On an spt function for the $4$-th symmetrized crank function

TL;DR

This paper introduces a smallest part function linked to the 4th symmetrized crank, establishing relationships with Garvan's spt functions, deriving congruences, asymptotics, and inequalities, and connecting to second crank and rank moments.

Contribution

It defines a new spt function associated with the 4th symmetrized crank and explores its properties, including congruences and asymptotic behavior, relating it to existing functions and moments.

Findings

Derived congruences for the spt function.

Established asymptotic formulas for the spt function.

Connected the spt function to second crank and rank moments.

Abstract

In this paper we find the smallest part function related to the $4$-th symmetrized crank function, corresponding to the one obtained in Patkowski [11] for the $4$-th symmetrized rank function. This provides us with a direct relationship with Garvan's second order smallest part function. We obtain some congruences for these $spt$ functions, as well as asymptotics and inequalities. A finite $q$-series which generates an $spt$ function related to the second crank moment is also stated. This identity has an important relationship to the one obtained by Patkowski [12] related to the second rank moment.