Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Combinatorial perspectives on identities for partitions with distinct even parts

Haijun Li

Abstract

Partitions with distinct even parts have long been the subject of extensive research. In this paper, We present some new perspectives on such partitions from a combinatorial viewpoint, and connect them with signed partitions and bicolored partitions, thereby obtaining several partition identities. We construct bijective proofs for each of our results. Furthermore, these bijections will partially answer the combinatorial problems posed by Andrews-El Bachraoui and K$\imath$l$\imath$ç-Kurşungöz. respectively.

Combinatorial perspectives on identities for partitions with distinct even parts

Abstract

Partitions with distinct even parts have long been the subject of extensive research. In this paper, We present some new perspectives on such partitions from a combinatorial viewpoint, and connect them with signed partitions and bicolored partitions, thereby obtaining several partition identities. We construct bijective proofs for each of our results. Furthermore, these bijections will partially answer the combinatorial problems posed by Andrews-El Bachraoui and Klç-Kurşungöz. respectively.
Paper Structure (4 sections, 9 theorems, 30 equations)

This paper contains 4 sections, 9 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:res1']}
  3. Proof of Theorem \ref{['thm:res2']}
  4. Proof of Theorem \ref{['thm:res3']}

Key Result

Theorem 1.1

For any nonnegative integer $n$, let $F(n)$ denote the number of partitions of $n$ where the smallest part is $1$, the first occurrence of $1$ may be overlined, each part is at most twice the number of the occurrences of $1$, and the remaining odd parts are not repeated. Then we have

Theorems & Definitions (18)

  • Theorem 1.1: cf. AB252
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • proof : Analytic proof of Theorem \ref{['thm:res1']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 8 more