The homology of partitions with an even number of

Let e 2n denote the subposet obtained by selecting even ranks in the partition lattice 2n : We show that the homology of e 2n has dimension (2n)! 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 Andr e or simsun number. Using the general theory of rank-selected homology representations developed in Su], we show that, for the special case of e 2n ; 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; deened 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 S i 2 S 2n?2i 1 ; with nonnegative integer multiplicity b i (n): The nonnegativity of the integers b i (n) would imply the existence of new reenements, into sums of powers of 2, of the tangent number and the Andr e 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) reene 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 e 2n ; 1 k n ? 1: We conjecture that these are all permutation modules for S 2n :