Coxeter’s Frieze Patterns Arising from Dyck Paths

Agustín Moreno Cañadas
Isaías David Marín Gaviria
Gabriel Bravo Rios
Pedro Fernando Fernández Espinosa

Abstract

Frieze patterns are defined by objects of a category of Dyck paths, to do
that, it is introduced the notion of diamond of Dynkin type $\mathbb{A}_{n}$.
Such diamonds constitute a tool to build integral frieze patterns.

Keywords and phrases : Diamond, Dyck path, Dyck paths category, frieze
pattern, seed vector, triangulation.

Mathematics Subject Classification 2010 : 16G20; 16G30; 16G60.

##  1 Introduction

Frieze patterns (as shown below) were introduced by Coxeter in the early 70s
[Coxeter]. According to Propp [Propp], they arose from Coxeter’s study of
metric properties of polytopes, and served as useful scaffolding for various
sorts of metric data.

$\textstyle{\dots}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{\dots}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{\dots}$$\textstyle{m_{-1,-1}}$$\textstyle{m_{00}}$$\textstyle{m_{11}}$$\textstyle{m_{22}}$$\textstyle{\dots}$$\textstyle{m_{-2,-1}}$$\textstyle{m_{-1,0}}$$\textstyle{m_{0,1}}$$\textstyle{m_{12}}$$\textstyle{m_{23}}$$\textstyle{\dots}$$\textstyle{\dots}$$\textstyle{\dots}$$\textstyle{\dots}$$\textstyle{\dots}$$\textstyle{\dots}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{\dots}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{\dots}$

Such patterns are defined as grids of numbers bounded from above by an
infinite row of 0s followed by a row of 1s and such that every four adjacent
numbers of the following form

$\textstyle{b}$$\textstyle{a}$$\textstyle{c}$$\textstyle{d}$

satisfy the arithmetic (or frieze) rule

$\begin{split}ac-bd=1.\end{split}$ (1)

The third line (i.e., the first nontrivial row) can be chosen in an arbitrary
way and then complete using the rule (1), which was named the modular equation
by Coxeter [Coxeter].

A frieze is called closed, if it is also bounded from below by a line of 1s
(followed by a line of 0s). A frieze is called integral if it consists of
positive integers. The sequence of integers in the first non-trivial row is
called quiddity sequence. This sequence completely determines the frieze
pattern. Each frieze pattern is also periodic since it is invariant under
glide reflection. The order of the frieze pattern is defined to be the number
of rows minus one. It follows that each frieze pattern of order $n$ is
$n$-periodic [Baur1]. Conway and Coxeter classified completely the frieze
patterns whose entries are positive integers, and show that these frieze
patterns constitute a manifestation of the Catalan numbers [Conway, Conway1]
giving a bijection between positive integer frieze patterns and triangulations
of regular polygons with labeled vertices.

Frieze patterns appear independently in the 70s in the context of quiver
representations, in such a case, the local arithmetic rule is an additive
analogue of Coxeter's unimodular rule. The generalization of the Coxeter's
unimodular rule on Auslander-Reiten quivers was found by Caldero and Chapoton
[Caldero1]. Assem, Reutenauer and Smith introduced also a generalization of
friezes by associating frieze patterns to Cartan matrices [Assem1].

According to Morier-Genoud [Morier] there are mainly three approaches for the
study of friezes:

  1. 1.

A representation theoretical and categorical approach, in deep connection with
the theory of cluster algebras, where entries in the friezes are rational
functions.

  2. 2.

A geometrical approach, in connection with moduli spaces of points in
projective space and Grassmannians, where entries in the friezes are more
often real or complex numbers.

  3. 3.

A combinatorial approach, focusing on friezes with positive integer entries.

Many authors have studied friezes from the different points of view finding
connections with different branches of mathematics [Morier]. For instance,
Baur et al [Baur1] studied mutation of friezes proving how mutation of a
cluster affects the associated frieze. On the other hand, Fontaine and
Plamondon [Fontaine] presented a formula for the number of friezes of type
$\mathbb{B}_{n},\mathbb{C}_{n},\mathbb{D}_{n}$, and $\mathbb{G}_{2}$. They
conjectured that the number of friezes of type
$\mathbb{E}_{6},\mathbb{E}_{7},\mathbb{E}_{8}$ and $\mathbb{F}_{4}$ is 868,
4400, 26592 and 112, respectively. In this way, the number of friezes can be
defined as a Dynkin function in the sense of Ringel [Ringel].

In this paper, frieze patterns are interpreted as objects of a novel category
of Dyck paths introduced recently by Cañadas and Rios [Canadas].

The following is a list of our main results, all of them dealing with integral
closed friezes.

  1. 1.

It is introduced the notion of diamond of Dynkin type $\mathbb{A}_{n}$, and
some of its properties are proved (see, Proposition 5, Proposition 6 and
Theorem 8). Such diamonds are used as a tool to build frieze patterns.

  2. 2.

It is proved that there is a bijective correspondence between the set of all
vectors associated to positive integral diamonds of Dynkin type
$\mathbb{A}_{n}$ and triangulations of a polygon with $n+3$ vertices. This
result is a consequence of a bijection between such triangulations and Dyck
paths of length $2(n+1)$ (see Lemma 16 and Theorem 17).

  3. 3.

It is proved that if $\mathscr{C}(A^{0},t)=\\{A^{t}\\}_{0\leq t\leq p-1}$ is
the minimal $p$-cycle generated by a diamond $A^{0}$ of Dynkin type
$\mathbb{A}_{n}$. Then $\mathscr{C}(A^{0},t)$ is in surjective correspondence
with a direct sum of $p$ indecomposable objects of a Dyck paths category (see
Theorem 21).

The paper is distributed as follows. In section 2, we recall main definitions
and notation to be used throughout the paper, in particular, it is reminded
the notion of a Dyck paths category. In section 3, we present the main
results, where it is introduced the notion of diamond of Dynkin type
$\mathbb{A}_{n}$ and some of its properties.

##  2 Preliminaries

In this section, we recall main definitions and notation to be used throughout
the paper [Barcucci, Canadas, Gunawan, Stanley].

###  2.1 Cluster Algebras From Quivers

For quivers, cluster algebras are defined as follows:

Fix an integer $n\geq 1$. In this case, a seed $(Q,u)$ consists of a finite
quiver $Q$ without loops or 2-cycles with vertex set $\\{1,\dots,n\\}$,
whereas $u$ is a free-generating set $\\{u_{1},\dots,u_{n}\\}$ of the field
$\mathbb{Q}(x_{1},\dots,x_{n})$.

Let $(Q,u)$ be a seed and $k$ a vertex of $Q$. The mutation $\mu_{k}(Q,u)$ of
$(Q,u)$ at $k$ is the seed $(Q^{\prime},u^{\prime})$, where;

  1. (a)

$Q^{\prime}$ is obtained from $Q$ as follows;

    1. (1)

reverse all arrows incident with $k$,

    2. (2)

for all vertices $i\neq j$ distinct from $k$, modify the number of arrows
between $i$ and $j$, in such a way that a system of arrows of the form
$(i\stackrel{{\scriptstyle r}}{{\longrightarrow}}j,i\stackrel{{\scriptstyle
s}}{{\longrightarrow}}k,k\stackrel{{\scriptstyle t}}{{\longrightarrow}}j)$ is
transformed into the system $(i\stackrel{{\scriptstyle
r+st}}{{\longrightarrow}}j,k\stackrel{{\scriptstyle
s}}{{\longrightarrow}}i,j\stackrel{{\scriptstyle t}}{{\longrightarrow}}k)$.
And the system $(i\stackrel{{\scriptstyle
r}}{{\longrightarrow}}j,j\stackrel{{\scriptstyle
t}}{{\longrightarrow}}k,k\stackrel{{\scriptstyle s}}{{\longrightarrow}}i)$ is
transformed into the system $(i\stackrel{{\scriptstyle
r-st}}{{\longrightarrow}}j,i\stackrel{{\scriptstyle
s}}{{\longrightarrow}}k,k\stackrel{{\scriptstyle t}}{{\longrightarrow}}j)$.
Where, $r$, $s$ and $t$ are non-negative integers, an arrow
$i\stackrel{{\scriptstyle l}}{{\longrightarrow}}j$, with $l\geq 0$ means that
$l$ arrows go form $i$ to $j$ and an arrow $i\stackrel{{\scriptstyle
l}}{{\longrightarrow}}j$, with $l\leq 0$ means that $-l$ arrows go from $j$ to
$i$.

  2. (b)

$u^{\prime}$ is obtained form $u$ by replacing the element $u_{k}$ with

$\begin{split}u_{k}&=\frac{1}{u_{k}}\underset{\mathrm{arrows}\hskip
2.84544pti\rightarrow k}{\prod}u_{i}+\underset{\mathrm{arrows}\hskip
2.84544ptk\rightarrow j}{\prod}u_{j}.\end{split}$ (2)

If there are no arrows from $i$ with target $k$, the product is taken over the
empty set and equals 1. It is not hard to see that
$\mu_{k}(\mu_{k}(Q,u))=(Q,u)$. In this case the matrix mutation $B^{\prime}$
has the form

$b^{\prime}_{ij}=\begin{cases}-b_{ij},&\hskip 5.69046pt\text{if}\hskip
2.84544pti=k\hskip 2.84544pt\text{or}\hskip 2.84544ptj=k,\\\
b_{ij}+sgn(b_{ik})[b_{ik}b_{kj}]_{+},&\hskip 5.69046pt\text{else},\end{cases}$

where $[x]_{+}=\mathrm{max(x,0)}$. Thus, if $Q$ is a finite quiver without
loops or 2-cycles with vertex set $\\{1,\dots,n\\}$, the following
interpretations take place:

  1. 1.

the clusters with respect to $Q$ are the sets $u$ appearing in seeds, $(Q,u)$
obtained from a initial seed $(Q,x)$ by iterated mutation,

  2. 2.

the cluster variables for $Q$ are the elements of all clusters,

  3. 3.

the cluster algebra $\mathscr{A}(Q)$ is the $\mathbb{Q}$-subalgebra of the
field $\mathbb{Q}(x_{1},\dots,x_{n})$ generated by all the cluster variables.

As example, the cluster variables associated to the quiver $Q=1\longrightarrow
2$ are:

$\\{x_{1},x_{2},\frac{1+x_{2}}{x_{1}},\frac{1+x_{1}+x_{2}}{x_{1}x_{2}},\frac{1+x_{1}}{x_{2}}\\}$.

Regarding cluster algebras arising from quivers, we recall that, Fomin and
Zelevinsky [Fomin1] proved that any cluster algebra $\mathcal{A}(Q)$ of finite
type has a finite set of cluster variables and the following result.

######  Theorem 1.

The cluster algebra $\mathcal{A}(Q)$ is of finite type if and only if $Q$ is
mutation-equivalent to an orientation of a simply-laced Dynkin diagram,
$\mathbb{A}_{n},n\geq 1$, $\mathbb{D}_{n},n\geq 4$, $\mathbb{E}_{6}$,
$\mathbb{E}_{7}$ and $\mathbb{E}_{8}$.

###  2.2 Friezes

An alternative way to define friezes is to say that they are ring
homomorphisms from a cluster algebra to the ring of integers such that all
cluster variables are sent to positive integers [Fontaine]. Let $Q$ be a
quiver without loops and $2$-cycles and let $\mathcal{A}(Q)$ be the
corresponding cluster algebra with trivial coefficients [Gunawan], then:

  1. (i)

A frieze of type $Q$ is a ring homomorphism
$\mathcal{F}:\mathcal{A}(Q)\rightarrow\mathbf{R}$ from the cluster algebra to
an integral domain $\mathbf{R}$. The frieze is called integral if
$\mathbf{R}=\mathbb{Z}$.

  2. (ii)

An integral frieze is said to be positive if every cluster variable in
$\mathcal{A}(Q)$ is mapped by $\mathcal{F}$ to a positive integer.

Let x$=(x_{1},\dots,x_{n})$ be a cluster of $\mathcal{A}(Q)$, then:

  1. (iii)

A vector $(a_{1},\dots,a_{n})\in\mathbf{R}^{n}$ is called a frieze vector
relative to x if the frieze $\mathcal{F}$ defined by the assignment
$\mathcal{F}(x_{i})=a_{i}$ has values in $\mathbf{R}$, $\mathcal{F}$ is said
to be unitary if there exists a cluster x such that $\mathcal{F}(x)$ is a unit
in R, for all $x\in\textbf{x}$. If the frieze $\mathcal{F}$ is unitary we say
that the frieze vector $(a_{1},\dots,a_{n})$ is unitary.

  2. (iv)

A vector $(a_{1},\dots,a_{n})\in\mathbb{Z}_{>0}^{n}$ is called a positive
frieze vector relative to x if the frieze $\mathcal{F}$ defined by
$\mathcal{F}(x_{i})=a_{i}$ is positive integral.

The following is an example of a frieze pattern. Hereinafter, frieze patterns
are assumed to be integral closed friezes.

$\textstyle{\dots}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{\dots}$$\textstyle{\dots}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{\dots}$$\textstyle{\dots}$$\textstyle{2}$$\textstyle{3}$$\textstyle{1}$$\textstyle{2}$$\textstyle{3}$$\textstyle{\dots}$$\textstyle{\dots}$$\textstyle{5}$$\textstyle{2}$$\textstyle{1}$$\textstyle{5}$$\textstyle{\dots}$$\textstyle{\dots}$$\textstyle{2}$$\textstyle{3}$$\textstyle{1}$$\textstyle{2}$$\textstyle{3}$$\textstyle{\dots}$$\textstyle{\dots}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{\dots}$$\textstyle{\dots}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{\dots}$

###  2.3 Dyck Paths Categories

In this section we recall the definition and main properties of the category
of Dyck paths as Cañadas and Rios describe in [Canadas].

A Dyck path is a lattice path in $\mathbb{Z}^{2}$ from $(0,0)$ to $(n,n)$ with
steps $(1,0)$ and $(0,1)$ such that the path never passes below the line
$y=x$. The number of Dyck paths of length $2n$ is equal to
$C_{n}=\frac{1}{n+1}\binom{2n}{n}$, the nth Catalan number [Stanley].

The set of Dyck words is the set of words $w$ in the free monoid
$X^{\ast}=\\{U,D\\}^{\ast}$ satisfying the following two conditions
[Barcucci]:

  * •

for any left factor $u$ of $w$ ( i.e., $w=uv$ for some suitable word $v$),
$|u|_{U}\geq|u|_{D}$,

  * •

$|w|_{U}=|w|_{D}$,

where $|w|_{a}$ is the number of occurrences of the letter $a\in X=\\{U,D\\}$
in the word $w$.

Henceforth, Dyck words defined as before are used to denote Dyck paths.

Let $\mathfrak{D}_{2n}$ be the set of all Dyck paths of length $2n$, let
$UWD=Uw_{1}\dots w_{n-1}D$ be a Dyck path in $\mathfrak{D}_{2n}$ with
$A=\\{UD,DU,UU,DD\\}$ being the set of all possible choices in $W$.

The support of $UWD$ (denoted by $\text{Supp
}UWD\subseteq\\{1,2,\dots,n-1\\}=\textbf{n-1}$) is a set of indices (of the
$w_{i}$s) such that

$\text{Supp}\hskip 2.84544ptUWD=\\{q\in\textbf{n-1}\text{ }|\text{
}w_{q}=UD\text{ or }w_{q}=UU\text{, }1\leq q\leq n-1\\}.$

A map $f:A\longrightarrow A$ such that for any $w\in A$, it holds that
$f(w)=f(ab)=w^{-1}=ba$, $a,b\in\\{U,D\\}$ is said to be a shift. An unitary
shift is a map $f_{i}:\mathfrak{D}_{2n}\longrightarrow\mathfrak{D}_{2n}$ such
that

$f_{i}(Uw_{1}\ldots w_{i-1}w_{i}w_{i+1}\ldots w_{n-1}D)=Uw_{1}\ldots
w_{i-1}f(w_{i})w_{i+1}\ldots w_{n-1}D.$

We will denote a unitary shift by a vector of maps from $\mathfrak{D}_{2n}$ to
itself of the form $(1_{1},\ldots,1_{i-1},f_{i},1_{i+1},\ldots,1_{n-1})$,
where $1_{k}$ is the identity map associated to the $i$-th coordinate.

An elementary shift is a composition of unitary shifts. A shift path of length
$m$ $UWD\longrightarrow UW_{1}D\longrightarrow\cdots\longrightarrow
UW_{m}D\longrightarrow UVD$ from $UWD$ to $UVD$ is a composition of $m$
elementary shifts. The set of all Dyck paths in a shift path between $UWD$ and
$UVD$ will be denoted by $J(W,V)$. For notation, we introduce the identity
shift as the elementary shift $(1_{1},\dots,1_{n-1})$.

Suppose that a map $R:\mathfrak{D}_{2n}\rightarrow\mathfrak{D}_{2n}$ is
defined by the application of successive elementary shifts to a given Dyck
path. Then $R$ is said to be an irreversible relation over $\mathfrak{D}_{2n}$
if and only if elementary shifts transforming Dyck paths (from one to the
other) are not reversible. In other words, if an elementary shift
$F=f_{p_{1}}\circ\dots\circ f_{p_{q}}$ transforms a Dyck path $UWD$ into a
Dyck path $UVD$ then there is not an elementary shift
$F^{\prime}=f_{p_{1}}\circ\dots\circ f_{p_{q}}$ transforming $UVD$ into $UWD$,
for some $p,q\in\mathbb{Z}^{+}$.

If there exist two paths $G\circ F$ and $G^{\prime}\circ F^{\prime}$ of
irreversible relations (of length 2) transforming a Dyck path $UWD$ into the
Dyck path $UVD$ over $R$ in the following form:

$UWD$$UW^{\prime}D$$UW^{\prime\prime}D$$UVD$,F${}_{F^{\prime}}$G${}_{G^{\prime}}$

with $W^{{}^{\prime}}\neq W^{{}^{\prime\prime}}$. Then $G\circ F$ is said to
be related with $G^{\prime}\circ F^{\prime}$ (denoted $G\circ
F\sim_{R}G^{\prime}\circ F^{\prime}$) whenever $G^{\prime}=F$ and
$G=F^{\prime}$.

As for the case of diagonals [Caldero], Cañadas and Rios [Canadas] defined a
$\mathbb{F}$-linear additive category $(\mathfrak{D}_{2n},R)$ based on Dyck
paths, in this case, objects are $\mathbb{F}$-linear combinations of Dyck
paths in $\mathfrak{D}_{2n}$ with space of morphisms from a Dyck path $UWD$ to
a Dyck path $UVD$ over $\mathbb{F}$ associated to $R$ being the vector space

$\mathrm{Hom}_{(\mathfrak{D}_{2n},R)}(UWD,UVD)=\langle\\{g\text{ }|\text{
}g\hskip 2.84544pt\text{is a shift path associated to
}R\\}\rangle/\langle\sim_{R}\rangle$.

$\mathrm{Hom}_{(\mathfrak{D}_{2n},R)}(UWD,UVD)\neq 0$ if and only if there are
shift paths transforming $UWD$ into $UVD$ and

$\bigcap_{UW_{i}D\in J(W,V)}\text{Supp }UW_{i}D\neq\varnothing,$

for each shift path, with $UWD$ and $UVD$ in $\mathfrak{D}_{2n}$.

Figure 1 shows the elementary shifts over $(\mathfrak{D_{6}},R)$ associated to
an irreversible relation $R$ defined over the set of all Dyck paths of length
$6$. And such that,

$R(UWD)=\begin{cases}f_{1}(UWD),&\mbox{ if }w_{1}=UD,\\\ f_{2}(UWD),&\mbox{ if
}w_{2}=UD.\end{cases}$

Figure 1:  Elementary shifts in $(\mathfrak{D}_{6},R)$.

$Q=$12345

Figure 2:  Quiver $Q$ and the Dyck path version of the Auslander-Reiten quiver
of $\mathrm{rep}\hskip 2.84544ptQ$ (see Corollary 3).

If $\textbf{n}=\\{1,2,\ldots,n\\}$ is an $n$-point chain then
$\mathscr{C}_{\textbf{(1,n)}}$ stands for all admissible subchains
$\mathscr{C}$ of n with $\text{min }\mathscr{C}=1$ and $\text{max
}\mathscr{C}=n$. Given an admissible subchain
$\mathscr{C}=\\{j_{1},\dots,j_{m},i_{1},\dots,i_{l}\\}$ then two Dyck paths
$D$ and $D^{\prime}$ of length $2n$ are said to be related by a relation of
type $R_{j_{1}\ldots j_{m}}^{i_{1}\ldots i_{l}}$ if there is an elementary
shift associated to the points of the subchain transforming one into the other
[Canadas].

We let $\mathfrak{C}_{2n}$ denote the subcategory of
$(\mathfrak{D}_{2n},R_{j_{1}\ldots j_{m}}^{i_{1}\ldots i_{l}})$ whose objects
are $\mathbb{F}$-linear combinations of Dyck paths with exactly $n-1$ peaks
and related by a relation of type $R_{j_{1}\ldots j_{m}}^{i_{1}\ldots i_{l}}$.

The following results describe the structure of the category of Dyck paths
[Canadas]. We recall that a point $x$ where a Dyck path $UWD$ changes from the
north to the east is said to be a peak of the path.

In the following theorem the symbol $Q$ is used to denote a quiver of type
$\mathbb{A}_{n-1}$ ($n\geq 2$) for which $\\{i_{1},\dots,i_{l}\\}$ and
$\\{j_{1},\dots,j_{m}\\}$ are the corresponding sets of sinks and sources.
$\mathrm{rep}\hskip 2.84544ptQ$ denotes the corresponding category of
representations.

######  Theorem 2 ([Canadas], Theorem 15).

There is a categorical equivalence between categories $\mathfrak{C}_{2n}$ and
$\mathrm{rep}\hskip 2.84544ptQ$.

######  Corollary 3 ([Canadas], Corollary 16).

There exists a bijection between the set of representatives of indecomposable
representations $\mathrm{Ind}\hskip 2.84544ptQ$ of $\mathrm{rep}\hskip
2.84544ptQ$ and the set of Dyck paths of length $2n$ with exactly $n-1$ peaks.

######  Corollary 4 ([Canadas], Corollary 17).

The category $\mathfrak{C}_{2n}$ is an abelian category.

##  3 Relationships Between Friezes and Dyck paths

In this section, we present the main results of the paper. In particular, it
is introduced the notion of integral diamond of Dynkin type $\mathbb{A}_{n}$,
which are integer arrays used to build frieze patterns associated to
triangulations of an $(n+3)$-polygon.

###  3.1 $\mathbb{A}_{n}$-Diamonds

Let $\mathbf{R}$ be an integral domain then a diamond of Dynkin type
$\mathbb{A}_{n}$ or $\mathbb{A}_{n}$-diamond is an array $A=(a_{i,j})$ with
entries in R, such entries satisfy conditions (D1) and (D2) associated to
arrays with the following shape:

---
$\textstyle{{}_{a_{2,0}}}$$\textstyle{{}_{a_{1,1}}}$$\textstyle{{}_{a_{2,1}}}$$\textstyle{{}_{a_{1,2}}}$⋱⋱$\textstyle{{}_{a_{2,n-1}}}$$\textstyle{{}_{a_{1,n}}}$$\textstyle{{}_{a_{2,n}}}$$\textstyle{{}_{a_{1,n+1}}}$

  1. (D1)

$a_{2,0}=a_{1,n+1}=1$,

  2. (D2)

$a_{1,j}a_{2,j}-a_{2,j-1}a_{1,j+1}=1$ for $1\leq j\leq n$,

where, $1$ is the identity element of $\mathbf{R}$.

If $\mathbf{R}=\mathbb{Z}$ then $A$ is said to be a positive integral diamond
of Dynkin type $\mathbb{A}_{n}$, if it also satisfies the following condition
(D3)

  1. (D3)

$a_{1,1}=a$ (or $a_{1,1}=a+m_{a}$), $a_{2,1}=a+m_{a}$ (or $a_{2,1}=a$) and
$a_{1,2}=a^{2}+am_{a}-1$, with $1\leq a\leq\lfloor\frac{n+2}{2}\rfloor$,
$1\leq m_{1}\leq n$ and $0\leq m_{a}\leq n+2(1-a)$ if $a>1$.

Henceforth, if it is not mentioned explicitly, $\mathbb{A}_{n}$-diamonds are
assumed to be positive integral diamonds of Dynkin type $\mathbb{A}_{n}$.

Two $\mathbb{A}_{n}$-diamonds $A$ and $B$ constitute a coupling, denoted
$A\models B$ if and only if $a_{2,j}=b_{1,j}$ for $1\leq j\leq n$.

A set $\\{A^{t}\\}_{t\geq 0}$ is an $\mathbb{A}_{n}$-sequence of couplings of
$\mathbb{A}_{n}$ if and only if $A^{r}\models A^{r+1}$ for $r\geq 0$
($X^{0}=X$ for any $\mathbb{A}_{n}$-diamond $X$). An $\mathbb{A}_{n}$-sequence
$\\{A^{t}\\}_{t\geq 0}$ of couplings is a $p$-cycle if there exists
$p\in\mathbb{N}$ such that $A^{t}=A^{t+p}$. If the $\mathbb{A}_{n}$-sequence
of couplings $S_{t}=\\{A^{t}\\}_{t\geq 0}$ constitute a frieze pattern
$\mathcal{F}$ then we will say that $S_{t}$ generates $\mathcal{F}$. In this
case, we point out that Lemma 7 proves that $A^{0}=A$ generates $S_{t}$,
meaning that $S_{t}$ can be obtained from a sequence of couplings starting
with $A$.

For example, let $\mathbf{R}=\mathbb{Z}$, the sets $\\{A^{t}\\}_{t\geq 0}$ and
$\\{B^{t}\\}_{t\geq 0}$ ($A^{0}$ and $B^{0}$ as shown below) are
$\mathbb{A}_{1}$-sequences which are $2$-cycles with $A^{2k}=B^{2k+1}=A$,
$A^{2k+1}=B^{2k}=B$ and $k\geq 0$.

$\textstyle{1}$$\textstyle{1}$$\textstyle{A=}$$\textstyle{1}$$\textstyle{2}$$\textstyle{B=}$$\textstyle{2}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$
(3)

In general, it can be written an $\mathbb{A}_{n}$-sequence $\\{A^{t}\\}_{t\geq
0}$ as an $\mathbb{A}_{n}$-array $C_{A^{t}}=(c_{i,j})$ such that
$c_{t+1,j}=a^{t}_{1,j}$ and $c_{t+1,0}=c_{t+1,n+1}=1$, for $t\geq 0$. For the
previous example,

$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{\dots}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{\dots}$$\textstyle{C_{A^{t}}=}$$\textstyle{1}$$\textstyle{2}$$\textstyle{1}$$\textstyle{2}$$\textstyle{\dots}$$\textstyle{C_{B^{t}}=}$$\textstyle{2}$$\textstyle{1}$$\textstyle{2}$$\textstyle{1}$$\textstyle{\dots}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{\dots}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{\dots}$

$C_{A^{t}}$ and $C_{B^{t}}$ are $\mathbb{A}_{1}$-arrays associated to
$\\{A^{t}\\}_{t\geq 0}$ and $\\{B^{t}\\}_{t\geq 0}$, respectively.

If the $\mathbb{A}_{n}$-sequence of couplings is finite of length $m$, then it
can be associated to an infinity $\mathbb{A}_{n}$-array,
$C^{m}_{A^{t}}=(c^{m}_{i,j})$ such that

$c^{m}_{(t+1)+km,j}=a^{t}_{1,j},\text{
}c^{m}_{(t+1)+km,0}=c^{m}_{(t+1)+km,n+1}=1,$ (4)

for $k\in\mathbb{Z}$. For any $\mathbb{A}_{n}$-sequence of couplings
$\\{A^{t}\\}_{t\geq 0}$, it is possible to choose an
$\mathbb{A}_{n}$-subsequence $\\{B^{z}\\}_{z\geq 0}$ with $B^{z}=A^{x+z}$ for
some $x\geq t$. In particular, if $\\{A^{t}\\}_{t\geq 0}$ is a $p$-cycle then
the subsequence $\\{B^{s_{0}}\\}_{0\leq s_{0}\leq p-1}$ such that
$B^{s_{0}}=A^{t}$ is called the minimal $p$-cycle of $\\{A^{t}\\}_{t\geq 0}$.

The following results give the main properties of diamonds of Dynkin type
$\mathbb{A}_{n}$.

######  Proposition 5.

Let $\\{A^{t}\\}_{t\geq 0}$ be a $p$-cycle and let $B=\\{B^{s_{0}}\\}_{0\leq
s_{0}\leq p-1}$ be its minimal $p$-cycle. Then, the array $C^{p}_{B}$ is a
frieze pattern of order $n+3$. In particular, $p$ divides $n+3$.

Proof. Let $C^{p}_{B}=(c^{p}_{ij})$ be the infinity $\mathbb{A}_{n}$-array
associated to $\\{B^{s_{0}}\\}_{0\leq s_{0}\leq p-1}$, identity (4) implies
that

$c^{p}_{(s_{0}+1)+kp,j}=a^{s_{0}}_{1,j},\quad
c^{p}_{(s_{0}+1)+kp,0}=c^{p}_{(s_{0}+1)+kp,n+1}=1,$

for $k\in\mathbb{Z}$, since that $\\{A^{t}\\}_{t\geq 0}$ is a $p$-cycle, then
$C^{p}_{B}$ is a frieze pattern. $\hfill\square$

######  Proposition 6.

Let $\\{A^{t}\\}_{t\geq 0}$ be a $p$-cycle of length $2p$, then the
subsequences $\\{B^{s_{i}}\\}$ with ${0\leq s_{i}\leq p-1}$ generate the same
frieze pattern of order $n+3$, for $0\leq i\leq p-1$, and
$B^{s_{i}}=A^{i+s_{i}}$.

Proof. Let $\\{A^{t}\\}_{t\geq 0}$ be a $p$-cycle of length $2p$, let
$C^{p}_{A}=(c^{p}_{ij})$ and $C^{p}_{B}=(c^{p^{\prime}}_{ij})$ be the infinity
arrays of the subsequences $A=\\{B^{s_{i}}\\}_{0\leq s_{i}\leq p-1}$ and
$B=\\{B_{s_{i^{\prime}}}\\}_{0\leq s_{i^{\prime}}\leq p-1}$ of
$\\{A^{t}\\}_{t\geq 0}$ for $0\leq i<i^{\prime}\leq p-1$. The following
identities (5) hold by applying the translation
$s_{i^{\prime}}=s_{i}-|i^{\prime}-i|$,

$\begin{split}c^{p}_{s_{i^{\prime}}+1+kp,j}=a_{1j}^{s_{i^{\prime}}+i^{\prime}}=a_{1j}^{s_{i}-|i^{\prime}-i|+i^{\prime}}=a_{ij}^{s_{i}+i}=c^{p}_{s_{i}+1+kp,j}.\end{split}$
(5)

We are done. $\hfill\square$

######  Lemma 7.

Let $\\{A^{t}\\}_{t\geq 0}$ be a sequence of couplings. Then
$\\{A^{t}\\}_{t\geq 0}$ is generated by $A^{0}$. In particular, $A^{0}$
generates a $p$-cycle for some $p>0$.

Proof. Let $\\{A^{t}\\}_{t\geq 0}$ be a sequence, then

$a^{x}_{2,j}=\frac{1+(a^{x}_{2,j-1})(a^{x-1}_{2,j+1})}{a^{x-1}_{2,j}},$

for $1\leq j\leq n$, and $x\geq t$, $a^{x}_{2,j}$ can be written by using the
set $\\{a^{0}_{2,j}\\}_{1\leq j\leq n}$ for $x>0$. In particular, the set
$\\{a^{0}_{2,1},\dots,a^{0}_{2,n}\\}$ is a seed of the cluster algebra
associated to the linearly oriented quiver of type $\mathbb{A}_{n}$. Since the
cluster variables are finite in the case $\mathbb{A}_{n}$, then there is
$p=n+3$ (in some cases, it is not minimal) such that $A^{0}=A^{p}$.
$\hfill\square$

######  Theorem 8.

Let $A$ be a diamond of Dynkin type $\mathbb{A}_{n}$ then $A$ generates a
frieze pattern.

Proof. It is a direct consequence of Lemma 7, and Proposition
5.$\hfill\square$

For instance, diamonds $A$ and $B$ given in (3) generate the following frieze
pattern.

$\textstyle{\ldots}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{\ldots}$$\textstyle{\ldots}$$\textstyle{1}$$\textstyle{2}$$\textstyle{1}$$\textstyle{2}$$\textstyle{\ldots}$$\textstyle{\ldots}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{\ldots}$

###  3.2 Seed Vectors

In this section, we give an algorithm to build a family of positive integral
frieze vectors associated to the linearly oriented quiver of type
$\mathbb{A}_{n}$. These vectors allow to find out a connection between the
positive integral diamonds of Dynkin type $\mathbb{A}_{n}$, triangulations,
and Dyck paths.

Let $A$ be a diamond of Dynkin type $\mathbb{A}_{n}$ then we can write its
first column as a vector with the form $v_{A}=(a_{1},\dots,a_{n})$, where
$a_{j}=a_{1,j}$. In such a case, we say that $v_{A}$ is associated to $A$ and
that $v_{A}$ generates $A$.

######  Proposition 9.

If $v=(a_{1},\dots,a_{n})$ is a vector associated to a positive integral
diamond of Dynkin type $\mathbb{A}_{n}$ with $a_{n}=1$, then the vector
$v^{\prime}=(a_{1},\dots,a_{i},a_{i}+a_{i+1},a_{i+1},\dots,a_{n-1})$ is also
associated to a positive integral diamond of Dynkin type $\mathbb{A}_{n}$, for
$1\leq i<n$.

Proof. Let $v_{A}=(a_{1},\dots,a_{n})$ be a vector associated to a positive
integral diamond $A=(a_{j,m})$ of Dynkin type $\mathbb{A}_{n}$, then we take
the vector $v_{A+i}=(a_{1},\dots,a_{i},a_{i}+a_{i+1},a_{i+1},\dots,a_{n-1})$
and the array $A+i$ of the following form:

$b_{1,m}=\begin{cases}a_{1,m},&\mbox{if $m\leq i$,}\\\
a_{1,i}+a_{1,i+1},&\mbox{if $m=i+1$,}\\\ a_{1,m-1},&\mbox{if
$m>i+1$,}\end{cases}$

and

$b_{2,m}=\begin{cases}a_{2,m},&\mbox{if $m\leq i-1$,}\\\
a_{2,i-1}+a_{2,i},&\mbox{if $m=i$,}\\\ a_{2,m-1},&\mbox{if $m\geq
i+1$,}\end{cases}$

then $b_{1,m}b_{2,m}-b_{2,m-1}b_{2,m+1}=1$, for $1\leq m\leq n$ and $1\leq
i<n$. Therefore $A+i$ is a positive integral diamond of Dynkin type
$\mathbb{A}_{n}$. $\hfill\square$

######  Proposition 10.

For each vector $v_{n,z}=(a_{1},\dots,a_{n})$ with

$a_{i}=\begin{cases}z+1-i,&\mbox{if $i<z$,}\\\ 1,&\mbox{if $i\geq
z$.}\end{cases}$ (6)

there is associated a unique positive integral diamond of Dynkin type
$\mathbb{A}_{n}$, for $z\in\\{1,\dots,n+1\\}$.

Proof. Let $v_{n,z}$ be a vector and let $z$ be a natural number between $1$
and $n+1$, we define a positive integral diamond $A$ with $a_{1,i}=a_{i}$ and
$a_{2,i}=b_{i}$ where

$b_{i}=\begin{cases}1,&\mbox{if $i<z$,}\\\ i+2-z,&\mbox{if $i\geq
z$,}\end{cases}$ (7)

then $a_{1,i}a_{2,i}-a_{2,i-1}a_{2,i+1}=1$ for $1\leq i\leq n$.
$\hfill\square$

######  Remark 11.

$v_{n,z}$ is called a seed vector. The vector $v^{n,z}=(b_{1},\dots,b_{n})$
defines a positive integral diamond $B$ of Dynkin type $\mathbb{A}_{n}$ such
that $b_{2,i}$ satisfies the following identity

$b_{2,i}=\begin{cases}i-1,&\mbox{if $i<z-1$,}\\\ (b_{1,i}+1)z-1,&\mbox{if
$z-2<i<n$,}\\\ z,&\mbox{if $i=n$.}\end{cases}$

and $b_{i}=b_{1,i}$ is defined as in (7).

######  Proposition 12.

The positive integral diamonds $A$ and $B$ of Dynkin type $\mathbb{A}_{n}$
generated by vectors $v_{n,z}$ and $v^{n,z}$ respectively constitute a
coupling.

Proof. It is a direct consequence of Proposition 10 and Lemma
7.$\hfill\square$

The number of ways of applying recursively Proposition 9 to a vector
$w_{A}=(a_{1},\dots,$ $a_{z-1},1,\dots,1)\in\mathbb{N}^{n}$ is given by the
next identity (denoted by $f_{n,z}$),

$f_{n,z}=\begin{cases}\sum_{i=z-1}^{n}f_{n-1,i},&\mbox{if $z>1$,}\\\
\sum_{i=1}^{n}f_{n-1,i},&\mbox{if $z=1$,}\end{cases}$

where it is included the trivial move $w_{A+0}=w_{A}$, for $n>1$, and any
$z\in\\{1,\dots,n+1\\}$. Actually, these numbers can be arranged as follows:

$\textstyle{f_{1,2}}$$\textstyle{f_{1,1}}$$\textstyle{f_{2,3}}$$\textstyle{f_{2,2}}$$\textstyle{f_{2,1}}$$\textstyle{f_{3,4}}$$\textstyle{f_{3,3}}$$\textstyle{f_{3,2}}$$\textstyle{f_{3,1}}$$\textstyle{f_{4,5}}$$\textstyle{f_{4,4}}$$\textstyle{f_{4,3}}$$\textstyle{f_{4,2}}$$\textstyle{f_{4,1}}$$\textstyle{\vdots}$$\textstyle{\ddots}$
(8)

for any of the vectors $w_{A}$. Since the first choices are $v_{1,1}=(1)$ and
$v_{1,2}=(2)$, then $f_{1,1}=1$ and $f_{1,2}=1$. The previous triangle (8)
appears in the On-Line Encyclopedia of Integer Sequences (OEIS) as A009766
(Catalan triangle [OEIS9]). In particular, we generate all positive integral
diamonds of Dynkin type $\mathbb{A}_{n}$ via the seed vectors $v_{n,z}$. For
example, for $n=3$, all vectors that generate positive integral diamonds of
Dynkin type $\mathbb{A}_{3}$ are:

$\textstyle{(1,1,1)}$$\textstyle{(1,1,2)}$$\textstyle{(1,2,1)}$$\textstyle{(1,2,3)}$$\textstyle{(1,3,2)}$$\textstyle{(2,1,1)}$$\textstyle{(2,1,2)}$$\textstyle{(2,3,1)}$$\textstyle{(2,3,4)}$$\textstyle{(2,5,3)}$$\textstyle{(3,2,1)}$$\textstyle{(3,2,3)}$$\textstyle{(3,5,2)}$$\textstyle{(4,3,2)}$

Let $G=UD\dots UD\dots$ be a Dyck path of length $2n$ and let $m_{i}$ be the
number of $U$s before the occurrence of the $i$-th $D$ in $G$ then $G$ can be
written as a vector $v_{G}=(v_{1},\dots,v_{n-1})$ where $v_{i}=m_{i}-i+1$. As
an example, consider the Dyck path $G$ shown in Figure 3, which has associated
the vector $v_{G}=(5,4,3,3,5,4,3,2)$.

Figure 3: A Dyck path of length $18$ associated to the vector
$v_{G}=(5,4,3,3,5,4,3,2)$.

In what follows, it is defined a map $T_{i}$ between vectors associated to
positive integral diamonds of Dynkin type $\mathbb{A}_{n}$ and Dyck paths by
using a relation over the coordinates of a vector $u=(a_{1},\ldots,a_{m})$.
$T_{i}$ is defined in such a way that,
$T_{i}:\mathbb{N}^{m}\rightarrow\mathbb{N}$ and:

  * •

If $a_{i}-a_{l}>0$ for some $l\in\\{1,\ldots,i\\}$, then we select the maximum
of such indexes $\mathrm{max}\hskip 2.84544pt\\{l\\}$ and write
$r_{1}=a_{i}-a_{l}$. In the same way, it is chosen $\mathrm{max}\hskip
2.84544pt\\{l\\}$ such that $r_{1}-a_{l}>0$ and write $r_{2}=r_{1}-a_{l}$,
this process ends if $\\{l\mid r_{t}-a_{l}>0\\}=\varnothing$. Thus, if
$u\in\mathbb{N}^{m}$ then $T_{i}(u)=r_{t}+t$, for some $t$.

  * •

If $a_{i}-a_{l}\leq 0$ for all $l\in\\{1,\ldots,i\\}$, then $T_{i}(u)=a_{i}$.

For instance, for the vector $u=(14,52,4,23,9,2)$, it holds that
$T_{1}(u)=14$, $T_{2}(u)=13$, $T_{3}(u)=4$, $T_{4}(u)=8$, $T_{5}(u)=3$, and
$T_{6}(u)=2$.

######  Proposition 13.

Let $v_{n,z}$ be a seed vector, then $(T_{1}(v_{n,z}),\dots,T_{n}(v_{n,z}))$
defines a Dyck path of length $2(n+1)$.

Proof. For any $z\in\\{1,\dots,n+1\\}$, $T_{i}(v_{n,z})=a_{i}$ with $a_{i}$
given by identity (6), there is a word $G_{v_{n,z}}=w_{1}\dots w_{2(n+1)}$ in
the free monoid $\\{U,D\\}^{\ast}$ such that

$G_{v_{n,z}}=\underbrace{U\dots U}_{z-1}\underbrace{D\dots D}_{z-1}UDUD\dots
UDUD,$

for any left factor $u_{s}$ in $G_{v_{n,z}}$ of length
$s\in\\{1,\dots,2(n+1)\\}$, $0\leq|u_{s}|_{U}-|u_{s}|_{D}\leq z-1$, therefore
$G_{v_{n,z}}\in\mathfrak{D}_{2(n+1)}$.$\hfill\square$

######  Proposition 14.

Let $v_{A}=(a_{1},\dots,a_{n})$ be a vector associated to a positive integral
diamond $A$ of Dynkin type $\mathbb{A}_{n}$ with $a_{n}=1$, such that
$(T_{1}(v_{A}),\dots,T_{n}(v_{A}))$ defines a Dyck path in
$\mathfrak{D}_{2(n+1)}$. Then $(T_{1}(v_{A+i}),\dots,T_{n}(v_{A+i}))$ also
defines a Dyck path in $\mathfrak{D}_{2(n+1)}$.

Proof. Let $v_{A}=(a_{1},\dots,a_{n})$ be a vector associated to a positive
integral diamond $A$ with $a_{n}=1$, then there exists a Dyck path
$G_{v_{A}}\in\mathfrak{D}_{2(n+1)}$ such that any left factor $u_{s}$ of
length $s$ satisfies $|u_{s}|_{U}\geq|u_{s}|_{D}$ for $1\leq s\leq 2(n+1)$.
Let $v_{A+i}$ be a vector associated to the positive integral diamond $A+i$
with

$T_{m}(v_{A+i})=\begin{cases}T_{m}(v_{A}),&\mbox{if $1\leq m\leq i$,}\\\
T_{m}(v_{A})+1,&\mbox{if $m=i+1$,}\\\ T_{m-1}(v_{A}),&\mbox{if $m\geq
i+1$,}\end{cases}$

then there is a word $G_{A+i}=w^{\prime}_{1},\dots,w^{\prime}_{2(n+1)}$ in
$\\{U,D\\}^{\ast}$, where we take the index $m_{1}$ of the $i$-th $D$ in
$G_{A+i}$, any left factor $u^{\prime}_{s}$ in $G_{A+i}$ satisfies the
identities

$|u^{\prime}_{s}|_{U}=\begin{cases}|u_{s}|_{U},&\mbox{if $1\leq s\leq
m_{1}$,}\\\ |u_{m_{1}}|_{U}+1,&\mbox{if $s=m_{1}+1$,}\\\
|u_{s-2}|_{U}+1,&\mbox{if $s\geq m_{1}+2$,}\end{cases}$

and

$|u^{\prime}_{s}|_{D}=\begin{cases}|u_{s}|_{D},&\mbox{if $1\leq s\leq
m_{1}$,}\\\ |u_{m_{1}}|_{D},&\mbox{if $s=m_{1}+1$,}\\\
|u_{s-2}|_{U}+1,&\mbox{if $s\geq m_{1}+2$,}\end{cases}$

then, we have the following possibilities:

  * •

If $1\leq s\leq m_{1}$,
$|u^{\prime}_{s}|_{U}=|u_{s}|_{U}\geq|u_{s}|_{D}=|u^{\prime}_{s}|_{D}$.

  * •

If $s=m_{1}+1$,
$|u^{\prime}_{m_{1}+1}|_{U}=|u_{m_{1}}|_{U}+1>|u_{m_{1}}|_{D}=|u^{\prime}_{m_{1}+1}|_{D}$.

  * •

If $m_{1}+2\leq s\leq 2(n+1)$,
$|u^{\prime}_{s}|_{U}=|u_{s-2}|_{U}+1\geq|u_{s-2}|_{D}+1=|u^{\prime}_{s}|_{D}$.

Therefore, $G_{A+i}\in\mathfrak{D}_{2(n+1)}$.$\hfill\square$

######  Lemma 15.

There is a bijective correspondence between the set of all vectors associated
to positive integral diamonds of Dynkin type $\mathbb{A}_{n}$ and the set of
all Dyck paths of length $2(n+1)$.

Proof. Let $\mathbb{D}_{\mathbb{A}_{n}}$ be the set of all vectors associated
to positive integral diamonds of Dynkin type $\mathbb{A}_{n}$ and let
$\mathfrak{D}_{2(n+1)}$ be the set of all Dyck paths of length $2(n+1)$ then
we define a map
$f:\mathbb{D}_{\mathbb{A}_{n}}\rightarrow\mathfrak{D}_{2(n+1)}$ with
$f(u_{A})=\big{(}T_{1}(u_{A}),\ldots,T_{n}(u_{A})\big{)}$, Propositions 13 and
14 allow us to establish that $f$ is well defined. In order to prove that the
map $f$ is injective, suppose that $u_{A}\neq v_{B}$, we take the minimum $l$
such that $u_{l}\neq v_{l}$. If $l=1$ then $T_{1}(u_{A})\neq T_{1}(v_{B})$. If
$l>1$, $u_{l}=m(u_{l-1})+a$ and $v_{l}=m^{\prime}(u_{l-1})+a$ with $m\neq
m^{\prime}$ is a consequence of Proposition 9 then $r_{t_{u_{l}}}\neq
r_{t_{v_{l}}}$, therefore $T_{l}(u_{A})\neq T_{k}(v_{B})$. $\hfill\square$

An alternative way of writing a Dyck path $G\in\mathfrak{D}_{2(n+1)}$ can be
defined by using a vector $\lambda_{G}=(\lambda_{1},\dots,\lambda_{n})$ where
$\lambda_{i}$ is the number of $D$s before the occurrence of the $(n+2-i)$-th
$U$ in $G$. For example, Dyck path in Figure 3 has associated the following
vector $\lambda_{G}=(4,4,4,3,0,0,0,0)$.

Let $\lambda$ be a vector associated to a Dyck path of length $2(n+1)$ then a
triangulation of an $(n+3)$-polygon can be realized by $\lambda$ as follows:

  * •

Fix a labeling for the vertices of polygon
$K_{0}^{n+3}=(v^{n+3}_{0},\dots,v^{n+3}_{n+2})$ with $v^{n+3}_{i}=i$, for
$0\leq i\leq n+2$.

  * •

For $\lambda_{i}$, we draw a diagonal $l_{i}^{\lambda_{i}}$ between
$\lambda_{i}$ and $\lambda_{i}+2$. Afterwards, we label the last polygon with
$n+3-i$ vertices $K_{i}^{n+3-i}=(v^{n+3-i}_{0},\dots,v^{n+3-i}_{n+2-i})$, and

$v^{n+3-i}_{j}=\begin{cases}v^{n+3-(i-1)}_{j},&\mbox{if
$j\leq\lambda_{i}$,}\\\ v^{n+3-(i-1)}_{j+1}-1,&\mbox{if
$j>\lambda_{i}$,}\end{cases}$

for $i=1,\dots,n$.

$2$$1$$5$$4$$0$$3$$l_{1}^{2}$$2$$1$$4$$3$$0$$l_{1}^{2}$$l_{2}^{2}$$2$$1$$3$$0$$l_{1}^{2}$$l_{2}^{2}$$l_{3}^{1}$$\longrightarrow$$\longrightarrow$
Figure 4:  Example of a triangulation realized by a Dyck path of length 8.

The previous algorithm describes that if $l_{i}^{\lambda_{i}}$ is a diagonal
then it does not cross the diagonals
$l_{1}^{\lambda_{1}},\dots,l_{i-1}^{\lambda_{i-1}}$ for $1\leq i\leq n$. For
instance, let $\lambda_{G}=(2,2,1)$ be the vector associated to $G=UDUDUUDD$,
then the triangulation of $\lambda_{G}$ is shown in Figure 4.

If we fix a labeling $K$ over all vertices of a polygon with $n+3$ vertices, a
triangulation $T$ is written as a sequence
$T=(l_{1}^{v_{1}},\dots,l_{n}^{v_{n}})$, where $v_{i}$ belongs to the set of
vertices.

######  Lemma 16.

There is a bijective correspondence between the set of all triangulations of a
polygon with $n+3$ vertices and the set of all Dyck paths of length $2(n+1)$.

Proof. Let $\mathcal{T}_{n}$ be the set of all triangulations of a polygon
with $n+3$ vertices then we can define a map
$g:\mathfrak{D}_{2(n+1)}\rightarrow\mathcal{T}_{n}$ with
$g(\lambda)=T_{\lambda}$. In order to prove that $g$ is one to one, we fix a
labeling $K$ and suppose that $g(\lambda_{G})=g(\sigma_{G^{\prime}})$, then
$(l_{1}^{\lambda_{1}},\dots,l_{n}^{\lambda_{n}})=(l_{1}^{\sigma_{1}},\dots,l_{n}^{\sigma_{n}})$,
provided that $l_{j}^{\lambda_{j}}=l_{j}^{\sigma_{j}}$. Since by definition
there are diagonals connecting vertices $\lambda_{j}$ with $(\lambda_{j}+2)$
and $\sigma_{j}$ with $(\sigma_{j}+2)$, therefore $\lambda_{j}=\sigma_{j}$ for
$j=1,\dots,n$. We are done.$\hfill\square$

The next theorem presents the main result regarding relationships between
positive integral diamonds of Dynkin type $\mathbb{A}_{n}$ and triangulations
of an $(n+3)$-polygon.

######  Theorem 17.

There is a bijective correspondence between the set of all vectors associated
to positive integral diamonds of Dynkin type $\mathbb{A}_{n}$ and
triangulations of a polygon with $n+3$ vertices.

Proof. We fix a labeling $K$ in a polygon with $n+3$ vertices, the map
$F:\mathbb{D}_{\mathbb{A}_{n}}\rightarrow\mathcal{T}_{n}$ defined by the
formula

$\begin{split}F(v_{A})&=(g\circ f)(v_{A})\end{split}$ (9)

is a bijection (see Lemmas 15 and 16). $\hfill\square$

Figure 5 presents an example of the bijective correspondence between a
positive integral diamond of Dynkin type $\mathbb{A}_{4}$, a Dyck path of
length $10$, and a triangulation of a polygon with $7$ vertices.

$\textstyle{1}$$\textstyle{2}$$\textstyle{2}$$\textstyle{3}$$\textstyle{3}$$\textstyle{4}$$\textstyle{1}$$\textstyle{1}$$\textstyle{2}$$\textstyle{1}$
Figure 5: Map $F$ defined in the proof of Theorem 17 allows to establish
identifications between an $\mathbb{A}_{4}$-diamond, a Dyck path of length 10
and a triangulation of a heptagon.

###  3.3 Frieze Patterns and Dyck Paths

In this section, we describe an algebraic interpretation of frieze patterns as
a direct sum of indecomposable objects of Dyck paths categories.

######  Lemma 18.

Vectors $v_{n,z}$ and $v^{n,z}$ realize the same triangulation except for one
anti-clockwise rotation.

Proof. Let $v_{n,z}$ and $v^{n,z}$ be frieze vectors, fixed a labeling $K_{1}$
in an $(n+3)$-polygon, then

1zn+1${}_{f(v_{n,z})}$${}_{f(v^{n,z})}$

the following identities hold by applying map $F$ (see (9)) as follows:

$\begin{split}F(v_{n,z})&=(\underbrace{n}_{1},\dots,\underbrace{z}_{n-z},\underbrace{0}_{n-z+1},\dots,\underbrace{0}_{n}),\\\
F(v^{n,z})&=(\underbrace{n-1}_{1},\dots,\underbrace{z-1}_{n-z},\underbrace{z-1}_{n-z+1},\dots,\underbrace{1}_{n}),\end{split}$

if we change $K_{1}$ for $K_{2}$ by making the following replacements:

  * •

Vertex $l\in K_{1}$ is changed for $l-1\in K_{2}$, $1\leq l\leq n+2$,

  * •

Vertex $0\in K_{1}$ is changed for $n+2\in K_{2}$,

then the diagonals from $0$ to $r_{1}$ in $K_{1}$ are diagonals from $r_{1}-1$
to $n+2$ in $K_{2}$, and the diagonals from $r_{2}$ in $K_{1}$ are diagonals
from $r_{2}-1$ in $K_{2}$, for $0\leq r_{1}\leq z\leq r_{2}\leq n$. Therefore
$F(v_{n,z})\in K_{1}$ coincides with $F(v^{n,z})\in K_{2}$.$\hfill\square$

Note that, there exists a permutation

$\sigma=\left(\begin{matrix}1&2&\dots&n-z-1&n-z&n-z+1&n-z+2&\dots&n-1&n\\\
1&2&\dots&n-z-1&n-z&n&n-1&\dots&n-z+2&n-z+1\end{matrix}\right)$

in $S_{n}$ that describes a bijection between the coordinates of the vector
$F(v_{n,z})=(u_{1},\dots,u_{n})$ and the vector
$F(v^{n,z})=(u^{\prime}_{1},\dots,u^{\prime}_{n})$ such that
$\sigma(F(v_{n,z}))=(u_{\sigma(1)},\dots,$
$u_{\sigma(n)})=(u^{\prime}_{1},\dots,u^{\prime}_{n})=F(v^{n,z})$ in $K_{2}$.
In general, if $v$ and $w$ realize the same triangulation except for one anti-
clockwise rotation, then there exists a permutation $\sigma^{\prime}\in S_{n}$
such that $\sigma^{\prime}(F(v))=F(w)$ in $K_{2}$.

######  Lemma 19.

Let $A$ and $B$ be a coupling of positive integral diamonds of Dynkin type
$\mathbb{A}_{n}$, and $v_{A}=(a_{1},\dots,a_{z},\dots,a_{n})$ a corresponding
associated vector with $a_{t}=1$ for $z\leq t\leq n$. If $v_{A}$ and $v_{B}$
realize the same triangulation except for one anti-clockwise rotation. Then
vectors

$\begin{split}v_{A+i}&=(a_{1},\dots,a_{i-1},a_{i-1}+a_{i},a_{i+1},\dots,a_{n-1})\quad\text{and}\\\
v_{B+i-1}&=(b_{1},\dots,b_{i-2},b_{i-2}+b_{i-1},b_{i-1},\dots,b_{n-1}),\end{split}$

realize the same triangulation except for one anti-clockwise rotation for
$z-1\leq i\leq n$, and $i\geq 2$.

Proof. Let $v_{A}$ and $v_{B}$ be associated vectors to the
$\mathbb{A}_{n}$-diamonds $A$ and $B$, respectively. Since $v_{A}$ and $v_{B}$
realize the same triangulation except for one anti-clockwise rotation then
there exists a permutation $\sigma\in S_{n}$ such that
$\sigma(F(v_{A}))=F(v_{B})$ in $K_{2}$. The following options arise from the
map $f$, such that:

  1. (1)

If $i>z\geq 1$, then

$\begin{split}f(v_{A})&=(\dots,\underbrace{1}_{i-1},\underbrace{1}_{i},\dots),\\\
f(v_{B})&=(\dots,\underbrace{d}_{i-2},\underbrace{2}_{i-1},\underbrace{2}_{i},\dots),\quad\text{
(see Figure \ref{figure4.5})}\end{split}$

    1. (1.1)

If $d=1$, then

$\begin{split}F(v_{A})&=(\dots,\underbrace{i}_{n-i},\underbrace{i-1}_{n+1-i},\dots),\\\
F(v_{B})&=(\dots,\underbrace{i-1}_{n-i},\underbrace{i-2}_{n+1-i},\dots),\end{split}$

and $\sigma$ satisfies the expression,

$\sigma(r)=\begin{cases}r,&\mbox{if $r\leq n+1-i$,}\\\
m,&\mbox{otherwise,}\end{cases}$

for some $m>n+1-i$. Applying $F$ to $v_{A+i}$ and $v_{B+i-1}$, it holds that

$\begin{split}F(v_{A+i})&=(\dots,\underbrace{i-1}_{n-i},\underbrace{i-1}_{n+1-i},\dots),\\\
F(v_{B+i-1})&=(\dots,\underbrace{i-2}_{n-i},\underbrace{i-2}_{n+1-i},\dots),\end{split}$

then there exits $\sigma^{\prime}\in S_{n}$ such that $\sigma^{\prime}=\sigma$
and $\sigma^{\prime}(F(v_{A+i}))=F(v_{B+i-1})$ in $K_{2}$ (see Figure 6).

${}_{{}_{n+1-i}}$iiid=1d=2iiid=1d=2
Figure 6: Dyck paths associated to vectors $v_{A}$, $v_{B}$ (left), $v_{A+i}$
and $v_{B+i-1}$ (right) for $i>z$.

    2. (1.2)

The case for $d=2$ is the same as the previous case.

    3. (1.3)

If $d=3$, then $A$ and $B$ do not realize the same triangulation.

Note that, if $z=1$, this case satisfies the condition (1.1) and (1.2) without
$d$.

  2. (2)

If $i=z\geq 2$, then

$\begin{split}f(v_{A})&=(\dots,\underbrace{2}_{i-1},\underbrace{1}_{i},\dots),\\\
f(v_{B})&=(\dots,\underbrace{b}_{i-2},\underbrace{a}_{i-1},\underbrace{2}_{i},\dots),\quad\text{
(see Figure \ref{figure4.7})}.\end{split}$

    1. (2.1)

If $a=1$ and $b=1$, then

$\begin{split}F(v_{A})&=(\dots,\underbrace{i}_{n-i},\dots),\\\
F(v_{B})&=(\dots,\underbrace{i-1}_{n-i},\underbrace{i-1}_{n+1-i},\dots),\end{split}$

and $\sigma_{1}$ is defined by the following cases:

$\sigma_{1}(r)=\begin{cases}r,&\mbox{if $r\leq n-i$,}\\\ n+1-i,&\mbox{if
$r=n$,}\\\ m,&\mbox{otherwise,}\end{cases}$

for some $m>n+1-i$. Applying $F$, we obtain

$\begin{split}F(v_{A+i})&=(\dots,\underbrace{i-1}_{n-i},\dots),\\\
F(v_{B+i-1})&=(\dots,\underbrace{i}_{n-i},\underbrace{i-2}_{n+1-i},\dots),\end{split}$

then there exits $\sigma^{\prime}_{1}\in S_{n}$ satisfying the following
cases:

$\sigma^{\prime}_{1}(r)=\begin{cases}n-i,&\mbox{if $r=n$,}\\\ n+1-i,&\mbox{if
$r=n-i$,}\\\ \sigma_{1}(r),&\mbox{otherwise,}\end{cases}$

therefore $\sigma^{\prime}_{1}(F(v_{A+i}))=F(v_{B+i-1})$ in $K_{2}$ (see
Figure 8).

${}_{{}_{n+1-i}}$iiiia=1a=1a=2b=1b=2b=3

Figure 7: Dyck paths associated to vectors $v_{A}$ and $v_{B}$ for $i=z$.

    2. (2.2)

If $a=1$ and $b=2$, then conditions defined in the case (2.1) hold.

    3. (2.3)

For $a=2$ and $b=1$ or $b=2$, we have only contradictions.

    4. (2.4)

If $a=2$ and $b=3$, $F(v_{B})=(\dots,\underbrace{i-1}_{n-i},\dots)$ and
$\sigma_{2}=\sigma$. Applying $F$ to $v_{B+i-1}$, it holds that
$F(v_{B+i-1})=(\dots,\underbrace{i-2}_{n-i},\dots)$ then there exits
$\sigma^{\prime}_{2}\in S_{n}$ such that $\sigma^{\prime}_{2}=\sigma$ and
$\sigma^{\prime}_{2}(F(v_{A+i}))=F(v_{B+i-1})$ in $K_{2}$ (see Figure 8).

    5. (2.5)

Case (2.3) holds for $a=3$ and $b=1,2,3$.

${}_{{}_{n+1-i}}$iiiia=1a=1a=2b=1b=2b=3

Figure 8: Dyck paths associated to vectors $v_{A+i}$ and $v_{B+i-1}$ for
$i=z$.

Note that, if $z=2$ then conditions for $a=1,2$ without $b$ hold.

  3. (3)

If $i=z-1\geq 3$, then

$\begin{split}f(v_{A})&=(\dots,\underbrace{2}_{i},\underbrace{1}_{i+1},\dots),\\\
f(v_{B})&=(\dots,\underbrace{b}_{i-1},\underbrace{a}_{i},\underbrace{2}_{i+1},\dots)\quad\text{
(see Figure \ref{figure4.9})}.\end{split}$

    1. (3.1)

If $a=1$ and $b=1$, then

$\begin{split}F(v_{A})&=(\dots,\underbrace{i+1}_{n-i-1},\dots),\\\
F(v_{B})&=(\dots,\underbrace{i}_{n-i-1},\underbrace{i}_{n-i},\underbrace{i-1}_{n+1-i},\dots),\quad\text{and}\\\
\end{split}$ $\sigma_{3}(r)=\begin{cases}r,&\mbox{if $r\leq n-i-1$,}\\\
n-i,&\mbox{if $r=n$,}\\\ n+1-i,&\mbox{if $r=n-1$,}\\\
m,&\mbox{otherwise,}\end{cases}$

for some $m>n+1-i$. Provided that

$\begin{split}F(v_{A+i})&=(\dots,\underbrace{i-1}_{n-i-1},\dots),\quad\text{
and }\\\
F(v_{B+i-1})&=(\dots,\underbrace{i+1}_{n-i-1},\underbrace{i}_{n-i},\underbrace{i-2}_{n+1-i},\dots),\end{split}$

then, there exits $\sigma^{\prime}_{3}\in S_{n}$ such that

$\sigma^{\prime}_{3}(r)=\begin{cases}n-i-1,&\mbox{if $r=n$,}\\\ n-i,&\mbox{if
$r=n-1$,}\\\ n+1-i,&\mbox{if $r=n-i-1$,}\\\
\sigma_{3}(r),&\mbox{otherwise,}\end{cases}$

then $\sigma^{\prime}_{3}(F(v_{A+i}))=F(v_{B+i-1})$ in $K_{2}$ (see Figure
10).

${}_{{}_{n+1-i}}$iiiia=1a=1a=2b=1b=2b=3

Figure 9: Dyck paths associated to vectors $v_{A}$ and $v_{B}$ for $i-1=z$.

    2. (3.2)

If $a=1$ and $b=2$. It holds that,

$\begin{split}F(v_{B})&=(\dots,\underbrace{i}_{n-i-1},\underbrace{i}_{n-i},\dots),\end{split}$

whereas, $\sigma_{4}$ is given by the identities

$\sigma_{4}(r)=\begin{cases}r,&\mbox{if $r\leq n-i-1$,}\\\ n-i,&\mbox{if
$r=n$,}\\\ m,&\mbox{otherwise,}\end{cases}$

for some $m>n-i$. Applying $F$ to $v_{B+i-1}$, we get

$\begin{split}F(v_{B+i-1})&=(\dots,\underbrace{i+1}_{n-i-1},\underbrace{i-2}_{n-i},\dots),\end{split}$

then there exists $\sigma^{\prime}_{4}$ with

$\sigma^{\prime}_{4}(r)=\begin{cases}n-i-1,&\mbox{if $r\leq n$,}\\\
n-i,&\mbox{if $r=n-i-1$,}\\\ \sigma_{4}(r),&\mbox{otherwise,}\end{cases}$

therefore $\sigma^{\prime}_{4}(F(v_{A+i}))=F(v_{B+i-1})$ in $K_{2}$ (see
Figure 10).

    3. (3.3)

If $a=2$ and $b=3$. $F(v_{B})=(\dots,\underbrace{i}_{n-i-1},\dots)$, provided
that

$\sigma_{5}(r)=\begin{cases}r,&\mbox{if $r\leq n-i-1$,}\\\
m,&\mbox{otherwise,}\end{cases}$

for some $m>n-i$. In this case,
$F(v_{B+i-1})=(\dots,\underbrace{i-2}_{n-i-1},\dots)$, and there is
$\sigma^{\prime}_{5}=\sigma_{5}$ such that
$\sigma^{\prime}_{5}(F(v_{A+i}))=F(v_{B+i-1})$ in $K_{2}$ (see Figure 10).

${}_{{}_{n+1-i}}$iiiia=1a=1a=2b=1b=2b=3

Figure 10: Dyck paths associated to vectors $v_{A+i}$ and $v_{B+i-1}$ for
$i-1=z$.

Same arguments are used for the remaining cases (see item (2) of this
proof).$\hfill\square$

######  Proposition 20.

Two positive integral diamonds of Dynkin type $\mathbb{A}_{n}$ are in the same
minimal $p$-cycle if their triangulations are in the same mutation class.

Proof. It is a direct consequence of Proposition 9, Lemmas 18 and
19$\hfill\square$

The following result gives a way to build frieze patterns.

######  Theorem 21.

Let $A^{0}$ be a positive integral diamond of Dynkin type $\mathbb{A}_{n}$ and
let $\\{A^{t}\\}_{0\leq t\leq p-1}$ be the minimal $p$-cycle generated by
$A^{0}$. Then:

  1. (i)

$A^{0}$ and $F(v_{A^{0}})$ generate the same frieze pattern (see
$\mathrm{(\ref{F})}$).

  2. (ii)

$\\{A^{t}\\}_{0\leq t\leq p-1}$ is in surjective correspondence with a direct
sum of $p$ indecomposable objects of a Dyck paths category.

Proof. Let $\mathbb{D}_{\mathbb{A}_{n}}$ be the set of all vectors associated
to positive integral diamonds of Dynkin type $\mathbb{A}_{n}$, $A^{0}$ a
positive integral diamond of Dynkin type $\mathbb{A}_{n}$, and
$\\{A^{t}\\}_{0\leq t\leq p-1}$ the minimal $p$-cycle generated by $A^{0}$.

  1. (i)

Let $K$ be a labeling of an $(n+3)$-polygon, Theorem 17 implies that

$\begin{array}[]{rcl}F(v_{A^{0}})&=&g((a^{0}_{11},T_{2}(v_{A^{0}}),\dots,T_{n}(v_{A^{0}})))\\\
&=&g(\lambda_{(a^{0}_{11},T_{2}(v_{A^{0}}),\dots,T_{n}(v_{A^{0}}))})\\\
&=&g((\lambda_{1},\dots,\lambda_{n+1-a^{0}_{11}},\underbrace{0,\dots,0}_{a^{0}_{11}}))\\\
&=&(l_{1}^{v_{1}},\dots,l_{n+1-a_{11}^{0}}^{v_{n+1-a^{0}_{11}}},l^{0}_{n-a^{0}_{11}},\dots,l^{0}_{n}),\end{array}$

then, there are $a^{0}_{11}-1$ diagonals from the vertex $0$ to other
vertices, i.e., there are $a^{0}_{11}$ triangles incident with vertex 0.
Proposition 20 allows us to establish that $a^{i}_{11}$ is the number of
triangles incident with the vertex $i$, for $1\leq i\leq n+3$, $i=pm$ and
$1\leq m\leq p\text{ }|\text{ }(n+3)$. Therefore $A^{0}$ and $F(v_{A^{0}})$
generate the same frieze pattern.

  2. (ii)

Let $(\mathfrak{D}_{2(n+1)},R)$ be any Dyck paths category, we take objects of
$(\mathfrak{D}_{2(n+1)},R)$ defined by the following identity

$\overline{Ob}\text{
}(\mathfrak{D}_{2n},R)=\bigg{\\{}\bigoplus_{G_{i}\in\mathfrak{D}_{2n}}G_{i}\text{
}\bigg{|}\text{ }g(\lambda_{G_{i}})\text{ and }g(\lambda_{G_{j}})\text{ are in
the same mutation class}\bigg{\\}},$

the map $\varphi:\mathbb{D}_{\mathbb{A}_{n}}\rightarrow\overline{Ob}\text{
}(\mathfrak{D}_{2n},R)$, such that

$\varphi(v_{A^{0}})=f(v_{A^{0}})\oplus\dots\oplus f(v_{A^{p-1}}),$

with $\\{A^{t}\\}_{0\leq t\leq p-1}$ is surjective as a consequence of Theorem
17 and Proposition 20. $\hfill\square$

$\oplus$$\oplus$ Figure 11: Examples of objects of a Dyck paths category.

As an example of the use of Theorem 21, we choose the object $D$ of a Dyck
paths category $(\mathfrak{D}_{2(n+1)},R)$ shown in Figure 11. Then $D$ has
associated the following frieze pattern where red numbers is a positive
integral diamond of Dynkin type $\mathbb{A}_{n}$.

$\textstyle{\dots}$$\textstyle{\color[rgb]{1,0,0}1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{\color[rgb]{1,0,0}1}$$\textstyle{1}$$\textstyle{\dots}$$\textstyle{\dots}$$\textstyle{\color[rgb]{1,0,0}1}$$\textstyle{{\color[rgb]{1,0,0}3}}$$\textstyle{2}$$\textstyle{{\color[rgb]{1,0,0}1}}$$\textstyle{{\color[rgb]{1,0,0}3}}$$\textstyle{2}$$\textstyle{\dots}$$\textstyle{\dots}$$\textstyle{{\color[rgb]{1,0,0}2}}$$\textstyle{{\color[rgb]{1,0,0}5}}$$\textstyle{1}$$\textstyle{\color[rgb]{1,0,0}2}$$\textstyle{{\color[rgb]{1,0,0}5}}$$\textstyle{\dots}$$\textstyle{\dots}$$\textstyle{1}$$\textstyle{{\color[rgb]{1,0,0}3}}$$\textstyle{{\color[rgb]{1,0,0}2}}$$\textstyle{1}$$\textstyle{{\color[rgb]{1,0,0}3}}$$\textstyle{{\color[rgb]{1,0,0}2}}$$\textstyle{\dots}$$\textstyle{\dots}$$\textstyle{1}$$\textstyle{\color[rgb]{1,0,0}1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{\color[rgb]{1,0,0}1}$$\textstyle{\dots}$

## References

Agustín Moreno Cañadas
<EMAIL_ADDRESS>
Department of Mathematics
Universidad Nacional de Colombia
Bogotá-Colombia

Isaías David Marín Gaviria
<EMAIL_ADDRESS>
Department of Mathematics
Universidad Nacional de Colombia
Bogotá-Colombia

Gabriel Bravo Rios
<EMAIL_ADDRESS>
Department of Mathematics
Universidad Nacional de Colombia
Bogotá-Colombia

Pedro Fernando Fernández Espinosa
<EMAIL_ADDRESS>
Department of Mathematics
Universidad Nacional de Colombia
Bogotá-Colombia