Holm-Jorgensen Frieze Patterns

• Holm–Jorgensen frieze patterns are generalized numerical arrays from p-angulations, where each quiddity entry is a positive integer multiple of 2cos(π/p) satisfying the unimodular diamond rule.
• They establish a bijection between p-angulations and p-Dyck paths, enabling refined combinatorial enumeration and symmetry analysis via cyclic sieving phenomena.
• These patterns reveal distinctive integrality properties and embed classical Conway–Coxeter friezes in special cases, while extending to infinite dissections with explicit determinant and Smith normal form results.

Holm–Jorgensen frieze patterns are a family of generalized frieze patterns arising from and parameterized by $p$-angulations of polygons, extending the foundational work of Conway–Coxeter on integral friezes and triangulations. The core structural feature of these patterns is their quiddity sequence, whose entries are positive integer multiples of $\lambda_p = 2\cos(\pi/p)$, with the unimodular (diamond) rule $ad - bc = 1$ governing local relations. This generalization connects the combinatorics of dissections to the algebraic and geometric frameworks of cluster algebras, moduli spaces, and integrable systems.

1. Definition and Structural Correspondence

For fixed $p \geq 3$, a Holm–Jorgensen frieze pattern of type $\Lambda_p$ is an array of numbers, commonly real, with rows and columns arranged such that every adjacent $2 \times 2$ minor (i.e., a diamond) satisfies

$ad - bc = 1$

where $a, b, c, d$ are the entries arranged in a square in the frieze.

Unlike the classical case (where the quiddity row consists of positive integers), the quiddity sequence for a pattern of type $\Lambda_p$ has the form

$$(\lambda_p q_0, \lambda_p q_1, \ldots, \lambda_p q_{n+2}),\qquad q_j \in \mathbb{Z}_{>0}$$

By Holm–Jorgensen’s bijection, these patterns correspond exactly to $p$-angulations of an $(n+3)$-gon, that is, dissections of the polygon into $p$-gons by noncrossing diagonals, with the combinatorial prescription that vertex $j$ receives weight $q_j$, the number of $p$-gons incident to $j$ (Andritsch, 2018).

An explicit formula relates the quiddity entry to the $p$-angulation: $F_T(i-1, i+1) = (\deg_T(v_i) + 1)\,\lambda_p$ where $\deg_T(v_i)$ is the number of diagonals of the angulation $T$ incident with $v_i$ (Adams et al., 21 Sep 2025).

2. Connections to Polygon Angulations and Dyck Paths

Underlying Holm–Jorgensen’s classification is a two-step factorization: via a classical bijection (Etherington’s mapping), every $p$-angulation of an $n$-gon corresponds to a $p$-Dyck path (an $m$-Dyck path with $m = p-2$), and hence so does every frieze of type $\Lambda_p$ (Adams et al., 21 Sep 2025).

Formally, for $\ell$ $p$-gons (so $n = m\ell + 2$), there is a bijection

$$\operatorname{brow} : \mathcal{T}^{(m)}_{n} \longrightarrow \mathcal{D}^{(m)}_{\ell}$$

assigning to each $(m+2)$-angulation a Dyck path. The entries of the associated Holm–Jorgensen frieze (after inverting the Dyck path encoding via the $\operatorname{rtn}$ map) are given explicitly by

$$F_{\operatorname{rtn}(D)}(i-1, i+1) = (\mathrm{up}_D(i) + \mathrm{bal}_D(i) + 1)\, \lambda_p$$

where $\mathrm{up}_D(i)$ is the number of up-steps and $\mathrm{bal}_D(i)$ counts certain balance lines at $x=i$ in the Dyck path $D$ (Adams et al., 21 Sep 2025).

This correspondence allows for refined enumeration and analysis of symmetries, including a translation of rotation actions on the polygon/dissection to explicit maps on Dyck paths and the associated friezes.

3. Unimodular Rule, Integrality, and Positivity

The Holm–Jorgensen patterns always satisfy the unimodular (diamond) relation $ad - bc = 1$, respecting the global periodicity and local determinantal structure. A central claim is that, apart from the exceptional cases $p=4$ and $p=6$, the entries of the frieze are never integral in all rows, because powers of $\lambda_p$ for $p \notin \{3,4,6\}$ are always irrational (Andritsch, 2018). For $p=4$ ($\lambda_4 = \sqrt{2}$) and $p=6$ ($\lambda_6 = \sqrt{3}$), the friezes of type $\Lambda_p$ contain, in every second row, classical Conway–Coxeter friezes. This embedding is formalized by constructing two triangulations associated to every $p$-angulation so that the Conway–Coxeter frieze agrees with the $\Lambda_p$ frieze pattern in specified rows (Andritsch, 2018).

Holm–Jorgensen frieze patterns are, by construction, positive in all entries, as every entry arises from sums of positive multiples of $\lambda_p$ assigned according to the $p$-angulation (Andritsch, 2018, Adams et al., 21 Sep 2025).

4. Cluster-Algebraic, Geometric, and Moduli Space Structures

Holm–Jorgensen patterns generalize the cluster structure of Conway–Coxeter friezes. In the case $p=3$, the frieze pattern corresponds to a cluster algebra of type $A_{n-3}$, with the cluster variables arising as entries defined by the local diamond rule and the positivity/Laurent phenomenon (Morier-Genoud et al., 2010). For $p>3$, Holm–Jorgensen friezes determine and are determined by higher angulations, and the associated combinatorial objects give rise to cluster–like manifolds, often with explicit determinant, Smith normal form, and mutation formulas (Bessenrodt et al., 2013, Bessenrodt, 2014).

Explicitly, in the context of cluster theory, the combinatorial data of a $p$-angulation induces a cluster structure via triangulations for $p=3$ and via $d$-angulations for general $p=d$, whose exchange relations and quiver structure control the algebraic structure of the frieze pattern and its mutation dynamics (Morier-Genoud et al., 2010). This perspective aligns Holm–Jorgensen patterns with the moduli space of genus zero curves with marked points: $M_{0,n} \cong \{\text{%%%%55%%%%–gons in } \mathbb{R}P^1\} / PSL_2$ with classical cluster coordinates reflecting different combinatorial parameterizations (Morier-Genoud et al., 2010).

5. Periodicity, Enumeration, and Cyclic Sieving

Friezes of type $\Lambda_p$ inherit global periodicity both from the combinatorics of $p$-angulations and the symmetry of the corresponding Dyck paths. Enumeration refinements rely on cyclic sieving phenomena: for example, the number of friezes (up to global row shift, i.e., cyclic equivalence) fixed under rotations is obtained by evaluating a $q$-analogue polynomial at appropriate roots of unity (Adams et al., 21 Sep 2025). These cyclic sieving polynomials—such as $a_{\mu}(q)$ for a composition $\mu$—organize the counts by orbit size, with the symmetry of the frieze or underlying dissection encoded in the combinatorics of $p$-Dyck paths or $p$-angulations.

Moreover, the principal growth coefficient (i.e., the growth rate of the entries along periodic directions in the frieze) distinguishes cases with nontrivial rotational symmetry, especially in the triangulated case. For such friezes corresponding to orbifolds (invariant under nontrivial rotation), unitary behavior is always observed (Adams et al., 21 Sep 2025).

6. Realization in Dissected Surfaces and Infinite Patterns

The theory extends naturally to classes of infinite friezes via dissections of annuli and once–punctured discs, with entries in $\mathbb{Z}[\lambda_{p_1}, \ldots, \lambda_{p_s}]$ (Banaian et al., 2021). Each vertex of the surface is assigned a “quiddity” number via

$m_{i-1,\, i+1} = \sum_{P \ni v_i} \lambda_{|P|}$

where the sum is over all subgons $P$ incident to $v_i$ in the dissection. There exist explicit combinatorial realizability algorithms—generalizing the “ear–removal” technique of polygon dissections—that decide whether a periodic frieze pattern comes from a dissection of an annulus or a quotient construction, by recursively “cutting” and “gluing” along subgon incidences (Banaian et al., 2021).

Combinatorial interpretations of entries (e.g., as sums over weighted walks or matchings, with weights given by normalized Chebyshev polynomials evaluated at $\lambda_{|P|}$) further generalize the classical arc–counting model of Broline–Crowe–Isaacs (Banaian et al., 2021).

7. Related Structures: Determinants, Smith Normal Forms, and Beyond

The matrices arising from generalized Holm–Jorgensen frieze patterns—encoded (for example) as the “arc matrix” of a $p$-angulation or as weight matrices for polynomially–weighted walks—are highly structured. Determinant and Smith normal form results are independent of the particular $p$-angulation; for a $d$-angulation of an $n$-gon, the determinant of the associated matrix $M_T$ is

$\det(M_T) = (-1)^{n-1} (d-1)^{m+1}$

where $n = d + m(d-2)$, and the SNF has $m+1$ entries equal to $d-1$ with all others unity (Bessenrodt et al., 2013). For polynomially weighted friezes, the determinant is an explicit multivariate polynomial depending only on the types of subgons in the dissection, generalizing earlier integral results (Bessenrodt, 2014).

Adjacent $2 \times 2$ minors in Holm–Jorgensen patterns (or their polynomial generalizations) are either zero or equal to a specific monomial, determined by the existence of zig–zag sequences connecting the boundary edges and by the exponents extracted from combinatorial paths (Bessenrodt, 2014).

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In summary, Holm–Jorgensen frieze patterns constitute a structural generalization of classical frieze patterns, built upon the combinatorics of $p$-angulations, cluster algebras, and the geometry of moduli spaces. Their integrality, positivity, determinantal and periodicity properties, and their relationships to Dyck paths, dissected surfaces, and cluster dynamics, render them a central object in modern algebraic, geometric, and combinatorial research on friezes (Morier-Genoud et al., 2010, Andritsch, 2018, Banaian et al., 2021, Adams et al., 21 Sep 2025).