Schur Positivity and the $q$-Log-convexity of the Narayana Polynomials

TL;DR

This paper proves conjectures on the $q$-log-convexity and log-convexity preservation of Narayana polynomials using Schur positivity and symmetric function identities, advancing understanding of their combinatorial properties.

Contribution

It introduces a novel proof of the $q$-log-convexity of Narayana polynomials and shows the Narayana transformation preserves log-convexity, utilizing Schur functions and Littlewood-Richardson rule.

Findings

Proves $q$-log-convexity of Narayana polynomials.

Shows Narayana transformation preserves log-convexity.

Establishes strong $q$-log-convexity and $q$-log-concavity results.

Abstract

Using Schur positivity and the principal specialization of Schur functions, we provide a proof of a recent conjecture of Liu and Wang on the $q$-log-convexity of the Narayana polynomials, and a proof of the second conjecture that the Narayana transformation preserves the log-convexity. Based on a formula of Br\"and$\mathrm{\acute{e}}$n which expresses the $q$-Narayana numbers as the specializations of Schur functions, we derive several symmetric function identities using the Littlewood-Richardson rule for the product of Schur functions, and obtain the strong $q$-log-convexity of the Narayana polynomials and the strong $q$-log-concavity of the $q$-Narayana numbers.