TY - UNPB

T1 - PED and POD partitions

T2 - combinatorial proofs of recurrence relations

AU - Ballantine, Cristina

AU - Welch, Amanda

PY - 2023/8/11

Y1 - 2023/8/11

N2 - 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.

AB - 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.

KW - math.CO

KW - math.NT

KW - 11P81, 11P84, 05A17, 05A19

M3 - Preprint

BT - PED and POD partitions

ER -