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

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 $k\ge 0$, Lin conjectured congruences for ${\text{PDO}}_t(8\cdot 3^{k}n)$ and ${\text{PDO}}_t(12\cdot 3^{k}n)$ modulo $3^{k+2}$. In this article, we develop a new approach to study these congruences. We study the generating functions of ${\text{PDO}}_t(8\cdot 3^{k}n)$ and ${\text{PDO}}_t(12\cdot 3^{k}n)$ 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.

17 pages. arXiv admin note: text overlap with arXiv:2110.14156