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 Brändén 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.

  AMS Classification:  05E05, 05E10

  Keywords:  q-log-concavity, q-log-convexity, q-Narayana number, Narayana polynomial, lattice permutation, Schur positivity, Littlewood-Richardson rule

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