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$\text {PDO}_t(n)$ BY POWERS OF 
$2$ AND 
$3$
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. Lin also proved some congruences modulo 3 and 9 for 
$\text {PDO}_t(n)$, and conjectured certain congruences modulo 
$3^{k+2}$ for 
$k\geq 0$. He proved the conjecture for 
$k=0$ and 
$k=1$ [‘The number of tagged parts over the partitions with designated summands’, J. Number Theory 184 (2018), 216–234]. We prove the conjecture for 
$k=2$. We also study the lacunarity of 
$\text {PDO}_t(n)$ modulo arbitrary powers of 2 and 3. Using nilpotency of Hecke operators, we prove that there exists an infinite family of congruences modulo any power of 2 satisfied by 
$\text {PDO}_t(n)$.
