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Combinatorial Expansions for Paths, Chung-Feller Theorem and Hankel Matrix

Combinatorial Expansions for Paths, Chung-Feller Theorem and Hankel Matrix. Speaker: Yeong-Nan Yeh Institute of mathemetics, Academia sinica. Online. Part I. Functions of uniform-partition type Part II. Combinatorial interpretations for a class

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Combinatorial Expansions for Paths, Chung-Feller Theorem and Hankel Matrix

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

  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. Part I. Functions of uniform-partition type

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

  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. 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. ((ab)c)d (ab)(cd) (a(bc))d a((bc)d) (ab)(cd) E.C. Catalan, Note surune equation aux differences finies, J. Math.Pures Appl. 3(1838), 508-515.

  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. 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.) • We say Chung-Feller theorem is an uniform partition of up-down type. K.L. Chung, W. Feller, On fluctuations in-coin tossing, Proc. Natl. Acad. Sci. USA 35 (1949) 605-608

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

  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. 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 theleft of the rightmost lowest point of a lattice path This uniform partition is of left-right type. • Eu, Liu andYeh 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. Another uniform partition of up-down type for Motzkin paths. The number of steps touching x-axis and under x-axis

  16. Our main results • 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 • Ma and Yeh gave a generalizations of Chung-Feller theoremsJ. Ma, Y.N. Yeh, Generalizations of Chung-Feller theorems, Bull. Inst. Math., Acad. Sin.(N.S.)4(2009) 299-332.

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

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

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

  20. Uniform-partition Extension

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

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

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

  24. 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. 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?

  26. 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?

  27. 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. Let . The recurrence relation which the sequence satisfies is independent on a0(z) . Hence, let a0(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) theweightof the setp s in the set S respectively,

  29. Let P be a sequence of vectors (x1,y1)…(xn,yn) in the set S such that y1+…yn=0, y1+…yi≥0 for all i. We call P anS-path. Let Ω(S) be the set of all S-paths. • Define the L-length of a S-path P= (x1,y1)…(xn,yn) , denoted by L(P), as L(P)=L(x1,y1)+…L(xn,yn). • Define the W-length of a S-path P= (x1,y1)…(xn,yn) , denoted by W(P), as W(P)=W(x1,y1)…W(xn,yn). • Define a generating function f(z) as

  30. A decomposition of a S-path. P=(0,1)P1(0,1)P2(0,1)P3…Pi-1(j,-i+1)Pi W(1,1)=1,W(j,-i+1)=ai,j

  31. Part III. Lattice paths and Fluctuation theory

  32. Fluctuation theory is the name given to that part of probability theory which deals with the fluctuations of the partial sums sn=x1+...+xn of a sequence of random variables x1,…,xn.

  33. Consider x=(r1,…rn). Let s0=0,si=r1+…+ri Let p(x) be the number of positive sums si Let m(x) be the index where the maximum is attained for the first time.

  34. r1=3,r2=1,r3=-2

  35. r1=1,r2=2,r3=-2

  36. Fix X=(r1,…rn). Let Xi=(ri,…rn,r1,…,ri-1) (cyclic permutations.) Let P(X)={p(Xi)| i=1,2,…,n} M(X)={m(Xi)| i=1,2,…,n}

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

  38. Remark Fix X=(r1,…rn). Suppose r1+…+rn=m. Let m=0.The conditions in the results of Spitzer arenecessary and sufficient conditions for P(X)=[0,n-1]The conditions in the results of Spitzer arenot necessary for M(X)=[0,n-1].

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

  40. 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<ri< 2 for all i=1,2,…,n such that sn=1. Then M(X)=[n].

  41. 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?

  42. Fix X=(r1,…rn). Given an index j=1,…,n, defineLP(X;j)={i|sj>si,i=1,…,j-1} and RP(X;j)={i|sj>=si i=j+1,…,n}

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

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

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

  46. Let M be a Motzkin path. LM: the set of the height of the level steps LM={0,3} PM: the set of the height of the peaks PM={2,1} VM: the set of the height of the valleys VM={0,1}

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

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