Elementary Proofs of Congruences for POND and PEND Partitions

Research output: Contribution to journalArticlepeer-review

Abstract

Recently, Ballantine and Welch considered various generalizations and refinements of POD and PED partitions. These are integer partitions wherein the odd parts must be distinct (in the case of POD partitions) or the even parts must be distinct (in the case of PED partitions). In the process, they were led to consider two classes of integer partitions which are, in some sense, the “opposite” of POD and PED partitions. They labeled these POND and PEND partitions, which are integer partitions wherein the odd parts cannot be distinct (in the case of POND partitions) or the even parts cannot be distinct (in the case of PEND partitions). In this work, we study these two types of partitions from an arithmetic perspective. Along the way, we are led to prove two infinite families of Ramanujan–like congruences modulo 3, one satisfied by the function pond(n), which counts the number of POND partitions of weight n, and the other satisfied by the function pend(n), which counts the number of PEND partitions of weight n, where n is a nonnegative integer. All of the proof techniques used herein are elementary, relying on classical q-series identities and generating function manipulations, along with mathematical induction.

Original languageEnglish (US)
Article number24.4.7
JournalJournal of Integer Sequences
Volume27
Issue number4
StatePublished - 2024

Bibliographical note

Publisher Copyright:
© 2024, University of Waterloo. All rights reserved.

Keywords

  • congruence
  • dissection
  • generating function
  • partition

Fingerprint

Dive into the research topics of 'Elementary Proofs of Congruences for POND and PEND Partitions'. Together they form a unique fingerprint.

Cite this