Estimates on binomial sums of partition functions

Let p(n) denote the partition function and define p(n, k) = Sigma (k)(j=0) ((n-j)(k-j)) p(j) where p(0) = 1.We prove that p(n, k) is unimodal and satisfies p(n, k) < 2.825/<root>n 2(n) for fixed n greater than or equal to 1 and all 1 less than or equal to k less than or equal to n. This result has an interesting application: the minimal dimension of a faithful module for a k-step nilpotent Lie algebra of dimension n is bounded by p(n, k) and hence by 3/rootn 2(n), independently of k. So far only the bound n(n-1) was known. We will also prove that p(n, n - 1) < <root>n exp(pi root 2n/3) for n greater than or equal to 1 and p(n - 1, n - 1) < exp(<pi>root 2n/3).