Let $d>1$ be an integer. If $n\geq 0$ is an integer we have a notion of $d$-dimensional *partitions* of $n$; the number of these, denoted $p_d(n)$, is the number of ways we can stack $n$ ($d$-dimensional) boxes in a corner of a $d$-dimensional "room". No closed formula is known for $p_d$, for *any* $d>1$. As far as I know, the generating function $\mathcal P_d$ for $p_d$ is known for $d=2,3$, but for no higher $d$'s:
\begin{align}
\mathcal P_2=\sum_{n\geq 0}p_2(n)t^n&=\prod_{k\geq 1}(1-t^k)^{-1},\notag\\
\mathcal P_3=\sum_{n\geq 0}p_3(n)t^n&=\prod_{k\geq 1}(1-t^k)^{-k}.\notag
\end{align}

However, it seems to me that to find $p_d(n)$ is to find the number of "higher dimensional Young Tableaux", and these correspond to monomial ideals in $\mathbb C^d$. So it should be true that $$p_d(n)=\chi(\textrm{Hilb}^n(\mathbb C^d)_0),$$ the topological Euler characteristic of the punctual Hilbert scheme. It is also true that, if $S$ is a smooth projective surface and $Y$ is a smooth projective threefold, then \begin{align} \sum_{n\geq 0}\chi(\textrm{Hilb}^nS)t^n&=\mathcal P_2^{\chi(S)}\,\,\,\,\,\,\,\textrm{(Göttsche's formula)}\notag\\ \sum_{n\geq 0}\chi(\textrm{Hilb}^nY)t^n&=\mathcal P_3^{\chi(Y)} \,\,\,\,\,\,\,\textrm{(Cheah's formula)}\notag \end{align}

**Question**: do we have such formulas for any $d$? in other words, do we have $$\sum_{n\geq 0}\chi(\textrm{Hilb}^nX)t^n=\mathcal P_d^{\chi(X)}$$ for any smooth projective $X$ of dimension $d$?