PED and POD partitions: combinatorial proofs of recurrence relations

TL;DR

This paper provides combinatorial proofs for recurrence relations involving PED and POD partitions, extending classical partition theorems and establishing new identities in partition theory.

Contribution

It offers the first combinatorial proofs for several recurrence relations and identities related to PED and POD partitions, including analogues of Euler's pentagonal number theorem.

Findings

Combinatorial proofs for PED and POD recurrence relations

New identities involving PED and POD partitions

Extension of Euler's pentagonal number theorem to PED partitions

Abstract

PED partitions are partitions with even parts distinct while odd parts are unrestricted. Similarly, POD partitions have distinct odd parts while even parts are unrestricted. Merca proved several recurrence relations analytically for the number of PED partitions of $n$. They are similar to the recurrence relation for the number of partitions of $n$ given by Euler's pentagonal number theorem. We provide combinatorial proofs for all of these theorems and also for the pentagonal number theorem for PED partitions proved analytically by Fink, Guy, and Krusemeyer. Moreover, we prove combinatorially a recurrence for POD partitions given by Ballantine and Merca, Beck-type identities involving PED and POD partitions, and several other results about PED and POD partitions.