An interesting $q$-series related to the 4-th symmetrized rank function

Abstract

This paper presents the methods to utilizing the $s$-fold extension of Bailey's lemma to obtain $spt$-type functions related to the symmetrized rank function $\eta_{2k}(n).$ We provide the $k=2$ example, but clearly illustrate how deep connections between higher-order spt functions exist for any integer $k>1,$ and provide several directions for possible research. In particular, we present why the function $spt_M^{*}(n),$ the total number of appearances of the smallest parts of partitions where parts greater than the smallest plus $M$ do not occur, is an $spt$ function that appears to have central importance.