Biases in Non-Unitary Partitions

TL;DR

The paper extends parity-bias phenomena to non-unitary partitions (parts $\\ge 2$) and to partitions with parts separated by parity, using $q$-series methods to produce analytical proofs of inequalities. It establishes that for non-unitary partitions the bias favors more even parts over odd parts: $q_o(n)<q_e(n)$ for $n\ge 8$, and it generalizes to classes $q_{j,k,m}(n)$ with $m\ge 2$, yielding $q_{0,1,m}(n)>q_{1,0,m}(n)$ for $n\ge 4m+3$; it also derives parity-based inequalities in the Andrews framework for partitions with parity-separated parts. The results hinge on generating-function identities and transformations (e.g., Euler-type products and Heine's transformation) to compare counting functions through analytic means, and they include non-unitary versions of the inequalities. The work suggests further avenues, including asymptotic analyses, stronger inequalities, and combinatorial proofs, with potential extensions to broader partition families studied by Andrews and collaborators.

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).

Theorems & Definitions (12)

• Theorem 1.1: Banerjee et al., zbMATH07524341
• Theorem 1.2
• Theorem 1.3
• Theorem 3.1: Theorem 1.5, zbMATH07524341
• Theorem 3.2
• proof : Proof of Theorem \ref{['thm:mm']}
• proof : Proof of Theorem \ref{['thm:kim-new']}
• Theorem 4.1
• proof
• Remark 4.1