Abstract:  We prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence . 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 is log-convexity preserving.

  AMS Classification:  05E05, 05E10

  Keywords:  q-log-convexity, Schur positivity, Pieri's rule, the Jacobi-Trudi identity, principal specialization

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