Computation of $P(n,m)$, the Number of Integer Partitions of $n$ into Exactly $m$ Parts

Abstract

Two algorithms for computing $P(n,m)$, the number of integer partitions of $n$ into exactly $m$ parts, are described, and using a combination of these two algorithms, the resulting algorithm is $O(n^{3/2})$. The second algorithm uses a list of $P(n)$, the number of integer partitions of $n$, which is cached and therefore needs to be computed only once. Computing this list is also $O(n^{3/2})$. With these algorithms also $Q(n,m)$, the number of integer partitions of $n$ into exactly $m$ distinct parts, and a list of $Q(n)$, the number of integer partitions of $n$ into distinct parts, can be computed in $O(n^{3/2})$. A list of $P(n,1)..P(n,n)$ and $P(m,m)..P(n,m)$ can be computed in $O(n^2)$. A computer algebra program is listed implementing these algorithms, and some timings of this program are provided.