q-ballot数的q-对数凹性

The q-log-concavity of q-ballot Numbers

Carlitz和Riordan引入了ballot数的q模拟f_(q)(n,k).本文利用f_(q)(n,k)的组合解释,通过构造单射的方法证明f_(q)(n,k)关于n和关于k都具有q-对数凹性,即关于q的多项式f_(q)(n,k)^(2)-f_(q)(n+1,k)f_(q)(n-1,k)和f_(q)(n,k)2-f_(q)(n,k+1)f_(q)(n,k-1)对于0<k<n都具有非负系数.

Carlitz and Riordan introduced a q-analogue of ballot numbers f_(q)(n,k).In this paper,by using the combinatorial interpretation of f_(q)(n,k)and constructing injections,we prove that f_(q)(n,k)is q-log-concave with respect to n and k,that is,all the coefficients of the polynomials f_(q)(n,k)2−f_(q)(n+1,k)f_(q)(n−1,k)and f_(q)(n,k)2−f_(q)(n,k+1)f_(q)(n,k−1)are nonnegative for 0<k<n.