A combinatorial proof of q-log-concavity of q-Eulerian numbers

Carlitz established a q-analog of the Eulerian numbers \(A_{n,k}(q)\) and defined the relationship \(A_{n,k}(q)=q^{\frac{(n-k)(n-k+1)}{2}}A_{n,k}^{*}(q)\). In this paper, by using the combinatorial interpretation of \(A_{n,k}^{*}(q)\) and constructing injective maps, we prove that \(A_{n,k}^{*}(q)\) and \(A_{n,k}(q)\) are q-log-concave, that is, all the coefficients of the polynomials \(( A_{n,k}^{*}(q)) ^{2}- A_{n,k-1}^{*}(q) A_{n,k+1}^{*}(q) \) and \((A_{n,k}(q)) ^{2}- A_{n,k-1}(q) A_{n,k+1}(q)\) are nonnegative for \(1< k <n\).