Combinatorial proof of an inequality on some partitions separated by parity

TL;DR

The paper addresses inequalities among parity-separated partition functions $p_{yz}^{wx}(n)$ and provides a concrete combinatorial proof that $p_{od}^{eu}(n)<p_{eu}^{od}(n)$ for $n\,\ge\,373$. It constructs a detailed 17-case injection from $A_{od}^{eu}(n)$ to $B_{eu}^{od}(n)$, partitioning the domain into $A_j(n)$ with corresponding images $B_j(n)$, and shows a bijection on each case for large $n$. Consequently, $p_{od}^{eu}(n)$ equals the sum of the $|A_j(n)|$, while $p_{eu}^{od}(n)$ counts a strictly larger set, establishing the inequality for $n\ge 373$. This work provides a full combinatorial proof addressing an open question and complements prior asymptotic results, though it notes that combinatorial proofs for two other related inequalities remain elusive.

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.

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Theorems & Definitions (2)

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