A Decomposition of Cylindric Partitions and Cylindric Partitions into Distinct Parts

TL;DR

This paper establishes a combinatorial correspondence between cylindric partitions and pairs of partitions, deriving their generating functions and proving part of a conjecture with elementary methods.

Contribution

It introduces a new combinatorial decomposition of cylindric partitions into ordinary and colored distinct parts, and derives their generating functions.

Findings

Established a one-to-one correspondence between cylindric partitions and pairs of partitions.

Derived the general form of the generating function for cylindric partitions into distinct parts.

Proved part of a conjecture by Corteel, Dousse, and Uncu.

Abstract

We show that cylindric partitions are in one-to-one correspondence with a pair which has an ordinary partition and a colored partition into distinct parts. Then, we show the general form of the generating function for cylindric partitions into distinct parts and give some examples. We prove part of a conjecture by Corteel, Dousse, and Uncu. The approaches and proofs are elementary and combinatorial.