On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with...
-
Upload
dominic-scattergood -
Category
Documents
-
view
214 -
download
1
Transcript of On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with...
![Page 1: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/1.jpg)
On Generalized Factor Order on Words
Sergey KitaevReykjaviacutek University
This is joint work with Jeff Liese Jeff Remmel and Bruce Sagan
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Outline of the talk
Basic definitions Automata Wilf equivalence Strong Wilf equivalence Explicit computations The Moumlbius function Open problems
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
u=322
v=12213431
For example for P = the set of the positive integers with the usual order relations we have ulePv where
embedding index = 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Eg if w=213221 then wt(w)=tx2txtx3tx2tx2tx=t6x11
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(123)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 2: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/2.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Outline of the talk
Basic definitions Automata Wilf equivalence Strong Wilf equivalence Explicit computations The Moumlbius function Open problems
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
u=322
v=12213431
For example for P = the set of the positive integers with the usual order relations we have ulePv where
embedding index = 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Eg if w=213221 then wt(w)=tx2txtx3tx2tx2tx=t6x11
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(123)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 3: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/3.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Outline of the talk
Basic definitions Automata Wilf equivalence Strong Wilf equivalence Explicit computations The Moumlbius function Open problems
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
u=322
v=12213431
For example for P = the set of the positive integers with the usual order relations we have ulePv where
embedding index = 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Eg if w=213221 then wt(w)=tx2txtx3tx2tx2tx=t6x11
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(123)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 4: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/4.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
u=322
v=12213431
For example for P = the set of the positive integers with the usual order relations we have ulePv where
embedding index = 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Eg if w=213221 then wt(w)=tx2txtx3tx2tx2tx=t6x11
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(123)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 5: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/5.jpg)
Sergey Kitaev On Generalized Factor Order on Words
u=322
v=12213431
For example for P = the set of the positive integers with the usual order relations we have ulePv where
embedding index = 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Eg if w=213221 then wt(w)=tx2txtx3tx2tx2tx=t6x11
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(123)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 6: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/6.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Eg if w=213221 then wt(w)=tx2txtx3tx2tx2tx=t6x11
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(123)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 7: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/7.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Eg if w=213221 then wt(w)=tx2txtx3tx2tx2tx=t6x11
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(123)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 8: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/8.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Eg if w=213221 then wt(w)=tx2txtx3tx2tx2tx=t6x11
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(123)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 9: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/9.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Eg if w=213221 then wt(w)=tx2txtx3tx2tx2tx=t6x11
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(123)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 10: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/10.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Eg if w=213221 then wt(w)=tx2txtx3tx2tx2tx=t6x11
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(123)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 11: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/11.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Eg if w=213221 then wt(w)=tx2txtx3tx2tx2tx=t6x11
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(123)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 12: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/12.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Eg if w=213221 then wt(w)=tx2txtx3tx2tx2tx=t6x11
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(123)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 13: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/13.jpg)
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(123)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 14: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/14.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 15: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/15.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 16: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/16.jpg)
Sergey Kitaev On Generalized Factor Order on Words
A DFA accepting S(132)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 17: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/17.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 18: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/18.jpg)
Sergey Kitaev On Generalized Factor Order on Words
The positive integers
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 19: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/19.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 20: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/20.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i S(utx) is recorded
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 21: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/21.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Wilf equivalences for |u|le3 and uile3 for all i
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 22: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/22.jpg)
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 4
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 23: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/23.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 24: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/24.jpg)
Sergey Kitaev On Generalized Factor Order on Words
The Wilf equivalences for permutations of length 5
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 25: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/25.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 26: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/26.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 27: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/27.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 28: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/28.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 29: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/29.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 30: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/30.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 31: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/31.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 32: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/32.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 33: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/33.jpg)
Sergey Kitaev On Generalized Factor Order on Words
2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 34: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/34.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 35: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/35.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 36: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/36.jpg)
Sergey Kitaev On Generalized Factor Order on Words
The Moumlbius Function of the Factor Order
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 37: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/37.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 38: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/38.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Question Is M(u)=w μ(uw) ne 0 regularAnswer No
This is surprising since this language is regular for ordinary subword order (Bjoumlrner and Reutenauer 1992) and the same holds for generalized subword order (Bjoumlrner and Sagan 2006)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 39: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/39.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 40: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/40.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 41: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/41.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 42: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/42.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Open problems
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer
![Page 43: On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.](https://reader036.fdocuments.in/reader036/viewer/2022062619/5516e523550346fe558b46d1/html5/thumbnails/43.jpg)
Sergey Kitaev On Generalized Factor Order on Words
Thank you for your attentionand have a nice summer