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 $_{2 n} ^{ e} \prod $_2n^e denote the subposet obtained by selecting even ranks in the partition lattice Õ $_{2 n} \prod $_2n . We show that the homology of Õ $_{2 n} ^{ e} \prod $_2n^e has dimension $frac(2 n)!2 ^2 n - 1 E _2 n - 1$ \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 $b _i( n), 2 S _2 ^ i xS _1 ^2 n - 2 i$ 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 $a _n(2 n)$. 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

Keywords: homology representation, permutation module, André permutations, simsun permutation, tangent and Genocchi number