Glaisher's Partition problem

TL;DR

This paper generalizes Glaisher's partition bijection, establishing a broad family of bijections between partitions with parts divisible by d and partitions with parts repeated d or more times, for integers n and d greater than 1.

Contribution

It introduces a comprehensive set of bijections extending Glaisher's classical result for arbitrary positive integers n and d.

Findings

Established a family of bijections for partitions with divisibility conditions.

Generalized Glaisher's bijection beyond the classical case.

Provides combinatorial tools for partition analysis.

Abstract

We generalize the well known Glaisher partition bijection result. For given positive integers n, d, both greater than 1, we provide a rich family of bijections between the set of partitions of n where at least one part is divisible by d, and the set of partitions where at least one part appears d or more times.