A Combinatorial Proof of the Log-Convexity of Catalan-Like Numbers

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.

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

Hua Sun and Yi Wang
School of Mathematical Sciences
Dalian University of Technology
Dalian 116024
PR China