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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

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Sergey Kitaev On Generalized Factor Order on Words

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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

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Sergey Kitaev On Generalized Factor Order on Words

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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

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words

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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

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Sergey Kitaev On Generalized Factor Order on Words A DFA accepting S(123)

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words A DFA accepting S(132)

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words The positive integers

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Sergey Kitaev On Generalized Factor Order on Words

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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.

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Sergey Kitaev On Generalized Factor Order on Words Wilf equivalences for |u|3 and u i 3 for all i

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Sergey Kitaev On Generalized Factor Order on Words The Wilf equivalences for permutations of length 4

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words The Wilf equivalences for permutations of length 5

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words 2143 is Wilf equivalent to 3412 but it is not strongly Wilf equivalent to it!

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words The Möbius Function of the Factor Order

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Sergey Kitaev On Generalized Factor Order on Words

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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)

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Sergey Kitaev On Generalized Factor Order on Words

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Sergey Kitaev On Generalized Factor Order on Words Open problems

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Sergey Kitaev On Generalized Factor Order on Words Open problems

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Sergey Kitaev On Generalized Factor Order on Words Open problems

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Sergey Kitaev On Generalized Factor Order on Words Thank you for your attention and have a nice summer!

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