The homology of partitions with an even number of blocks

Sundaram, Sheila

Journal of Algebraic Combinatorics, Tome 4 (1995) no. 1, pp. 69-92.

Résumé

Summary: Let Õ ${}_{2n}^e\prod$_2n^e denote the subposet obtained by selecting even ranks in the partition lattice Õ ${}_{2n}\prod$_2n . We show that the homology of Õ ${}_{2n}^e\prod$_2n^e has dimension $frac(2n)!2^{2}n-1E_{2}n-1$ \frac{{$(2n)$!}}{{2^{2n - 1} }}E\_{2n - 1} , where $E_{2}n-1$ 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 [22], we show that, for the special case of \~O ${}_2{n}^{e}prod${\_{2n}^e } , the character of the symmetric group $S_{2}n$ on the homology is supported on the set of involutions. Our proof techniques lead to the discovery of a family of integers $b_i(n),2S_2^{i}x{S}_1^{2}n-2i$ S\_2^i xS\_1^{2n - 2i} , 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 Andr\'e or simsun number $a_n(2n)$. Similarly, the restriction of this homology module to $S_{2}n-1$ yields a family of integers $d_i(n)$, 1 \~O ${}_2{n}^{e}prod${\_{2n}^e } , 1 $leklen-1$. We conjecture that these are all permutation modules for $S_{2}n$.$

Détail

Mots-clés : homology representation, permutation module, André permutations, simsun permutation, tangent and Genocchi number