1. Combinatorial Expansions for Paths, Chung-Feller Theorem and Hankel Matrix Speaker: Yeong-Nan Yeh Institute of mathemetics, Academia sinica

2. 2 Online Part I. Functions of uniform-partition type Part II. Combinatorial interpretations for a class of function equations Part III. Lattice paths and Fluctuation theory Part IV Paths with some avoiding sets shift equivalence Part V. Addition formulas of polynomials and Hankel determinants

3. 3 Part I. Functions of uniform-partition type

4. 4 Catalan paths An n-Catalan path is a lattice path in the first quadrant starting at (0,0) and ending at (2n,0) with only two kinds of steps---up-step: U=(1,1) and down- step: D=(1,-1).

5. 5 Catanlan number 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, …,

6. 6 Catanlan number The Catalan sequence was first described in the 18th century by Leonhard Euler, who was interested in the number of different ways of dividing a polygon into triangles.

7. 7 Eugène Charles Catalan (May 30, 1814 – February 14, 1894) was a French and Belgian mathematician. The sequence is named after Eugène Charles Catalan, who discovered the connection to parenthesized expressions during his exploration of the Towers of Hanoi puzzle. E.C. Catalan, Note surune equation aux di ff erences finies, J. Math.Pures Appl. 3(1838), 508-515. ((ab)c)d (ab)(cd) (a(bc))d a((bc)d) (ab)(cd)

8. 8 Catanlan number The counting trick for Catalan words was found by D. André in 1887 D. André, Solution directe du problème résolu par M. Bertrand, Comptes Rendus de l’Académie des Sciences, Paris 105 (1887) 436–437.

9. 9 Chung-Feller Theorem (The number of Dyck path of semi-length n with m nonpositive up-steps is the n-th Catalan number and independent on m.) K.L. Chung, W. Feller, On fluctuations in-coin tossing, Proc. Natl. Acad. Sci. USA 35 (1949) 605-608 We say Chung- Feller theorem is an uniform partition of up- down type.

10. 10 The classical Chung-Feller theorem was proved by Macmahon. MacMahon, P. A. Memoir on the theory of the partitions of numbers, Philos. Trans. Roy. Soc. London, Ser. A, 209 (1909), 153-175. Chung and Feller reproved the theorem by analytic method. Chung, K. L. and Feller, W. On fluctuations in-coin tossing, Proc. Natl. Acad. Sci. USA 35 (1949) 605-608. A combinatorial proof. Narayana, T. V. Cyclic permutation of lattice paths and the Chung-Feller theorem, Skand. Aktuarietidskr. (1967) 23-30 Eu, Liu and Yeh proved the Chung-Feller theorem by using the Taylor expansions of generating functions. Eu, S. P. Liu, S. C. and Yeh, Y. N. Taylor expansions for Catalan and Motzkin numbers, Adv. Appl. Math. 29 (2002) 345-357 Eu, Fu and Yeh gave a strengthening of the Chung-Feller Theorem and a weighted version for schroder paths. Eu, S. P. Fu, T. S. and Yeh, Y. N. Refined Chung-Feller theorems for lattice paths, J. Combin. Theory Ser. A 112 (2005) 143-162

11. 11 Bijection proofs. D. Callan, Pair them up! A visual approach to the Chung-Feller theorem, Coll. Math. J. 26(1995)196-198. R.I. Jewett, K. A. Ross, Random walk on Z, Coll. Math. J. 26(1995)196-198. Mohanty’s book devotes an entire section to exploring the Chung-Feller theorem. Mohanty, S. G. Lattice path counting and applications, NewYork : Academic Press, 1979. Narayana's book introduced a refinement of this theorem. T.V. Narayana, Lattice path combinatorics, with statistical applications,Toronto;Buffalo : University of Toronto Press, c1979. Callan reviewed and compared combinatorial interpretations of three different expressions for the Catalan number by cycle method. D. Callan, Why are these equal? http://www.stat.wisc.edu/~callan/notes/ Huq developed generalized versions of this theorem for lattice paths. A. Huq, Generalized Chung-Feller Theorems for Lattice Paths(Thesis), http://arxiv.org/abs/0907.3254

12. 12 W.J. Woan, Uniform partitions of lattice paths and Chung-Feller Generalizations, Amer. Math. Monthly 108(2001) 556-559. Another uniform partition for Dyck paths The number of up-steps at the left of the rightmost lowest point of a dyck path We say this uniform partition is of left-right type.

13. 13 Motzkin paths An n-Motizkin path is a lattice path in the first quadrant starting at (0,0) and ending at (n,0) with only two kinds of steps---level-step: (1,0), up-step: U=(1,1) and down- step: D=(1,-1).

14. 14 An uniform partition for Motzkin paths Shapiro found an uniform partition for Motzkin path. L. Shapiro, Some open questions about random walks, involutions, limiting distributions, and generating functions, Advances in Applied Math. 27 (2001), 585- 596. The number of steps at the left of the rightmost lowest point of a lattice path This uniform partition is of left- right type. Eu, Liu and Yeh proved this proposition. Eu, S. P. Liu, S. C. and Yeh, Y. N. Taylor expansions for Catalan and Motzkin numbers, Adv. Appl. Math. 29 (2002) 345-357

15. 15 Another uniform partition of up-down type for Motzkin paths. The number of steps touching x-axis and under x-axis

16. 第 16 页 Our main results 1.Eu, Liu and Yeh proved the Chung-Feller theorem by using the Taylor expansions of generating functions. Eu, S. P. Liu, S. C. and Yeh, Y. N. Taylor expansions for Catalan and Motzkin numbers, Adv. Appl. Math. 29 (2002) 345-357 2.Eu, Fu and Yeh gave a strengthening of the Chung-Feller Theorem and a weighted version for schroder paths. Eu, S. P. Fu, T. S. and Yeh, Y. N. Refined Chung-Feller theorems for lattice paths, J. Combin. Theory Ser. A 112 (2005) 143-162 3.Ma and Yeh gave a generalizations of Chung-Feller theorems J. Ma, Y.N. Yeh, Generalizations of Chung-Feller theorems, Bull. Inst. Math., Acad. Sin.(N.S.)4(2009) 299-332.

17. 第 17 页 Our main results 4.Ma and Yeh gave a characterization for uniform partitions of cyclic permutations of a sequence of real number J. Ma, Y.N. Yeh, Cyclic permutations ofsequences and uniform partitions, The electronic journal ofcombinatorics 17 (2010), #R117. 5.Liu, Wang, Yeh gave the concepts of functions of Chung-Feller type S.C. Liu, Y. Wang, Y.N. Yeh, Chung-Feller Property in View of Generating Functions, Electron. J. Comb. 18(2011), #P104. 6.Ma and Yeh gave a refinement of Chung-Feller theorems J. Ma, Y.N. Yeh, Refinements of (n,m)-Dyck paths, European. J. Combin. 32(2011) 92-99.

18. 第 18 页 Our main results 7.Ma and Yeh generalized the cycle lemma. J. Ma, Y.N. Yeh, Generalizations of the cycle lemma, (Accepted 2014). 8.Ma and Yeh gave a characterization for uniform partitions of cyclic permutations of a sequence of real number J. Ma, Y.N. Yeh, Rooted cyclic permutations of a lattice paths and uniform partitions, submitted. 9.Ma and Yeh studied a class of generating functions and their functions of Chung-Feller type J.Ma, Y.N.Yeh, Combinatorial interpretations for a class of functions of Chung-Feller theorem. submitted

19. 19 Part II. Combinatorial interpretations for a class of function equations

20. 20 Uniform-partition Extension

21. 21 Liu, S. C. Wang, Y. and Yeh, Y. N. The function of uniform-partition type, submitted the function of uniform-partition type for :

22. 22 An example for catalan sequence (up-down type)

23. 23 An example for Motzkin sequence ( left-right type ) rightmost lowest point

24. 第 24 页 In general, given a combinatorial structure, let f(z) be a generating function correspoding with this combinatorial structure. We can obtain a functional equation which f(z) satisfies.

25. 第 25 页 Combinatorial structureGenerating function f(z) Functional equation which f(z) satisfies Catalan path:(1,1),(1,-1) in the first quadrant C(z) Motzkin path:(1,1),(1,-1),(1,0) in the first quadrant M(z) Schroder path:(1,1),(1,-1),(2,0) in the first quadrant S(z)

26. 第 26 页 Combinatorial structureGenerating function f(z) Functional equation which f(z) satisfies Catalan path:(1,1),(1,-1) in the first quadrant C(z)C(z)=1+z[C(z)]2 Motzkin path:(1,1),(1,-1),(1,0) in the first quadrant M(z)M(z)=1+zM(z)+z 2 [M(z)] 2 Schroder path:(1,1),(1,-1),(2,0) in the first quadrant S(z)S(z)=1+zS(z)+z[S(z)] 2

27. 第 27 页 Combinatorial structureGenerating function f(z) Functional equation which f(z) satisfies Catalan path:(1,1),(1,-1) in the first quadrant C(z)C(z)=1+z[C(z)]2 Motzkin path:(1,1),(1,-1),(1,0) in the first quadrant M(z)M(z)=1+zM(z)+z 2 [M(z)] 2 Schroder path:(1,1),(1,-1),(2,0) in the first quadrant S(z)S(z)=1+zS(z)+z[S(z)] 2 Given a functional equation, how to find a combinatorial structure and its corresponding generating function f(z) such that f(z) satisfies this functional equation?

28. 第 28 页 Combinatorial structureGenerating function f(z) Functional equation which f(z) satisfies Catalan path:(1,1),(1,-1) in the first quadrant C(z)C(z)=1+z[C(z)]2 Motzkin path:(1,1),(1,-1),(1,0) in the first quadrant M(z)M(z)=1+zM(z)+z 2 [M(z)] 2 Schroder path:(1,1),(1,-1),(2,0) in the first quadrant S(z)S(z)=1+zS(z)+z[S(z)] 2 Given a functional equation, how to find a combinatorial structure and its corresponding generating function f(z) such that f(z) satisfies this functional equation?

29. 第 29 页 Combinatorial structureGenerating function f(z) Functional equation which f(z) satisfies Catalan path:(1,1),(1,-1) in the first quadrant C(z)C(z)=1+z[C(z)]2 Motzkin path:(1,1),(1,-1),(1,0) in the first quadrant M(z)M(z)=1+zM(z)+z 2 [M(z)] 2 Schroder path:(1,1),(1,-1),(2,0) in the first quadrant S(z)S(z)=1+zS(z)+z[S(z)] 2 ??? ??f(z) Given a functional equation, how to find a combinatorial structure and its corresponding generating function f(z) such that f(z) satisfies this functional equation?

30. 第 30 页 Let. The recurrence relation which the sequence satisfies is independent on a 0 (z). Hence, let a 0 (z) =1.. We focus on the following functional equation. Let S be a set of vector in the plane Z×Z. We also call the set S step set and vectors in S steps. Let L be a function from S to N, where N is the set of nonnegative integers. We call L a step-length function of the set S and L(s) the step length of the step s in the set S repectively. Let W be a function from S to R, where R is the set of real numbers. We call W a weight function of the set S and W(s) the weight of the setp s in the set S respectively,

31. 第 31 页 Let P be a sequence of vectors (x 1,y 1 )…(x n,y n ) in the set S such that y 1 +…y n =0, y 1 +…y i ≥0 for all i. We call P an S-path. Let Ω(S) be the set of all S-paths. Define the L-length of a S-path P= (x 1,y 1 )…(x n,y n ), denoted by L(P), as L(P)=L(x 1,y 1 )+…L(x n,y n ). Define the W-length of a S-path P= (x 1,y 1 )…(x n,y n ), denoted by W(P), as W(P)=W(x 1,y 1 )…W(x n,y n ). Define a generating function f(z) as

32. 第 32 页

33. 第 33 页 A decomposition of a S-path. P=(0,1)P 1 (0,1)P 2 (0,1)P 3 …P i-1 (j,-i+1)P i W(1,1)=1,W(j,-i+1)=a i,j

34. 第 34 页 Part III. Lattice paths and Fluctuation theory

35. 35 Fluctuation theory is the name given to that part of probability theory which deals with the fluctuations of the partial sums s n =x 1 +...+x n of a sequence of random variables x 1, …,x n.

36. 36 Consider x=(r 1, … r n ). Let s 0 =0,s i =r 1 + … +r i Let p(x) be the number of positive sums s i Let m(x) be the index where the maximum is attained for the first time.

37. 37 x partial sump(x)m(x) (1,2,3)(3,1,-2)(3,4,2)32 (1,3,2)(3,-2,1)(3,1,2)31 (2,1,3)(1,3,-2)(1,4,2)32 (3,1,2)(-2,3,1)(-2,1,2)23 (2,3,1)(1,-2,3)(1,-1,2)23 (3,2,1)(-2,1,3)(-2,-1,2)13 r 1 =3,r 2 =1,r 3 =-2

38. 38 x partial sump(x)m(x) (1,2,3)(1,2,-2)(1,3,1)32 (1,3,2)(1,-2,2)(1,-1,1)21 (2,1,3)(2,1,-2)(2,3,1)32 (3,1,2)(-2,1,2)(-2,-1,1)13 (2,3,1)(2,-2,1)(2,0,1)21 (3,2,1)(-2,2,1)(-2,0,1)13 r 1 =1,r 2 =2,r 3 =-2

39. 39 Fix X=(r 1, … r n ). Let X i =(r i, … r n,r 1, …,r i-1 ) (cyclic permutations.) Let P(X)={p(X i )| i=1,2, …,n} M(X)={m(X i )| i=1,2, …,n}

40. 40 F. Spitzer, (1956) Let X be a sequence of real numbers of length n such that s n =0 and no other partial sum of distinct elements vanishes. Then P(X)=M(X)=[0,n-1].

41. 41 Remark Fix X=(r 1, … r n ). Suppose r 1 + … +r n =m. Let m=0. The conditions in the results of Spitzer are necessary and sufficient conditions for P(X)=[0,n-1] The conditions in the results of Spitzer are not necessary for M(X)=[0,n-1].

42. 42 T.V. Narayana, (1967) Let n be a positive integer and X be a sequence of integers with -n<r i < 2 for all i=1,2, …,n such that s n =1. Then P(X)=[n].

43. 43 J. Ma, Y.N. Yeh, Generalizations of The Chung-Feller Theorem II, submitted. Let n be a positive integer and X be a sequence of integers with -n<r i < 2 for all i=1,2, …,n such that s n =1. Then M(X)=[n].

44. 44 Two natural problems What are necessary and sufficient conditions for M(X)=[n] and P(X)=[n] if m>0? What are necessary and sufficient conditions for M(X)=[0,n-1] and P(X)=[0,n-1] if m<=0?

45. 45 Fix X=(r 1, … r n ). Given an index j=1, …,n, define LP(X;j)={i|s j >s i, i=1, …,j-1} and RP(X;j)={i|s j >=s i i=j+1, …,n}

46. 46 Let m>0. The necessary and sufficient conditions for M(X)=[n] are s m(X) -s i >=m for all i in LP(X;m(X)) The necessary and sufficient conditions for P(X)=[n] are s j -s i >=m for any j in [n] and any all i in [0,j-1]\LP(X;j)

47. 47 Let m<=0. The necessary and sufficient conditions for M(X)=[0,n-1] are s i -s m(X) <m for all i in RP(X;m(X)) The necessary and sufficient conditions for P(X)=[0,n-1] are s j -s i <m for any j in [n] and any all i in [0,j-1]\LP(X;j)

48. 48 Part IV. Paths with some avoiding sets shift equivalence

49. 49 Let M be a Motzkin path. L M : the set of the height of the level steps L M ={0,3} P M : the set of the height of the peaks P M ={2,1} V M : the set of the height of the valleys V M ={0,1}

50. 50 Motzkin paths from (0,0) to (2(n-1),0) without level of height larger than 0

51. 51 Peaks-, Valleys- and Level-avoiding Sets we consider the Motzkin path such that Given the sets A: level-avoiding set B: peak-avoiding sest C: valley-avoiding set A: level-restricting set B: peak-restricting set C: valley-restricting set (1): (2):

52. 52 Generating Functions the number of the Motzkin path of length n with k levles, l peaks and s valleys

53. 53

54. 54 Some results E. Deutsch, Dyck path enumeration, Discrete Math. 204 (1999), 167--202. P. Peart and W-J. Woan, Dyck paths with no peaks at height k, J. Integer Seq. 4 (2001), Article 01.1.3. the n-th Fine number the (n-1)-th Catalan number

55. 55 Shu-Chung Liu, Jun Ma, Yeong-Nan Yeh, Dyck Paths with Peak- and Valley-Avoiding Sets, Stud. Appl. Math. 121:263-289 S.-P. Eu, S.-C. Liu, and Y.-N. Yeh, Dyck Paths with Peaks Avoiding or Restricted to a Given Set, Stud. Appl. Math. 111 Iss 4 (2003), 453--465. Continued fractions Close forms Shift equivalence

56. 56 Supposeand Then if and only if there is a positive integer m such that is a polynomial. If there exist non-negative integers p and q such that

57. 57 Some interesting shift equivalence

58. 58 Some interesting shift equivalence

59. 59 Continued fractions It is difficult to represent as a continued fractions

60. 60 Close form Matrix methods We just consider

61. 61

62. 62

63. 63 Let Then where

64. 64

65. 65 (A,B) being Congruence classes Define the congruence classes Let Then where

66. 66 then we say that F(x) is algebraic If F(x) is a solution of an equation The algebraic degree of F(x):

67. 67 is algebraic since it is a solution of a quadratic Equations

68. 68 Problem I Characterize the set

69. 69 Problem II Given a sequence a1,a2, …,an, …, find a pair (A,B) of the sets such that

70. 70 We consider the coefficients in

71. 71 Let Suppose Then if and only if (1) (2) (3) Such that

72. 72 Bijection methods Suppose Then In fact, if

73. 73 Problem III If the sequences then we say that (A 1,B 1 ) and (A 2,B 2 ) shift equivalent, denoted by Give a characterization of

74. 74 is shift equivalent to the Fibonacci numbers The sequence m n+2i;{i},{1} has Chung-Feller property, i.e., m n+2i;{i},{1} =Fn is independent on i, where F n is the n-th Fibonacci number.

75. 75 is shift equivalent with the Central binomial coefficients Replace valleys(DU) of height 0 and level into peak DU and U respectively Remove the first and final steps (i=1) Left factor of Dyck path

76. 第 76 页 Addition formulas and Hankel matrix

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81. 第 81 页 Theorem

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84. 第 84 页 Corollary

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86. 第 86 页 Theorem

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88. 第 88 页 Thank you for your attention!