introducing 05a06 patterns in permutations and words

Introducing 05A06: Patterns in Permutations and Words Eric S. Egge - PowerPoint PPT Presentation

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Introducing 05A06: Patterns in Permutations and Words Eric S. Egge Carleton College September 20, 2014 Eric S. Egge (Carleton College) Introducing 05A06: Patterns in Permutations and WordsSeptember 20, 2014 1 / 34 The Case There are

  1. Introducing 05A06: Patterns in Permutations and Words Eric S. Egge Carleton College September 20, 2014 Eric S. Egge (Carleton College) Introducing 05A06: Patterns in Permutations and WordsSeptember 20, 2014 1 / 34

  2. The Case There are connections with many other areas. Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 2 / 34

  3. The Case There are connections with many other areas. There are already numerous cool results. Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 2 / 34

  4. The Case There are connections with many other areas. There are already numerous cool results. We’ve answered some deep questions. Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 2 / 34

  5. The Case There are connections with many other areas. There are already numerous cool results. We’ve answered some deep questions. Even more open problems remain, some just as deep. Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 2 / 34

  6. The Case There are connections with many other areas. There are already numerous cool results. We’ve answered some deep questions. Even more open problems remain, some just as deep. Surprising and exciting new ideas and approaches surface regularly. Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 2 / 34

  7. The Case There are connections with many other areas. There are already numerous cool results. We’ve answered some deep questions. Even more open problems remain, some just as deep. Surprising and exciting new ideas and approaches surface regularly. There’s room for all, from undergraduates to wily veterans. Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 2 / 34

  8. The Definition Definition Suppose π and σ are permutations, written in one-line notation. An occurrence of σ in π is a subsequence of π whose entries are in the same relative order as the entries of σ . Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 3 / 34

  9. The Definition Definition Suppose π and σ are permutations, written in one-line notation. An occurrence of σ in π is a subsequence of π whose entries are in the same relative order as the entries of σ . Example 3561274 contains 9 occurrences of 21. (inversions) Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 3 / 34

  10. The Definition Definition Suppose π and σ are permutations, written in one-line notation. An occurrence of σ in π is a subsequence of π whose entries are in the same relative order as the entries of σ . Example 3561274 contains 12 occurrences of 12. (coinversions) Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 3 / 34

  11. The Definition Definition Suppose π and σ are permutations, written in one-line notation. An occurrence of σ in π is a subsequence of π whose entries are in the same relative order as the entries of σ . Example 3561274 contains 7 occurrences of 312. Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 3 / 34

  12. The Definition Definition Suppose π and σ are permutations, written in one-line notation. An occurrence of σ in π is a subsequence of π whose entries are in the same relative order as the entries of σ . Example 3561274 contains 7 occurrences of 312. 3561274 3561274 3561274 3561274 3561274 3561274 3561274 Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 3 / 34

  13. The Definition in Pictures Definition Suppose π and σ are permutations, written in one-line notation. An occurrence of σ in π is a subsequence of π whose entries are in the same relative order as the entries of σ . r r r r r r r 3 5 6 1 2 7 4 Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 4 / 34

  14. The Definition in Pictures Definition Suppose π and σ are permutations, written in one-line notation. An occurrence of σ in π is a subsequence of π whose entries are in the same relative order as the entries of σ . r r r r r r r 3 5 6 1 2 7 4 Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 4 / 34

  15. The Definition in Pictures Definition Suppose π and σ are permutations, written in one-line notation. An occurrence of σ in π is a subsequence of π whose entries are in the same relative order as the entries of σ . r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 4 / 34

  16. The Definition in Pictures r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r Observation Every symmetry f of the square is a bijection between occurrences of σ in π and occurrences of σ f in π f . Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 4 / 34

  17. Enumeration Questions σ [ π ] := number of occurrences of σ in π Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 5 / 34

  18. Enumeration Questions σ [ π ] := number of occurrences of σ in π Theorem (Rodrigues, 1839) q 21[ π ] = 1(1 + q )(1 + q + q 2 ) · · · (1 + q + · · · + q n − 1 ) � π ∈ S n Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 5 / 34

  19. Enumeration Questions σ [ π ] := number of occurrences of σ in π Theorem (Rodrigues, 1839) q 21[ π ] = 1(1 + q )(1 + q + q 2 ) · · · (1 + q + · · · + q n − 1 ) � π ∈ S n Problem � q σ [ π ] . For each σ , find π ∈ S n Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 5 / 34

  20. Enumeration Questions σ [ π ] := number of occurrences of σ in π Theorem (Rodrigues, 1839) q 21[ π ] = 1(1 + q )(1 + q + q 2 ) · · · (1 + q + · · · + q n − 1 ) � π ∈ S n Ambition � q σ [ π ] . For each σ , find π ∈ S n Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 5 / 34

  21. Enumeration Questions σ [ π ] := number of occurrences of σ in π Theorem (Rodrigues, 1839) q 21[ π ] = 1(1 + q )(1 + q + q 2 ) · · · (1 + q + · · · + q n − 1 ) � π ∈ S n Dream � q σ [ π ] . For each σ , find π ∈ S n Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 5 / 34

  22. Enumeration Questions σ [ π ] := number of occurrences of σ in π Theorem (Rodrigues, 1839) q 21[ π ] = 1(1 + q )(1 + q + q 2 ) · · · (1 + q + · · · + q n − 1 ) � π ∈ S n Opium-Induced Fever Dream � q σ [ π ] . For each σ , find π ∈ S n Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 5 / 34

  23. Pattern Avoidance: Reining in Our Ambitions Definition We say π avoids σ whenever σ [ π ] = 0. Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 6 / 34

  24. Pattern Avoidance: Reining in Our Ambitions Definition We say π avoids σ whenever σ [ π ] = 0. Av n ( σ ) = S n ( σ ) := set of permutations in S n which avoid σ Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 6 / 34

  25. Pattern Avoidance: Reining in Our Ambitions Definition We say π avoids σ whenever σ [ π ] = 0. Av n ( σ ) = S n ( σ ) := set of permutations in S n which avoid σ Question For each n and each σ , what is | Av n ( σ ) | ? Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 6 / 34

  26. Pattern Avoidance: Reining in Our Ambitions Definition We say π avoids σ whenever σ [ π ] = 0. Av n ( R ) = S n ( R ) := set of permutations in S n which avoid all σ ∈ R Question For each n and each R , what is | Av n ( R ) | ? Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 6 / 34

  27. Pattern Avoidance: Reining in Our Ambitions Definition We say patterns σ 1 and σ 2 are Wilf-equivalent whenever | Av n ( σ 1 ) | = | Av n ( σ 2 ) | for all n . Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 7 / 34

  28. Pattern Avoidance: Reining in Our Ambitions Definition We say patterns σ 1 and σ 2 are Wilf-equivalent whenever | Av n ( σ 1 ) | = | Av n ( σ 2 ) | for all n . Question Which patterns of each length are Wilf-equivalent? Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 7 / 34

  29. Enumerative Results | Av n ( σ ) | OGF σ 1 − √ 1 − 4 x � 2 n � 123 1 C n = 132 n + 1 n 2 x Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 8 / 34

  30. More Enumerative Results | Av n ( R ) | OGF R 1 − x 2 n − 1 123, 132 1 − 2 x 1 − 2 x + 2 x 2 � n � 123, 231 1 + (1 − x ) 3 2 1 + x + 2 x 2 + 4 x 3 + 4 x 4 123, 321 0 for n ≥ 5 1 123, 132, 213 F n +1 1 − x − x 2 1 123, 132, 231 n (1 − x ) 2 Eric S. Egge (Carleton College) 05A06: Patterns in Permutations and Words September 20, 2014 9 / 34

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