Combinatorial proofs of inequalities involving the number of partitions with parts separated by parity

TL;DR

This paper provides combinatorial proofs for inequalities involving the counts of partitions of integers with parts separated by parity, including new results and a conjecture related to multiplicity restrictions.

Contribution

It introduces combinatorial proofs for inequalities between partition counts with parity-based restrictions, including a conjecture by Fu and Tang.

Findings

Proved that for n≥5, p_{od}^{eu}(n)<p_{ed}^{ou}(n).

Established combinatorial inequalities between partitions with parity restrictions.

Proved a conjecture involving partitions with multiplicity restrictions.

Abstract

We consider the number of various partitions of $n$ with parts separated by parity and prove combinatorially several inequalities between these numbers. For example, we show that for $n\geq 5$ we have $p_{od}^{eu}(n)<p_{ed}^{ou}(n)$, where $p_{od}^{eu}(n)$ is the number of partitions of $n$ with odd parts distinct and even parts unrestricted and all odd parts less than all even parts and $p_{ed}^{ou}(n)$ is the number of partitions of $n$ with even parts distinct and odd parts unrestricted and all even parts less than all odd parts. We also prove a conjectural inequality of Fu and Tang involving partitions with parts separated by parity with restrictions on the multiplicity of parts.