Answering a question of Geoff Robinson, we compute the large n limiting proportion of i(n, q)/qbn 2/2c, where i(n, q) denotes the number of involutions in GL(n, q). We give similar results for the finite unitary, symplectic, and orthogonal groups, in both odd and even characteristic. At the heart of this work are certain new “sum=product” identities. Our self-contained treatment of the enumerat...

We introduce a new class of combinatorial objects, which we call “bipieri tableaux,” which arise in a natural way from the evaluation of products consisting of commutative or noncommutative complete homogeneous symmetric functions and elementary symmetric functions through repeated applications of Pieri rules. We prove using sign-reversing involutions on bipieri tableaux an elegant coproduct fo...