1. Combinatorics of Paths and Permutations William Y. C. Chen Center for Combinatorics, LPMC Nankai University, Tianjin 300071, P. R. China Email: chen@nankai.edu.cn Joint work with Eva Y. P. Deng, Rosena R. X. Du, Toufik Mansour, Sherry H. F. Yan and Laura L. M. Yang

2. Permutation Permutation Containing Certain Patterns Restricted Permutation Restricted Involution Restricted Matching Restricted Partition Restricted Permutation

3. Let be the set of permutations on For example Permutation

4. Pattern For a permutation of positive integers, the pattern of is defined as a permutation on obtained from by substituting the minimum element by 1, the second minimum element by 2, ..., and the maximum element by .

5. For example The pattern of 914 is 312. The pattern of 37925 is 24513.

6. Restricted Permutation For a permutation and a permutation , we say that is -avoiding if and only if there is no subsequence whose pattern is . We write for the set of -avoiding permutations of .

7. For example 512673849 avoids 321 pattern. But 512673849 contains 3412 pattern, since 512673849; 512673849; 512673849.

8. For example

9. Stack Sorting Problem (Knuth, 1960’s) 312-avoiding 8 7 6 5 4 3 2 1

10. How many permutations of length do avoid a given subsequence of length k? Question (Herbert Wilf, 1990’s)

11. In 1972, Hammersley gave the first explicit enumeration for • In 1973, Knuth first proved that • is enumerated by Catalan numbers. For k=3

12. For k=4 • J. West (1990), Z. Stankova (1990’s) classified • the permutations with forbidden patterns of length 4, i.e. • 1234, 1243, 2143, 1432 • 1342, 2413 • 1324

13. For k=4 • 1234, 1243, 2143, 1432 • In 1990, Ira M. Gessel gave the generating function by using symmetric functions. • 1342, 2413 • In 1997, M. Bόna gave the exactly formula. • 1324 • D. Marinov & R. Radoicic (2003) gave the first few numbers.

14. Open Problems

15. For each pattern , there is an absolute constant so that holds. Conjecture ( Stanley and Wilf, 1990’s)

16. M. Bόna, The solution of a conjecture of Stanley and Wilf for all layered patterns, JCTA 85 (1999). • Richard Arratia, On the Stanley-Wilf conjecture for the number of permutations avoiding a given pattern, Electron. J. Combin. 6 (1999). • Noga Alon, Ehud Friedgut, On the number of permutations avoiding a given pattern, JCTA 89 (2000).

17. M. Klazar, The Fueredi-Hajnal conjecture implies the Stanley-Wilf conjecture. Formal power series and algebraic combinatorics (Moscow, 2000), 250-255, Springer, Berlin, 2000. • A. Marcus and G. Tardos, Excluded permutation matrices and the Stanley-Wilf conjecture, JCTA 107 (2004) 1, 153-160.

18. Combinatorics for Restricted Permutation Since Catalan numbers have more than 60 kinds of combinatorial descriptions, it is a question to give restricted permutations some combinatorial correspondings.

19. Dyck Path • A Dyck path of semilength n is a lattice path • in the plane from the origin (0,0) to (2n,0) • consisting of up steps (1,1) and down steps (1,-1) • that never run below the x-axis.

20. n=1 • For example • n=2 • n=3

21. Krattenthaler (2001) gave bijections between • and Dyck paths respectively. Restricted Permutationand Dyck Path • Banlow and Killpatrick (2001) gave bijections between 123 (132)-avoiding permutations and Dyck paths. • (with Eva Y.P. Deng & Rosena R.X. Du) Labelling Schemes for Lattice Paths

22. For example

23. For example

24. Schröder Path • A Schröder path of semilength n is a lattice • path in the plane from the origin (0,0) to (2n,0) • consisting of up steps (1,1), down steps (1,-1) • and double horizontal steps (2,0) that never run • below the x-axis.

25. Bandlow-Egge-Killpatrick (2002) gave a bijection between Schröder paths and • In 2003, Egge-Mansour gave a bijection between Schröder paths and Restricted Permutation and Schröder Path • Kremer (2000) proved that for ten pairs of patterns of length 4, permutations avoiding these patterns are all enumerated by the Schröder numbers.

26. Motzkin Path • A Motzkin path of length n is a lattice path in • the plane from the origin (0,0) to (n,0) consisting • of up steps (1,1), down steps (1,-1) and horizontal • steps (1,0) that never run below the x-axis.

27. In 1993, S. Gire discovered that • and are enumerated by the Motzkin numbers. Restricted Permutation and Motzkin Path • Barcucci-Del Lungo-Pergola-Pinzani (2000’s), Guibert (1995) studied the two kinds of resricted permutations by using generating trees.

28. Restricted Permutation and Motzkin Path • (with Eva Y.P. Deng & Laura L.M. Yang) Motzkin Paths and Reduced Decompositions for Permutations with Forbidden Patterns, Elect. J. Combin. 9(2) (2003), R15

29. Discrete Continuity • Barcucci-Del Lungo-Pergola-Pinzani (2000) • provided a ``discrete continuity" between the • Motzkin and the Catalan sequences. And they posed a question of searching for a combinatorial description. • (with Eva Y.P. Deng & Rosena R.X. Du & Sherry H.F. Yan & Laura L.M. Yang) Discrete Continuity, give combinatorial descriptions from Motzkin to Catalan permutations and from Catalan to Schröder permutations.

30. Let We say is an involution if and • only if The set of involutions in is denoted by . Involution • For example I3={123, 132, 213, 321}.

31. The set of involutions in which avoid the pattern is denoted by . Restricted Involution • For example • 65782134 is an involution avoiding 3214.

32. How many involutions length do avoid a given subsequence of length k? Question:

33. For k=3 Simion and Schmidt (1985) gave explicit expressions, i.e. They also gave the formulas for the number of involutions avoiding several patterns of length 3.

34. For k=4 • Guibert, Phd. Thesis, 1995. • Guibert-Pergola-Pinzani, Ann. Combin. 5 (2001).

35. Guibert et. al. conjectured that For k=4 • A.D. Jaggard (Elect. J. Combin. 9 (2003)) • gave an affirmative answer to this conjecture by introducing the equivalence of In(1234) and In(3214). • But it is still interesting to find a bijection between In(3214) • and the set of Motzkin paths of length n.

36. For k=4 • (with Sherry H.F. Yan & Laura L.M. Yang) • 3214-Avoiding Involutions, 321-Avoiding Involutions and Motzkin Paths

37. For k≥5 • A. Regev (1981) obtained an asymptotic formula for the number of 12··· k-avoiding involutions by using Young diagrams.

38. Open Problems • How about the others involutions avoiding a pattern of length 4 ? • How about the involutions avoiding a pattern of length greater than 4 ? • Is there others restricted involutions that can be corresponding to lattice paths or other simpler combinatorial objections ?

39. Partition • A partition of is a collection of nonempty disjoint subsets of called blocks, whose union is • Any partition P can be expressed by its canonical sequential form.

40. For Example

41. Question: How many partitions of length n do avoid a given subsequence ?

42. Noncrossing Partition

43. For example

44. Noncrossing Partition • Davenport-Schinzel sequence • RNA secondary structures They can be corresponding to some special cases of noncrossing partitions.

45. Noncrossing Partition • R. Simion and D. Ullman (1991) • M. Klazar (1990’s) gave enumerations and some combinatorial descriptions for noncrossing partitions.

46. Noncrossing Partition • (with Eva Y.P. Deng & Rosena R.X. Du) Reduction of Regular Noncrossing Partitions, European J. Combin., to appear.

47. Noncrossing Partition • (with Sherry H.F. Yan & Laura L.M. Yang) • Colored combinatorial objects • This paper defines and studies colored Dyck paths, • plane trees, hilly poor noncrossing partitions • and Motkzin paths, and answers two problems • posed by C. Coker (2003).

48. Nonnesting Partition

49. Open Problems • How about k-noncrossing parititions? (k≥3) • How about k-nonnesting partitions? (k≥2)

50. Matching A matching is a special case of partition with each block of cardinality two.