Let $\alpha = (\alpha_1 , \dots , \alpha_n)$ be a vector with $\alpha_i \geq 0$ for all $1 \leq i \leq n$ and $|\alpha| := \sum_i \alpha_i = d$. Define the order $\leq$ by saying $\alpha ' \leq \alpha$ if $\alpha'_i \leq \alpha_i$ for all $1 \leq i \leq n$.
My question is the following: let $0 \leq \ell \leq d$. How can I count the number of $\alpha'$ such that $\alpha' \leq \alpha$ and $|\alpha'| = \ell$? If each $\alpha_i \geq \ell$, then this number is of course $\binom{n+\ell-1}{\ell}$ ("stars and bars"), but it seems to be a bit more subtle to count when some components are $< \ell$.
My suspicion is that this is a well known counting problem, but I am not particularly well versed in combinatorics. Thanks.
This is the problem of distributing balls in bins with limited capacity. It can be solved using inclusion–exclusion. The result is
$$ \sum_{S\subseteq[n]}(-1)^{|S|}\binom{n+\ell-\sum_{j\in S}(\alpha_j+1)-1}{n-1}\;, $$
where $[n]=\{1,\ldots,n\}$ and where, contrary to the usual convention, the binomial coefficient is taken to be zero if the upper index is negative.