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On Generalized Factor Order on Words

On Generalized Factor Order on Words. Sergey Kitaev Reykjavík University. This is joint work with Jeff Liese, Jeff Remmel and Bruce Sagan. Outline of the talk. Basic definitions Automata Wilf equivalence Strong Wilf equivalence Explicit computations The Möbius function Open problems.

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On Generalized Factor Order on Words

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

  2. On Generalized Factor Order on Words

  3. Outline of the talk • Basic definitions • Automata • Wilf equivalence • Strong Wilf equivalence • Explicit computations • The Möbius function • Open problems On Generalized Factor Order on Words

  4. On Generalized Factor Order on Words

  5. For example, for P = the set of the positive integers with the usual order relations, we have u≤Pv where u=322 v=12213431 embedding index = 5 On Generalized Factor Order on Words

  6. On Generalized Factor Order on Words

  7. On Generalized Factor Order on Words

  8. On Generalized Factor Order on Words

  9. On Generalized Factor Order on Words

  10. On Generalized Factor Order on Words

  11. On Generalized Factor Order on Words

  12. E.g., if w=213221 then wt(w)=tx2·tx·tx3·tx2tx2·tx=t6x11 On Generalized Factor Order on Words

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

  14. On Generalized Factor Order on Words

  15. On Generalized Factor Order on Words

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

  17. On Generalized Factor Order on Words

  18. The positive integers On Generalized Factor Order on Words

  19. On Generalized Factor Order on Words

  20. Wilf equivalences for |u|≤3 and ui≤3 forall i; S(u;t,x) is recorded. On Generalized Factor Order on Words

  21. Wilf equivalences for |u|≤3 and ui≤3 forall i On Generalized Factor Order on Words

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

  23. On Generalized Factor Order on Words

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

  25. On Generalized Factor Order on Words

  26. On Generalized Factor Order on Words

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  28. On Generalized Factor Order on Words

  29. On Generalized Factor Order on Words

  30. On Generalized Factor Order on Words

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  32. On Generalized Factor Order on Words

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

  34. On Generalized Factor Order on Words

  35. On Generalized Factor Order on Words

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

  37. On Generalized Factor Order on Words

  38. Question: Is M(u)={w : μ(u,w) ≠ 0} regular? Answer:No! This is surprising since this language isregular for ordinary subword order (Björner and Reutenauer, 1992) and the same holds for generalized subword order (Björner and Sagan, 2006) On Generalized Factor Order on Words

  39. On Generalized Factor Order on Words

  40. Open problems On Generalized Factor Order on Words

  41. Open problems On Generalized Factor Order on Words

  42. Open problems On Generalized Factor Order on Words

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

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