Schur positivity and the q-log-convexity of the Narayana polynomials

Authors:
William Y. Chen;Larry X. Wang;Arthur L. Yang
Affiliations:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin, P.R. China 300071;Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin, P.R. China 300071;Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin, P.R. China 300071
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2010

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Abstract

We prove two recent conjectures of Liu and Wang by establishing the strong q-log-convexity of the Narayana polynomials, and showing that the Narayana transformation preserves log-convexity. We begin with a formula of Brändén expressing the q-Narayana numbers as a specialization of Schur functions and, by deriving several symmetric function identities, we obtain the necessary Schur-positivity results. In addition, we prove the strong q-log-concavity of the q-Narayana numbers. The q-log-concavity of the q-Narayana numbers N q (n,k) for fixed k is a special case of a conjecture of McNamara and Sagan on the infinite q-log-concavity of the Gaussian coefficients.