The Homology of Partitions with an Even Number of Blocks

Abstract

Let \Pi_{2n}^e denote the subposet obtained by selecting even ranks in the partition lattice \Pi_{2n}. We show that the homology of \Pi_{2n}^e has dimension {{(2n)!}\over {2^{2n-1}}} E_{2n-1}, where E_{2n-1} is the tangent number. It is thus an integral multiple of both the Genocchi number and an Andre or simsun number. Using the general theory of rank-selected homology representations developed in l22r, we show that, for the special case of \Pi_{2n}^e, the character of the symmetric group S2n 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 S^i_2 \times S^{2n-2i}_1, 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 Andre or simsun number an(2n). Similarly, the restriction of this homology module to S2n−1 yields a family of integers di(n), 1 ≤ i ≤ n − 1, such that the numbers 2−idi(n) refine the Genocchi number G2n. 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 \Pi_{2n}^e, 1 ≤ k ≤ n − 1. We conjecture that these are all permutation modules for S2n.