Parity of parts and excludant statistics in partitions

TL;DR

This paper explores the relationship between parity-based excludant statistics in partitions and quantum modular forms, providing new generating functions, asymptotic analysis, and connections to Ramanujan's work.

Contribution

It introduces new parity-dependent excludant statistics and links their generating functions to quantum modular forms and Nahm-type sums, advancing partition theory.

Findings

Generated new partition statistics related to parity.

Connected generating functions to quantum modular forms.

Derived asymptotic formulas for the sequences.

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 $\sigma^*(q)$ introduced by Ramanujan, and $v_2(q)$, a Nahm-type sum, introduced by Andrews. Utilizing Tauberian method, we obtain asymptotics of such sequences.