Residue class biases in unrestricted partitions, partitions into distinct parts, and overpartitions

TL;DR

This paper investigates residue class biases in various partition sets, proving inequalities and asymptotic formulas that reveal underlying distribution patterns of parts in partitions.

Contribution

It establishes specific biases and asymptotic formulas for residue class distributions in unrestricted, distinct, and overpartition sets, advancing understanding of partition residue behaviors.

Findings

Proves biases in residue class occurrences for partition sets.

Derives inequalities for residue-weighted partition functions.

Provides asymptotic formulas for symmetric residue class biases.

Abstract

We prove specific biases in the number of occurrences of parts belonging to two different residue classes $a$ and $b$, modulo a fixed non-negative integer $m$, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size $n$ that belong to these sets of partitions and have a symmetric residue class bias (i.e., for $1\le a<m/2$ and $b=m-a$), as $n$ tends to infinity.