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Schur positivity and the \(q\)-log-convexity of the Narayana polynomials. (English) Zbl 1230.05277

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.

MSC:

05E05 Symmetric functions and generalizations
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
05A15 Exact enumeration problems, generating functions

Software:

Maple; ACE
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References:

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