Journal of Integer Sequences, Vol. 27 (2024), Article 24.4.7 |

Department of Mathematics and Statistics

University of Minnesota Duluth

Duluth, MN 55812

USA

**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 Ramanujanâ€“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.

(Concerned with sequences A001935 A006950 A265254 A265256.)

Received September 5 2023; revised version received April 7 2024.
Published in *Journal of Integer Sequences*,
April 11 2024.

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