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

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

2008-06-10

arxiv.org/abs/0806.1561v1

Mathematics

Combinatorics

38 pages, 6 figures

Scientific paper

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.

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