Distributions of parity differences and biases in partitions into distinct parts

TL;DR

This work addresses the distribution of parity differences in partitions into distinct parts by examining the statistic $pd(\lambda)$ and its normalized form $n^{-1/4}pd(\lambda)$. The authors deploy a circle method with a carefully widened major arc via Euler–Maclaurin expansions to obtain an explicit asymptotic expansion for $d_{\alpha,\beta;N;c_0n^{1/4}}(n)$, revealing a predominantly $e^{\pi\sqrt{n/3}} n^{-3/4}$ scaling with coefficients that depend on residue data and thresholds. They prove that $n^{-1/4}pd_{\alpha,\beta;N}(\lambda)$ converges to a normal distribution with mean $0$ and variance $\frac{2\sqrt{3}}{\pi N}$, and they provide a detailed description of parity biases, including exact asymptotics and a limiting bias distribution for $N=2$ and $N\ge5$. The results extend to generalised parity differences modulo $N$, including explicit coefficient evaluations and a robust computational framework, thereby quantifying parity phenomena in restricted partitions and enriching the understanding of partition statistics in modular settings.

Abstract

For a partition $λ\vdash n$, we let $\operatorname{pd}(λ)$, the parity difference of $λ$, be the number of odd parts of $λ$ minus the number of even parts of $λ$. We prove for $c_0\in\mathbb{R}$ an asymptotic expansion for the number of partitions of $n$ into distinct parts with normalised parity difference $n^{- 1/4}\operatorname{pd}(λ)$ greater than $c_0$ as $n\to \infty$. As a corollary, we find the distribution of the parity differences and parity biases for partitions of $n$ into distinct parts. We also establish analogous results for generalised parity differences modulo $N$.

Figures (4)

• Figure 1: The ratios between asymptotic estimates of $d_{1,2;2;c_0n^{1/4}}(n)$ and $d_{2,1;2;c_0n^{1/4}}(n)$ to their true values, $c_0=1$, $1\le n\le 50000$.
• Figure 2: The ratios between asymptotic estimates of $d_{1,2;2;c_0n^{1/4}}(n)$ and $d_{2,1;2;c_0n^{1/4}}(n)$ to their true values, $c_0=2$, $1\le n\le 50000$.
• Figure 3: $\#\left\{{\lambda \in \mathcal{D}(50000)}\; \middle|\; {\operatorname{pd}_{1,2;2}(\lambda)=k}\right\}$ for $-50\le k \le 50$, normalised.
• Figure 4: $\#\{\lambda \in \mathcal{D}(50000)\;|\;\operatorname{pd}_{1,2;2}(\lambda)=k\} - \#\{\lambda \in \mathcal{D}(50000)\;|\:\operatorname{pd}_{2,1;2}(\lambda)=k\}$ for $0\le k \le 50$, normalised.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark
• Proposition 2.1
• Remark
• Remark
• Proposition 2.2: BMRS24AP
• Proposition 2.3: BMRS24AP