# On the q-log-concavity of Gaussian binomial coefficients

Abstract. We give a combinatorial proof that \qbinom {a}{k} \qbinom {b}{l} - \qbinom {a}{k-1} \qbinom {b}{l+1} is a polynomial in q with nonnegative coefficients for nonnegative integers a, b, k, l with a>=b and l>=k. In particular, for a=b=n and l=k, this implies the q-log-concavity of the Gaussian binomial coefficient \qbinom {n}{k}, which was conjectured by Butler (Proc. Amer. Math. Soc. 101 (1987), 771-775).

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