An Overpartition Analogue of Bressoud's Theorem of Rogers-Ramanujan Type

TL;DR

This paper extends Bressoud's Rogers-Ramanujan type theorem to overpartitions, establishing new combinatorial identities and relations between overpartition counts with specific restrictions.

Contribution

It provides a general overpartition analogue of Bressoud's theorem for all valid i, generalizing previous results limited to i=1.

Findings

Established an overpartition analogue of Bressoud's theorem for general i.

Proved the equality C_{k,i}(n)=D_{k,i}(n) for overpartition counts.

Extended Rogers-Ramanujan-Gordon type identities to overpartitions.

Abstract

For $k\geq i\geq 1$, let $B_{k,i}(n)$ denote the number of partitions of $n$ such that part 1 appears at most $i-1$ times, two consecutive integers l and $l+1$ appear at most $k-1$ times and if l and $l+1$ appear exactly $k-1$ times then the total sum of the parts l and $l+1$ is congruent to $i-1$ modulo 2. Let $A_{k,i}(n)$ denote the number of partitions with parts not congruent to $i$, $2k-i$ and $2k$ modulo $2k$. Bressoud's theorem states that $A_{k,i}(n)=B_{k,i}(n)$. Corteel, Lovejoy, and Mallet found an overpartition analogue of Bressoud's theorem for $i=1$, that is, for partitions not containing nonoverlined part 1. We obtain an overpartition analogue of Bressoud's theorem in the general case. For $k\geq i\geq 1$, let $D_{k,i}(n)$ denote the number of overpartitions of $n$ such that the nonoverlined part 1 appears at most $i-1$ times, for any integer $l$, $l$ and nonoverlined $l+1$ appear at most $k-1$ times and if the parts $l$ and the nonoverlined part $l+1$ appear exactly $k-1$ times then the total sum of the parts $l$ and nonoverlined part $l+1$ is congruent to the number of overlined parts that are less than $l+1$ plus $i-1$ modulo 2. Let $C_{k,i}(n)$ denote the number of overpartitions with the nonoverlined parts not congruent to $\pm i$ and $2k-1$ modulo $2k-1$. We show that $C_{k,i}(n)=D_{k,i}(n)$. This relation can also be considered as a Rogers-Ramanujan-Gordon type theorem for overpartitions.