Schur positivity and the q-log-convexity of the Narayana polynomials
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- Chen, W.Y.C., Wang, L.X.W. & Yang, A.L.B. J Algebr Comb (2010) 32: 303. doi:10.1007/s10801-010-0216-x
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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 Nq(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.