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A Combinatorial Proof of the Log-Convexity
of Catalan-Like Numbers Hua Sun and Yi Wang
School of Mathematical Sciences
Dalian University of Technology
Dalian 116024
PR China
mailto:hanch.sun@gmail.comhanch.sun@gmail.com
mailto:wangyi@dlut.edu.cnwangyi@dlut.edu.cn

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Abstract:

The Catalan-like numbers c_n,0, defined by
$$\begin{align*}&c_{n+1,k}=r_{k-1}c_{n,k-1}+s_kc_{n,k}+t_{k+1}c_{n,k+1}\text{ for $n,k\geq 0$ },\\ &c_{0,0}=1, c_{0,k}=0 \text{ for $k\neq 0$ }, \end{align*}$$
unify a substantial amount of well-known counting coefficients. Using an algebraic approach, Zhu showed that the sequence $(c_{n,0})_{n\geq 0}$ is log-convex if $r_{k}t_{k+1}\leq s_{k}s_{k+1}$ for all $k\geq 0$. Here we give a combinatorial proof of this result from the point of view of weighted Motzkin paths.