DIVISIBILITY OF THE PARTITION FUNCTION BY POWERS OF AND

Abstract

AbstractLin 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 Theory184 (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 $\text {PDO}_t(n)$ .