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

## TL;DR

This paper shows that certain number patterns (partitions) have biases in how often specific remainders appear when divided by a fixed number.

## Contribution

The paper proves new biases in residue classes for three types of partitions and provides asymptotic formulas for these biases.

## Key findings

- Biases in residue classes for unrestricted partitions are proven.
- Asymptotic formulas for symmetric residue class biases are established.
- Results apply to partitions into distinct parts and overpartitions.

## Abstract

We prove specific biases in the number of occurrences of parts belonging to two different residue classes a and b, modulo a fixed nonnegative 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 \documentclass[12pt]{minimal}
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\begin{document}$$1\le a<m/2$$\end{document}1≤a<m/2 and \documentclass[12pt]{minimal}
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\begin{document}$$b=m-a$$\end{document}b=m-a), as n tends to infinity.