PED and POD partitions: Combinatorial proofs of recurrence relations

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. In [16] several recurrence relations for the number of PED partitions of n are proved analytically. 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 in [11], which motivated the theorems in [16]. Moreover, we prove combinatorially a recurrence for POD partitions given in [7], Beck-type identities involving PED and POD partitions, and several other results about PED and POD partitions.