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arXiv · 2408.14365

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

Michael J. Schlosser·Nian Hong Zhou
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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.

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Michael J. Schlosser, Nian Hong Zhou. 2024-08-26. Residue class biases in unrestricted partitions, partitions into distinct parts, and overpartitions. https://arxiv.org/abs/2408.14365

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