Post on 17-Jul-2020
Rich Words
Amy Glen
CRM-ISM Postdoctoral Fellow
LaCIM, Université du Québec à Montréal
amy.glen@gmail.comhttp://www.lacim.uqam.ca/∼glen
Le Séminaire du LaCIM
September 12, 2008
Amy Glen (LaCIM) Rich Words September 2008 1 / 22
Outline
1 Rich Words: A Brief Overview
2 Properties & Examples
3 Recent Results
4 Further Work
Amy Glen (LaCIM) Rich Words September 2008 2 / 22
Rich Words: A Brief Overview
Outline
1 Rich Words: A Brief Overview
2 Properties & Examples
3 Recent Results
4 Further Work
Amy Glen (LaCIM) Rich Words September 2008 3 / 22
Rich Words: A Brief Overview
What Are Rich Words?
Vague Answer: finite and infinite words that are “rich” in palindromesin the utmost sense.
Amy Glen (LaCIM) Rich Words September 2008 4 / 22
Rich Words: A Brief Overview
What Are Rich Words?
Vague Answer: finite and infinite words that are “rich” in palindromesin the utmost sense.
A palindrome is a finite word that reads the same backwards asforwards.
Amy Glen (LaCIM) Rich Words September 2008 4 / 22
Rich Words: A Brief Overview
What Are Rich Words?
Vague Answer: finite and infinite words that are “rich” in palindromesin the utmost sense.
A palindrome is a finite word that reads the same backwards asforwards.
English examples: eye, civic, radar, glenelg (Aussie suburb).
Amy Glen (LaCIM) Rich Words September 2008 4 / 22
Rich Words: A Brief Overview
What Are Rich Words?
Vague Answer: finite and infinite words that are “rich” in palindromesin the utmost sense.
A palindrome is a finite word that reads the same backwards asforwards.
English examples: eye, civic, radar, glenelg (Aussie suburb).
Droubay-Justin-Pirillo, 2001: any finite word w of length |w | containsat most |w | + 1 distinct palindromes (including the empty word ε).
Amy Glen (LaCIM) Rich Words September 2008 4 / 22
Rich Words: A Brief Overview
What Are Rich Words?
Vague Answer: finite and infinite words that are “rich” in palindromesin the utmost sense.
A palindrome is a finite word that reads the same backwards asforwards.
English examples: eye, civic, radar, glenelg (Aussie suburb).
Droubay-Justin-Pirillo, 2001: any finite word w of length |w | containsat most |w | + 1 distinct palindromes (including the empty word ε).
G.-Justin, 2007: initiated a unified study of finite and infinite wordsthat are characterized by containing the maximal number of distinctpalindromes, called rich words.
Amy Glen (LaCIM) Rich Words September 2008 4 / 22
Rich Words: A Brief Overview
What Are Rich Words?
Vague Answer: finite and infinite words that are “rich” in palindromesin the utmost sense.
A palindrome is a finite word that reads the same backwards asforwards.
English examples: eye, civic, radar, glenelg (Aussie suburb).
Droubay-Justin-Pirillo, 2001: any finite word w of length |w | containsat most |w | + 1 distinct palindromes (including the empty word ε).
G.-Justin, 2007: initiated a unified study of finite and infinite wordsthat are characterized by containing the maximal number of distinctpalindromes, called rich words.
Ambrož-Frougny-Masáková-Pelantová, 2005: independent work on “fullwords”, following earlier work of Brlek-Hamel-Nivat-Reutenauer, 2004.
Amy Glen (LaCIM) Rich Words September 2008 4 / 22
Rich Words: A Brief Overview
Formal Definitions & Examples
Definition
A finite word w is rich iff w contains exactly |w | + 1 distinct palindromes.
Amy Glen (LaCIM) Rich Words September 2008 5 / 22
Rich Words: A Brief Overview
Formal Definitions & Examples
Definition
A finite word w is rich iff w contains exactly |w | + 1 distinct palindromes.
Examples
abac is rich, whereas abca is not rich.
Amy Glen (LaCIM) Rich Words September 2008 5 / 22
Rich Words: A Brief Overview
Formal Definitions & Examples
Definition
A finite word w is rich iff w contains exactly |w | + 1 distinct palindromes.
Examples
abac is rich, whereas abca is not rich.
The word rich is rich . . . and poor is rich too!
Amy Glen (LaCIM) Rich Words September 2008 5 / 22
Rich Words: A Brief Overview
Formal Definitions & Examples
Definition
A finite word w is rich iff w contains exactly |w | + 1 distinct palindromes.
Examples
abac is rich, whereas abca is not rich.
The word rich is rich . . . and poor is rich too!
But plentiful is not rich.
Amy Glen (LaCIM) Rich Words September 2008 5 / 22
Rich Words: A Brief Overview
Formal Definitions & Examples
Definition
A finite word w is rich iff w contains exactly |w | + 1 distinct palindromes.
Examples
abac is rich, whereas abca is not rich.
The word rich is rich . . . and poor is rich too!
But plentiful is not rich.
Definition
An infinite word is rich iff all of its factors are rich.
Amy Glen (LaCIM) Rich Words September 2008 5 / 22
Rich Words: A Brief Overview
Formal Definitions & Examples
Definition
A finite word w is rich iff w contains exactly |w | + 1 distinct palindromes.
Examples
abac is rich, whereas abca is not rich.
The word rich is rich . . . and poor is rich too!
But plentiful is not rich.
Definition
An infinite word is rich iff all of its factors are rich.
Examples
aω = aaaaaa · · · and abω = abbb · · · are rich.
Amy Glen (LaCIM) Rich Words September 2008 5 / 22
Rich Words: A Brief Overview
Formal Definitions & Examples
Definition
A finite word w is rich iff w contains exactly |w | + 1 distinct palindromes.
Examples
abac is rich, whereas abca is not rich.
The word rich is rich . . . and poor is rich too!
But plentiful is not rich.
Definition
An infinite word is rich iff all of its factors are rich.
Examples
aω = aaaaaa · · · and abω = abbb · · · are rich.
(ab)ω = abababab · · · and (aba)ω = abaabaaba · · · are rich.
Amy Glen (LaCIM) Rich Words September 2008 5 / 22
Rich Words: A Brief Overview
Formal Definitions & Examples
Definition
A finite word w is rich iff w contains exactly |w | + 1 distinct palindromes.
Examples
abac is rich, whereas abca is not rich.
The word rich is rich . . . and poor is rich too!
But plentiful is not rich.
Definition
An infinite word is rich iff all of its factors are rich.
Examples
aω = aaaaaa · · · and abω = abbb · · · are rich.
(ab)ω = abababab · · · and (aba)ω = abaabaaba · · · are rich.
abc is rich, but (abc)ω = abcabcabc · · · is not rich.
Amy Glen (LaCIM) Rich Words September 2008 5 / 22
Properties & Examples
Outline
1 Rich Words: A Brief Overview
2 Properties & Examples
3 Recent Results
4 Further Work
Amy Glen (LaCIM) Rich Words September 2008 6 / 22
Properties & Examples
Characteristic Properties
Characteristic Property 1 (Droubay-Justin-Pirillo, 2001)
A finite word w is rich if and only if every prefix (resp. suffix) of w has aunioccurrent palindromic suffix (resp. prefix).
Amy Glen (LaCIM) Rich Words September 2008 7 / 22
Properties & Examples
Characteristic Properties
Characteristic Property 1 (Droubay-Justin-Pirillo, 2001)
A finite word w is rich if and only if every prefix (resp. suffix) of w has aunioccurrent palindromic suffix (resp. prefix).
To see this . . .
Let P(w) denote the number of palindromic factors of w .
Amy Glen (LaCIM) Rich Words September 2008 7 / 22
Properties & Examples
Characteristic Properties
Characteristic Property 1 (Droubay-Justin-Pirillo, 2001)
A finite word w is rich if and only if every prefix (resp. suffix) of w has aunioccurrent palindromic suffix (resp. prefix).
To see this . . .
Let P(w) denote the number of palindromic factors of w .
For any word u and letter x ,
P(ux) =
{P(u) if ux does not have a unioccurrent pali. suffix,P(u) + 1 if ux has a unioccurrent pali. suffix.
Amy Glen (LaCIM) Rich Words September 2008 7 / 22
Properties & Examples
Characteristic Properties
Characteristic Property 1 (Droubay-Justin-Pirillo, 2001)
A finite word w is rich if and only if every prefix (resp. suffix) of w has aunioccurrent palindromic suffix (resp. prefix).
To see this . . .
Let P(w) denote the number of palindromic factors of w .
For any word u and letter x ,
P(ux) =
{P(u) if ux does not have a unioccurrent pali. suffix,P(u) + 1 if ux has a unioccurrent pali. suffix.
Therefore, by induction, P(w) is the number of prefixes of w thathave a unioccurrent palindromic suffix.
Amy Glen (LaCIM) Rich Words September 2008 7 / 22
Properties & Examples
Characteristic Properties
Characteristic Property 1 (Droubay-Justin-Pirillo, 2001)
A finite word w is rich if and only if every prefix (resp. suffix) of w has aunioccurrent palindromic suffix (resp. prefix).
To see this . . .
Let P(w) denote the number of palindromic factors of w .
For any word u and letter x ,
P(ux) =
{P(u) if ux does not have a unioccurrent pali. suffix,P(u) + 1 if ux has a unioccurrent pali. suffix.
Therefore, by induction, P(w) is the number of prefixes of w thathave a unioccurrent palindromic suffix.
Hence P(w) ≤ |w | + 1.
Amy Glen (LaCIM) Rich Words September 2008 7 / 22
Properties & Examples
Characteristic Properties
Characteristic Property 1 (Droubay-Justin-Pirillo, 2001)
A finite word w is rich if and only if every prefix (resp. suffix) of w has aunioccurrent palindromic suffix (resp. prefix).
To see this . . .
Let P(w) denote the number of palindromic factors of w .
For any word u and letter x ,
P(ux) =
{P(u) if ux does not have a unioccurrent pali. suffix,P(u) + 1 if ux has a unioccurrent pali. suffix.
Therefore, by induction, P(w) is the number of prefixes of w thathave a unioccurrent palindromic suffix.
Hence P(w) ≤ |w | + 1.
In particular P(w) = |w | + 1 (i.e., w is rich) if and only if each prefixof w has a unioccurrent palindromic suffix.
Amy Glen (LaCIM) Rich Words September 2008 7 / 22
Properties & Examples
Characteristic Properties
Infinite case of Characteristic Property 1:
An infinite word w is rich if and only if every prefix of w has aunioccurrent palindromic suffix.
Amy Glen (LaCIM) Rich Words September 2008 8 / 22
Properties & Examples
Characteristic Properties
Infinite case of Characteristic Property 1:
An infinite word w is rich if and only if every prefix of w has aunioccurrent palindromic suffix.
A new palindrome is introduced at each position in a rich word.
Amy Glen (LaCIM) Rich Words September 2008 8 / 22
Properties & Examples
Characteristic Properties
Infinite case of Characteristic Property 1:
An infinite word w is rich if and only if every prefix of w has aunioccurrent palindromic suffix.
A new palindrome is introduced at each position in a rich word.
Example: abaabaaabaaaabaaaaab · · ·
Amy Glen (LaCIM) Rich Words September 2008 8 / 22
Properties & Examples
Characteristic Properties
Infinite case of Characteristic Property 1:
An infinite word w is rich if and only if every prefix of w has aunioccurrent palindromic suffix.
A new palindrome is introduced at each position in a rich word.
Example: abaabaaabaaaabaaaaab · · ·
Amy Glen (LaCIM) Rich Words September 2008 8 / 22
Properties & Examples
Characteristic Properties
Infinite case of Characteristic Property 1:
An infinite word w is rich if and only if every prefix of w has aunioccurrent palindromic suffix.
A new palindrome is introduced at each position in a rich word.
Example: abaabaaabaaaabaaaaab · · ·
Amy Glen (LaCIM) Rich Words September 2008 8 / 22
Properties & Examples
Characteristic Properties
Infinite case of Characteristic Property 1:
An infinite word w is rich if and only if every prefix of w has aunioccurrent palindromic suffix.
A new palindrome is introduced at each position in a rich word.
Example: abaabaaabaaaabaaaaab · · ·
Amy Glen (LaCIM) Rich Words September 2008 8 / 22
Properties & Examples
Characteristic Properties
Infinite case of Characteristic Property 1:
An infinite word w is rich if and only if every prefix of w has aunioccurrent palindromic suffix.
A new palindrome is introduced at each position in a rich word.
Example: abaabaaabaaaabaaaaab · · ·
Amy Glen (LaCIM) Rich Words September 2008 8 / 22
Properties & Examples
Characteristic Properties
Infinite case of Characteristic Property 1:
An infinite word w is rich if and only if every prefix of w has aunioccurrent palindromic suffix.
A new palindrome is introduced at each position in a rich word.
Example: abaabaaabaaaabaaaaab · · ·
Amy Glen (LaCIM) Rich Words September 2008 8 / 22
Properties & Examples
Characteristic Properties
Infinite case of Characteristic Property 1:
An infinite word w is rich if and only if every prefix of w has aunioccurrent palindromic suffix.
A new palindrome is introduced at each position in a rich word.
Example: abaabaaabaaaabaaaaab · · ·
Amy Glen (LaCIM) Rich Words September 2008 8 / 22
Properties & Examples
Characteristic Properties
Infinite case of Characteristic Property 1:
An infinite word w is rich if and only if every prefix of w has aunioccurrent palindromic suffix.
A new palindrome is introduced at each position in a rich word.
Example: abaabaaabaaaabaaaaab · · ·
Amy Glen (LaCIM) Rich Words September 2008 8 / 22
Properties & Examples
Characteristic Properties
Infinite case of Characteristic Property 1:
An infinite word w is rich if and only if every prefix of w has aunioccurrent palindromic suffix.
A new palindrome is introduced at each position in a rich word.
Example: abaabaaabaaaabaaaaab · · ·
Amy Glen (LaCIM) Rich Words September 2008 8 / 22
Properties & Examples
Characteristic Properties
Infinite case of Characteristic Property 1:
An infinite word w is rich if and only if every prefix of w has aunioccurrent palindromic suffix.
A new palindrome is introduced at each position in a rich word.
Example: abaabaaabaaaabaaaaab · · ·
Amy Glen (LaCIM) Rich Words September 2008 8 / 22
Properties & Examples
Characteristic Properties
Infinite case of Characteristic Property 1:
An infinite word w is rich if and only if every prefix of w has aunioccurrent palindromic suffix.
A new palindrome is introduced at each position in a rich word.
Example: abaabaaabaaaabaaaaab · · ·
Amy Glen (LaCIM) Rich Words September 2008 8 / 22
Properties & Examples
Characteristic Properties
Infinite case of Characteristic Property 1:
An infinite word w is rich if and only if every prefix of w has aunioccurrent palindromic suffix.
A new palindrome is introduced at each position in a rich word.
Example: abaabaaabaaaabaaaaab · · ·
Characteristic Property 2 (Droubay-Justin-Pirillo, 2001)
A finite or infinite word w is rich if and only if for each factor u of w , everyprefix (resp. suffix) of u has a unioccurrent palindromic suffix (resp. prefix).
Amy Glen (LaCIM) Rich Words September 2008 8 / 22
Properties & Examples
Characteristic Properties
Let u be a factor of a finite or infinite word w .A complete return to u in w is a factor of w having exactly twooccurrences of u, one as a prefix and one as a suffix.
Amy Glen (LaCIM) Rich Words September 2008 9 / 22
Properties & Examples
Characteristic Properties
Let u be a factor of a finite or infinite word w .A complete return to u in w is a factor of w having exactly twooccurrences of u, one as a prefix and one as a suffix.Example: aabcbaa is a complete return to aa in aabcbaaba (rich).
Amy Glen (LaCIM) Rich Words September 2008 9 / 22
Properties & Examples
Characteristic Properties
Let u be a factor of a finite or infinite word w .A complete return to u in w is a factor of w having exactly twooccurrences of u, one as a prefix and one as a suffix.Example: aabcbaa is a complete return to aa in aabcbaaba (rich).
Characteristic Property 3 (G.-Justin, 2007)
A finite or infinite word w is rich if and only if for each palindromic factor p
of w , every complete return to p in w is a palindrome.
Amy Glen (LaCIM) Rich Words September 2008 9 / 22
Properties & Examples
Characteristic Properties
Let u be a factor of a finite or infinite word w .A complete return to u in w is a factor of w having exactly twooccurrences of u, one as a prefix and one as a suffix.Example: aabcbaa is a complete return to aa in aabcbaaba (rich).
Characteristic Property 3 (G.-Justin, 2007)
A finite or infinite word w is rich if and only if for each palindromic factor p
of w , every complete return to p in w is a palindrome.
Proof:
(⇒): Suppose w is rich, but contains a non-palindromic completereturn r to a palindrome p.
Amy Glen (LaCIM) Rich Words September 2008 9 / 22
Properties & Examples
Characteristic Properties
Let u be a factor of a finite or infinite word w .A complete return to u in w is a factor of w having exactly twooccurrences of u, one as a prefix and one as a suffix.Example: aabcbaa is a complete return to aa in aabcbaaba (rich).
Characteristic Property 3 (G.-Justin, 2007)
A finite or infinite word w is rich if and only if for each palindromic factor p
of w , every complete return to p in w is a palindrome.
Proof:
(⇒): Suppose w is rich, but contains a non-palindromic completereturn r to a palindrome p.Then r = pup for some non-palindromic word u.
Amy Glen (LaCIM) Rich Words September 2008 9 / 22
Properties & Examples
Characteristic Properties
Let u be a factor of a finite or infinite word w .A complete return to u in w is a factor of w having exactly twooccurrences of u, one as a prefix and one as a suffix.Example: aabcbaa is a complete return to aa in aabcbaaba (rich).
Characteristic Property 3 (G.-Justin, 2007)
A finite or infinite word w is rich if and only if for each palindromic factor p
of w , every complete return to p in w is a palindrome.
Proof:
(⇒): Suppose w is rich, but contains a non-palindromic completereturn r to a palindrome p.Then r = pup for some non-palindromic word u.But then r does not have a unioccurrent palindromic suffix, acontradiction.
Amy Glen (LaCIM) Rich Words September 2008 9 / 22
Properties & Examples
Characteristic Properties
Let u be a factor of a finite or infinite word w .A complete return to u in w is a factor of w having exactly twooccurrences of u, one as a prefix and one as a suffix.Example: aabcbaa is a complete return to aa in aabcbaaba (rich).
Characteristic Property 3 (G.-Justin, 2007)
A finite or infinite word w is rich if and only if for each palindromic factor p
of w , every complete return to p in w is a palindrome.
Proof:
(⇐): Suppose not. Let u be a factor of w of minimal length such thatu is not rich.
Amy Glen (LaCIM) Rich Words September 2008 9 / 22
Properties & Examples
Characteristic Properties
Let u be a factor of a finite or infinite word w .A complete return to u in w is a factor of w having exactly twooccurrences of u, one as a prefix and one as a suffix.Example: aabcbaa is a complete return to aa in aabcbaaba (rich).
Characteristic Property 3 (G.-Justin, 2007)
A finite or infinite word w is rich if and only if for each palindromic factor p
of w , every complete return to p in w is a palindrome.
Proof:
(⇐): Suppose not. Let u be a factor of w of minimal length such thatu is not rich.Then u = xvy with x , y letters. By minimality xv is rich, and thelongest palindromic suffix p of u occurs more than once in u.
Amy Glen (LaCIM) Rich Words September 2008 9 / 22
Properties & Examples
Characteristic Properties
Let u be a factor of a finite or infinite word w .A complete return to u in w is a factor of w having exactly twooccurrences of u, one as a prefix and one as a suffix.Example: aabcbaa is a complete return to aa in aabcbaaba (rich).
Characteristic Property 3 (G.-Justin, 2007)
A finite or infinite word w is rich if and only if for each palindromic factor p
of w , every complete return to p in w is a palindrome.
Proof:
(⇐): Suppose not. Let u be a factor of w of minimal length such thatu is not rich.Then u = xvy with x , y letters. By minimality xv is rich, and thelongest palindromic suffix p of u occurs more than once in u.Since all complete returns to palindromes are palindromes, we reach acontradiction to the maximality of p.Amy Glen (LaCIM) Rich Words September 2008 9 / 22
Properties & Examples
Characteristic Properties
Let v denote the reversal of a word v . Example: v = abc , v = cba.
Characteristic Property 4 (Bucci-De Luca-G.-Zamboni, 2008)
For any finite or infinite word w , the following conditions are equivalent:
i) w is rich;
ii) for each factor v of w , every factor of w beginning with v and endingwith v and containing no other occurrences of v or v is a palindrome.
Amy Glen (LaCIM) Rich Words September 2008 10 / 22
Properties & Examples
Characteristic Properties
Let v denote the reversal of a word v . Example: v = abc , v = cba.
Characteristic Property 4 (Bucci-De Luca-G.-Zamboni, 2008)
For any finite or infinite word w , the following conditions are equivalent:
i) w is rich;
ii) for each factor v of w , every factor of w beginning with v and endingwith v and containing no other occurrences of v or v is a palindrome.
Proof:
i) ⇒ ii): Let u = v · · · v . If v is a palindrome, then u is a palindrome.
Amy Glen (LaCIM) Rich Words September 2008 10 / 22
Properties & Examples
Characteristic Properties
Let v denote the reversal of a word v . Example: v = abc , v = cba.
Characteristic Property 4 (Bucci-De Luca-G.-Zamboni, 2008)
For any finite or infinite word w , the following conditions are equivalent:
i) w is rich;
ii) for each factor v of w , every factor of w beginning with v and endingwith v and containing no other occurrences of v or v is a palindrome.
Proof:
i) ⇒ ii): Let u = v · · · v . If v is a palindrome, then u is a palindrome.Otherwise, for non-palindromic v , suppose u is not a palindrome . . .
Amy Glen (LaCIM) Rich Words September 2008 10 / 22
Properties & Examples
Characteristic Properties
Let v denote the reversal of a word v . Example: v = abc , v = cba.
Characteristic Property 4 (Bucci-De Luca-G.-Zamboni, 2008)
For any finite or infinite word w , the following conditions are equivalent:
i) w is rich;
ii) for each factor v of w , every factor of w beginning with v and endingwith v and containing no other occurrences of v or v is a palindrome.
Proof:
i) ⇒ ii): Let u = v · · · v . If v is a palindrome, then u is a palindrome.Otherwise, for non-palindromic v , suppose u is not a palindrome . . .
u =
v v
p p
· · ·
· · · · · ·
(p is longest palindromic suffix of u)
︸ ︷︷ ︸
complete return to p → palindrome
Amy Glen (LaCIM) Rich Words September 2008 10 / 22
Properties & Examples
Characteristic Properties
Let v denote the reversal of a word v . Example: v = abc , v = cba.
Characteristic Property 4 (Bucci-De Luca-G.-Zamboni, 2008)
For any finite or infinite word w , the following conditions are equivalent:
i) w is rich;
ii) for each factor v of w , every factor of w beginning with v and endingwith v and containing no other occurrences of v or v is a palindrome.
Proof:
i) ⇒ ii): Let u = v · · · v . If v is a palindrome, then u is a palindrome.Otherwise, for non-palindromic v , suppose u is not a palindrome . . .
u =
v v
p p
· · ·
· · · · · ·
(p is longest palindromic suffix of u)
︸ ︷︷ ︸
complete return to p → palindrome
. . . contradiction!
Amy Glen (LaCIM) Rich Words September 2008 10 / 22
Properties & Examples
Characteristic Properties
Let v denote the reversal of a word v . Example: v = abc , v = cba.
Characteristic Property 4 (Bucci-De Luca-G.-Zamboni, 2008)
For any finite or infinite word w , the following conditions are equivalent:
i) w is rich;
ii) for each factor v of w , every factor of w beginning with v and endingwith v and containing no other occurrences of v or v is a palindrome.
Proof:
Conversely, ii) ⇒ every complete return to a palindromic factor v
(= v) is a palindrome.
Amy Glen (LaCIM) Rich Words September 2008 10 / 22
Properties & Examples
Characteristic Properties
Let v denote the reversal of a word v . Example: v = abc , v = cba.
Characteristic Property 4 (Bucci-De Luca-G.-Zamboni, 2008)
For any finite or infinite word w , the following conditions are equivalent:
i) w is rich;
ii) for each factor v of w , every factor of w beginning with v and endingwith v and containing no other occurrences of v or v is a palindrome.
Proof:
Conversely, ii) ⇒ every complete return to a palindromic factor v
(= v) is a palindrome.
Thus w is rich by Characteristic Property 3.
Amy Glen (LaCIM) Rich Words September 2008 10 / 22
Properties & Examples
Rich Examples
Purely Periodic Rich Infinite Words
(abcba)ω = abcbaabcbaabcba · · ·
Amy Glen (LaCIM) Rich Words September 2008 11 / 22
Properties & Examples
Rich Examples
Purely Periodic Rich Infinite Words
(abcba)ω = abcbaabcbaabcba · · ·
(aabkaabab)ω = aabkaababaabkaababaabkaabab · · · with k ≥ 0
Amy Glen (LaCIM) Rich Words September 2008 11 / 22
Properties & Examples
Rich Examples
Purely Periodic Rich Infinite Words
(abcba)ω = abcbaabcbaabcba · · ·
(aabkaabab)ω = aabkaababaabkaababaabkaabab · · · with k ≥ 0
vω = vvv · · · is rich ⇔ v2 is rich
Amy Glen (LaCIM) Rich Words September 2008 11 / 22
Properties & Examples
Rich Examples
Purely Periodic Rich Infinite Words
(abcba)ω = abcbaabcbaabcba · · ·
(aabkaabab)ω = aabkaababaabkaababaabkaabab · · · with k ≥ 0
vω = vvv · · · is rich ⇔ v = pq & all circular shifts are rich
Amy Glen (LaCIM) Rich Words September 2008 11 / 22
Properties & Examples
Rich Examples
Purely Periodic Rich Infinite Words
(abcba)ω = abcbaabcbaabcba · · ·
(aabkaabab)ω = aabkaababaabkaababaabkaabab · · · with k ≥ 0
vω = vvv · · · is rich ⇔ v = pq & all circular shifts are rich
Other Rich Infinite Words
abcdω = abcddd · · ·
Amy Glen (LaCIM) Rich Words September 2008 11 / 22
Properties & Examples
Rich Examples
Purely Periodic Rich Infinite Words
(abcba)ω = abcbaabcbaabcba · · ·
(aabkaabab)ω = aabkaababaabkaababaabkaabab · · · with k ≥ 0
vω = vvv · · · is rich ⇔ v = pq & all circular shifts are rich
Other Rich Infinite Words
abcdω = abcddd · · ·
aba2ba3ba4ba5b · · ·
Amy Glen (LaCIM) Rich Words September 2008 11 / 22
Properties & Examples
Rich Examples
Purely Periodic Rich Infinite Words
(abcba)ω = abcbaabcbaabcba · · ·
(aabkaabab)ω = aabkaababaabkaababaabkaabab · · · with k ≥ 0
vω = vvv · · · is rich ⇔ v = pq & all circular shifts are rich
Other Rich Infinite Words
abcdω = abcddd · · ·
aba2ba3ba4ba5b · · ·
limn→∞ σn(a) = ababbababbbbababbababbbbbbbbababbaba · · ·where σ : a 7→ aba, b 7→ bb (Cassaigne, 1997).
Amy Glen (LaCIM) Rich Words September 2008 11 / 22
Properties & Examples
Rich Examples
Purely Periodic Rich Infinite Words
(abcba)ω = abcbaabcbaabcba · · ·
(aabkaabab)ω = aabkaababaabkaababaabkaabab · · · with k ≥ 0
vω = vvv · · · is rich ⇔ v = pq & all circular shifts are rich
Other Rich Infinite Words
abcdω = abcddd · · ·
aba2ba3ba4ba5b · · ·
limn→∞ σn(a) = ababbababbbbababbababbbbbbbbababbaba · · ·where σ : a 7→ aba, b 7→ bb (Cassaigne, 1997).
Fibonacci word: f = limn→∞ ϕn(a) = abaababaabaababaaba · · ·where ϕ : a 7→ ab, b 7→ a.
Amy Glen (LaCIM) Rich Words September 2008 11 / 22
Properties & Examples
Rich Examples
Purely Periodic Rich Infinite Words
(abcba)ω = abcbaabcbaabcba · · ·
(aabkaabab)ω = aabkaababaabkaababaabkaabab · · · with k ≥ 0
vω = vvv · · · is rich ⇔ v = pq & all circular shifts are rich
Other Rich Infinite Words
abcdω = abcddd · · ·
aba2ba3ba4ba5b · · ·
limn→∞ σn(a) = ababbababbbbababbababbbbbbbbababbaba · · ·where σ : a 7→ aba, b 7→ bb (Cassaigne, 1997).
Fibonacci word: f = limn→∞ ϕn(a) = abaababaabaababaaba · · ·where ϕ : a 7→ ab, b 7→ a.
Tribonacci word: r = limn→∞ θn(a) = abacabaabacababacabaaba · · ·where θ : a 7→ ab, b 7→ ac , c 7→ a.
Amy Glen (LaCIM) Rich Words September 2008 11 / 22
Properties & Examples
Rich Examples
Purely Periodic Rich Infinite Words
(abcba)ω = abcbaabcbaabcba · · ·
(aabkaabab)ω = aabkaababaabkaababaabkaabab · · · with k ≥ 0
vω = vvv · · · is rich ⇔ v = pq & all circular shifts are rich
Other Rich Infinite Words
abcdω = abcddd · · ·
aba2ba3ba4ba5b · · ·
limn→∞ σn(a) = ababbababbbbababbababbbbbbbbababbaba · · ·where σ : a 7→ aba, b 7→ bb (Cassaigne, 1997).
Fibonacci word: f = limn→∞ ϕn(a) = abaababaabaababaaba · · ·where ϕ : a 7→ ab, b 7→ a.
Tribonacci word: r = limn→∞ θn(a) = abacabaabacababacabaaba · · ·where θ : a 7→ ab, b 7→ ac , c 7→ a.
ψk(f) where ψk : a 7→ aabkaabab, b 7→ bab, k ≥ 0.
Amy Glen (LaCIM) Rich Words September 2008 11 / 22
Properties & Examples
More General Examples
Rich words have appeared in many different contexts; they include:
Sturmian and episturmian wordsDroubay-Justin-Pirillo, 2001: characteristic property 1
Anne-Zamboni-Zorca, 2005: characteristic property 3
Bucci-De Luca-G.-Zamboni, 2008: characterization of recurrent rich infinite words
Complementation-symmetric Rote sequencesAllouche-Baake-Cassaigne-Damanik, 2003 + Bucci-De Luca-G.-Zamboni, 2008
Symbolic codings of trajectories of symmetric interval exchangetransformations – Ferenczi-Zamboni, 2008
A certain class of words associated with β-expansions where β is asimple Parry numberAmbrož-Frougny-Masáková-Pelantová, 2006 + Bucci-De Luca-G.-Zamboni, 2008
Infinite words with “abundant palindromic prefixes”Introduced by Fischler in 2006 in relation to Diophantine approximation
Amy Glen (LaCIM) Rich Words September 2008 12 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:(race)+ =
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:(race)+ = race
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:(race)+ = race car
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:(race)+ = race car
(tops)+ =
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:(race)+ = race car
(tops)+ = top s
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:(race)+ = race car
(tops)+ = top spot
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:(race)+ = race car
(tops)+ = top spot
(party)+ =
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:(race)+ = race car
(tops)+ = top spot
(party)+ = party
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:(race)+ = race car
(tops)+ = top spot
(party)+ = party trap
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:(race)+ = race car
(tops)+ = top spot
(party)+ = party trap
(tie)+ =
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:(race)+ = race car
(tops)+ = top spot
(party)+ = party trap
(tie)+ = tie
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:(race)+ = race car
(tops)+ = top spot
(party)+ = party trap
(tie)+ = tie it
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:(race)+ = race car
(tops)+ = top spot
(party)+ = party trap
(tie)+ = tie it
(abac)+ =
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:(race)+ = race car
(tops)+ = top spot
(party)+ = party trap
(tie)+ = tie it
(abac)+ = abac
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Basic Properties & Results (G.-Justin, 2007)
If a finite word w is rich, then its reversal w is also rich.
Example: w = aabac and w = cabaa are both rich.
If w and w ′ are rich with the same set of palindromic factors, thenthey are abelianly equivalent, i.e., |w |x = |w ′|x for all letters x .
Palindromic closure preserves richness.
The palindromic closure of a word v , denoted by v+, is the uniqueshortest palindrome beginning with v .
Examples:(race)+ = race car
(tops)+ = top spot
(party)+ = party trap
(tie)+ = tie it
(abac)+ = abac aba
Amy Glen (LaCIM) Rich Words September 2008 13 / 22
Properties & Examples
Sturmian Words Are Rich
Theorem (de Luca, 1997)
An infinite word s over {a, b} is a standard Sturmian word if and only ifthere exists an infinite word ∆ = x1x2x3 · · · over {a, b} (not of the formuaω or ubω), called the directive word of s , such that
s = limn→∞
Pal(x1x2 · · · xn).
Amy Glen (LaCIM) Rich Words September 2008 14 / 22
Properties & Examples
Sturmian Words Are Rich
Theorem (de Luca, 1997)
An infinite word s over {a, b} is a standard Sturmian word if and only ifthere exists an infinite word ∆ = x1x2x3 · · · over {a, b} (not of the formuaω or ubω), called the directive word of s , such that
s = limn→∞
Pal(x1x2 · · · xn).
Pal is the iterated palindromic closure function:
Pal(ε) = ε (empty word) and Pal(wx) = (Pal(w)x)+
for any word w and letter x . Example: Pal(aba) =
Amy Glen (LaCIM) Rich Words September 2008 14 / 22
Properties & Examples
Sturmian Words Are Rich
Theorem (de Luca, 1997)
An infinite word s over {a, b} is a standard Sturmian word if and only ifthere exists an infinite word ∆ = x1x2x3 · · · over {a, b} (not of the formuaω or ubω), called the directive word of s , such that
s = limn→∞
Pal(x1x2 · · · xn).
Pal is the iterated palindromic closure function:
Pal(ε) = ε (empty word) and Pal(wx) = (Pal(w)x)+
for any word w and letter x . Example: Pal(aba) =a
Amy Glen (LaCIM) Rich Words September 2008 14 / 22
Properties & Examples
Sturmian Words Are Rich
Theorem (de Luca, 1997)
An infinite word s over {a, b} is a standard Sturmian word if and only ifthere exists an infinite word ∆ = x1x2x3 · · · over {a, b} (not of the formuaω or ubω), called the directive word of s , such that
s = limn→∞
Pal(x1x2 · · · xn).
Pal is the iterated palindromic closure function:
Pal(ε) = ε (empty word) and Pal(wx) = (Pal(w)x)+
for any word w and letter x . Example: Pal(aba) =a b
Amy Glen (LaCIM) Rich Words September 2008 14 / 22
Properties & Examples
Sturmian Words Are Rich
Theorem (de Luca, 1997)
An infinite word s over {a, b} is a standard Sturmian word if and only ifthere exists an infinite word ∆ = x1x2x3 · · · over {a, b} (not of the formuaω or ubω), called the directive word of s , such that
s = limn→∞
Pal(x1x2 · · · xn).
Pal is the iterated palindromic closure function:
Pal(ε) = ε (empty word) and Pal(wx) = (Pal(w)x)+
for any word w and letter x . Example: Pal(aba) =a b a
Amy Glen (LaCIM) Rich Words September 2008 14 / 22
Properties & Examples
Sturmian Words Are Rich
Theorem (de Luca, 1997)
An infinite word s over {a, b} is a standard Sturmian word if and only ifthere exists an infinite word ∆ = x1x2x3 · · · over {a, b} (not of the formuaω or ubω), called the directive word of s , such that
s = limn→∞
Pal(x1x2 · · · xn).
Pal is the iterated palindromic closure function:
Pal(ε) = ε (empty word) and Pal(wx) = (Pal(w)x)+
for any word w and letter x . Example: Pal(aba) =a b a a
Amy Glen (LaCIM) Rich Words September 2008 14 / 22
Properties & Examples
Sturmian Words Are Rich
Theorem (de Luca, 1997)
An infinite word s over {a, b} is a standard Sturmian word if and only ifthere exists an infinite word ∆ = x1x2x3 · · · over {a, b} (not of the formuaω or ubω), called the directive word of s , such that
s = limn→∞
Pal(x1x2 · · · xn).
Pal is the iterated palindromic closure function:
Pal(ε) = ε (empty word) and Pal(wx) = (Pal(w)x)+
for any word w and letter x . Example: Pal(aba) =a b a a b a
Amy Glen (LaCIM) Rich Words September 2008 14 / 22
Properties & Examples
Sturmian Words Are Rich
Theorem (de Luca, 1997)
An infinite word s over {a, b} is a standard Sturmian word if and only ifthere exists an infinite word ∆ = x1x2x3 · · · over {a, b} (not of the formuaω or ubω), called the directive word of s , such that
s = limn→∞
Pal(x1x2 · · · xn).
Pal is the iterated palindromic closure function:
Pal(ε) = ε (empty word) and Pal(wx) = (Pal(w)x)+
for any word w and letter x . Example: Pal(aba) =a b a a b a
The Fibonacci word is directed by ∆ = (ab)ω = ababab · · · .
That is: f = Pal(ababab · · · ) = abaababaaba · · · .
Amy Glen (LaCIM) Rich Words September 2008 14 / 22
Properties & Examples
Sturmian Words Are Rich
Theorem (de Luca, 1997)
An infinite word s over {a, b} is a standard Sturmian word if and only ifthere exists an infinite word ∆ = x1x2x3 · · · over {a, b} (not of the formuaω or ubω), called the directive word of s , such that
s = limn→∞
Pal(x1x2 · · · xn).
Pal is the iterated palindromic closure function:
Pal(ε) = ε (empty word) and Pal(wx) = (Pal(w)x)+
for any word w and letter x . Example: Pal(aba) =a b a a b a
The Fibonacci word is directed by ∆ = (ab)ω = ababab · · · .
That is: f = Pal(ababab · · · ) = abaababaaba · · · .
Palindromic closure preserves richness ⇒ Pal does too ⇒ Sturmianwords are RICH.Amy Glen (LaCIM) Rich Words September 2008 14 / 22
Properties & Examples
Episturmian Words Are Rich Too
{a, b} −→ A (finite alphabet) gives standard episturmian words.
Theorem (Droubay-Justin-Pirillo, 2001)
An infinite word s over A is a standard episturmian word if and only ifthere exists an infinite word ∆ = x1x2x3 · · · over A such that
s = limn→∞
Pal(x1x2 · · · xn).
Amy Glen (LaCIM) Rich Words September 2008 15 / 22
Properties & Examples
Episturmian Words Are Rich Too
{a, b} −→ A (finite alphabet) gives standard episturmian words.
Theorem (Droubay-Justin-Pirillo, 2001)
An infinite word s over A is a standard episturmian word if and only ifthere exists an infinite word ∆ = x1x2x3 · · · over A such that
s = limn→∞
Pal(x1x2 · · · xn).
Example
∆ = (abc)ω = abcabcabc · · · directs the Tribonacci word:
r = abacabaabacababacabaabacabacabaabaca · · ·
Amy Glen (LaCIM) Rich Words September 2008 15 / 22
Recent Results
Outline
1 Rich Words: A Brief Overview
2 Properties & Examples
3 Recent Results
4 Further Work
Amy Glen (LaCIM) Rich Words September 2008 16 / 22
Recent Results
A Connection Between Palindromic & Factor Complexity
Let w be a finite or infinite word.
Amy Glen (LaCIM) Rich Words September 2008 17 / 22
Recent Results
A Connection Between Palindromic & Factor Complexity
Let w be a finite or infinite word.
Palindromic complexity function Pw (n): counts the number of distinctpalindromic factors of w of length n for each n ∈ N.
Amy Glen (LaCIM) Rich Words September 2008 17 / 22
Recent Results
A Connection Between Palindromic & Factor Complexity
Let w be a finite or infinite word.
Palindromic complexity function Pw (n): counts the number of distinctpalindromic factors of w of length n for each n ∈ N.
Factor complexity function Cw (n): counts the number of distinctfactors of w of length n for each n ∈ N.
Amy Glen (LaCIM) Rich Words September 2008 17 / 22
Recent Results
A Connection Between Palindromic & Factor Complexity
Let w be a finite or infinite word.
Palindromic complexity function Pw (n): counts the number of distinctpalindromic factors of w of length n for each n ∈ N.
Factor complexity function Cw (n): counts the number of distinctfactors of w of length n for each n ∈ N.
Allouche-Baake-Cassaigne-Damanik, 2003: for any aperiodic infiniteword w,
Pw(n) ≤16
nCw
(n +
⌊n
4
⌋)for all n ∈ N.
Amy Glen (LaCIM) Rich Words September 2008 17 / 22
Recent Results
A Connection Between Palindromic & Factor Complexity
Let w be a finite or infinite word.
Palindromic complexity function Pw (n): counts the number of distinctpalindromic factors of w of length n for each n ∈ N.
Factor complexity function Cw (n): counts the number of distinctfactors of w of length n for each n ∈ N.
Allouche-Baake-Cassaigne-Damanik, 2003: for any aperiodic infiniteword w,
Pw(n) ≤16
nCw
(n +
⌊n
4
⌋)for all n ∈ N.
Baláži-Masáková-Pelantová, 2007: for any uniformly recurrent infinite wordw with F (w) closed under reversal,
Pw(n) + Pw(n + 1) ≤ Cw(n + 1) − Cw(n) + 2 for all n ∈ N. (∗)
Amy Glen (LaCIM) Rich Words September 2008 17 / 22
Recent Results
A Connection Between Palindromic & Factor Complexity
Let w be a finite or infinite word.
Palindromic complexity function Pw (n): counts the number of distinctpalindromic factors of w of length n for each n ∈ N.
Factor complexity function Cw (n): counts the number of distinctfactors of w of length n for each n ∈ N.
Allouche-Baake-Cassaigne-Damanik, 2003: for any aperiodic infiniteword w,
Pw(n) ≤16
nCw
(n +
⌊n
4
⌋)for all n ∈ N.
Baláži-Masáková-Pelantová, 2007: for any uniformly recurrent infinite wordw with F (w) closed under reversal,
Pw(n) + Pw(n + 1) ≤ Cw(n + 1) − Cw(n) + 2 for all n ∈ N. (∗)
Bucci-De Luca-G.-Zamboni, 2008: infinite words w for whichPw(n) +Pw(n + 1) reaches the upper bound in (∗) for every n are rich . . .
Amy Glen (LaCIM) Rich Words September 2008 17 / 22
Recent Results
A Connection Between Palindromic & Factor Complexity
Theorem A (Bucci-De Luca-G.-Zamboni, 2008)
For any infinite word w with set of factors F (w) closed under reversal, thefollowing conditions are equivalent:
(I) all complete returns to any palindrome in w are palindromes;
(II) Pw(n) + Pw(n + 1) = Cw(n + 1) − Cw(n) + 2 for all n ∈ N.
Amy Glen (LaCIM) Rich Words September 2008 18 / 22
Recent Results
A Connection Between Palindromic & Factor Complexity
Theorem A (Bucci-De Luca-G.-Zamboni, 2008)
For any infinite word w with set of factors F (w) closed under reversal, thefollowing conditions are equivalent:
(I) all complete returns to any palindrome in w are palindromes;
(II) Pw(n) + Pw(n + 1) = Cw(n + 1) − Cw(n) + 2 for all n ∈ N.
Complementation-symmetric Rote sequences:
Infinite words over {a, b} with factors closed under bothcomplementation and reversal, and such that C(n) = 2n for all n.
Amy Glen (LaCIM) Rich Words September 2008 18 / 22
Recent Results
A Connection Between Palindromic & Factor Complexity
Theorem A (Bucci-De Luca-G.-Zamboni, 2008)
For any infinite word w with set of factors F (w) closed under reversal, thefollowing conditions are equivalent:
(I) all complete returns to any palindrome in w are palindromes;
(II) Pw(n) + Pw(n + 1) = Cw(n + 1) − Cw(n) + 2 for all n ∈ N.
Complementation-symmetric Rote sequences:
Infinite words over {a, b} with factors closed under bothcomplementation and reversal, and such that C(n) = 2n for all n.
Allouche-Baake-Cassaigne-Damanik, 2003: P(n) = 2 for all n.
Amy Glen (LaCIM) Rich Words September 2008 18 / 22
Recent Results
A Connection Between Palindromic & Factor Complexity
Theorem A (Bucci-De Luca-G.-Zamboni, 2008)
For any infinite word w with set of factors F (w) closed under reversal, thefollowing conditions are equivalent:
(I) all complete returns to any palindrome in w are palindromes;
(II) Pw(n) + Pw(n + 1) = Cw(n + 1) − Cw(n) + 2 for all n ∈ N.
Complementation-symmetric Rote sequences:
Infinite words over {a, b} with factors closed under bothcomplementation and reversal, and such that C(n) = 2n for all n.
Allouche-Baake-Cassaigne-Damanik, 2003: P(n) = 2 for all n.
Hence P(n) +P(n + 1) = 4 = C (n + 1)−C (n) + 2 for all n ⇒ RICH.
Amy Glen (LaCIM) Rich Words September 2008 18 / 22
Recent Results
A Connection Between Palindromic & Factor Complexity
Theorem A (Bucci-De Luca-G.-Zamboni, 2008)
For any infinite word w with set of factors F (w) closed under reversal, thefollowing conditions are equivalent:
(I) all complete returns to any palindrome in w are palindromes;
(II) Pw(n) + Pw(n + 1) = Cw(n + 1) − Cw(n) + 2 for all n ∈ N.
Sturmian words:
Morse-Hedlund, 1940: C(n) = n + 1 for all n
Amy Glen (LaCIM) Rich Words September 2008 18 / 22
Recent Results
A Connection Between Palindromic & Factor Complexity
Theorem A (Bucci-De Luca-G.-Zamboni, 2008)
For any infinite word w with set of factors F (w) closed under reversal, thefollowing conditions are equivalent:
(I) all complete returns to any palindrome in w are palindromes;
(II) Pw(n) + Pw(n + 1) = Cw(n + 1) − Cw(n) + 2 for all n ∈ N.
Sturmian words:
Morse-Hedlund, 1940: C(n) = n + 1 for all n
Droubay-Pirillo, 1999: P(n) = 1 for n even, P(n) = 2 for n odd
Amy Glen (LaCIM) Rich Words September 2008 18 / 22
Recent Results
A Connection Between Palindromic & Factor Complexity
Theorem A (Bucci-De Luca-G.-Zamboni, 2008)
For any infinite word w with set of factors F (w) closed under reversal, thefollowing conditions are equivalent:
(I) all complete returns to any palindrome in w are palindromes;
(II) Pw(n) + Pw(n + 1) = Cw(n + 1) − Cw(n) + 2 for all n ∈ N.
Sturmian words:
Morse-Hedlund, 1940: C(n) = n + 1 for all n
Droubay-Pirillo, 1999: P(n) = 1 for n even, P(n) = 2 for n odd
Hence P(n) + P(n + 1) = 3 = C(n + 1) − C(n) + 2 for all n ⇒ RICH.
Amy Glen (LaCIM) Rich Words September 2008 18 / 22
Recent Results
Finite Case of Theorem A
Using completely different methods . . .
Theorem (de Luca-G.-Zamboni, 2008)
For any finite word w , the following two conditions are equivalent:
i) w is a rich palindrome;
ii) Pw (n) + Pw (n + 1) = Cw (n + 1) − Cw (n) + 2 for all n, 0 ≤ n ≤ |w |.
Amy Glen (LaCIM) Rich Words September 2008 19 / 22
Recent Results
Finite Case of Theorem A
Using completely different methods . . .
Theorem (de Luca-G.-Zamboni, 2008)
For any finite word w , the following two conditions are equivalent:
i) w is a rich palindrome;
ii) Pw (n) + Pw (n + 1) = Cw (n + 1) − Cw (n) + 2 for all n, 0 ≤ n ≤ |w |.
We also explored various interconnections between rich words,Sturmian words, and trapezoidal words.
Amy Glen (LaCIM) Rich Words September 2008 19 / 22
Recent Results
Finite Case of Theorem A
Using completely different methods . . .
Theorem (de Luca-G.-Zamboni, 2008)
For any finite word w , the following two conditions are equivalent:
i) w is a rich palindrome;
ii) Pw (n) + Pw (n + 1) = Cw (n + 1) − Cw (n) + 2 for all n, 0 ≤ n ≤ |w |.
We also explored various interconnections between rich words,Sturmian words, and trapezoidal words.
A finite word w is trapezoidal if the graph of Cw (n) as a function of n
(for 0 ≤ n ≤ |w |) is that of a regular trapezoid.
Amy Glen (LaCIM) Rich Words September 2008 19 / 22
Recent Results
Finite Case of Theorem A
Using completely different methods . . .
Theorem (de Luca-G.-Zamboni, 2008)
For any finite word w , the following two conditions are equivalent:
i) w is a rich palindrome;
ii) Pw (n) + Pw (n + 1) = Cw (n + 1) − Cw (n) + 2 for all n, 0 ≤ n ≤ |w |.
We also explored various interconnections between rich words,Sturmian words, and trapezoidal words.
A finite word w is trapezoidal if the graph of Cw (n) as a function of n
(for 0 ≤ n ≤ |w |) is that of a regular trapezoid.Introduced by de Luca in 1999 when studying the factor complexity offinite Sturmian words.
Amy Glen (LaCIM) Rich Words September 2008 19 / 22
Recent Results
Finite Case of Theorem A
Using completely different methods . . .
Theorem (de Luca-G.-Zamboni, 2008)
For any finite word w , the following two conditions are equivalent:
i) w is a rich palindrome;
ii) Pw (n) + Pw (n + 1) = Cw (n + 1) − Cw (n) + 2 for all n, 0 ≤ n ≤ |w |.
We also explored various interconnections between rich words,Sturmian words, and trapezoidal words.
A finite word w is trapezoidal if the graph of Cw (n) as a function of n
(for 0 ≤ n ≤ |w |) is that of a regular trapezoid.Introduced by de Luca in 1999 when studying the factor complexity offinite Sturmian words.
Every finite Sturmian word is trapezoidal, but not conversely.E.g., aabb is trapezoidal, but not Sturmian.
Amy Glen (LaCIM) Rich Words September 2008 19 / 22
Recent Results
Finite Case of Theorem A
Using completely different methods . . .
Theorem (de Luca-G.-Zamboni, 2008)
For any finite word w , the following two conditions are equivalent:
i) w is a rich palindrome;
ii) Pw (n) + Pw (n + 1) = Cw (n + 1) − Cw (n) + 2 for all n, 0 ≤ n ≤ |w |.
We also explored various interconnections between rich words,Sturmian words, and trapezoidal words.
A finite word w is trapezoidal if the graph of Cw (n) as a function of n
(for 0 ≤ n ≤ |w |) is that of a regular trapezoid.Introduced by de Luca in 1999 when studying the factor complexity offinite Sturmian words.
Every finite Sturmian word is trapezoidal, but not conversely.E.g., aabb is trapezoidal, but not Sturmian.
Every trapezoidal word is rich, but not conversely. E.g., aabbaa.
Amy Glen (LaCIM) Rich Words September 2008 19 / 22
Further Work
Outline
1 Rich Words: A Brief Overview
2 Properties & Examples
3 Recent Results
4 Further Work
Amy Glen (LaCIM) Rich Words September 2008 20 / 22
Further Work
More Stuff on Rich Words
G.-Justin-Widmer-Zamboni, Palindromic richness, 2008
almost rich words: only a finite number of prefixes do not have aunioccurrent palindromic suffix
Amy Glen (LaCIM) Rich Words September 2008 21 / 22
Further Work
More Stuff on Rich Words
G.-Justin-Widmer-Zamboni, Palindromic richness, 2008
almost rich words: only a finite number of prefixes do not have aunioccurrent palindromic suffix
Example: (pq)ω = pqpqpq · · · where p, q are palindromes
Amy Glen (LaCIM) Rich Words September 2008 21 / 22
Further Work
More Stuff on Rich Words
G.-Justin-Widmer-Zamboni, Palindromic richness, 2008
almost rich words: only a finite number of prefixes do not have aunioccurrent palindromic suffix
Example: (pq)ω = pqpqpq · · · where p, q are palindromes
weakly rich words: all complete returns to letters are palindromes
Amy Glen (LaCIM) Rich Words September 2008 21 / 22
Further Work
More Stuff on Rich Words
G.-Justin-Widmer-Zamboni, Palindromic richness, 2008
almost rich words: only a finite number of prefixes do not have aunioccurrent palindromic suffix
Example: (pq)ω = pqpqpq · · · where p, q are palindromes
weakly rich words: all complete returns to letters are palindromes
Example: (aacbccbcacbc)ω = aacbccbcacbcaacbccbcacbc · · ·
Amy Glen (LaCIM) Rich Words September 2008 21 / 22
Further Work
More Stuff on Rich Words
G.-Justin-Widmer-Zamboni, Palindromic richness, 2008
almost rich words: only a finite number of prefixes do not have aunioccurrent palindromic suffix
Example: (pq)ω = pqpqpq · · · where p, q are palindromes
weakly rich words: all complete returns to letters are palindromes
Example: (aacbccbcacbc)ω = aacbccbcacbcaacbccbcacbc · · ·
action of morphisms on (almost) rich words
Amy Glen (LaCIM) Rich Words September 2008 21 / 22
Further Work
More Stuff on Rich Words
G.-Justin-Widmer-Zamboni, Palindromic richness, 2008
almost rich words: only a finite number of prefixes do not have aunioccurrent palindromic suffix
Example: (pq)ω = pqpqpq · · · where p, q are palindromes
weakly rich words: all complete returns to letters are palindromes
Example: (aacbccbcacbc)ω = aacbccbcacbcaacbccbcacbc · · ·
action of morphisms on (almost) rich words
morphisms that preserve (almost) richness
Amy Glen (LaCIM) Rich Words September 2008 21 / 22
Further Work
More Stuff on Rich Words
G.-Justin-Widmer-Zamboni, Palindromic richness, 2008
almost rich words: only a finite number of prefixes do not have aunioccurrent palindromic suffix
Example: (pq)ω = pqpqpq · · · where p, q are palindromes
weakly rich words: all complete returns to letters are palindromes
Example: (aacbccbcacbc)ω = aacbccbcacbcaacbccbcacbc · · ·
action of morphisms on (almost) rich words
morphisms that preserve (almost) richness
Further Work
Characterization of morphisms that preserve (almost) richness
Enumeration of rich words
Amy Glen (LaCIM) Rich Words September 2008 21 / 22
Thank You!
Dammit, I’m mad!
U R 2, R U?
* Both phrases are rich palindromes! *
Amy Glen (LaCIM) Rich Words September 2008 22 / 22