Set superpartitions and superspace duality modules

The superspace ring ${\mathrm{\Omega }}_n$ is a rank $n$ polynomial ring tensor a rank $n$ exterior algebra. Using an extension of the Vandermonde determinant to ${\mathrm{\Omega }}_n$, the authors previously defined a family of doubly graded quotients ${\mathbb{W}}_{n,k}$ of ${\mathrm{\Omega }}_n$ which carry an action of the symmetric group ${\mathfrak{S}}_n$ and satisfy a bigraded version of Poincar'e Duality. In this paper, we examine the duality modules ${\mathbb{W}}_{n,k}$ in greater detail. We describe a monomial basis of ${\mathbb{W}}_{n,k}$ and give combinatorial formulas for its bigraded Hilbert and Frobenius series. These formulas involve new combinatorial objects called {\em ordered superpartitions}. These are ordered set partitions $(B_1\mid \cdots \mid B_k)$ of $\{1,\dots ,n\}$ in which the non-minimal elements of any block $B_i$ may be barred or unbarred.