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 $d$?



On the generating functions for Euler characteristic of Hilbert schemes of points
Yes.


Write $\mathcal P\_d= 1 + p\_d$, so $\mathcal P\_d^{\chi(X)}= \sum\_{k=0}^{\infty} \left( \begin{array}{c} \chi( X) \\ k \end{array}\right) p\_d^k$. 


I will show that $\left( \begin{array}{c} \chi( X) \\ k \end{array}\right) p\_d^k$ is the generating function for the stratum of $Hilb^n X$ consisting of subschemas that are supported on $k$ distinct points.


This subscheme is a fibration over the variety $\left( \begin{array}{c} X \\ k \end{array}\right)$, the variety of all sets of $k$ distinct points in $X$. We can easily check that the Euler characteristic of $\left( \begin{array}{c} X \\ k \end{array}\right)$ is $\left( \begin{array}{c} \chi(X) \\ k \end{array}\right)$. The Euler characteristic of a fibration is the Euler characteristic of the base times the Euler characteristic of the fiber. So we must show that the Euler characteristic of the fiber is $p\_d^k$. But this is clear - it's just the Hilbert scheme of subschemes supported exactly at $k$ distinct fixed points, which is just a $k$-fold product of the hilbert scheme of nonempty subschemes supported at a single point, which is $p\_d$.


