  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.

Number Theory