Arithmetic Properties of Overcolored Odd Partitions

TL;DR

This paper investigates the arithmetic properties of overcolored odd partitions, establishing new infinite families of congruences modulo powers of 2 using advanced mathematical techniques.

Contribution

It introduces novel congruences for overcolored odd partitions and extends the understanding of their arithmetic behavior using generating functions and modular form theory.

Findings

Established new congruences modulo powers of 2 for overcolored odd partitions.

Proved these congruences hold for infinitely many values of s.

Extended previous results using Hecke eigenform theory and Newman’s results.

Abstract

Let $\bar{a}_s(n)$ denote the number of partitions of $n$, wherein each odd part is multicolored (atmost $s\ge 1$ colors) and the first appearance of parts may be overlined. In this paper, we establish new families of congruences modulo powers of $2$ satisfied by $\bar{a}_s(n)$ for infinitely many $s$. Our approach builds upon generating function manipulations, Hecke eigenform theory and results of Newman.