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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 twoalgorithms, the resulting algorithm is $O(n^{3/2})$. The second algorithm usesa list of $P(n)$, the number of integer partitions of $n$, which is cached andtherefore 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 integerpartitions of $n$ into exactly $m$ distinct parts, and a list of $Q(n)$, thenumber 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 computedin $O(n^2)$. A computer algebra program is listed implementing thesealgorithms, and some timings of this program are provided.

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