Permutation resolutions for Specht modules

For every composition λ of a positive integer r, we construct a finite chain complex whose terms are direct sums of permutation modules M ^μ for the symmetric group (mathfrak{S}_{r}) with Young subgroup stabilizers (mathfrak{S}_{mu}). The construction is combinatorial and can be carried out over every commutative base ring k. We conjecture that for every partition λ the chain complex has homology concentrated in one degree (at the end of the complex) and that it is isomorphic to the dual of the Specht module S ^λ . We prove the exactness in special cases.