Some methods for finding the number of partitions of n with m−parts, P(n, m)

P(n, m) denotes the number of partitions of n with m parts, while Pm(n) denotes the number of partitions of n with parts at most m. It is a well-known result that the number of partitions of n ≥ 0 into m or fewer parts is equal to Pm(n). In the literature, there are some results for P(n, m) and Pm(n) for some values of m. In this work, we give more compact results for P(n, m) for the values of m ≤ 12. Furthermore, we obtain a method for determining P(n, m) when we know the expression for P(n, m − 1).