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Combinatorial Proofs of Some Results of Andrews and El Bachraoui

Pankaj Jyoti Mahanta, Manjil P. Saikia

Abstract

Recently, Andrews and El Bachraoui (2024) proved three very interesting $q$-series identities, from which three simple looking identities involving certain restricted partitions into distinct even parts and $4$-regular partitions follow. In this short note, we give combinatorial proofs of these identities. We also prove the counterpart identities for the restricted partitions into distinct odd parts.

Combinatorial Proofs of Some Results of Andrews and El Bachraoui

Abstract

Recently, Andrews and El Bachraoui (2024) proved three very interesting -series identities, from which three simple looking identities involving certain restricted partitions into distinct even parts and -regular partitions follow. In this short note, we give combinatorial proofs of these identities. We also prove the counterpart identities for the restricted partitions into distinct odd parts.

Paper Structure

This paper contains 3 sections, 6 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Combinatorial Proofs of Theorems \ref{['cor1']}, \ref{['cor2']} and \ref{['cor3']}
  3. Proof of Theorem \ref{['pod-cor2']}

Key Result

Theorem 1.1

For $n>0$, we have

Theorems & Definitions (9)

  • Theorem 1.1: Corollary 1, AndrewsElBachraoui
  • Theorem 1.2: Corollary 2, AndrewsElBachraoui
  • Theorem 1.3: Corollary 3, AndrewsElBachraoui
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • proof : Proof of Theorem \ref{['cor1']}
  • proof : Proof of Theorem \ref{['cor2']}
  • proof : Proof of Theorem \ref{['cor3']}