Let $\Phi$ denote a root system and let $\mathfrak P$ denote the associated Kostant partition function. Thus $\mathfrak P(\lambda)$ is the number of ways of writing $\lambda$ as a sum of elements of $\Phi^+$. For example, if $\Phi=A_3$ then the longest root $\alpha_0$ can be written as $\alpha_0=(\alpha_1+\alpha_2+\alpha_3)$, $(\alpha_1)+(\alpha_2)+(\alpha_3)$, $(\alpha_1+\alpha_2)+\alpha_3$ or $\alpha_1 + (\alpha_2+\alpha_3)$. Thus $\mathfrak P(\alpha_0)=4$. There is a recursion $$\mathfrak P(\mu)=-\sum_{1\neq w\in W} (-1)^{l(w)}\mathfrak P(\mu+w(\rho)-\rho)$$ due to Kostant.

I can't find very many general results on $\mathfrak P$.

One knows that $\mathfrak P((p-1)\rho)$ is the dimension of the $0$ weight space of the Steinberg module (see Jantzen, RAGS, II.10.12). I conjecture that one also has $\mathfrak P(r\rho)$ is the dimension of the $0$ weight space in $L_{\mathbb C}(r\rho)$, where $L_{\mathbb C}(r\rho)$ is the irreducible representation of high weight $r\rho$ for a complex Lie algebra with root system $\Phi$ (which may be zero if $r$ is odd). A reference even for this basic statement would be good!

[RE-EDIT: This paragraph, including the conjecture, is completely wrong. See Chuck Hague's and Jim Humphreys' remarks below. Of course one must have $\dim H^0(\lambda)_0\leq \mathfrak P(\lambda)$ for all $\lambda$, so this gives a lower bound.]

I'd also like some data on asymptotics. I have used the Maple `Coxeter/Weyl' package to compute $\mathfrak P(2\rho)$ for $A_n$ up to $n=5$. The numbers are $1,3,15,219,7834\dots$ and correspond to sequence A007081 on the OEIS.

[EDIT: These calculated values are in fact the dimensions of the $0$ weight space of $H^0(2\rho)$. Since the conjecture that $\mathfrak P(2\rho)=\dim H^0(2\rho)_0$ is incorrect these do *not* coincide with the values of $\mathfrak P(2\rho)$ as claimed.]

I guess $\mathfrak P(2\rho) < n^n$. [EDIT: This looks very optimistic---suspect it's wrong for $n=4$.] Any ideas? What about $\mathfrak P (2r\rho)<(nr)^{nr}$ or something better? [EDIT: Similarly.] Lower bounds?