Parity of the Partition Function p(n,k)

TL;DR

This paper investigates the parity properties of the partition function p(n,k), establishing its periodicity modulo m and analyzing the distribution of odd values for fixed k.

Contribution

It proves the periodicity of p(n,k) modulo m and provides bounds on the frequency of odd values for fixed k, advancing understanding of partition function parity.

Findings

p(n,k) is periodic modulo m for fixed k and m

Bounds are established for the number of odd p(n,k) values

Insights into the distribution of parity in partition functions

Abstract

Let p(n, k) denote the number of partitions of n into parts less than or equal to k. We show several properties of this function modulo 2. First, we prove that for fixed positive integers k and m, p(n,k) is periodic modulo m. Using this, we are able to find lower and upper bounds for the number of odd values of the function for a fixed k.