = (Pal(w)x)+.
For example:Pal(abab) = aba
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
For example: (glen)+ = glenelg
(race)+ = racecar
We define Pal(ε) = ε, and for any word w and letter x ,
Pal(wx) = (Pal(w)x)+.
For example:Pal(abab) = abaa
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
For example: (glen)+ = glenelg
(race)+ = racecar
We define Pal(ε) = ε, and for any word w and letter x ,
Pal(wx) = (Pal(w)x)+.
For example:Pal(abab) = abaaba
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
For example: (glen)+ = glenelg
(race)+ = racecar
We define Pal(ε) = ε, and for any word w and letter x ,
Pal(wx) = (Pal(w)x)+.
For example:Pal(abab) = abaabab
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
For example: (glen)+ = glenelg
(race)+ = racecar
We define Pal(ε) = ε, and for any word w and letter x ,
Pal(wx) = (Pal(w)x)+.
For example:Pal(abab) = abaababaaba
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Lyndon Words More on Factorisations
Theorem (Melançon 1999, Berstel-de Luca 1997)
Suppose w is finite word over {a, b}. Then w is a Lyndon word with the propertythat its left and right standard factorisations coincide if and only if w = aub (whena ≺ b) or w = bua (when b ≺ a) where u = Pal(v) for some word v over {a, b}.
Pal is the iterated palindromic closure operator [Justin 2005] defined asfollows.
For a given word v, let v+ denote the unique shortest palindrome beginningwith v, called the (right-)palindromic closure of v.
For example: (glen)+ = glenelg
(race)+ = racecar
We define Pal(ε) = ε, and for any word w and letter x ,
Pal(wx) = (Pal(w)x)+.
For example:Pal(abab) = abaababaaba
−→ coincidental Lyndon word: aPal(abab)b = aabaabab · aabab
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Lyndon Words More on Factorisations
The previous theorem says that the “coincidental Lyndon words” over {a, b} witha ≺ b are precisely the words aPal(v)b where v ∈ {a, b}∗.
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Lyndon Words More on Factorisations
The previous theorem says that the “coincidental Lyndon words” over {a, b} witha ≺ b are precisely the words aPal(v)b where v ∈ {a, b}∗.
These words are known as (lower) Christoffel words (named after E. Christoffel1800’s)
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Lyndon Words More on Factorisations
The previous theorem says that the “coincidental Lyndon words” over {a, b} witha ≺ b are precisely the words aPal(v)b where v ∈ {a, b}∗.
These words are known as (lower) Christoffel words (named after E. Christoffel1800’s) – they can be constructed geometrically by the coding the horizontal andvertical steps of certain lattice paths . . .
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Lyndon Words More on Factorisations
The previous theorem says that the “coincidental Lyndon words” over {a, b} witha ≺ b are precisely the words aPal(v)b where v ∈ {a, b}∗.
These words are known as (lower) Christoffel words (named after E. Christoffel1800’s) – they can be constructed geometrically by the coding the horizontal andvertical steps of certain lattice paths . . .
Consider a line (call it ℓ) of the form:
y =p
qx
where p, q are positive integers with gcd(p, q) = 1.
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Lyndon Words More on Factorisations
The previous theorem says that the “coincidental Lyndon words” over {a, b} witha ≺ b are precisely the words aPal(v)b where v ∈ {a, b}∗.
These words are known as (lower) Christoffel words (named after E. Christoffel1800’s) – they can be constructed geometrically by the coding the horizontal andvertical steps of certain lattice paths . . .
Consider a line (call it ℓ) of the form:
y =p
qx
where p, q are positive integers with gcd(p, q) = 1.
Let P denote the path along the integer lattice below the line ℓ that startsat the point (0, 0) and ends at the point (q, p) with the property that theregion in the plane enclosed by P and ℓ contains no other points in Z × Z
besides those of the path P .
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 31
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Lyndon Words More on Factorisations
The previous theorem says that the “coincidental Lyndon words” over {a, b} witha ≺ b are precisely the words aPal(v)b where v ∈ {a, b}∗.
These words are known as (lower) Christoffel words (named after E. Christoffel1800’s) – they can be constructed geometrically by the coding the horizontal andvertical steps of certain lattice paths . . .
Consider a line (call it ℓ) of the form:
y =p
qx
where p, q are positive integers with gcd(p, q) = 1.
Let P denote the path along the integer lattice below the line ℓ that startsat the point (0, 0) and ends at the point (q, p) with the property that theregion in the plane enclosed by P and ℓ contains no other points in Z × Z
besides those of the path P .
The so-called lower Christoffel word of slope p/q, denoted by L(p, q), isobtained by coding the steps of the path P .
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 31
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Lyndon Words More on Factorisations
The previous theorem says that the “coincidental Lyndon words” over {a, b} witha ≺ b are precisely the words aPal(v)b where v ∈ {a, b}∗.
These words are known as (lower) Christoffel words (named after E. Christoffel1800’s) – they can be constructed geometrically by the coding the horizontal andvertical steps of certain lattice paths . . .
Consider a line (call it ℓ) of the form:
y =p
qx
where p, q are positive integers with gcd(p, q) = 1.
Let P denote the path along the integer lattice below the line ℓ that startsat the point (0, 0) and ends at the point (q, p) with the property that theregion in the plane enclosed by P and ℓ contains no other points in Z × Z
besides those of the path P .
The so-called lower Christoffel word of slope p/q, denoted by L(p, q), isobtained by coding the steps of the path P .
– A horizontal step is denoted by the letter a.
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 31
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Lyndon Words More on Factorisations
The previous theorem says that the “coincidental Lyndon words” over {a, b} witha ≺ b are precisely the words aPal(v)b where v ∈ {a, b}∗.
These words are known as (lower) Christoffel words (named after E. Christoffel1800’s) – they can be constructed geometrically by the coding the horizontal andvertical steps of certain lattice paths . . .
Consider a line (call it ℓ) of the form:
y =p
qx
where p, q are positive integers with gcd(p, q) = 1.
Let P denote the path along the integer lattice below the line ℓ that startsat the point (0, 0) and ends at the point (q, p) with the property that theregion in the plane enclosed by P and ℓ contains no other points in Z × Z
besides those of the path P .
The so-called lower Christoffel word of slope p/q, denoted by L(p, q), isobtained by coding the steps of the path P .
– A horizontal step is denoted by the letter a.
– A vertical step is denoted by the letter b.
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 31
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
a
L(3, 5) = a
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
a a
L(3, 5) = aa
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
a a
b
L(3, 5) = aab
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
a a
ba
L(3, 5) = aaba
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
a a
ba a
L(3, 5) = aabaa
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
a a
ba a
b
L(3, 5) = aabaab
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
a a
ba a
ba
L(3, 5) = aabaaba
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Construction by Lattice Paths
Lower Christoffel word of slope 3
5
a a
ba a
ba
b
L(3, 5) = aabaabab
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 32
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Lyndon Words More on Factorisations
Christoffel Words
Lower & Upper Christoffel words of slope 3
5
a a
a a
a
b
b
b
a
a a
a a
b
b
b
L(3, 5) = aabaabab U (3, 5) = babaabaa
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 33
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Lyndon Words More on Factorisations
Remarks
Christoffel words recently came in handy for obtaining a completedescription of the minimal intervals containing all the fractional parts {x2n},n ≥ 0, for some positive real number x .
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 34
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Lyndon Words More on Factorisations
Remarks
Christoffel words recently came in handy for obtaining a completedescription of the minimal intervals containing all the fractional parts {x2n},n ≥ 0, for some positive real number x . [Allouche-Glen 2009]
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 34
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Lyndon Words More on Factorisations
Remarks
Christoffel words recently came in handy for obtaining a completedescription of the minimal intervals containing all the fractional parts {x2n},n ≥ 0, for some positive real number x . [Allouche-Glen 2009]
Christoffel words appeared in the literature as early as 1771 in JeanBernoulli’s study of continued fractions.
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 34
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Lyndon Words More on Factorisations
Remarks
Christoffel words recently came in handy for obtaining a completedescription of the minimal intervals containing all the fractional parts {x2n},n ≥ 0, for some positive real number x . [Allouche-Glen 2009]
Christoffel words appeared in the literature as early as 1771 in JeanBernoulli’s study of continued fractions.
Since then, many relationships between these particular Lyndon words andother areas of mathematics have been revealed.
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 34
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Lyndon Words More on Factorisations
Remarks
Christoffel words recently came in handy for obtaining a completedescription of the minimal intervals containing all the fractional parts {x2n},n ≥ 0, for some positive real number x . [Allouche-Glen 2009]
Christoffel words appeared in the literature as early as 1771 in JeanBernoulli’s study of continued fractions.
Since then, many relationships between these particular Lyndon words andother areas of mathematics have been revealed.
For instance, the words in {a, b}∗ that are conjugates of Christoffel wordsare exactly the positive primitive elements of the free group F2 = 〈a, b〉.[Kassel-Reutenauer 2007]
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 34
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Lyndon Words More on Factorisations
Remarks
Christoffel words recently came in handy for obtaining a completedescription of the minimal intervals containing all the fractional parts {x2n},n ≥ 0, for some positive real number x . [Allouche-Glen 2009]
Christoffel words appeared in the literature as early as 1771 in JeanBernoulli’s study of continued fractions.
Since then, many relationships between these particular Lyndon words andother areas of mathematics have been revealed.
For instance, the words in {a, b}∗ that are conjugates of Christoffel wordsare exactly the positive primitive elements of the free group F2 = 〈a, b〉.[Kassel-Reutenauer 2007]
Christoffel words also have connections to the theory of continued fractionsand Markoff numbers.
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 34
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Lyndon Words More on Factorisations
Remarks
Christoffel words recently came in handy for obtaining a completedescription of the minimal intervals containing all the fractional parts {x2n},n ≥ 0, for some positive real number x . [Allouche-Glen 2009]
Christoffel words appeared in the literature as early as 1771 in JeanBernoulli’s study of continued fractions.
Since then, many relationships between these particular Lyndon words andother areas of mathematics have been revealed.
For instance, the words in {a, b}∗ that are conjugates of Christoffel wordsare exactly the positive primitive elements of the free group F2 = 〈a, b〉.[Kassel-Reutenauer 2007]
Christoffel words also have connections to the theory of continued fractionsand Markoff numbers. [Markoff (1879, 1880), Cusick and Flahive (1989),Reutenauer (2006), Berstel et al. (2008)]
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 34
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Lyndon Words More on Factorisations
Remarks
Christoffel words recently came in handy for obtaining a completedescription of the minimal intervals containing all the fractional parts {x2n},n ≥ 0, for some positive real number x . [Allouche-Glen 2009]
Christoffel words appeared in the literature as early as 1771 in JeanBernoulli’s study of continued fractions.
Since then, many relationships between these particular Lyndon words andother areas of mathematics have been revealed.
For instance, the words in {a, b}∗ that are conjugates of Christoffel wordsare exactly the positive primitive elements of the free group F2 = 〈a, b〉.[Kassel-Reutenauer 2007]
Christoffel words also have connections to the theory of continued fractionsand Markoff numbers. [Markoff (1879, 1880), Cusick and Flahive (1989),Reutenauer (2006), Berstel et al. (2008)]
Nowadays they are studied in the context of Sturmian words
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 34
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Lyndon Words More on Factorisations
Remarks
Christoffel words recently came in handy for obtaining a completedescription of the minimal intervals containing all the fractional parts {x2n},n ≥ 0, for some positive real number x . [Allouche-Glen 2009]
Christoffel words appeared in the literature as early as 1771 in JeanBernoulli’s study of continued fractions.
Since then, many relationships between these particular Lyndon words andother areas of mathematics have been revealed.
For instance, the words in {a, b}∗ that are conjugates of Christoffel wordsare exactly the positive primitive elements of the free group F2 = 〈a, b〉.[Kassel-Reutenauer 2007]
Christoffel words also have connections to the theory of continued fractionsand Markoff numbers. [Markoff (1879, 1880), Cusick and Flahive (1989),Reutenauer (2006), Berstel et al. (2008)]
Nowadays they are studied in the context of Sturmian words, but that’s astory for another day . . .
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 34
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Thank You!
Amy Glen (MU, Perth) Combinatorics of Lyndon words February 2012 35