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arXiv · 2306.11909

Distributions of parity differences and biases in partitions into distinct parts

Siu Hang Man
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Abstract

For a partition $\lambda \vdash n$, we let $\operatorname{pd}(\lambda)$, the parity difference of $\lambda$, be the number of odd parts of $\lambda$ minus the number of even parts of $\lambda$. 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 $n^{- 1/4}\operatorname{pd}(\lambda)$ 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 $N$.

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Siu Hang Man. 2023-06-20. Distributions of parity differences and biases in partitions into distinct parts. https://doi.org/10.1016/j.ejc.2025.104157

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