---
title: "Triangulated polygons and Y-frieze patterns"
authors: ["Hin Chung Henry Tsang", "Jon Wilson"]
year: 2026
venue: "arXiv:math.CO"
arxiv: "2608.29655"
doi: null
source: "https://arxiv.org/abs/2608.29655"
topics: ["curiosity", "maths-foundations"]
seed_rank: 1549
seed_batch: "curiosity-2026-09-03"
reviewed: "2026-09-03"
pool: "maths-foundations"
relevance_score: 9
lineage: cluster-frieze
cites:
  - title: "Triangulated polygons and Y-frieze patterns"
    url: "https://arxiv.org/abs/2608.29655"
    year: 2026
    arxiv: "2608.29655"
    doi: null
see: []
---

# Triangulated polygons and Y-frieze patterns

## One-sentence takeaway

All Y-frieze patterns of type A_n arise from Conway–Coxeter friezes (settling de Saint Germain), and they are counted by Fuss–Catalan numbers.

## Why it's lovely

Why you might love this: Conway–Coxeter friezes are the diamond-rule number patterns that secretly encode triangulated polygons. Y-friezes are their cluster-y cousins; this paper classifies them all, proves they come from classical friezes, and counts them with Fuss–Catalan numbers. Pure combinatorial sparkle with a settled conjecture.

## Key ideas

- Classify all Y-frieze patterns of type A_n in the spirit of Conway–Coxeter.
- Settle de Saint Germain's conjecture: every Y-frieze arises from a Conway–Coxeter frieze.
- Enumeration via Fuss–Catalan numbers.
- Bridge between triangulated polygons, classical friezes, and cluster-adjacent Y-friezes.

## Caveats

Type A_n only; not a classification for other Dynkin / cluster types. "Arise from" is a precise combinatorial map — read the paper for which frieze data determines the Y-pattern. Distinct from aperiodic-frieze decorations of Penrose tilings (2607.21348, left as near-keeper).

## Implications for Broadside

Combinatorics delight on the cluster / Catalan shelf. No immediate Anoptic build, but frieze ↔ triangulation dictionaries are the same shape as mesh connectivity invariants you might want to hash.

## Links

- arXiv abs: https://arxiv.org/abs/2608.29655
- PDF: https://arxiv.org/pdf/2608.29655.pdf