A combinatorial proof of the log-convexity of sequences in Riordan arrays

Abstract

A Riordan array \(R=[r_{n,k}]_{n,k\ge 0}\) can be characterized by two sequences \(A=(a_n)_{n\ge 0}\) and \(Z=(z_n)_{n\ge 0}\) such that \(r_{0,0}=1, r_{0,k}=0~(k\ge 1)\) and $$\begin{aligned} r_{n+1,0}=\sum _{j\ge 0} z_j r_{n,j}, \quad r_{n+1,k+1}=\sum _{j\ge 0} a_j r_{n,k+j} \end{aligned}$$for \(n,k\ge 0\). Using an algebraic approach, Chen, Liang and Wang showed that the sequence \((r_{n,0})_{n\ge 0}\) is log-convex if the production matrix \([\zeta ,A]\) is TP\(_2\), where \(\zeta =[z_0,z_1,z_2,\ldots ]'\) and \(A=[a_{n-k}]_{n,k\ge 0}\) is the Toeplitz matrix of the A-sequence. In this paper, we present an injective proof of this result from the point of view of weighted Łukasiewicz paths, which gives a combinatorial proof of the log-convexity of many combinatorial sequences, including the Catalan numbers, the Motzkin numbers, the Schröder numbers, the central binomial coefficients, and the restricted hexagonal numbers, in a unified approach. This method also present an injective proof of the strong q-log-convexity of many well-known combinatorial polynomials.