Arithmetic properties for a partition function related to the Ramanujan/Watson mock theta function $$\omega (q)$$ ω ( q )

Abstract

Recently, Andrews, Dixit, and Yee introduced a new partition function $$p_{\omega }(n)$$ that denotes the number of partitions of n in which each odd part is less than twice the smallest part. The generating function of $$p_{\omega }(n)$$ is associated with the third-order mock theta function $$\omega (q)$$ . Andrews, Passary, Sellers, and Yee proved three infinite families of congruences modulo 4 and 8 for $$p_{\omega }(n)$$ and provided elementary proofs of congruences modulo 5 for $$p_{\omega }(n)$$ which were first discovered by Waldherr. In this paper, we prove some new congruences modulo 5 and powers of 2 for $$p_{\omega }(n)$$ . In particular, we obtain some non-standard congruences for $$p_{\omega }(n)$$ . For example, we prove that for $$k\ge 0$$ , $$ p_{\omega }\left( \frac{7\times 5^{2k+1}+1 }{3}\right) \equiv (-1)^k \ (\mathrm{mod}\ 5) $$ and $$ p_\omega \left( \frac{2^{2k+7}+1}{3}\right) \equiv 1251 \times (-1)^k \ (\mathrm{mod}\ 2^{11})$$ .