the coordinate ring $s = \mathbb{c}[x_{i,j}]$ of the space of $m \times n$ matrices carries an action of the group $gl_m \times gl_n$ via row and column operations on the matrix entries. if we consider any $gl_m \times gl_n$-invariant ideal $i$ in $s$, the syzygy modules $\mathrm{tor}_i(i,\mathbb{c})$ will carry a natural action of $gl_m \times gl_n$. by the bgg correspondence, they also carry an action of $\bigwedge^{\bullet}(\mathbb{c}^m \otimes \mathbb{c}^n)$. it turns out that we can combine these actions together and make them modules over the general linear lie superalgebra $\mathfrak{gl}(m \mid n)$. we will explain how this works and how it enables us to commute all betti number of any $gl_m \times gl_n$-invariant ideal $i$. this latter part will involve combinatorics of dyck paths.