Generalized higher order spt-functions

TL;DR

This paper introduces a new generalized spt-function, extending previous concepts with combinatorial interpretations involving Durfee squares, and further generalizes higher order spt-functions to a two-fold framework.

Contribution

It presents a novel generalization of the spt-function and higher order spt-functions, along with their combinatorial interpretations, expanding the theoretical framework of partition functions.

Findings

Introduces $ extup{Spt}_j(n)$ with combinatorial interpretation.

Generalizes $ extup{spt}_k(n)$ to ${}_j extup{spt}_k(n)$.

Provides combinatorial insights into the new functions.

Abstract

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