Andrews, Lewis, and Lovejoy introduced partitions with designated summands. Later on, 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 \(\text {PDO}_t(8\cdot 3^kn)\) and \(\text {PDO}_t(12\cdot 3^kn)\) modulo \(3^{k+2}\) . In this article, we study these congruences. Firstly, we study the generating functions of \(\text {PDO}_t(8\cdot 3^kn)\) and \(\text {PDO}_t(12\cdot 3^kn)\) modulo \(3^{k+3}\) for certain values of k. Next, we 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.
