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Generalization of the Extended Minimal Excludant of Andrews and Newman

Aritram Dhar, Avi Mukhopadhyay, Rishabh Sarma

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.

Generalization of the Extended Minimal Excludant of Andrews and Newman

Abstract

In a recent pioneering work, Andrews and Newman defined an extended function of their minimal excludant or "mex" of a partition function. By considering the special cases and , 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.
Paper Structure (20 sections, 31 theorems, 45 equations, 3 tables)

This paper contains 20 sections, 31 theorems, 45 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Proof of the main results
  3. Proof of Theorem \ref{['mex3132']}
  4. Proof of Theorem \ref{['mexrecur']}
  5. Proof of Theorem \ref{['mexrank1']}
  6. Proof of Theorem \ref{['mexcrank1']}
  7. Proof of Theorem \ref{['AnNew1']}
  8. Proof of Theorem \ref{['AnNew2']}
  9. Proof of Theorem \ref{['mexrankmom']}
  10. Proof of Theorem \ref{['mexspt']}
  11. Proof of Theorem \ref{['mexcongJTP']}
  12. Proof of Theorem \ref{['AnNew3']}
  13. Proof of Theorem \ref{['mexcong1']}
  14. Proof of Theorem \ref{['mexcong2']}
  15. Proof of Theorem \ref{['mexcong3']}
  16. ...and 5 more sections

Key Result

Theorem 1

If $n$ is a non-negative integer, then $p_{1,1}(n)$ equals the number of partitions of $n$ with non-negative crank.

Theorems & Definitions (55)

  • Theorem 1: An-New20, Theorem $2$
  • Theorem 2: An-New20, Theorem $3$
  • Theorem 3: An-New20, Theorem $4$
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 45 more