# On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan.

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On Generalized Factor Order on Words Sergey Kitaev Reykjavík 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 Möbius 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 u P v 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 E.g., if w=213221 then wt(w)=tx 2 ·tx·tx 3 ·tx 2 tx 2 ·tx=t 6 x 11

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|3 and u i 3 for all i; S(u;t,x) is recorded.

Sergey Kitaev On Generalized Factor Order on Words Wilf equivalences for |u|3 and u i 3 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 Möbius 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(u,w) 0} regular? Answer: No! This is surprising since this language is regular for ordinary subword order (Björner and Reutenauer, 1992) and the same holds for generalized subword order (Björner 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 attention and have a nice summer!

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