Residue class biases in unrestricted partitions, partitions into distinct parts, and overpartitions

TL;DR

The paper investigates residue-class biases in unrestricted partitions, partitions into distinct parts, and overpartitions by introducing weighted counts $p_n(a,b,m;x,y)$ and deriving their generating functions. It establishes general bias inequalities $p_n(a,b,m;x,y)\ge p_n(b,a,m;x,y)$ for $x\ge1$, $y\ge0$, and $1\le a<b\le m$, and develops a residue-class version of Ingham's Tauberian theorem to obtain precise asymptotics in symmetric residue cases, linking to the standard partition asymptotics. The authors provide explicit generating functions for symmetric-residue biases, derive their near-root-of-unity asymptotics, and prove an overpartition analogue of Andrews’ result, along with a conjecture for distinct partitions that guides future work. Together, these results give detailed nonnegativity and growth patterns for residue-weighted partition statistics and expand the toolkit for studying modular-residue phenomena in partition theory. The work has potential implications for understanding residue-class distributions in partition-related combinatorial structures and their analytic behavior.

Abstract

We prove specific biases in the number of occurrences of parts belonging to two different residue classes $a$ and $b$, modulo a fixed non-negative integer $m$, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size $n$ that belong to these sets of partitions and have a symmetric residue class bias (i.e., for $1\le a<m/2$ and $b=m-a$), as $n$ tends to infinity.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• proof
• Theorem 2.4