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arXiv · 1910.08224

Overpartitions and Bressoud's conjecture, I

Thomas Y. He·Kathy Q. Ji·Alice X. H. Zhao
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Abstract

In 1980, Bressoud conjectured a combinatorial identity $A_j=B_j$ for $j=0$ or $1$, where the function $A_j$ counts the number of partitions with certain congruence conditions and the function $B_j$ counts the number of partitions with certain difference conditions. Bressoud's conjecture specializes to a wide variety of well-known theorems in the theory of partitions. Special cases of his conjecture have been subsequently proved by Bressoud, Andrews, Kim and Yee. Recently, Kim resolved Bressoud's conjecture for the case $j=1$. In this paper, we introduce a new partition function $\bar{B}_j$ which can be viewed as an overpartition analogue of the partition function $B_j$ introduced by Bressoud. By means of Gordon markings, we build bijections to obtain a relationship between $\bar{B}_1$ and $B_0$ and a relationship between $\bar{B}_0$ and $B_1$. Based on these former relationships, we further give overpartition analogues of many classical partition theorems including Euler's partition theorem, the Rogers-Ramanujan-Gordon identities, the Bressoud-Rogers-Ramanujan identities, the Andrews-G\"ollnitz-Gordon identities and the Bressoud-G\"ollnitz-Gordon identities.

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BibTeXRIS

Thomas Y. He, Kathy Q. Ji, Alice X. H. Zhao. 2019-10-18. Overpartitions and Bressoud's conjecture, I. https://arxiv.org/abs/1910.08224

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