New congruences modulo 2, 4, and 8 for the number of tagged parts over the partitions with designated summands

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. 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.