Is this integer partition studied? First, this is my firt post here; I'm not a rigorously trained mathematician so apologies for the abuse of language, or if the problem is too trivial :)
In a finite-system problem in statistical physics, we ended up with an equivalent problem of integer partition:
Given $k, m, n \in \mathbf{N}\cup \left\{0\right\}$ and $k\le m\times n$, find the number of solutions of the following equation
\begin{equation}
k = \sum_{i=1}^{m} x_i,
\end{equation}
where $x_i \in \mathbf{N}\cup \left\{0\right\}$, and is restricted by
\begin{equation}
n \ge x_1 \ge x_2 \ldots x_{m-1} \ge x_m \ge 0. 
\end{equation}
The solution of such a problem turned out to be the core part of our calculations. We have found a recursive relation of the desired number of solutions, and it will be nice if we can cite / refer to related references, if they exist, when writing up our manuscript.
Thanks very much in advance!

You can find any information about such problems in follow source (see chapter $3$):
George E. Andrews, The Theory of Partitions, Addison-Wesley Publishing Company. 
