Taylor expansions for Catalan and Motzkin numbers

Eu, Sen-Peng; Liu, Shu-Chung; Yeh, Yeong‐Nan

doi:10.1016/s0196-8858(02)00018-0

Cited by 20 publications

(24 citation statements)

References 19 publications

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“…By the same technique, similar results are proved for Catalan paths of order d (Theorem 3.1) and a more general case (Theorem 3.3). Our bijection is inspired by the previous study of Taylor expansions of generating functions [8]. By a Taylor expansion, we mean that a generating function is expanded in a form the remainder of which is expressed as a function of the generating function itself.…”

Section: Namementioning

confidence: 99%

“…[8,Theorem 2.1]). Since the remainder x n F n (C) is both the generating function for the number of Catalan paths of length m, for m n, and the generating function for the number of flawed paths of length m with n flaws, it suggests that other information may be available linking the two sets of paths.…”

Section: Namementioning

confidence: 99%

“…Note that the point (4, 6) is on the line y = x+2 and that the last north step N that rises from the line y = x +1 to the line y = x +2 is labeled with 5. The blue section − 6 ( ) (from (0, 0) to (4, 6)) has a factorization − 6 ( ) = N , where = (1, 2, 3, 4) and = (6,7,8). On the right of Fig.…”

Section: A Weighted Chung-feller Theorem For Schröder Pathsmentioning

confidence: 99%

“…10 is a path ∈ A 8 (4, 6). The first diagonal step D 1 on the line y = x of is labeled with 3 and the blue section − 6 ( ) (from (0, 0) to (4,6)) is factorized as − 6 ( ) = D 1 , where = (1, 2) and = (4,5,6,7,8). On the right of Fig.…”

Section: A Weighted Chung-feller Theorem For Schröder Pathsmentioning

confidence: 99%

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Refined Chung–Feller theorems for lattice paths

Yeh

2005

Journal of Combinatorial Theory, Series A

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In this paper we prove a strengthening of the classical Chung-Feller theorem and a weighted version for Schröder paths. Both results are proved by refined bijections which are developed from the study of Taylor expansions of generating functions. By the same technique, we establish variants of the bijections for Catalan paths of order d and certain families of Motzkin paths. Moreover, we obtain a neat formula for enumerating Schröder paths with flaws.

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Section: Namementioning

confidence: 99%

Section: Namementioning

confidence: 99%

Section: A Weighted Chung-feller Theorem For Schröder Pathsmentioning

confidence: 99%

Section: A Weighted Chung-feller Theorem For Schröder Pathsmentioning

confidence: 99%

See 2 more Smart Citations

Refined Chung–Feller theorems for lattice paths

Yeh

2005

Journal of Combinatorial Theory, Series A

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“…One also attempted to generalize Chung-Feller Theorem for finding uniformly partitions of other combinatorial structures, see also [15,10,9,17,19]. Woan [25] proved the number of lattice paths in D n in which the leftmost point with maximal y-coordinate is exactly the ith point which is equal to 1 2n+1  2n+1 n  for every i ∈ {1, 2, .…”

Section: Introductionmentioning

confidence: 99%

Rooted cyclic permutations of lattice paths and uniform partitions

Shen

Yeh

2015

Discrete Mathematics

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a b s t r a c tA partition of a given set is said to be uniform if all the partition classes have the same cardinality. In this paper, we will introduce the concepts of rooted n-lattice path and rooted cyclic permutation and prove some fundamental theorems concerning the actions of rooted cyclic permutations on rooted lattice n-paths. The main results obtained have important applications in finding new uniform partitions. Many uniform partitions of combinatorial structures are special cases or consequences of our main theorems.

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A Chung–Feller theorem for lattice paths with respect to cyclically shifting boundaries

Guo

Wang

2018

J Algebr Comb

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Irving and Rattan gave a formula for counting lattice paths dominated by a cyclically shifting piecewise linear boundary of varying slope. Their main result may be considered as a deep extension of well-known enumerative formulas concerning lattice paths from (0, 0) to (kn, n) lying under the line x = ky (e.g., the Dyck paths when k = 1). On the other hand, the classical Chung-Feller theorem tells us that the number of lattice paths from (0, 0) to (n, n) with exactly 2k steps above the line x = y is independent of k and is therefore the Catalan number 1 n+1 2n n . In this paper, we study the number of lattice path boundary pairs (P, a) with k flaws, where P is a lattice path from (0, 0) to (n, m), a is a weak m-part composition of n, and a flaw is a horizontal step of P above the boundary ∂a. We prove bijectively, for a given a, that summing these numbers over all cyclic shifts of the boundary ∂a is equal to n+m m−1 . That is, we generalize the Irving-Rattan formula to a Chung-Feller-type theorem. We also give a refinement of this result by taking the number of double ascents of lattice paths into account.

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Taylor expansions for Catalan and Motzkin numbers

Cited by 20 publications

References 19 publications

Refined Chung–Feller theorems for lattice paths

Refined Chung–Feller theorems for lattice paths

Rooted cyclic permutations of lattice paths and uniform partitions

A Chung–Feller theorem for lattice paths with respect to cyclically shifting boundaries

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