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On theq-log-concavity of Gaussian binomial coefficients

Authors:	Christian Krattenthaler

Affiliation:	(1) Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria

Abstract:	We give a combinatorial proof that $left[ {begin{array}{{20}c} a k end{array} } right]_q left[ {begin{array}{{20}c} b l end{array} } right]_q - left[ {begin{array}{{20}c} a {k - 1} end{array} } right]_q left[ {begin{array}{{20}c} b {l + 1} end{array} } right]_q$ is a polynomial inq with nonnegative coefficients for nonnegative integersa, b, k, l withab andlk. In particular, fora=b=n andl=k, this implies theq-log-concavity of the Gaussian binomial coefficients $left[ {begin{array}{*{20}c} n k end{array} } right]_q$ , which was conjectured byButler (Proc. Amer. Math. Soc. 101 (1987), 771–775).

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