Binomial transforms and integer partitions into parts of \(k\) different magnitudes (Q1656810)

Given a positive integer \(n\), let \(v_k(n)\) denote the number of partitions of \(n\) with exactly \(k\) distinct parts, and let \((-1)^k\mu_k(n)\) denote the difference between the number of partitions of \(n\) into an even number of parts and odd number parts, where the partitions must have exactly \(k\) distinct parts. For example, \(\mu_3(8) = 1\) since, while there are five partitions of \(8\) using exactly three distinct parts, three of the partitions have an odd number of parts and two have an even number of parts. Now, for a classical group \(G = G(n)\) acting on an \(n\)-dimensional vector space over a finite field with \(m\) elements, let \(P_{G,n}(k,m)\) be the probability that an element of \(G\) fixes a \(k\)-dimensional subspace. Denote by \(P_{G,\infty}(k,m)\) the limit of \(P_{G,n}(k,m)\) as \(n \to \infty\). Finally, let \(S_p\) denote the symplectic group. In the article under review, the author derives a formula for \(P_{S_p,\infty}(k,m)\) in terms of \(\mu_k\). As a consequence, the author derives a formula for \(P_{G,\infty}(k,m)\) in terms of \(v_k\).