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?

To use Sage to compute these numbers, start with the following lines:

```
h = SymmetricFunctions(QQ).homogeneous()
e = SymmetricFunctions(QQ).elementary()
```

Now, the number of monomial ideals in three variables up to codimension 7 may be computed using

```
for n in range(7):
print sum([1 if h[a].itensor(h[b].itensor(h[c])).scalar(e[n]) == 1 else 0\
for a in Partitions(n)\
for b in Partitions(n)\
for c in Partitions(n)])
```

which lists the numbers $1, 1, 3, 6, 13, 24, 48,$ an initial segment of OEIS A000219. Similarly, monomial ideals in four variables may be counted

```
for n in range(5):
print sum([1 if h[a].itensor(h[b].itensor(h[c].itensor(h[d])))\
.scalar(e[n]) == 1 else 0\
for a in Partitions(n)\
for b in Partitions(n)\
for c in Partitions(n)\
for d in Partitions(n)])
```

The resulting sequence, $1, 1, 4, 10, 26$, is an initial segment of OEIS A000293. It is a famous open problem in combinatorics to find a generating function for this sequence.

EDIT for the curious, the paper has been posted: https://arxiv.org/abs/1701.05277