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

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

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

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 is

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
avoids 321 pattern.
But contains 3412 pattern,
since ; ;

8. For example

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

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

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

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

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), , Springer, Berlin, 2000.
A. Marcus and G. Tardos, Excluded permutation matrices and the Stanley-Wilf conjecture, JCTA 107 (2004) 1,

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. For example
n=1
n=2
n=3

21. Restricted Permutation and Dyck Path
Krattenthaler (2001) gave bijections between
and Dyck paths respectively.
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. 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.
Bandlow-Egge-Killpatrick (2002) gave a bijection between Schröder paths and
In 2003, Egge-Mansour gave a bijection between Schröder paths and

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. Restricted Permutation and Motzkin Path
In 1993, S. Gire discovered that
and are enumerated by the Motzkin numbers.
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. Involution Let We say is an involution if and only if
The set of involutions in is denoted by
For example
I3={123, 132, 213, 321}.

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

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

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. For k=4 Guibert et. al. conjectured that
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.

51. k-Nonnesting Matching
A. Regev (1981) gave asymptotic values
for k-nonnesting matchings.
D. Gouyou-Beauchamps (1989) gave the
enumeration for 3-nonnesting matchings
by using Young tableaux.

52. k-Noncrossing Matching
M. Klazar (1990’s, 2003) studied 2-noncrossing matchings, and said that even for k=3, the enumerations is still open.
(with Eva Y.P. Deng & Rosena R.X. Du)
Matching with forbidden substructure
k-nonnesting matching is equivalent to
k-noncrossing mathing.

53. Open Problems
The exactly formula for k-noncrossing (k-nonnesting) matchings? (k≥4)
How about matchings avoiding the pattern which is not noncrossing nor nonnesting.

54. Permutations containing a given number of a certain patterns
Question:
How many permutations of length are there that contain exactly r occurrences of pattern ?

55. Permutations containing a given number of a certain patterns
Noonan (1996) enumerated permutations
contain exactly one 123 pattern.
Bόna (1998) enumerated permutations contain
exactly one 132 pattern.
Fulmek (2003) enumerated permutations contain
exactly two subsequences of 132 pattern.

56. Conjecture (Noonan, Zeilberger, 1996)
For any given subsequence and for any given r, then number of n-permutations containing exactly r subsequence is a
P-recursive function of n.

57. Bόna (1997) proved that permutations containing a fix number of subsequence
132 is P-recursive.

58. Involutions containing a given number of a certain patterns
Question:
How many involutions of length do contain exactly r occurrences of pattern ?

59. Involutions containing a given number of a certain patterns
Guibert and Mansour (2002) gave an explicit expression for involutions containing one 132.
Deutsch, Robertson and Saracion (2004)
enumerated involutions containing one 231.

60. Involutions containing a given number of a certain patterns
(with Toufik Mansour & Sherry H.F. Yan)
P-recursiveness of the number of involutions with a fixed number of 132-subsequences

61. Open Problems
How about the enumerations for the other cases in permutations (involutions) containing a fixed number of certain pattern ?
How about partitions or matchings ?

62. Thanks !