The Homology of Partitions with an Even Number of Blocks

Abstract

Let (Formula presented.) denote the subposet obtained by selecting even ranks in the partition lattice (Formula presented.). We show that the homology of (Formula presented.) has dimension (Formula presented.), where (Formula presented.) is the tangent number. It is thus an integral multiple of both the Genocchi number and an André or simsun number. Using the general theory of rank-selected homology representations developed in [22], we show that, for the special case of (Formula presented.), the character of the symmetric group S2 n on the homology is supported on the set of involutions. Our proof techniques lead to the discovery of a family of integers bi(n), 2 ≤ i ≤ n, defined recursively. We conjecture that, for the full automorphism group S2n, the homology is a sum of permutation modules induced from Young subgroups of the form (Formula presented.), with nonnegative integer multiplicity bi(n). The nonnegativity of the integers bi(n) would imply the existence of new refinements, into sums of powers of 2, of the tangent number and the André or simsun number an(2 n). Similarly, the restriction of this homology module to S2 n−1 yields a family of integers di(n), 1 ≤ i ≤ n − 1, such that the numbers 2−idi(n) refine the Genocchi number G2 n. We conjecture that 2−idi(n) is a positive integer for all i. Finally, we present a recursive algorithm to generate a family of polynomials which encode the homology representations of the subposets obtained by selecting the top k ranks of (Formula presented.), 1 ≤ k ≤ n − 1. We conjecture that these are all permutation modules for S2 n. © 1995, Kluwer Academic Publishers. All rights reserved.