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Separable overpartition classes and the $r$-chain excludant sizes of an overpartition

Y. H. Chen, Thomas Y. He, H. X. Huang, X. Zhang

TL;DR

This paper addresses overpartitions by studying the $r$-chain excludant sizes, denoted $mes(\pi; r)$ and $maes(\pi; r)$, and by introducing two separable overpartition classes, $L_k$-overpartitions and $F_k$-overpartitions. It connects $mes$ and $maes$ to the $(r+1)$-repeating size through conjugation and derives generating-function formulas for the aggregated statistics $\sigma_r{mes}(n)$ and $\sigma_r{maes}(n)$. It then defines $L_k$- and $F_k$-overpartitions as separable classes, proves their separability, and provides explicit generating functions that yield the $L_k$ and $F_k$ formulas. Overall, the work extends the separable-class framework to overpartitions and supplies closed-form generating functions for these new overpartition families.

Abstract

An overpartition is a partition such that the first occurrence (equivalently, the last occurrence) of a number may be overlined. In this article, we will investigate two contents of overpartitions. We first consider the $r$-chain minimal and maximal excludant sizes of an overpartition. Then, we introduce $L_k$-overpartitions and $F_k$-overpartitions, which are two types of separable overpartition classes.

Separable overpartition classes and the $r$-chain excludant sizes of an overpartition

TL;DR

This paper addresses overpartitions by studying the -chain excludant sizes, denoted and , and by introducing two separable overpartition classes, -overpartitions and -overpartitions. It connects and to the -repeating size through conjugation and derives generating-function formulas for the aggregated statistics and . It then defines - and -overpartitions as separable classes, proves their separability, and provides explicit generating functions that yield the and formulas. Overall, the work extends the separable-class framework to overpartitions and supplies closed-form generating functions for these new overpartition families.

Abstract

An overpartition is a partition such that the first occurrence (equivalently, the last occurrence) of a number may be overlined. In this article, we will investigate two contents of overpartitions. We first consider the -chain minimal and maximal excludant sizes of an overpartition. Then, we introduce -overpartitions and -overpartitions, which are two types of separable overpartition classes.
Paper Structure (10 sections, 13 theorems, 101 equations)

This paper contains 10 sections, 13 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. The $r$-chain minimal and maximal excludant sizes
  3. $L_k$-overpartitions and $F_k$-overpartitions
  4. Proofs of \ref{['gen-mes-r']} and \ref{['gen-maes-r']}
  5. Conjugate and $(r+1)$-repeating size
  6. Proof of \ref{['gen-mes-r']}
  7. Proof of \ref{['gen-maes-r']}
  8. Proofs of \ref{['eqn-L-k']} and \ref{['eqn-F-k']}
  9. Proof of \ref{['eqn-L-k']}
  10. Proof of \ref{['eqn-F-k']}

Key Result

Theorem 2.1

For $j\geq 0$, $k\geq 1$ and $n\geq 0$, the number of overpartitions $\pi$ of $n$ such that ${mes}(\pi;r)=k$ and there are $j$ parts of size greater than $k$ in $\pi$ equals the number of overpartitions $\lambda$ of $n$ such that the largest $(r+1)$-repeating size of $\lambda$ is $j$ and there are $

Theorems & Definitions (20)

  • Definition 1.1: $r$-chain minimal excludant size
  • Definition 1.2: $r$-chain maximal excludant size
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 2.1
  • Theorem 2.2
  • Definition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • ...and 10 more