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The Cyclic Sieving Phenomenon and frieze patterns

Ashleigh Adams, Esther Banaian

TL;DR

This work develops refined cyclic sieving phenomena for polygon dissections, including fixed-type dissections and dissections of punctured polygons, and connects these CSPs to frieze patterns and orbifold geometries. It introduces explicit $q$-analogues $a_\mu(q)$ and $t_{n,s}^{(m)}(q)$ that realize CSPs under rotations, and provides exact enumeration and symmetry-counting formulas via Fuss–Catalan numbers and related generating functions. A central thread links dissections to frieze patterns (finite and infinite) of Type $\Lambda_{m+2}$, with many results expressed through cluster-algebraic language and periodic/AR-translation perspectives, including unitary orbifold friezes for orders 2 and 3. The paper also establishes a Dyck-path framework (via $m$-Dyck paths and the Brow/rtn maps) that mirrors the rotation of angulations, giving a combinatorial bridge among dissections, friezes, and Dyck-path dynamics and suggesting several directions for future work and conjectures. Collectively, these results yield refined enumerative tools, deepen the interplay between combinatorics and representation theory, and illuminate how symmetries organize a broad web of Catalan-like objects.

Abstract

We exhibit two instances of the cyclic sieving phenomenon - one on dissections of a polygon of a fixed type and one on triangulations of a once-punctured polygon. We use these results to give refined enumerations of certain families of frieze patterns. We also give an interpretation of finite, positive integral frieze patterns fixed under nontrivial rotations as frieze patterns from a family of orbifolds and show that these are always unitary. Finally, we give a bijection between Holm-Jorgensen frieze patterns and p-Dyck paths, extending a recent construction of Canadas, Espinosa, Gaviria, and Rios, and discuss an induced rotation map on Dyck paths. Several conjectures and questions for future study are highlighted throughout the article.

The Cyclic Sieving Phenomenon and frieze patterns

TL;DR

This work develops refined cyclic sieving phenomena for polygon dissections, including fixed-type dissections and dissections of punctured polygons, and connects these CSPs to frieze patterns and orbifold geometries. It introduces explicit -analogues and that realize CSPs under rotations, and provides exact enumeration and symmetry-counting formulas via Fuss–Catalan numbers and related generating functions. A central thread links dissections to frieze patterns (finite and infinite) of Type , with many results expressed through cluster-algebraic language and periodic/AR-translation perspectives, including unitary orbifold friezes for orders 2 and 3. The paper also establishes a Dyck-path framework (via -Dyck paths and the Brow/rtn maps) that mirrors the rotation of angulations, giving a combinatorial bridge among dissections, friezes, and Dyck-path dynamics and suggesting several directions for future work and conjectures. Collectively, these results yield refined enumerative tools, deepen the interplay between combinatorics and representation theory, and illuminate how symmetries organize a broad web of Catalan-like objects.

Abstract

We exhibit two instances of the cyclic sieving phenomenon - one on dissections of a polygon of a fixed type and one on triangulations of a once-punctured polygon. We use these results to give refined enumerations of certain families of frieze patterns. We also give an interpretation of finite, positive integral frieze patterns fixed under nontrivial rotations as frieze patterns from a family of orbifolds and show that these are always unitary. Finally, we give a bijection between Holm-Jorgensen frieze patterns and p-Dyck paths, extending a recent construction of Canadas, Espinosa, Gaviria, and Rios, and discuss an induced rotation map on Dyck paths. Several conjectures and questions for future study are highlighted throughout the article.

Paper Structure

This paper contains 11 sections, 34 theorems, 38 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Dissections of Polygons
  3. Relationship to Previous Work
  4. $(m+2)$-angulations of Punctured Polygons
  5. Enumeration
  6. Cyclic Sieving
  7. Frieze Patterns
  8. Frieze Patterns from Dissections
  9. Counting Frieze Patterns up to Symmetry
  10. Frieze patterns from orbifolds
  11. Frieze Patterns and Dyck Paths

Key Result

Lemma 1

Let $\mu = (\mu_1,\mu_2,\ldots,\mu_n), k = \sum_{i=1}^n \mu_i$, and $n = \sum_{i=1}^n i\mu_i$. Let $d$ be a positive integer such that $d \vert (n+2)$

Figures (7)

  • Figure 1: Two triangulations of $P_6^\bullet$. Arc $(v_1,v_5)$ in the triangulation on the right is dashed for emphasis.
  • Figure 2: Associating a smaller $4$-angulation to a $4$-angulation with 2-fold symmetry, as in the proof of \ref{['prop:CSPPuncturedPolygon']}.
  • Figure 3: Representatives for the three symmetry classes of infinite frieze patterns of type $\Lambda_5$ with one 1 in any fundamental domain and minimal period 12.
  • Figure 4: An outline of the choices for building an $m$-Dyck path from an $(m+2)$-angulation of a polygon.
  • Figure 5: On the left, we have a 2-Dyck path of order 5 and on the right, we have the corresponding 4-angulation of a 12-gon. These are related by the maps $\mathsf{brow}$ and $\mathsf{rtn}$.
  • ...and 2 more figures

Theorems & Definitions (80)

  • Definition 1: reiner2004cyclic
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 1
  • Conjecture 1: Stanton's Conjecture
  • Definition 2
  • Lemma 2
  • proof
  • ...and 70 more