Geodesic
Schur positivity and the $q$-log-convexity of the Narayana polynomials
Journal of Algebraic Combinatorics, Tome 32 (2010) no. 3, pp. 303-338.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Keywords: keywords $q$-log-concavity, $q$-log-convexity, $q$-narayana number, narayana polynomial, lattice permutation, Schur positivity, Littlewood-Richardson rule 
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     author = {Chen, William Y.C. and Wang, Larry X.W. and Yang, Arthur L.B.},
     title = {Schur positivity and the $q$-log-convexity of the {Narayana} polynomials},
     journal = {Journal of Algebraic Combinatorics},
     pages = {303--338},
     publisher = {mathdoc},
     volume = {32},
     number = {3},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2010__32_3_a7/}
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Chen, William Y.C.; Wang, Larry X.W.; Yang, Arthur L.B. Schur positivity and the $q$-log-convexity of the Narayana polynomials. Journal of Algebraic Combinatorics, Tome 32 (2010) no. 3, pp. 303-338. http://geodesic.mathdoc.fr/item/JAC_2010__32_3_a7/