  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.
