Generalization of the Extended Minimal Excludant of Andrews and Newman

TL;DR

This paper generalizes the extended minimal excludant function of Andrews and Newman, linking it to partition statistics like rank and crank, and introduces new identities and a minimal excludant for overpartitions.

Contribution

It extends the extended mex function to arbitrary bounds, connects it with partition rank and crank, and introduces a new minimal excludant for overpartitions.

Findings

Generalized extended mex function for arbitrary bounds

Derived identities relating mex to partition rank and crank

Introduced a minimal excludant for overpartitions

Abstract

In a recent pioneering work, Andrews and Newman defined an extended function $p_{A,a}(n)$ of their minimal excludant or "mex" of a partition function. By considering the special cases $p_{k,k}(n)$ and $p_{2k,k}(n)$, they unearthed connections to the rank and crank of partitions and some restricted partitions. In this paper, we build on their work and obtain more general results associating the extended mex function with the number of partitions of an integer with arbitrary bound on the rank and crank. We also derive a new result expressing the smallest parts function of Andrews as a finite sum of the extended mex function in consideration with a curious coefficient. We also obtain a few restricted partition identities with some reminiscent of shifted partition identities. Finally, we define and explore a new minimal excludant for overpartitions.