New Congruences Modulo 2, 4, and 8 for the Number of Tagged Parts Over the Partitions with Designated Summands

TL;DR

This paper proves new congruences modulo 8 for the partition functions counting tagged parts with designated summands, extending previous results and discovering many new congruences related to powers of 2.

Contribution

It establishes the conjectured modulo 8 congruences and uncovers numerous new congruences and infinite families for these partition functions.

Findings

Proved the conjectured congruences modulo 8.

Discovered new congruences modulo small powers of 2.

Identified infinite families of congruences for the partition functions.

Abstract

Recently, Lin introduced two new partition functions PD$_t(n)$ and PDO$_t(n)$, which count the total number of tagged parts over all partitions of $n$ with designated summands and the total number of tagged parts over all partitions of $n$ with designated summands in which all parts are odd. Lin also proved some congruences modulo 3 and 9 for PD$_t(n)$ and PDO$_t(n)$, and conjectured some congruences modulo 8. Very recently, Adansie, Chern, and Xia found two new infinite families of congruences modulo 9 for PD$_t(n)$. In this paper, we prove the congruences modulo 8 conjectured by Lin and also find many new congruences and infinite families of congruences modulo some small powers of 2.