Let $x_{ij}$ be a matrix of non-negative integers, with $i\in\{1,\dots,N\}$ and $j\in\{1,\dots,M\}$. I need to evaluate the following sum: $$ \sum_{\{x_{ij}\}} \frac{\prod_{j}\left( \sum_{i}x_{ij} \right)!}{\prod_{ij}x_{ij}!} \prod_{i}\delta\left( \sum_{j}x_{ij}-c_{i} \right) $$ where the $c_i$ are given non-negative integers and $\delta(\cdot)$ is the Kronecker delta. Can we evaluate this in closed-form, or simplify it somehow?
Relation to multinomials. The difficulty stems from the presence of the factorials $\left( \sum_{i}x_{ij} \right)!$ in the numerator of the summand. Indeed, if these terms were not present, the sum can be carried out with the Multinomial Theorem, $$ \sum_{\{x_{ij}\}} \frac{1}{\prod_{ij}x_{ij}!} \prod_{i}\delta\left( \sum_{j}x_{ij}-X_{i} \right) = \prod_i\sum_{\{x_{ij}\}} \frac{\delta\left( \sum_{j}x_{ij}-X_{i} \right)}{\prod_{j}x_{ij}!} = \prod_i \frac{M^{X_i}}{X_i!} $$
