My coauthors and I are writing a (mostly expository) paper in which we construct the Specht module. Our proof that the Specht module is irreducible in characteristic zero implies the following stronger fact:

The number of monomial ideals of $\mathbb{Q}[x_1, \ldots, x_k]$ of codimension $n$ is equal to the number of $k$-tuples of partitions, $(\lambda^1, \ldots, \lambda^k)$, $\lambda^i \vdash n$ , for which $\langle h_{\lambda^1} \ast \cdots \ast h_{\lambda^k} \;,\; e_n \rangle = 1$.

Here, $\ast$ denotes Kronecker product, $h$ denotes the homogeneous symmetric function, $e$ denotes the elementary symmetric function, and $\langle \;\;, \;\; \rangle$ denotes the Hall inner product.

It seems likely that this fact is known. However, the literature on planar partitions (and higher-dimensional partitions) focuses on finding a generating function, which is a different, much harder question. Does anyone know a reference for the easier fact above?