Let $\vec k=(k_1,...,k_d)$ be a $d$-tuple with $k_i$ being non-negative integers satisfying $\sum_i k_i=n$; $\vec k$ is then referred to as a weak $d$-composition of $n$. It is clear that if $\vec a$ is a weak $d$-composition of $l$, and $\vec b$ is a weak $d$-composition of $m$, then $\vec k\equiv\vec a +\vec b=(a_1+b_1,...,a_d+b_d)$ is a weak $d$-composition of $l+m\equiv n$.
I want to know, for a given $\vec k$ and $l$, how many pairs $\vec a$ and $\vec b$ exist? Alternatively, we are looking for all $\vec a$ which are weak $d$-composition of $l$ satisfying $a_i\leq k_i$ for all $i$. Let $\mathbb{A}^l(\vec k)$ denote the set of all such $\vec a$, for a fixed $l$ and $\vec k$.
Q1: Is this concept and notation already in the literature?
Q2: What is the cardinality of this set?
It is clear that the cardinality is symmetric under $l\to n-l$. Please provide references, as I am quite new to combinatorics. This problem emerges when writing Vandermonde's identity for multinomials (e.g. Eq. 4 here, different notation), where we implicitly sum over all $\vec a \in \mathbb{A}^l(\vec k)$. I wrote code on Mathematica that counts the elements for individual cases of $\vec k$, but I didn't succeed in conjecturing a formula.
