Congruences for the partition function $\text{PDO}_t(n)$ modulo powers of $2$ and $3$

TL;DR

This paper investigates congruences of the partition function $ ext{PDO}_t(n)$ modulo powers of 2 and 3, developing new methods to prove existing conjectures and establish new infinite families of congruences.

Contribution

The authors introduce a novel approach to study $ ext{PDO}_t(n)$ congruences, proving new results modulo powers of 2 and 3, and extending Lin's conjectures.

Findings

Established infinitely many congruences modulo 8 and 32.

Developed new methods for analyzing generating functions of $ ext{PDO}_t(n)$.

Proved several congruences modulo small powers of 2.

Abstract

Lin introduced the partition function $\text{PDO}_t(n)$, which counts the total number of tagged parts over all the partitions of $n$ with designated summands in which all parts are odd. For $k\geq0$, Lin conjectured congruences for $\text{PDO}_t(8\cdot3^kn)$ and $\text{PDO}_t(12\cdot3^kn)$ modulo $3^{k+2}$. In this article, we develop a new approach to study these congruences. We study the generating functions of $\text{PDO}_t(8\cdot3^kn)$ and $\text{PDO}_t(12\cdot3^kn)$ modulo $3^{k+3}$ for certain values of $k$. We also study $\text{PDO}_t(n)$ modulo powers of $2$. We establish infinitely many congruences for $\text{PDO}_t(n)$ modulo $8$ and $32$. We prove several congruences modulo small powers of $2$ and discuss the existence of congruences modulo arbitrary powers of $2$ similar to those in Lin's conjecture. In reference to this, we also pose some problems for future work.