Partitions With Designated Summands Not Divisible by $2^l$, $2$, and $3^l$ Modulo $2$, $4$, and $3$
Combinatorics
2024-07-30 v1
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 , , and working modulo and , 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.
Keywords
Cite
@article{arxiv.2101.04058,
title = {Partitions With Designated Summands Not Divisible by $2^l$, $2$, and $3^l$ Modulo $2$, $4$, and $3$},
author = {Daniel Herden and Mark R. Sepanski and Jonathan Stanfill and Cordell Hammon and Joel Henningsen and Henry Ickes and Indalecio Ruiz},
journal= {arXiv preprint arXiv:2101.04058},
year = {2024}
}