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Enumerating tame friezes over $\mathbb{Z}/n\mathbb{Z}$

Sammy Benzaira, Ian Short, Matty van Son, Andrei Zabolotskii

Abstract

We use a class of Farey graphs introduced by the final three authors to enumerate the tame friezes over $\mathbb{Z}/n\mathbb{Z}$. Using the same strategy we enumerate the tame regular friezes over $\mathbb{Z}/n\mathbb{Z}$, thereby reproving a recent result of Böhmler, Cuntz, and Mabilat.

Enumerating tame friezes over $\mathbb{Z}/n\mathbb{Z}$

Abstract

We use a class of Farey graphs introduced by the final three authors to enumerate the tame friezes over . Using the same strategy we enumerate the tame regular friezes over , thereby reproving a recent result of Böhmler, Cuntz, and Mabilat.

Paper Structure

This paper contains 4 sections, 15 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Farey graphs
  3. Proof of Theorem A
  4. Proof of Theorem B

Key Result

Theorem A

The number of tame friezes of width $m$ over the ring $\mathbb{Z}/n\mathbb{Z}$ is

Figures (2)

  • Figure 1.1: A tame frieze over $\mathbb{Z}/5\mathbb{Z}$ of width 6 (left) and a diamond of four entries (right)
  • Figure 4.1: Graph $G$ (left) and its adjacency matrix (right)

Theorems & Definitions (24)

  • Theorem A
  • Theorem B
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • ...and 14 more