The q-log-concavity of q-binomial coefficients

AbstractThe number [nk]q of k-dimensional subspaces of an n-dimensional vector space over the field with q elements is a polynomial in q with nonnegative coefficients. We establish that [nk]q, 0 ⩽ k ⩽ n, is a log-concave sequence of polynomials. That is, the polynomial [nk]q2−[nk−1]q[nk+1]q has nonnegative coefficients for 0 < k < n.Our proof is simple and combinatorial. Our result generalizes the easily seen fact that [nk]q, 0 ⩽ k ⩽ n, is a log-concave sequence of numbers when q ⩾ 0, and it strengthens our two year old observation that the polynomial [nk]q2−q[nk−1]q[nk+1]q has nonnegative coefficients for 0 < k < n. We discuss related results and questions