Parity of parts and excludant statistics in partitions

Abstract

In this paper, we study restricted excludant statistics depending on its parity in partitions where parts with same parity are distinct. Using $q$-series transformations, we show that generating functions of these partition statistics are related to the quantum modular forms $\s(q)$, its companion ${\sigma }^{\ast }(q)$ introduced by Ramanujan, and $v_2(q)$, a Nahm-type sum, introduced by Andrews. Utilizing Tauberian method, we obtain asymptotics of such sequences.