Frieze patterns over finite commutative local rings

TL;DR

This paper provides formulas for counting tame frieze patterns over finite commutative local rings, specifically for rings of the form Z/p^rZ, linking these counts to relations in quotient modular groups.

Contribution

It introduces explicit formulas for counting tame frieze patterns over Z/p^rZ, extending understanding of their structure over finite local rings.

Findings

Closed formulas for counts of frieze patterns over Z/p^rZ

Connections between frieze pattern counts and modular group relations

Extension of frieze pattern enumeration to finite local rings

Abstract

We count numbers of tame frieze patterns with entries in a finite commutative local ring. For the ring $\mathbb{Z}/p^r\mathbb{Z}$, $p$ a prime and $r\in\mathbb{N}$ we obtain closed formulae for all heights. These may be interpreted as formulae for the numbers of certain relations in quotients of the modular group.