The homology of partitions with an even number of blocks (Q1344191)

Let \(\Pi_ n^ e\) be the subposet of the partition lattice \(\Pi_ n\), consisting of partitions with an even number of blocks. It is proved that the homology of \(\Pi_{2n}^ e\) has dimension \(\frac{(2n)!}{2^{2n- 1}} E_{2n-1}\), where \(E_{2n-1}\) are tangent numbers. It is shown that the character values of \(S_{2n}\) acting on the top homology of \(\Pi_{2n}^ e\) are supported on involutions, so the Frobenius characteristic \(R_{2n}\) is a polynomial in the homogeneous symmetric functions \(h_ 2, h_ 1: R_{2n}= \sum_{i= 2}^ n b_ i(n) h_ 2^ i h_ 1^{2n- 2i}\). It is conjectured that the homology of \(\Pi_{2n}^ e\) is in fact a permutation module for \(S_{2n}\), so the integers \(b_ i(n)\) are positive. This would imply the existence of new refinements, into sums of powers of 2, of the tangent numbers and the simsun numbers. Similar conjectures about \(S_{2n-1}\) refine the Genocchi numbers.