Generalized higher order spt-functions

We give a new generalization of the spt-function of G.E. Andrews, namely ${\text{Spt}}_j(n)$, and give its combinatorial interpretation in terms of successive lower-Durfee squares. We then generalize the higher order spt-function ${\text{spt}}_k(n)$, due to F.G. Garvan, to ${}_j{\text{spt}}_k(n)$, thus providing a two-fold generalization of $\text{spt}(n)$, and give its combinatorial interpretation.