Identities from weighted Motzkin paths

Abstract

Based on a weighted version of the bijection between Dyck paths and 2-Motzkin paths, we find combinatorial interpretations of two identities related to the Narayana polynomials and the Catalan numbers. These interpretations answer two questions posed recently by Coker.

γ-positivity and partial γ-positivity of descent-type polynomials

“…It is well known that n−1 k=0 N (n, k + 1)x k is the rank-generating function of the lattice of noncrossing partition lattice with cardinality C n (see [36]). There are several combinatorial interpretations of the numbers N (n, k) (see [6,13]). A 231-avoiding permutation π is a permutation with no triple of indices i < j < k such that π(k) < π(i) < π(j).…”

Moments of the transmission eigenvalues, proper delay times and random matrix theory II

“…The n-dependent part of the factors in the summands of (21) and (25) are ratios of Gamma functions, whose asymptotic expansion is 71…”

“…Proof. For the negative Laguerre moments, we set b = (w − 1)n and insert the term of order O(n −k ) in equation (40) into (25) leading to…”

“…As in previous proofs, we begin by inserting (59) into (58) and extract the component of order O(n −k−1 ). Then, by substituting it into (25) we arrive at…”

Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs

“…There is a relation well known between the numbers m 2 i and Narayana numbers N (n, k) = 1 n n k n k−1 with 1 ⩽ k ⩽ n which enumerate a large variety of combinatorial objects, see sequence A001263. In particular, there is the following identity, [7]…”