Biases in Non-Unitary Partitions

TL;DR

This paper extends the study of parity biases in integer partitions to non-unitary partitions with parts greater than one, providing new results and inequalities between classes of partitions based on parity.

Contribution

It introduces analogous results for non-unitary partitions and explores inequalities between classes of partitions separated by parity, expanding prior work on parity bias.

Findings

Proves new results for non-unitary partitions similar to previous studies.

Establishes inequalities between classes of partitions based on parity.

Extends the concept of parity bias to a broader class of partitions.

Abstract

Recently, the concept of parity bias in integer partitions has been studied by several authors. We continue this study here, but for non-unitary partitions (namely, partitions with parts greater than $1$). We prove analogous results for these restricted partitions to those that have been obtained by Kim, Kim, and Lovejoy (2020) and Kim and Kim (2021). We also look at inequalities between two classes of partitions studied by Andrews (2019), where the parts are separated by parity (either all odd parts are smaller than all even parts or vice versa).