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

TL;DR

This paper introduces efficient algorithms for computing the number of integer partitions of n into m parts and into distinct parts, achieving a time complexity of O(n^{3/2}) and providing practical implementation details.

Contribution

The paper presents two novel algorithms for calculating partition counts with improved efficiency and combines them to compute related partition functions in optimal time.

Findings

Algorithms run in O(n^{3/2}) time for partition counts.

A combined algorithm computes all P(n,m) efficiently.

Practical implementation and timing results are provided.

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.