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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