Stanley--Elder--Fine theorems for colored partitions

TL;DR

This paper provides a new proof of Elder's theorem and extends it to various types of colored partitions, including those with odd parts, distinct parts, and overpartitions, broadening the theorem's applicability.

Contribution

It introduces a novel proof of Elder's theorem and generalizes it to multiple classes of colored partitions, including overpartitions.

Findings

New proof of Elder's theorem established

Extended theorem to b-colored partitions and overpartitions

Results applicable to partitions with specific coloring constraints

Abstract

We give a new proof of a partition theorem popularly known as Elder's theorem, but which is also credited to Stanley and Fine. We extend the theorem to the context of colored partitions (or prefabs). More specifically, we give analogous results for $b$-colored partitions, where each part occurs in $b$ colors; for $b$-colored partitions with odd parts (or distinct parts); for partitions where the part $k$ comes in $k$ colors; and, overpartitions.