A combinatorial model for the decomposition of multivariate polynomials rings as an $S_n$-module

We consider the symmetric group S-n-module of the polynomial ring with m sets of n commuting variables and m' sets of n anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition lambda (a partition of n) is the number of multiset tableaux of shape lambda satisfying certain column and row strict conditions. We also present a finite generating set for the ring of S-n invariant polynomials of this ring.