Partitions With Designated Summands Not Divisible by $2^l$, $2$, and $3^l$ Modulo $2$, $4$, and $3$

TL;DR

This paper characterizes partitions with designated summands avoiding divisibility by powers of 2 and 3 modulo 2, 4, and 3, extending previous results and revealing new connections and applications.

Contribution

It explicitly describes the counts of such partitions under specific divisibility constraints and introduces a new link to partitions with odd multiplicities.

Findings

Derived explicit formulas for partition counts under divisibility restrictions.

Established new congruences and recurrence relations for these partitions.

Discovered a novel connection between designated summand partitions and odd multiplicity partitions.

Abstract

Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with designated summands whose parts are not divisible by $2^\ell$, $2$, and $3^\ell$ working modulo $2,\ 4,$ and $3$, respectively, greatly extending previous results on the subject. We provide a few applications of our characterizations throughout in the form of congruences and a computationally fast recurrence. Moreover, we illustrate a previously undocumented connection between the number of partitions with designated summands and the number of partitions with odd multiplicities.