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Indexed by:期刊论文

Date of Publication:2020-12-01

Journal:DISCRETE MATHEMATICS

Volume:343

Issue:12

ISSN No.:0012-365X

Key Words:Motzkin number; Total positivity; Polynomial with only real roots; Asymptotic normality; Riordan array

Abstract:The Motzkin numbers count the number of lattice paths which go from (0, 0) to (n, 0) using steps (1, 1), (1, 0) and (1,-1) and never go below the x-axis. Let M-n,M-k be the number of such paths with exactly k horizontal steps. We investigate the analytic properties of various combinatorial triangles related to the Motzkin triangle [M-n,M-k](n,k >= 0), including their total positivity, the real-rootedness and interlacing property of the generating functions of their rows, and the asymptotic normality (by central and local limit theorems) of these triangles. We also prove several identities related to these triangles. (c) 2020 Elsevier B.V. All rights reserved.