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Fixed perimeter analogues of some partition results

Gabriel Gray, Emily Payne, Holly Swisher, Ren Watson

TL;DR

The paper extends classical partition identities to the fixed-perimeter setting, introducing fixed-perimeter analogues FO_{j,k}(n) and FD_{j,k}(n) and proving a central Franklin-type result FO_{j,2}(n)=FD_{j,2}(n). It develops generating-function and bijective proofs, and derives Beck-type companion identities, S-T type inequalities, and Kang-Kim–style asymptotics within this framework. The results provide a coherent fixed-perimeter theory that parallels and extends traditional partition theory, with tools for further identities and inequalities. Overall, the work offers new structural insights and techniques for analyzing partitions by perimeter rather than by total size, with potential applications to related combinatorial identities and asymptotics.

Abstract

Euler's partition identity states that the number of partitions of $n$ into odd parts is equal to the number of partitions of $n$ into distinct parts. Strikingly, Straub proved in 2016 that this identity also holds when counting partitions of any size with largest hook (perimeter) $n$. This has inspired further investigation of partition identities and inequalities in the fixed perimeter setting. Here, we explore fixed perimeter analogues of some well-known partition results inspired by Euler's partition identity.

Fixed perimeter analogues of some partition results

TL;DR

The paper extends classical partition identities to the fixed-perimeter setting, introducing fixed-perimeter analogues FO_{j,k}(n) and FD_{j,k}(n) and proving a central Franklin-type result FO_{j,2}(n)=FD_{j,2}(n). It develops generating-function and bijective proofs, and derives Beck-type companion identities, S-T type inequalities, and Kang-Kim–style asymptotics within this framework. The results provide a coherent fixed-perimeter theory that parallels and extends traditional partition theory, with tools for further identities and inequalities. Overall, the work offers new structural insights and techniques for analyzing partitions by perimeter rather than by total size, with potential applications to related combinatorial identities and asymptotics.

Abstract

Euler's partition identity states that the number of partitions of into odd parts is equal to the number of partitions of into distinct parts. Strikingly, Straub proved in 2016 that this identity also holds when counting partitions of any size with largest hook (perimeter) . This has inspired further investigation of partition identities and inequalities in the fixed perimeter setting. Here, we explore fixed perimeter analogues of some well-known partition results inspired by Euler's partition identity.

Paper Structure

This paper contains 10 sections, 4 theorems, 102 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. Franklin-type identities
  3. Alder-type identities and inequalities
  4. Beck-type companion identities
  5. S-T type inequalities
  6. Kang-Kim type asymptotics
  7. Fixed Perimeter Generating Functions and Proofs of Theorem \ref{['k=2']}
  8. Proofs of Theorem \ref{['perimST']}
  9. Proof of Theorem \ref{['beck']}
  10. Proof of Proposition \ref{['FPKKcase']}

Key Result

Theorem 1.1

For all $n$, $j \geq 0$,

Theorems & Definitions (12)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Proposition 1.6
  • proof : Proof of Theorem \ref{['k=2']} via generating functions
  • proof : Proof of Theorem \ref{['k=2']} via combinatorial bijection
  • proof : Proof of Theorem \ref{['perimST']} via counting functions
  • proof : Proof of Theorem \ref{['perimST']} via set injection
  • ...and 2 more