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Congruences for the partition function $\text{PDO}_t(n)$ modulo powers of $2$ and $3$ (2307.04687v1)

Published 10 Jul 2023 in math.NT

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 $\text{PDO}_t(8\cdot3kn)$ and $\text{PDO}_t(12\cdot3kn)$ modulo $3{k+2}$. In this article, we develop a new approach to study these congruences. We study the generating functions of $\text{PDO}_t(8\cdot3kn)$ and $\text{PDO}_t(12\cdot3kn)$ 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.

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