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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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Presentation on theme: "On Generalized Factor Order on Words Sergey Kitaev Reykjavík University This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan."— Presentation transcript:

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

2 Sergey Kitaev On Generalized Factor Order on Words

3 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

4 Sergey Kitaev On Generalized Factor Order on Words

5 Sergey Kitaev On Generalized Factor Order on Words u=322 v= For example, for P = the set of the positive integers with the usual order relations, we have u P v where embedding index = 5

6 Sergey Kitaev On Generalized Factor Order on Words

7 Sergey Kitaev On Generalized Factor Order on Words

8 Sergey Kitaev On Generalized Factor Order on Words

9 Sergey Kitaev On Generalized Factor Order on Words

10 Sergey Kitaev On Generalized Factor Order on Words

11 Sergey Kitaev On Generalized Factor Order on Words

12 Sergey Kitaev On Generalized Factor Order on Words E.g., if w= then wt(w)=tx 2 ·tx·tx 3 ·tx 2 tx 2 ·tx=t 6 x 11

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

14 Sergey Kitaev On Generalized Factor Order on Words

15 Sergey Kitaev On Generalized Factor Order on Words

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

17 Sergey Kitaev On Generalized Factor Order on Words

18 Sergey Kitaev On Generalized Factor Order on Words The positive integers

19 Sergey Kitaev On Generalized Factor Order on Words

20 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.

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

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

23 Sergey Kitaev On Generalized Factor Order on Words

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

25 Sergey Kitaev On Generalized Factor Order on Words

26 Sergey Kitaev On Generalized Factor Order on Words

27 Sergey Kitaev On Generalized Factor Order on Words

28 Sergey Kitaev On Generalized Factor Order on Words

29 Sergey Kitaev On Generalized Factor Order on Words

30 Sergey Kitaev On Generalized Factor Order on Words

31 Sergey Kitaev On Generalized Factor Order on Words

32 Sergey Kitaev On Generalized Factor Order on Words

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

34 Sergey Kitaev On Generalized Factor Order on Words

35 Sergey Kitaev On Generalized Factor Order on Words

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

37 Sergey Kitaev On Generalized Factor Order on Words

38 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)

39 Sergey Kitaev On Generalized Factor Order on Words

40 Sergey Kitaev On Generalized Factor Order on Words Open problems

41 Sergey Kitaev On Generalized Factor Order on Words Open problems

42 Sergey Kitaev On Generalized Factor Order on Words Open problems

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


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