##### The corresponding article has been published in
Monatshefte Math. **107** (1989), 333-339.

## Christian Krattenthaler

# 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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