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Chen, Xi; Wang, Yi; Zheng, Sai-Nan

Analytic properties of combinatorial triangles related to Motzkin numbers. (English) Zbl 1448.05214

Discrete Math. 343, No. 12, Article ID 112133, 11 p. (2020).
Summary: 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 \(M_{n,k}\) 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 \([M_{n,k}]_{n,k\geq 0}\), 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.

Cited in 3 Documents

MSC:

05E10 Combinatorial aspects of representation theory

Keywords:

Motzkin number; total positivity; polynomial with only real roots; asymptotic normality; Riordan array

Software:

OEIS
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Full Text: DOI

References:

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