The Refined Lecture Hall Theorem via Abacus Diagrams

TL;DR

This paper provides an elementary bijective proof of a refined version of the lecture hall theorem, which generalizes Euler's partition theorem, using abacus diagrams for clarity and simplicity.

Contribution

It introduces a new, elementary bijection-based proof of the refined lecture hall theorem leveraging abacus diagrams, simplifying previous approaches.

Findings

Elementary proof of the refined lecture hall theorem

Use of abacus diagrams for bijective combinatorics

Enhanced understanding of partition identities

Abstract

Bousquet-M\'elou & Eriksson's lecture hall theorem generalizes Euler's celebrated distinct-odd partition theorem. We present an elementary and transparent proof of a refined version of the lecture hall theorem using a simple bijection involving abacus diagrams.