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

Chen, William Y. C.; Wang, Larry X. W.; Yang, Arthur L. B.

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

Authors

Chen, William Y. C.

Wang, Larry X. W.

Yang, Arthur L. B.

Description

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 Bränd$\mathrm{\acute{e}}$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.
38 pages, 6 figures