The q-log-concavity of q-ballot numbers

Abstract

Carlitz and Riordan introduced the q-analogue f_q\left(n,k\right) of ballot numbers. In this paper, using the combinatorial interpretation of f_q\left(n,k\right) and constructing injections, we prove that f_q\left(n,k\right) is q-log-concave with respect to n and k, i.e., all coefficients of the polynomials f_q{\left(n,k\right)}^2-f_q\left(n+1,k\right)f_q\left(n-1,k\right) and f_q{\left(n,k\right)}^2-f_q\left(n,k+1\right)f_q\left(n,k-1\right) are non-negative for .

Graphical abstract

Keywords

q-log-concavity / q-ballot number / lattice path / inversion

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