The q-log-convexity of the Narayana polynomials of type B

Abstract

We prove a conjecture of Liu and Wang on the q-log-convexity of the Narayana polynomials of type B. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. By the principal specialization this, leads to q-log-convexity. We also show that the linear transformation with respect to the triangular array of Narayana numbers of type B is log-convexity preserving.

Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences

“…There is a similar result for log-convexity. The following result follows from Liu and Wang [13,Conjecture 5.3], which has been shown by Chen et al [5]. For the sake of brevity we here omit the details of the proof.…”

Two integer sequences related to Catalan numbers

“…Ordinary Narayana polynomials correspond to a root system of type A. For their combinatorial study we refer to [2,3] and references therein.…”

“…The values of a n for 1 ≤ n ≤ 16 are given by 2,1,2,8,52,495,6470,111034,2419928,65269092,2133844440,83133090480,3805035352536,202147745618247,12336516593999598,857054350280418290.…”

Narayana polynomials and Hall–Littlewood symmetric functions

“…Ordinary Narayana polynomials correspond to a root system of type A. For their combinatorial study we refer to [3,4] It is an open problem whether the polynomials W r (z) can be obtained by specialization of some classical symmetric function. The Hall-Littlewood polynomial of type B [16] might be a good candidate.…”