Combinatorial proof of an inequality on some partitions separated by parity

TL;DR

This paper provides a combinatorial proof for an inequality involving integer partitions separated by parity, addressing a question from recent research and establishing the inequality for all sufficiently large n.

Contribution

It offers the first combinatorial proof of a parity-separated partition inequality, extending the understanding of partition function inequalities beyond asymptotic results.

Findings

Proves the inequality for n ≥ 373 using combinatorial methods

Answers a question posed by Bringmann, Craig, and Nazaroglu

Establishes a concrete bound for the inequality's validity

Abstract

In 2019, Andrews investigated integer partitions in which all parts of a given parity are smaller than those of the opposite parity and introduced eight partition functions based on the parity of the smaller parts and parts of a given parity appearing at most once or an unlimited number of times. Recently, Bringmann, Craig and Nazaroglu studied the asymptotic behavior of the eight partition functions proved several inequalities for sufficiently large $n$. At the end of their paper, they asked for combinatorial proofs of those inequalities. In this paper, we prove that an inequality on partitions separated by parity holds for $n\geq 373$ by a combinatorial method. This answers a question posed by Bringmann, Craig and Nazaroglu.