Some identities on the Catalan, Motzkin and Schröder numbers

“…For instance, the 01-filling in left side of Fig. 1 has two SE-chains of length 3, one chain containing 1's in cells (4, 1), (6, 3), (7,5) and the other chain containing 1's in cells (4, 1), (6,3), (8,4), where only the latter chain is proper.…”

Section: Via 01-filling Of Triangular Shapementioning

confidence: 99%

“…This Motzkin-Catalan identity was first discovered by Donaghey [7], who interpreted it a generating function counting plane trees by number of branches. Many other interpretations in terms of different models are found later in [5,6,8] and lately in [9].…”

Section: Introductionmentioning

confidence: 99%

A combinatorial bijection on $k$-noncrossing partitions

Zhao¹,

Kim²

2019

Preprint

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For any integer k ≥ 2, we prove combinatorially the following Euler (binomial) transformation identitywhere) is the sum of weights, t number of blocks , of partitions of {1, . . . , m} without k-crossings (resp. enhanced k-crossings). The special k = 2 and t = 1 case, asserting the Euler transformation of Motzkin numbers are Catalan numbers, was discovered by Donaghey 1977. The result for k = 3 and t = 1, arising naturally in a recent study of pattern avoidance in ascent sequences and inversion sequences, was proved only analytically.

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“…Note that (3.20), (3.22) and (4.4) can be regarded as companion ones of an identity obtained by Deng and Yan [5],…”

Section: It Is Clear That the Weights Of The Setsmentioning

confidence: 99%

Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths

Sun

2013

Preprint

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A partial Motzkin path is a path from (0, 0) to (n, k) in the XOY -plane that does not go below the X-axis and consists of up steps U = (1, 1), down steps D = (1, −1) and horizontal steps H = (1, 0). A weighted partial Motzkin path is a partial Motzkin path with the weight assignment that all up steps and down steps are weighted by 1, the horizontal steps are endowed with a weight x if they are lying on X-axis, and endowed with a weight y if they are not lying on X-axis. Denote by M n,k (x, y) to be the weight function of all weighted partial Motzkin paths from (0, 0) to (n, k), and M = (M n,k (x, y)) n≥k≥0 to be the infinite lower triangular matrices. In this paper, we consider the sums of minors of second order of the matrix M, and obtain a lot of interesting determinant identities related to M, which are proved by bijections using weighted partial Motzkin paths. When the weight parameters (x, y) are specialized, several new identities are obtained related to some classical sequences involving Catalan numbers. Besides, in the alternating cases we also give some new explicit formulas for Catalan numbers.

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Some identities on the Catalan, Motzkin and Schröder numbers

Cited by 13 publications

References 10 publications

Some Inverse Relations Determined by Catalan Matrices

Some Inverse Relations Determined by Catalan Matrices

A combinatorial bijection on $k$-noncrossing partitions

Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths

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