Combinatorial proofs and refinements of three partition theorems of Andrews

TL;DR

This paper provides bijective proofs and refinements of three partition theorems by Andrews, focusing on two-color partitions with difference restrictions, and introduces refinements considering the number of parts in each color.

Contribution

It offers new bijective proofs and detailed refinements of Andrews's theorems, enhancing understanding of two-color partition structures.

Findings

Bijective proofs of Andrews's three theorems.

Refinements considering the number of parts in each color.

Deeper insights into two-color partition differences.

Abstract

In his recent work, Andrews revisited two-color partitions with certain restrictions on the differences between consecutive parts, and he established three theorems linking these two-color partitions with more familiar kinds of partitions. In this note, we provide bijective proofs as well as refinements of those three theorems of Andrews. Our refinements take into account the numbers of parts in each of the two colors.