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arXiv · 2406.19823

Separable integer partition classes and partitions with congruence conditions

Thomas Y. He·C. S. Huang·H. X. Li·X. Zhang
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Abstract

In this article, we first investigate the partitions whose parts are congruent to $a$ or $b$ modulo $k$ with the aid of separable integer partition classes with modulus $k$ introduced by Andrews. Then, we introduce the $(k,r)$-overpartitions in which only parts equivalent to $r$ modulo $k$ may be overlined and we will show that the number of $(k,k)$-overpartitions of $n$ equals the number of partitions of $n$ such that the $k$-th occurrence of a part may be overlined. Finally, we extend separable integer partition classes with modulus $k$ to overpartitions and then give the generating function for $(k,r)$-modulo overpartitions, which are the $(k,r)$-overpartitions satisfying certain congruence conditions.

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Thomas Y. He, C. S. Huang, H. X. Li, X. Zhang. 2024-06-28. Separable integer partition classes and partitions with congruence conditions. https://arxiv.org/abs/2406.19823

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