A Bijection for Partitions with Initial Repetitions

TL;DR

This paper provides a bijective proof of Andrews' theorem relating two classes of integer partitions with specific repetition constraints, including some generalizations.

Contribution

It offers a bijective proof of Andrews' theorem and explores related generalizations of the partition conditions.

Findings

Established a bijection between the two partition classes

Extended the theorem to include new generalizations

Confirmed the equivalence of the partition conditions through bijection

Abstract

A theorem of Andrews equates partitions in which no part is repeated more than 2k-1 times to partitions in which, if j appears at least k times, all parts less than j also do so. This paper proves the theorem bijectively, with some of the generalizations that usually arise from such proofs.