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 (n-k)(n-k+1)/ 2 A(n,k)(& lowast;)(q). In this paper, by using the combinatorial interpretation of A(n,k)(& lowast;)(q) and constructing injective maps, we prove that A(n,k)(& lowast;)(q) and A(n,k)(& lowast;)(q) are q-log-concave, that is, all the coefficients of the polynomials A(n,k)(& lowast;)(q))(2 )- A(n,k-1)(& lowast;)(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.