The Homology of Partitions with an Even Number of Blocks

Let denote the subposet obtained by selecting even ranks in the partition lattice . We show that the homology of has dimension , where 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 , the character of the symmetric group S _2n on the homology is supported on the set of involutions. Our proof techniques lead to the discovery of a family of integers b _i(n), 2 i n, defined recursively. We conjecture that, for the full automorphism group S _2n, the homology is a sum of permutation modules induced from Young subgroups of the form , with nonnegative integer multiplicity b _i(n). The nonnegativity of the integers b _i(n) would imply the existence of new refinements, into sums of powers of 2, of the tangent number and the André or simsun number a _n(2n).Similarly, the restriction of this homology module to S _2n–1 yields a family of integers d _i(n), 1 i n – 1, such that the numbers 2^–i d _i(n) refine the Genocchi number G _2n . We conjecture that 2^–i d _i(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 , 1 k n – 1. We conjecture that these are all permutation modules for S _2n .