Anti-lecture Hall Compositions and Overpartitions

TL;DR

This paper establishes a new combinatorial identity linking anti-lecture hall compositions and overpartitions, extending the anti-lecture hall theorem with refined identities and bijections, and providing multiple proofs including Rogers-Ramanujan type identities.

Contribution

It introduces a refined identity connecting anti-lecture hall compositions and overpartitions, along with new Rogers-Ramanujan type identities and alternative proofs for the case when k is odd.

Findings

Established a bijection between anti-lecture hall compositions and overpartitions.

Derived Rogers-Ramanujan type identities for overpartitions.

Provided multiple proofs including bijective and analytic methods.

Abstract

We show that the number of anti-lecture hall compositions of n with the first entry not exceeding k-2 equals the number of overpartitions of n with non-overlined parts not congruent to $0,\pm 1$ modulo k. This identity can be considered as a refined version of the anti-lecture hall theorem of Corteel and Savage. To prove this result, we find two Rogers-Ramanujan type identities for overpartition which are analogous to the Rogers-Ramanjan type identities due to Andrews. When k is odd, we give an alternative proof by using a generalized Rogers-Ramanujan identity due to Andrews, a bijection of Corteel and Savage and a refined version of a bijection also due to Corteel and Savage.