Computation of q-Binomial Coefficients with the $P(n,m)$ Integer Partition Function

TL;DR

This paper presents efficient algorithms for computing q-binomial coefficients and related partition functions, including formulas for partitions with constraints, using properties of the $P(n,m)$ and $P(n,m,p)$ functions.

Contribution

It introduces new algorithms with polynomial time complexity for computing q-binomial coefficients and related partition counts, expanding the computational tools for partition theory.

Findings

q-binomial coefficient can be computed in O(n^3) time

Formulas for partitions with parts constrained by size or distinctness

Implementation of algorithms in computer algebra system

Abstract

Using $P(n,m)$, the number of integer partitions of $n$ into exactly $m$ parts, which was the subject of an earlier paper, $P(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ parts with each part at most $p$, can be computed in $O(n^2)$, and the q-binomial coefficient can be computed in $O(n^3)$. Using the definition of the q-binomial coefficient, some properties of the q-binomial coefficient and $P(n,m,p)$ are derived. The q-multinomial coefficient can be computed as a product of q-binomial coefficients. A formula for $Q(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ distinct parts with each part at most $p$, is given. Some formulas for the number of integer partitions with each part between a minimum and a maximum are derived. A computer algebra program is listed implementing these algorithms using the computer algebra program of the earlier paper.