Combinatorial interpretations for a class of algebraic equations and uniform partitions Speaker: Yeong-Nan Yeh Institute of mathemetics, Academia sinica Aug. 21, 2012

第2页第2页第2页第2页 Catalan paths An n-Catalan path is a lattice path from (0,0) to (2n,0) in the first quadrant consisting of up- step (1,1) and down-step (1,-1).

第3页第3页第3页第3页 Catanlan number 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, …,

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第5页第5页第5页第5页 Motzkin paths An n-Motizkin path is a lattice path from (0,0) to (n,0) in the first quadrant consisting of up- step (1,1), level-step (1,0) and down-step (1,-1). Motzkin number:1, 1, 2, 4, 9, 21, 51, 127, 323, 835, …,

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第7页第7页第7页第7页 Combinatorial structureGenerating function f(z)Algebaric equation Catalan path:(1,1),(1,-1) in the first quadrant C(z)C(z)=1+z[C(z)] 2 =1+zC(z)·C(z) Motzkin path:(1,1),(1,- 1),(1,0) in the first quadrant M(z)M(z)=1+zM(z)+z 2 [M(z)] 2 =1+zM(z)·[1+zM(z)] ?????f(z) Given an algebaric equation for arbitrary polynomial F(z,y), how to construct a combinatorial structure such that its generating function f(z) satisfies this equation?

第8页第8页 Lattice paths A lattice path is a sequence (x 1,y 1 )(x 2,y 2 )…(x k,y k ) of vectors in the plane with (x i,y i ) ∈ Z ≥0 ×Z\{(0,0)}, where Z and Z ≥0 are the sets of integers and nonnegative integers respectively.

第9页第9页 Weight of a lattice path Let w be a function from Z ≥0 ×Z to R, where R is the set of real numbers. For any lattice path P=(x 1,y 1 )(x 2,y 2 )…(x k,y k ) define the weight of P, denoted by w(P), as

第 10 页 S-path and S-nonnegative path Let S be a finite subset of Z ≥0 ×Z\{(0,0)}. An S-path is a lattice path (x 1,y 1 )(x 2,y 2 )…(x k,y k ) with (x i,y i ) ∈ S. An S-nonnegative path is an S-path in the first quadrant

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第 12 页 A decomposition of a S-nonnegative path. P=(0,1)P 1 (0,1)P 2 (0,1)P 3 …P i-1 (j,-i+1)P i w(0,1)=1,w(j,-i+1)=a i,j

第 13 页 More general cases Let λ be a function from Z ≥0 ×Z to Z ≥0. For any lattice path P=(x 1,y 1 )(x 2,y 2 )…(x k,y k ) define the λ-length of P, denoted by λ(P), as

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第 15 页 Uniform partition An n-Dyck path is a lattice path from (0,0) to (2n,0) in the plane integer lattice Z×Z consisting of up-step (1,1) and down-step (1,-1). The number of n-Dyck paths is

第 16 页 K.L. Chung, W. Feller, On fluctuations in-coin tossing, Proc. Natl. Acad. Sci. USA 35 (1949) Chung-Feller theorem: The number of Dyck path of semi-length n with m up-steps under x-axis is the n-th Catalan number and independent on m.

第 17 页 Uniform partition (An uniform partition for Dyck paths) The number of up- steps (1,1) lying below x-axis

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第 19 页 Lifted Motzkin paths A lifted n-Motizkin path is a lattice path from (0,0) to (n+1,1) in the plane integer lattice Z×Z consisting of up-step (1,1), level-step (1,0) and down-step (1,-1), which never passes below the line y=1 except (0,0).

第 20 页 Free Lifted Motzkin paths A free lifted n-Motizkin path is a lattice path from (0,0) to (n+1,1) in the plane integer lattice Z×Z consisting of up-step (1,1), level-step (1,0) and down-step (1,-1).

第 21 页 The number of free lifted n-Motzkin path with m steps at the left of the rightmost lowest point is the n-th Motzkin number and independent on m.

第 22 页 An uniform partition for free lifted 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), The number of steps at the left of the rightmost lowest point of a lattice path 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)

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第 24 页 Function of uniform partition type For any generating function f(x), the form is called the function of uniform partition type for f(x).

第 25 页 Combinatorial structure Generating functionCombinatorial structure Function of uniform partition Catalan pathC(z) C(z)=1+z[C(z)] 2 n-Dyck path: Motzkin pathM(z) M(z)=1+zM(z)+z 2 [M(z)] 2 lifted n- Motizkin path ???f(z) f(z)=1+yzF(y,f(z)) ??? Function of uniform partition type

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第 28 页 A decomposition of a S-nonnegative path. P=(1,1)P 1 (1,1)P 2 (1,1)P 3 …P i-1 (j-i+1,-i+1)P i w(1,1)=1, w(j-i+1,-i+1)=a i,j (j-i+1,-i+1)

第 29 页 Function of uniform partition type

第 30 页 Combinatorial interpretations for H(y,z) and G(y,z)?

第 31 页 Combinatorial interpretation for H(y,z) Recall that an S-path is a lattice path P=(x 1,y 1 )(x 2,y 2 )…(x k,y k ) with (x i,y i ) ∈ S. Define the nonpositive length of P, denoted by nl(P), as the sum of x-coordinate of steps touching or going below x-axis.

第 32 页 Combinatorial interpretation for H(y,z) Define the nonpositive length of P, denoted by nl(P), as the sum of x-coordinate of steps touching or going below x-axis, nl(P)= =7

第 33 页 Combinatorial interpretation for H(y,z)

第 34 页 f(z) f(yz) H(y,z) A decomposition of a S-path.

第 35 页 A rooted S-nonnegative path is a pair [P;k] consisting of an S-nonnegative path P=(x 1,y 1 )(x 2,y 2 )…(x n,y n ) with x n ≥1 and a nonnegative integer k with 0≤ k≤ x n - 1. For example, P=(1,1)(1,1)(1,-2)(1,0)(1,1)(1,1)(1,1)(1,-1)(2,-1). [P;0],[P;1] and [P;2] are rooted S-nonnegative path. Combinatorial interpretation for G(y,z)

第 36 页 Combinatorial interpretation for G(y,z)

第 37 页 f(z) k A decomposition of a rooted S-nonnegative path.

第 38 页 A lifted S-path is an S-path in the plane starting at (0,0) and ending at a point in the line y=1. A rooted lifted S-path is a pair [P;k] consisting of a lifted S-path P=(x 1,y 1 )(x 2,y 2 )…(x n,y n ) with x n ≥1 and a nonnegative integer k with 0≤ k≤ x n -1. Combinatorial interpretation for CF(y,z)

第 39 页 Combinatorial interpretation for CF(y,z)

第 40 页 The last step (1,1) from y=0 to y=1 (n+1-k,0) A decomposition for an rooted lifted S-path G(y,z) H(y,z) (n+1,1) (0,0)

第 41 页 More general cases Let λ be a function from Z ≥0 ×Z to Z ≥0. For any lattice path P=(x 1,y 1 )(x 2,y 2 )…(x k,y k ) define the λ-length of P, denoted by λ(P), as

第 42 页 A λ-rooted lifted S-path is a pair [P;k] consisting of a lifted S-path P=(x 1,y 1 )(x 2,y 2 )…(x n,y n ) with λ( x n,y n ) ≥1 and a nonnegative integer k with 0≤ k≤ λ( x n,y n ) -1.

第 43 页 For any S-path P=(x 1,y 1 )(x 2,y 2 )…(x n,y n ), define the λ- nonpositive length of P, denoted by nl λ (P), as the sum of λ- length of steps touching or going below x-axis. For example, let λ(1,1)=0, λ(x,y)=x for any (x,y)≠ （ 1,1 ） nl λ (P)= =5 For any rooted lifted S-path [P;k], define the rootedλ- nonpositive length of P as nl λ (P)+k.

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第 63 页 Thank you for your attention!