The arithmetical combinatorics of $k,l$-regular partitions

TL;DR

This paper investigates the arithmetical combinatorics behind the Little Glaisher theorem on partitions, providing a new arithmetical proof and exploring connections to Schur-type companions for the case when l=2.

Contribution

It offers an arithmetical approach to establish a canonical correspondence for the Little Glaisher theorem, addressing an open problem in partition theory.

Findings

Provides an arithmetical proof of the Little Glaisher theorem

Discusses the construction of a Schur-type companion for l=2

Establishes a new perspective on partition combinatorics

Abstract

For all positive integers $k,l,n$, the Little Glaisher theorem states that the number of partitions of $n$ into parts not divisible by $k$ and occurring less than $l$ times is equal to the number of partitions of $n$ into parts not divisible by $l$ and occurring less than $k$ times. While this refinement of Glaisher theorem is easy to establish by computation of the generating function, there is still no one-to-one canonical correspondence explaining it. Our paper brings an answer to this open problem through an arithmetical approach. Furthermore, in the case $l=2$, we discuss the possibility of constructing a Schur-type companion of the Little Glaisher theorem via the weighted words.