The q-Log-convexity of the Generating Functions of the Squares of Binomial Coefficients

Details The q-Log-convexity of the Generating Functions of the Squares of Binomial Coefficients The q-Log-convexity of the Generating Functions of the Squares of Binomial Coefficients

2008-10-13

arxiv.org/abs/0810.2247v1

Mathematics

Combinatorics

29 pages

Scientific paper

We prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence $\{\sum_{k=0}^n{n\choose k}^2q^k\}_{n\geq 0}$. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. Then the principal specialization leads to the q-log-convexity. We also prove that a technical condition of Liu and Wang holds for the squares of the binomial coefficients. Hence we deduce that the linear transformation with respect to the triangular array $\{{n\choose k}^2\}_{0\leq k\leq n}$ is log-convexity preserving.

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