Matrix identities on weighted partial Motzkin paths

Abstract

We give a combinatorial interpretation of a matrix identity on Catalan numbers and the sequence (1, 4, 4 2 , 4 3 , . . .) which has been derived by Shapiro, Woan and Getu by using Riordan arrays. By giving a bijection between weighted partial Motzkin paths with an elevation line and weighted free Motzkin paths, we find a matrix identity on the number of weighted Motzkin paths and the sequence (1, k, k 2 , k 3 , . . .) for any k ≥ 2. By extending this argument to partial Motzkin paths with multiple elevation lines, we give a combinatorial proof of an identity recently obtained by Cameron and Nkwanta. A matrix identity on colored Dyck paths is also given, leading to a matrix identity for the sequence (1, t 2 + t, (t 2 + t) 2 , . . .).

Some New Binomial Sums Related to the Catalan Triangle

“…Many properties of the Catalan numbers can be generalized easily to the ballot numbers, which have been studied intensively by Gessel [25]. The combinatorial interpretations of the ballot numbers can be found in [5,8,10,11,14,18,19,21,22,28,31,35,39,41,44,50,51]. It was shown by Ma [33] that the Catalan triangle C can be generated by context-free grammars in three variables.…”

“…Some alternating sum identities on the Catalan triangle B were established by Zhang and Pang [53], who showed that the Catalan triangle B can be factorized as the product of the Fibonacci matrix and a lower triangular matrix, which makes them build close connections among C n , B n,k and the Fibonacci numbers. Motivated by a matrix identity related to the Catalan triangle B [46], Chen et al [14] derived many nice matrix identities on weighted partial Motzkin paths. n/ k 0 1 2 3 4 5 6 0 1 1 1 1 2 2 3 1 3 5 9 5 1 4 14 28 20 7 1 5 42 90 75 35 9 1 6 132 297 275 154 54 11 1 Table 1.2.…”

Approximate substitutions and the normal ordering problem

“…We define here the space of transformation matrices and its topology, and then we concentrate on the Riordan subgroup [7] (i.e transformations which are substitutions with prefactor functions) [4,9].…”

Counting Lattice Paths With Four Types of Steps

“…These paths are connected with weighted free (t, l)-Motzkin paths [2]. A weighted free (t, l)-Motzkin path is a lattice path from (0, 0) to (n, 0) consisting of horizontal steps (1, 0), down steps (1, −1), and up steps (1, 1), and for which each of horizontal and down steps have been assigned a number from the sets {1, 2, .…”

“…We follow the notation of Lehner [9] and Chen et al [2]. A lattice path is a sequence of points in the integer lattice Z 2 .…”