On congruence properties of the restricted partition functions $$p_{\omega }(n,k)$$ p ω ( n , k ) and $$p_{\nu }(n,k)$$ p ν ( n , k )

Abstract

In a recent work by Andrews, Dixit, and Yee the partition functions $$p_{\omega }(n)$$ and $$p_{\nu }(n)$$ were introduced in connection with the third order mock theta functions $$\omega (q)$$ and $$\nu (q)$$ , respectively. The function $$p_{\omega }(n)$$ counts the number of partitions of n in which each odd part is less than twice the smallest part, and $$p_{\nu }(n)$$ counts the number of partitions of n under the same conditions as $$p_{\omega }(n)$$ and having all parts distinct. In this paper, we consider restrictions of these functions, namely $$p_{\omega }(n,k)$$ and $$p_{\nu }(n,k)$$ , where k is the number of parts. We present congruence properties for these restricted partition functions and we obtain classes of Ramanujan-type congruences for them.