Let $\bar{n} = (n_1,\dots, n_k)$ be a fixed tuple of positive integers. Define $\mathcal P_m ({\bar{n}}) $ to be the set of all partitions of $m$ with exactly $k$ parts and the i-th part is bounded by $n_i$.
How to find the cardinality of the set $\mathcal P_m ({\bar{n}}) $?
Kindly share your thoughts.