# On Dyck Path Expansion Formulas for Rank 2 Cluster Variables

Amanda Burcroff

###### Abstract.

In this paper, we simplify and generalize formulas for the expansion of rank
$2$ cluster variables. In particular, we prove an equivalent, but simpler,
description of the colored Dyck subpaths framework introduced by Lee and
Schiffler. We then prove the conjectured bijectivity of a map constructed by
Feiyang Lin between collections of colored Dyck subpaths and compatible pairs,
objects introduced by Lee, Li, and Zelevinsky to study the greedy basis. We
use this bijection along with Rupel’s expansion formula for quantum greedy
basis elements, which sums over compatible pairs, to provide a quantum
generalization of Lee and Schiffler’s colored Dyck subpaths formula.

###### Contents

  1. 1 Introduction
  2. 2 Statement of Results
  3. 3 Preliminaries
  4. 4 Simplification of the Colored Dyck Subpaths Conditions
    1. 4.1 Lee-Schiffler Expansion Formula
    2. 4.2 Vertices and Slopes in $\mathcal{D}_{n}$
    3. 4.3 Proof of the Simplification
  5. 5 Bijection between Compatible Pairs and Colored Subpaths of Dyck Paths
    1. 5.1 Compatible Pairs and Lin’s Map
    2. 5.2 Proof of Bijectivity
  6. 6 Quantization of Colored Dyck Subpaths
  7. 7 Further Directions
  8. 8 Acknowledgements

##  1\. Introduction

The theory of _cluster algebras_ , introduced twenty years ago by Fomin and
Zelevinsky [7], gives us a combinatorial framework for understanding the
previously opaque nature of certain algebras. Each cluster algebra is
generated by its _cluster variables_ , which can be obtained via the recursive
process of _mutation_. The _Laurent phenomenon_ says that each cluster
variable in a rank-$n$ cluster algebra can be expressed as a Laurent
polynomial in the $n$ _initial_ cluster variables. While in general finding
explicit formulas for the Laurent expansions of arbitrary cluster variables is
difficult, there has been significant progress in understanding the expansions
in low-rank cluster algebras. In this work, we attempt to unify and simplify
some of the existing expansion formulas for rank-$2$ cluster variables and
their quantum generalizations.

In 2011, Lee and Schiffler provided the first combinatorial formula for the
Laurent expansion of arbitrary skew-symmetric rank-$2$ cluster variables [12,
Theorem 9]. They expressed the coefficients as sums over certain collections
of non-overlapping _colored subpaths_ of a _maximal Dyck path_. This
established the positivity of the Laurent expansion in skew-symmetric rank $2$
cluster algebras. Lee and Schiffler [13, Theorem 11] and Rupel [18, Theorem 6]
then generalized this formula (in the skew-symmetric and skew-symmetrizable
cases, respectively) to the non-commutative rank-2 setting, giving each
collection a weight expressed as an ordered product of two non-commuting
initial cluster variables. In 2012, Lee, Li, and Zelevinsky [11] defined the
_greedy basis_ for rank-$2$ cluster algebras, which includes the cluster
variables. They provided a combinatorial formula for the Laurent expansion of
each greedy basis elements as a sum over _compatible pairs_ , certain
collections of edges of a maximal Dyck path [11, Theorem 11]. Rupel later gave
a non-commutative analogue of this formula, which specializes to a formula for
the coefficients in the _quantum rank- $2$ cluster algebra_ setting [19,
Corollary 5.4]. In particular, each compatible pair is weighted by a
corresponding power of a quantum parameter $q$, where the exponent is computed
as a sum over all pairs of edges in the maximal Dyck path. Rupel [20, Theorem
1.2] has also provided a quantum analogue to the Caldero-Chapoton expansion
formula [2] for rank-$2$ cluster variables expressed as a sum over
indecomposable valued-quiver representations.

To summarize, there are two different combinatorial formulas for the Laurent
expansion of skew-symmetric rank-$2$ cluster variables: one in terms of
collections of colored Dyck subpaths [12] and the other in terms of compatible
pairs [11]. The combinatorics of collections of colored Dyck subpaths and
compatible pairs are somewhat similar, suggesting that there might be a nice
correspondence between them. A correspondence was known to Lee, Li, and
Zelevinsky [11] and they suggested in their 2012 paper that they planned to
provide details in the future, but this has not yet appeared in the
literature. In 2021, Feiyang Lin constructed a map between a superset of the
collections of colored Dyck subpaths and compatible pairs and conjectured that
the map restricts to a bijection in [16, Conjecture 3]. Lin made partial
progress toward proving this, reducing the conjecture to a technical statement
[16, Conjecture 4].

In this work, we start by providing a simplification of Lee-Schiffler’s
formula for rank-$2$ cluster variables in terms of colored Dyck subpath
conditions. We then use our simpler formula to prove Lin’s conjectures [16,
Conjectures 3 & 4] that the map constructed between collections of colored
Dyck subpaths and compatible pairs is indeed a bijection. (Our methods do not
rely on the technical reformulation presented by Lin.) This bijection gives an
efficient method for generating all compatible pairs in the cluster variable
case. We then use the bijection along with Rupel’s quantum weighting of
compatible pairs [19] to provide a quantum version of Lee and Schiffer’s
rank-$2$ expansion formula for cluster variables. This new formula has the
advantage of requiring less computation than that in [19, Corollary 5.4] and
explicitly calculating the coefficients in the quantum case, rather than
expressing each term as an ordered product as in [18]. It is also more
elementary than the expansion formula in [20], which is based on the theory of
valued quiver representations.

The paper is organized as follows. In Section 2, we give an overview of the
results. Section 3 contains some preliminaries concerning maximal Dyck paths.
The proof of the simplification of Lee and Schiffler’s [12] colored Dyck
subpath conditions is the focus of Section 4. Section 5 contains the proof of
Lin’s conjectures [16, Conjectures 3 & 4], establishing a bijection between
collections of colored Dyck subpaths from the Lee-Schiffler [12] setting and
compatible pairs from the Lee-Li-Zelevinsky [11] setting. This bijection is
applied to Rupel’s [19] quantum weighting on compatible pairs to yield a
quantum analogue of Lee-Schiffler’s [12] expansion formula in Section 6. We
conclude with a discussion of further directions in Section 7.

##  2\. Statement of Results

For a positive integer $r$ and variables $X_{1},X_{2}$, we consider the
sequence $\\{X_{n}\\}_{n\in\mathbb{Z}}$ of expressions recursively defined by

(2.1) $X_{n+1}=\frac{X_{n}^{r}+1}{X_{n-1}}\,.$

This sequence is precisely the set of variables of the rank-$2$ cluster
algebra $\mathcal{A}(r,r)$ associated to the $r$-Kronecker quiver, which
consists of two vertices with $r$ arrows between them. The sequence is
periodic when $r=1$, and otherwise all $X_{n}$ are distinct. For background on
cluster algebras, see [6].

######  Definition 2.1.

The _maximal Dyck path_ $\mathcal{P}(a,b)$ is the path proceeding by unit
north and east steps from $(0,0)$ to $(a,b)$ that is closest to the line
segment between $(0,0)$ and $(a,b)$ without crossing strictly above it. For
two vertices $u,w$ along such a path, let $s(u,w)$ denote the slope of the
line segment between them.

Let $\\{c_{n}\\}_{n=1}^{\infty}$ be the sequence of non-negative integers
defined recursively by:

(2.2) $c_{1}=0,c_{2}=1,\text{ and }c_{n}=rc_{n-1}-c_{n-2}\text{ for }r\geq
2\,.$

Let $\mathcal{C}_{n}=\mathcal{P}(c_{n-1},c_{n-2})$ and
$\mathcal{D}_{n}=\mathcal{P}(c_{n-1}-c_{n-2},c_{n-2})$. We label the leftmost
vertex at each height of $\mathcal{D}_{n}$ by $v_{i}$ for $i=0,\dots,c_{n-2}$,
with the subindex increasing from south to north; such vertices are called
_northwest corners_. Let $\gamma(i,k)$ be the subpath spanning from $v_{i}$ to
$v_{k}$ for any $0\leq i<k\leq b$. An example of the maximal Dyck path
$\mathcal{D}_{5}=\mathcal{P}(5,3)$ is shown in Figure 1.

######  Theorem 2.2.

Let $t(i)$ be the minimum integer greater than $i$ such that
$s(v_{i},v_{t(i)})>s$. Then we have $t(i)-i=c_{m}-wc_{m-1}$ for a unique
choice of $2\leq w\leq r-1$ and $3\leq m\leq n-2$.

This result allows us to simplify the expansion formula of Lee and Schiffler,
which is briefly described below. The next few definitions and 2.4 emulate the
results of Lee and Schiffler, except for slight modifications due to the
simplified coloring conditions above.

######  Definition 2.3 (cf. 4.1).

For any $0\leq i<k\leq c_{n-2}$, let $\gamma(i,k)$ be the subpath of
$\mathcal{D}_{n}$ from $v_{i}$ to $v_{k}$, which is assigned a color as
follows:

  1. (1)

If $s(v_{i},v_{t})\leq s$ for all $t$ such that $i<t\leq k$, then
$\gamma(i,k)$ is  blue.

  2. ($2^{*}$)

Otherwise, let $m,w$ be chosen with respect to $i$ as in 2.2. Then we say
$\gamma(i,k)$ is  $(m,w)$-brown.111The color brown was chosen because it is
the combination of Lee and Schiffler’s red and green cases.

A _subpath_ of $\mathcal{D}_{n}$ is a path of the form $\gamma(i,k)$ or a
single edge $\alpha_{i}$. We denote the set of such subpaths by
$\mathcal{P}^{\prime}(\mathcal{D}_{n})$. We define the set
$\mathcal{F}^{\prime}(\mathcal{D}_{n})$ to contain any collection of subpaths
in $\mathcal{P}^{\prime}(\mathcal{D}_{n})$ satisfying that no two subpaths
share an edge, two subpaths share a vertex only if at least one of them is a
single edge, and at least one of the $c_{m-1}-2c_{m-2}$ edges preceding each
$(m,w)$-brown subpath is contained in another subpath. Given
$\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{n})$, the quantity $|\beta|_{1}$ is
defined additively over the subpaths, taking value $k-i$ on $\gamma(i,k)$ and
value $0$ on single edges. The quantity $|\beta|_{2}$ is the total number of
edges in $\beta$. This yields the following expansion formula for the cluster
variables.

######  Corollary 2.4 (analogue of [12, Theorem 9]).

Consider the cluster algebra $\mathcal{A}(r,r)$ with cluster variables $X_{i}$
for $i\in\mathbb{Z}$. For $n\geq 4$, we have

$X_{n}=X_{1}^{-c_{n-1}}X_{2}^{-c_{n-2}}\sum_{\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{n})}X_{1}^{r|\beta|_{1}}X_{2}^{r\left(c_{n-1}-|\beta|_{2}\right)}$

and

$X_{3-n}=X_{2}^{-c_{n-1}}X_{1}^{-c_{n-2}}\sum_{\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{n})}X_{2}^{r|\beta|_{1}}X_{1}^{r\left(c_{n-1}-|\beta|_{2}\right)}\,.$

A generalization of the above expansion result to the case of skew-symmetric
rank-$2$ cluster algebras with coefficients is presented at the end of
Subsection 4.3 (see 4.18).

######  Example 2.5.

The collection of colored subpaths
$\beta=\\{\gamma(0,1),\alpha_{6},\gamma(2,3)\\}$ is in
$\mathcal{F}^{\prime}(\mathcal{D}_{5})$. The subpath $\gamma(0,1)$ is blue,
while the subpath $\gamma(2,3)$ is $(3,2)$-brown. This collection is depicted
in Figure 1 for $r=3$, where the single edge $\alpha_{6}$ is represented by an
orange edge. Note that in this case we have $|\beta|_{1}=2$ and
$|\beta|_{2}=6$.

We now discuss the bijection between colored Dyck subpaths of
$\mathcal{D}_{n}$ and compatible pairs in $\mathcal{C}_{n}$. Given two
vertices $u,w$ in a maximal Dyck path $\mathcal{P}(a,b)$, let
$\overrightarrow{uw}$ denote the subpath proceeding east from $u$ to $w$,
continuing cyclically around $\mathcal{P}(a,b)$ if $u$ is to the east of $w$.
Let $|uw|_{1}$ (resp., $|uw|_{2}$) denote the number of horizontal (resp.,
vertical) edges of $\overrightarrow{uw}$. Given a set of horizontal edges
$S_{1}$ and vertical edges $S_{2}$ in $\mathcal{P}(a,b)$, the pair
$(S_{1},S_{2})$ is _compatible_ if, for every edge in $S_{1}$ with left vertex
$u$ and every edge $S_{2}$ with top vertex $w$, there exists a lattice point
$t\neq u,w$ in the subpath $\overrightarrow{uw}$ such that

$|tw|_{1}=r|\overrightarrow{tw}\cap S_{2}|_{2}\text{ or
}|ut|_{2}=r|\overrightarrow{ut}\cap S_{1}|_{1}\,.$

Let the horizontal (resp., vertical) edges of $\mathcal{C}_{n}$ be labeled by
$\eta_{i}$ (resp., $\nu_{i}$), increasing to the east (resp., north). Lin
defined the following map $\Phi$ from $\mathcal{F}^{\prime}(\mathcal{D}_{n})$
to pairs $(S_{1},S_{2})$ in $\mathcal{C}_{n}$. Note that while we define
$\Phi$ as a map on blue/brown colored subpaths, it was originally defined for
Lee-Schiffler’s blue/green/red colored subpaths, and the two definitions are
essentially identical.

Figure 1. The leftmost image shows the maximal Dyck path $\mathcal{D}_{5}$ for
$r=3$, along with the corresponding $5\times 3$ grid and main diagonal. The
northwest corners are labeled and depicted as filled vertices. The center
image is a collection $\beta$ of colored Dyck subpaths in
$\mathcal{F}^{\prime}(\mathcal{D}_{5})$, and the rightmost image is the
compatible pair on $\mathcal{C}_{5}$ that $\beta$ maps to under $\Phi$, where
an edge is thickened whenever it is included in the compatible pair.

######  Definition 2.6 ([16]).

Given $\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{n})$, let
$\Phi(\beta)=(\Phi_{1}(\beta),\Phi_{2}(\beta))$, where

$\displaystyle\Phi_{1}(\beta)$ $\displaystyle=\\{\eta_{s}:\alpha_{s}\text{ is
not a part of any subpath of }\beta\\}\,,$ $\displaystyle\Phi_{2}(\beta)$
$\displaystyle=\\{\nu_{s}:\gamma(i,k)\in\beta\text{ for some }i<s\leq k\\}\,.$

######  Example 2.7.

The compatible pair
$\left(\\{\eta_{4},\eta_{5}\\},\\{\nu_{1},\nu_{3}\\}\right)$ obtained by
applying $\Phi$ to the collection of subpaths
$\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{5})$ from 2.5 is shown in Figure 1.

######  Theorem 2.8.

The map $\Phi$ is a bijection between collections of colored subpaths in
$\mathcal{F}^{\prime}(\mathcal{D}_{n})$ and compatible pairs in
$\mathcal{C}_{n}$.

Switching to the quantum setting, we will now work inside the quantum torus
$\mathcal{T}:=\mathbb{Z}[q^{\pm 1}]\langle Z_{1}^{\pm 1},Z_{2}^{\pm
1}:Z_{1}Z_{2}=q^{2}Z_{2}Z_{1}\rangle$. The quantum rank-$2$ $r$-Kronecker
cluster algebra $\mathcal{A}_{q}(r,r)$ is the $\mathbb{Z}[q^{\pm 1}]$
subalgebra of the skew field of fractions of $\mathcal{T}$ generated by the
quantum cluster variables $\\{Z_{n}\\}_{n\in\mathbb{Z}}$, which follow the
recursion $Z_{n+1}Z_{n-1}=q^{-r}Z_{n}^{r}+1$ (cf. Equation 2.1). We use the
bijectivity of $\Phi$ along with Rupel’s quantum weighting on compatible pairs
[19] to construct a quantum weighting of collections of colored subpaths.

For
$\beta=\\{\beta_{1},\dots,\beta_{t}\\}\in\mathcal{F}^{\prime}(\mathcal{D}_{n})$,
where $\beta_{i}$ appears to the left of $\beta_{i+1}$, we define the set of
_complimentary subpaths_
$\overline{\beta_{0}},\overline{\beta_{1}},\dots,\overline{\beta_{t}}$ such
that $\overline{\beta_{i}}$. For $1\leq i\leq t-1$, let $\overline{\beta_{i}}$
contain all edges and northwest corners between the end of $\beta_{i}$ and the
start of $\beta_{i+1}$, where a northwest corner on the boundary of a path is
included in $\overline{\beta_{i}}$ unless it is the right endpoint of a brown
or blue subpath. We set $\overline{\beta_{0}}$ to be the portion of the path
before $\beta_{1}$ excluding $v_{0}$, and we set $\overline{\beta_{t}}$ to be
the portion of the path after $\beta_{t}$ including $v_{c_{n-2}}$. Note that
it is possible for some $\overline{\beta_{i}}$ to be empty. Let
$|\overline{\beta_{i}}|_{1}$ (resp., $|\overline{\beta_{i}}|_{2}$) denote the
number of northwest corners (resp., edges) in $\overline{\beta_{i}}$.

######  Definition 2.9.

For
$\beta=\\{\beta_{1},\dots,\beta_{t}\\}\in\mathcal{F}^{\prime}(\mathcal{D}_{n})$,
we let

$w_{q}(\beta)=(c_{n-1}+c_{n-2}-1)+\sum_{j=0}^{t}r|\overline{\beta_{j}}|_{2}\left(\sum_{i=1}^{t}(-1)^{\mathbbm{1}_{i<j}}|\beta_{i}|_{2}\right)+\left(r|\overline{\beta_{j}}|_{1}-r^{2}|\overline{\beta_{j}}|_{2}\right)\left(\sum_{i=1}^{t}(-1)^{\mathbbm{1}_{i<j}}|\beta_{i}|_{1}\right),$

where $\mathbbm{1}_{i<j}$ takes value $1$ when $i<j$ and $0$ otherwise. We
then set

$u_{q}(\beta)=w_{q}(\beta)-(c_{n-1}+c_{n-2}-1)+(c_{n-1}-r|\beta|_{1})(c_{n}-r|\beta|_{2})\,.$

######  Example 2.10.

For the collection $\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{5})$ from 2.5,
we have $\overline{\beta_{0}}=\overline{\beta_{3}}=\emptyset$,
$\overline{\beta_{1}}=\\{\alpha_{4},\alpha_{5}\\}$, and
$\overline{\beta_{2}}=\\{v_{2}\\}$. We thus have $w_{q}(\beta)=10$.

We prove that the quantum cluster variable Laurent coefficients can be
expressed as a sum over weighted collections of subpaths in
$\mathcal{F}^{\prime}(\mathcal{D}_{n})$.

######  Theorem 2.11.

Consider the quantum cluster algebra $\mathcal{A}_{q}(r,r)$ with quantum
cluster variables $Z_{i}$ for $i\in\mathbb{Z}$. For $n\geq 4$, we have

$Z_{n}=Z_{1}^{-c_{n-1}}Z_{2}^{-c_{n-2}}\sum_{\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{n})}q^{u_{q}(\beta)}Z_{1}^{r|\beta|_{1}}Z_{2}^{r\left(c_{n-1}-|\beta|_{2}\right)}$

and

$Z_{3-n}=Z_{2}^{-c_{n-1}}Z_{1}^{-c_{n-2}}\sum_{\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{n})}q^{u_{q}(\beta)}Z_{2}^{r|\beta|_{1}}Z_{1}^{r\left(c_{n-1}-|\beta|_{2}\right)}\,.$

##  3\. Preliminaries

Let $r\geq 2$ be fixed throughout this paper. Both the Lee-Schiffler and Lee-
Li-Zelevinsky expansion formulas involve sums over certain collections of
edges in a maximal Dyck path. Moreover, the width and height of these Dyck
paths have certain recursive properties that are used in the proofs of both
formulas. We begin by setting up a framework for studying these paths and
describing their recursive behavior.

Recall the sequence $c_{n}$ defined recursively by Equation 2.2. While the
indexing of the sequence $\\{c_{n}\\}_{n\geq 1}$ is identical to the indexing
in the work of Lee and Schiffler [12], the indexing is shifted by one from
that defined by Lin [16], i.e., it is equivalent to $\\{c_{n-1}\\}_{n\geq 1}$
in Lin’s work. It is straightforward to check that for $n>1$, the quantities
$c_{n}$ and $c_{n+1}$ are relatively prime, hence so are $c_{n}$ and
$c_{n+1}-c_{n}$. Thus the only vertices of $\mathcal{D}_{n}$ and
$\mathcal{C}_{n}$ that lie on the main diagonal are the first and last.

Fix $a,b\in\mathbb{N}$. Consider a rectangle with vertices $(0,0)$, $(0,b)$,
$(a,0)$, and $(a,b)$ having a designated diagonal from $(0,0)$ to $(a,b)$.

######  Definition 3.1.

A _Dyck path_ is a lattice path in $\mathbb{Z}^{2}$ starting at $(0,0)$ and
ending at a lattice point $(a,b)$ where $a,b\geq 0$, proceeding by only unit
north and east steps and never passing strictly above the diagonal. Given a
Dyck path $P$, we denote the number of east steps by $|P|_{1}$ and the number
of north steps by $|P|_{2}$. The _length_ of the Dyck path $P$ is the quantity
$|P|_{1}+|P|_{2}$. We denote the set of lattice points contained in the Dyck
path $P$, including the left and right endpoints, by $V(P)$.

The Dyck paths from $(0,0)$ to $(a,b)$ form a partially ordered set by
comparing the heights at all vertices. The maximal Dyck path
$\mathcal{P}(a,b)$, as defined in 2.1, is the maximal element under this
partial order. We focus on the following two classes of maximal Dyck paths,
defined for $n\geq 3$, $\mathcal{C}_{n}=\mathcal{P}(c_{n-1},c_{n-2})$ and
$\mathcal{D}_{n}=\mathcal{P}(c_{n-1}-c_{n-2},c_{n-2})$.

Recall that a vertex of a maximal Dyck path $P$ is a _northwest corner_ if
there are no vertices directly north of (equivalently, to the east of) it. In
$\mathcal{D}_{n}$, these are precisely the vertices labeled by $v_{i}$ for
some $0\leq i\leq c_{n-2}$.

Figure 2. The maximal Dyck paths $\mathcal{D}_{6}$ (above) and
$\mathcal{C}_{6}$ (below) are shown with some of their vertex and edge labels
for $r=3$. Each northwest corner of $\mathcal{D}_{6}$ is labeled with both its
corresponding $w_{i}$ and $v_{j}$ label. Some edges of $\mathcal{C}_{6}$ are
labeled; the $\nu_{i}$’s refer to the vertical edge left of the label, and the
$\eta_{j}$’s refer to the horizontal edge below the label.

When $a$ and $b$ are relatively prime, as is the case for $\mathcal{C}_{n}$
and $\mathcal{D}_{n}$, we can associate to this Dyck path the _(lower)
Christoffel word_ of slope $\frac{b}{a}$ on the alphabet $\\{E,N\\}$. This
word can be constructed by reading the edges of the maximal Dyck path from
$(0,0)$ to $(a,b)$, recording an $E$ for each east step and an $N$ for each
north step. For further background on Christoffel words, see [1].

######  Example 3.2.

Let $r=3$. The Christoffel words corresponding to the maximal Dyck paths
$\mathcal{D}_{6}$ and $\mathcal{C}_{6}$ depicted in Figure 2 are

$E^{2}NE^{2}NENE^{2}NE^{2}NENE^{2}NEN\text{ and
}E^{3}NE^{3}NE^{2}NE^{3}NE^{3}NE^{2}NE^{3}NE^{2}N\,,$

respectively.

######  Remark 3.3.

The Christoffel word corresponding to $\mathcal{C}_{n}$ is obtained by
applying the morphism $\theta=\\{N\mapsto EN\\}$, i.e., the map that replaces
each instance of the letter $N$ with the string $EN$, to the Christoffel word
corresponding to $\mathcal{D}_{n}$. This follows directly from, for example,
[1, Lemma 2.2].

######  Observation 3.4.

It is straightforward to calculate that

$V(\mathcal{C}_{3})=\\{(0,0),(1,0)\\}\text{ and
}V(\mathcal{C}_{4})=\\{(0,0),(1,0),\dots,(r,0),(r,1)\\}$

As we shall see, both of these families of Dyck paths have a recursive
structure. The following lemma is a special case of a result of Rupel.

######  Lemma 3.5 ([18, Lemma 3]).

For all $n\geq 4$, the maximal Dyck path $\mathcal{D}_{n}$ consists of $r-1$
copies of $D_{n-1}$ followed by a copy of $D_{n-1}$ with a prefix $D_{n-2}$
removed. In particular, $\mathcal{D}_{n-1}$ (resp., $\mathcal{C}_{n-1}$) is a
subpath of $\mathcal{D}_{n}$ (resp., $\mathcal{C}_{n}$).

This allows us to define the limit of these paths, which can be realized by
taking a union of finite subpaths.

######  Definition 3.6.

Let $\mathcal{C}$ (resp., $\mathcal{D}$) be the infinite path on
$\mathbb{Z}^{2}$ formed by the union $\bigcup_{n\geq 3}\mathcal{C}_{n}$
(resp., $\bigcup_{n\geq 3}\mathcal{D}_{n}$). We identify the paths
$\mathcal{C}_{n}$ and $\mathcal{D}_{n}$ with the prefix of the same length of
$\mathcal{C}$ and $\mathcal{D}$, respectively. Thus, the vertices of
$\mathcal{D}$ are labeled by $w_{i}$ and the northwest corners by $v_{j}$, as
described after 2.1 in Section 2. Similarly, horizontal edges of $\mathcal{C}$
are labeled by $\eta_{i}$ and the vertical edges are labeled by $\nu_{j}$.

##  4\. Simplification of the Colored Dyck Subpaths Conditions

###  4.1. Lee-Schiffler Expansion Formula

We first recall the original expansion formula given by Lee and Schiffler [12]
for rank-two skew-symmetric cluster variables. This requires us to set up the
language of colored subpaths in a Dyck path via Lee and Schiffler’s
conventions, which differs from that in Section 2. We then describe the map
between certain non-overlapping collections of colored subpaths, namely from
$\mathcal{F}(\mathcal{D}_{n})$ as defined by Lee-Schiffler to the set
$\mathcal{F}^{\prime}(\mathcal{D}_{n})$ which we defined after 2.3 in Section
2.

Let $s$ denote the slope of the main diagonal of $\mathcal{D}_{n}$, so
$s=\frac{c_{n-2,r}}{c_{n-1,r}-c_{n-2,r}}$.

######  Definition 4.1 ([12], cf. 2.3).

For any $0\leq i<k\leq c_{n-2}$, let $\alpha(i,k)$ be the subpath of
$\mathcal{D}_{n}$ defined as follows:

  1. (1)

If $s(v_{i},v_{t})\leq s$ for all $t$ such that $i<t\leq k$, then
$\alpha(i,k)$ is defined to be the subpath from $v_{i}$ to $v_{k}$; each such
subpath is called  blue.

  2. (2)

If $s(v_{i},v_{t})>s$ for some $i<t\leq k$, then

    1. (2-a)

if the smallest such $t$ is of the form $i+c_{m}-wc_{m-1}$ for some $3\leq
m\leq n-2$ and $1\leq w\leq r-2$, then $\alpha(i,k)$ is defined to be the
subpath from $v_{i}$ to $v_{k}$; each such subpath is called  $(m,w)$-green.

    2. (2-b)

otherwise, $\alpha(i,k)$ is set to be the subpath from the vertex immediately
below $v_{i}$ to $v_{k}$; each such subpath is called  red.

Each such pair $i,k$ corresponds to precisely one subpath of
$\mathcal{D}_{n}$. Denote the single edges of $\mathcal{D}_{n}$ be by
$\alpha_{1},\dots,\alpha_{c_{n-1}}$ proceeding from southwest to northeast,
and let

$\mathcal{P}(\mathcal{D}_{n})=\\{\alpha(i,k):0\leq i<k\leq
c_{n-2}\\}\cup\\{\alpha_{1},\dots,\alpha_{c_{n-1}}\\}\,.$

The formula involves sums over collections of subsets of
$\mathcal{P}(\mathcal{D}_{n})$ satisfying certain non-overlapping
requirements. In particular, Lee and Schiffler set

$\displaystyle\mathcal{F}(\mathcal{D}_{n})=\\{\\{\beta_{1},\dots,\beta_{t}\\}:\;$
$\displaystyle t\geq 0,\beta_{j}\in\mathcal{P}(\mathcal{D}_{n})\text{ for all
}1\leq j\leq t,$ if $j\neq j^{\prime}$ then $\beta_{j}$ and
$\beta_{j^{\prime}}$ have no common edge, if $\beta_{j}=\alpha(i,k)$ and
$\beta_{j^{\prime}}=\alpha(i^{\prime},k^{\prime})$ then $i\neq k^{\prime}$ and
$i^{\prime}\neq k$, and if $\beta_{j}$ is $(m,w)$-green then at least one of
the $(c_{m-1}-wc_{m-2})$ $\displaystyle\;\;\;\;\;\;\;\;\text{preceding edges
of $v_{i}$ is contained in some $\beta_{j^{\prime}}$}\\}\,.$

For any collection of subpaths $\beta$, we associate two non-negative integers
$|\beta|_{1}$ and $|\beta|_{2}$. The first quantity $|\beta|_{1}$ is defined
to be $0$ on single edges and $k-i$ on $\alpha(i,k)$, then extended additively
on unions of these subpaths. The second quantity $|\beta|_{2}$ is the total
number of edges $\alpha_{i}$ covered by the subpaths in $\beta$. We can now
state the original formulation of Lee and Schiffler’s expansion result.

######  Theorem 4.2 ([12, Theorem 9]).

For $n\geq 4$, we have

$X_{n}=X_{1}^{-c_{n-1}}X_{2}^{-c_{n-2}}\sum_{\beta\in\mathcal{F}(\mathcal{D}_{n})}X_{1}^{r|\beta|_{1}}X_{2}^{r\left(c_{n-1}-|\beta|_{2}\right)}$

and

$X_{3-n}=X_{2}^{-c_{n-1}}X_{1}^{-c_{n-2}}\sum_{\beta\in\mathcal{F}(\mathcal{D}_{n})}X_{2}^{r|\beta|_{1}}x_{1}^{r\left(c_{n-1}-|\beta|_{2}\right)}\,.$

In order to show that our expansion formula 2.4 is equivalent to 4.2, we
define the map $\chi$ which connects the colored subpaths in both settings.

######  Definition 4.3.

We define a map
$\chi:\mathcal{F}(\mathcal{D}_{n})\to\mathcal{F}^{\prime}(\mathcal{D}_{n})$
that modifies the colored subpaths of $\beta\in\mathcal{F}(\mathcal{D}_{n})$
via the following rules:

  * •

replace each red subpath in $\beta$ with the two subpaths obtained by
splitting into its first edge and the remainder of the path,

  * •

change the color of $(m,w)$-green subpaths to become $(m,w)$-brown subpaths

  * •

the blue subpaths and single edges remain unchanged.

Figure 3. The top image depicts a collection of colored Dyck subpaths in
$\mathcal{F}(\mathcal{D}_{6})$, namely,
$\\{\alpha(0,1),\alpha_{4},\alpha(2,5),\alpha_{16},\alpha(6,8)\\}$. The bottom
image depicts the corresponding collection of colored Dyck subpaths in
$\mathcal{F}^{\prime}(\mathcal{D}_{6})$, namely
$\\{\gamma(0,1),\alpha_{4},\alpha_{6},\gamma(2,5),\alpha_{16},\gamma(6,8)\\}$,
obtained by applying the map $\chi$ to the top collection.

Note that the set of edges covered by $\beta$ is preserved under $\chi$. 4.17
establishes that $\chi$ is well-defined and is in fact a weight-preserving
bijection with respect to $|\beta|_{1}$ and $|\beta|_{2}$. The statement and
proof of 4.17 appear in Subsection 4.3.

###  4.2. Vertices and Slopes in $\mathcal{D}_{n}$

We now prove several results relating to position of vertices in
$\mathcal{D}_{n}$ and the slopes of the line segments between northwest
corners of $\mathcal{D}_{n}$. To help illuminate the recursive structure of
the infinite path $\mathcal{D}$, which contains each $\mathcal{D}_{n}$ as a
prefix, we define a map taking vertices to northwest corners in $\mathcal{D}$.

######  Definition 4.4.

Let $\mu:V(\mathcal{D})\to V(\mathcal{D})$ be the map sending $w_{i}$, the
$i^{\text{th}}$ vertex of $\mathcal{D}$, to $v_{i}$, the $i^{\text{th}}$
northwest corner of $\mathcal{D}$.

The following result describes the behavior of $\mu$ in terms of coordinates.

######  Lemma 4.5.

If $w_{i}\in V(\mathcal{D})$ has coordinates $(x,y)$, then
$\mu(w_{i})=\left((r-1)x+(r-2)y,x+y\right)$.

###### Proof.

Fix $n$ large enough such that $(x,y)\in V(\mathcal{D}_{n})$. For each
$(x,y)\in V(\mathcal{D}_{n})$, the claim is equivalent to showing that the
following inequalities hold:

$\frac{x+y}{(r-1)x+(r-2)y}\leq\frac{c_{n-1}}{c_{n}-c_{n-1}}<\frac{x+y+1}{(r-1)x+(r-2)y}\,.$

In order to prove the first inequality, we first note that

$\frac{c_{n-2}}{c_{n-1}-c_{n-2}}\geq\frac{y}{x}\,,$

which holds since $(x,y)\in V(\mathcal{D}_{n})$. Cross multiplying and adding
$yc_{n-2}$ to both sides yields

$\frac{c_{n-2}}{c_{n-1}}\geq\frac{y}{x+y}\,.$

Hence we have

$\frac{c_{n-1}}{c_{n}-c_{n-1}}=\frac{c_{n-1}}{(r-1)c_{n-1}-c_{n-2}}=\frac{1}{(r-1)-\frac{c_{n-2}}{c_{n-1}}}\geq\frac{1}{(r-1)-\frac{y}{x+y}}=\frac{x+y}{(r-1)x+(r-2)y}\,,$

as desired.

We can prove the second inequality similarly. Since we have
$\frac{c_{n-2}}{c_{n-1}-c_{n-2}}<\frac{y+1}{x},$ then cross multiplying and
adding $(y+1)c_{n-2}$ to both sides yields

$\frac{c_{n-2}}{c_{n-1}}<\frac{y+1}{x+y+1}\,.$

Thus, we can conclude

$\frac{c_{n-1}}{c_{n}-c_{n-1}}=\frac{1}{(r-1)-\frac{c_{n-2}}{c_{n-1}}}<\frac{1}{(r-1)-\frac{y+1}{x+y+1}}\leq\frac{x+y+1}{(r-1)x+(r-2)y}\,.\qed$

We now study the slopes of the line segments between northwest corners of
$\mathcal{D}$, as these play a central role in the definition of colored Dyck
subpaths in both our setting and the Lee–Schiffler setting. We will utilize
several classical results concerning maximal Dyck paths and Christoffel words,
which can be found, for example, in [1].

######  Definition 4.6.

For $n\geq 3$, we define the function
$\pi_{n}:\\{0,1,\dots,c_{n-1}\\}\to\mathbb{Z}$ by

$\pi_{n}(i):=xc_{n-2}-(i-x)(c_{n-1}-c_{n-2})\text{ where
}w_{i}=(x,i-x)\in\mathcal{D}_{n}\,.$

######  Remark 4.7.

Note that we have $\pi_{n}(0)=\pi_{n}(c_{n-1})=0$. It is a standard result
from the theory of Christoffel words (see, for example, [1, Lemma 1.3]) that
the sequence $\pi_{n}(1),\pi_{n}(2),\dots,\pi_{n}(c_{n-1})$ is a permutation
of the elements $\\{0,1,\dots,c_{n-1}-1\\}$, and this is order-isomorphic to
sequence of distances from each vertex to the line segment between $w_{0}$ and
$w_{c_{n-1}}$. Thus, $s(w_{i},w_{j})\geq s=s(w_{0},w_{c_{n-1}})$ in
$\mathcal{D}_{n}$ if and only if $\pi_{n}(i)\leq\pi_{n}(j)$.

######  Example 4.8.

For $r=3$, the values $\pi_{6}(0),\pi_{6}(1),\dots,\pi_{6}(21)$ are given by
the following sequence:

$0,8,16,3,11,19,6,14,1,9,17,4,12,20,7,15,2,10,18,5,13,0\,.$

We now prove several relations between the sequences
$\\{\pi_{n}(i)\\}_{i=0}^{c_{n-1}}$ and $\\{\pi_{n-1}(i)\\}_{i=0}^{c_{n-2}}$.

######  Lemma 4.9.

For any vertex with coordinates $(x,y)$ in $\mathcal{D}_{n-1}$ where $n\geq
4$, we have

$\pi_{n}(rx+(r-1)y)=\pi_{n-1}(x+y)\,.$

###### Proof.

By 3.5 and 4.5, both $(x,y)$ and $\mu(w_{x+y})=((r-1)x+(r-2)y,x+y)$ are
vertices of $\mathcal{D}_{n}$. Hence we can expand the right-hand side using
4.6 and apply the relation $c_{n}=rc_{n-1}-c_{n-2}$ to obtain

$\displaystyle\pi_{n}(rx+(r-1)y)$
$\displaystyle=((r-1)x+(r-2)y)c_{n-2}-(x+y)(c_{n-1}-c_{n-2})$
$\displaystyle=((r-1)x+(r-2)y)c_{n-2}-(x+y)((r-1)c_{n-2}-c_{n-3})$
$\displaystyle=-yc_{n-2}+(x+y)c_{n-3}$
$\displaystyle=xc_{n-3}-y(c_{n-2}-c_{n-3})\;.$

Comparing with 4.6, we see that the final quantity is precisely
$\pi_{n-1}(x+y)$, as desired. ∎

######  Lemma 4.10.

For $n\geq 3$, the set $\left\\{(x,y)\in
V(\mathcal{D}_{n}):\pi_{n}(x+y)\in\\{0,\dots,c_{n-1}-1\\}\right\\}$ is
precisely the set of northwest corners of $\mathcal{D}_{n}$.

###### Proof.

We proceed inductively on $n$. The base case $n=3$ is straightforward to
check. 4.7 implies that $\pi_{n}^{-1}(0)=\\{v_{0},v_{c_{n-1}}\\}$ and that
$\pi_{n}(i)$ takes each value of $\\{1,2,\dots,c_{n-1}-1\\}$ exactly once for
$i\in\\{2,3,\dots,c_{n-1}\\}$. For each vertex $(x,y)\in\mathcal{D}_{n-1}$, we
can evaluate

$\pi_{n}(rx+(r-1)y)=\pi_{n-1}(x+y)\in\\{0,1,2,\dots,c_{n-2}-1\\}\,.$

Moreover, the unique vertex $(x^{\prime},y^{\prime})\in\mathcal{D}_{n}$ such
that $x^{\prime}+y^{\prime}=rx+(r-1)y$ is $\mu(w_{x+y})$. Since the image of
$\mu$ applied to the set $V(\mathcal{D}_{n-1})$ is the set of northwest
corners of $\mathcal{D}_{n}$, the desired set equality holds. ∎

######  Corollary 4.11.

Applying the morphism $\lambda=\\{E\mapsto E^{r-1}N,N\mapsto E^{r-2}N\\}$ to
the Christoffel word for $\mathcal{D}_{n}$ yields the Christoffel word for
$\mathcal{D}_{n+1}$.

Applying the morphism $\theta\circ\lambda\circ\theta^{-1}$ to the Christoffel
word for $\mathcal{C}_{n}$ yields the Christoffel word for
$\mathcal{C}_{n+1}$.

###### Proof.

Let $w_{i}$ denote the Christoffel word for $\mathcal{D}_{i}$. Applying the
map $\mu$ to $V(\mathcal{D}_{n})$ takes a vertex $(x,y)$ to
$\left((r-1)x+(r-2)y,x+y\right)$. Hence by 4.10, the index $j$ of the
$k^{\text{th}}$ vertical edge $\alpha_{j}$ in $\mathcal{D}_{n+1}$, i.e., the
positions of $y$’s in $w_{n+1}$, is equal to

$r|\\{\alpha_{i}^{n}:i<j,\alpha_{i}^{n}\text{ is
horizontal}\\}|+(r-1)|\\{\alpha_{i}^{n}:i<j,\alpha_{i}^{n}\text{ is
vertical}\\}|+1\,.$

This is precisely the position of $y$’s in the word $\lambda(w_{n})$, hence we
can conclude that $\lambda(w_{n})=w_{n+1}$. The second statement follows from
3.3. ∎

Finally, we can determine the rest of the values of $\pi_{n}$ from its values
on the northwest corners.

######  Observation 4.12.

Suppose that $(i,j)\in\mathcal{D}_{n}$ is not a northwest corner. Let
$(i^{\prime},j)$ be the corner vertex immediately preceding $(i,j)$. Then it
follows from the definition of $\pi_{n}$ that

$\pi_{n}(i+j)=\pi_{n}(i^{\prime}+j)+(i-i^{\prime})c_{n-2}\;.$

###  4.3. Proof of the Simplification

We now show that the map $\chi$ (see 4.3) is well-defined and weight-
preserving with respect to $|\beta|_{1}$ and $|\beta|_{2}$. This shows that
our definition of $\mathcal{F}^{\prime}(\mathcal{D}_{n})$ is, in a sense,
equivalent to that of Lee-Schiffler.

######  Observation 4.13.

4.9 states that the values of $\pi_{n}$ on the northwest corners of
$\mathcal{D}_{n}$ lie in the set $\\{0,1,\dots,c_{n-2}-1\\}$. Since $\pi_{n}$
is injective on the interior of $\mathcal{D}_{n}$, this implies that the
values of $\pi_{n}$ of the vertices which are not northwest corners constitute
the set $\\{c_{n-2},c_{n-2}+1,\dots,c_{n-1}-1\\}$.

The colors of paths in Lee and Schiffler’s setting depends on the number of
northwest corners that one needs to traverse from the starting endpoint of the
path until the slope to the northwest corner was at least the slope of the
diagonal. We also consider the slopes between vertices on $\mathcal{D}_{n}$
that are not necessarily northwest corners, which will help us to derive
results about the northwest corners. We in turn use this to determine which
vertex $w_{d(i)}$ to the east of a given vertex $w_{i}$ is the first such that
the slope between $w_{i}$ and $w_{d(i)}$ is at least that of the diagonal.

######  Definition 4.14.

For $0\leq i<c_{n-1}$, we define

$d(i)=\min\left(\\{j\in\\{i+1,i+2,\dots,c_{n-1}\\}:s(w_{i},w_{j})\geq
s(0,c_{n-1})\\}\right)\,.$

Note that $d(i)$ is well-defined since $s(i,c_{n-1})\geq s(0,c_{n-1})$ for all
$0\leq i<c_{n-1}$. We now show some properties of how the functions $d$ and
$\mu$ interact.

######  Corollary 4.15.

Suppose $w_{i}$ is a northwest corner of $\mathcal{D}_{n}$ for $n\geq 4$. Let
$w_{i}=\mu\left(w_{i^{\prime}}\right)$ for some
$w_{i^{\prime}}\in\mathcal{D}_{n-1}$, and let $(x,y)=w_{d(i^{\prime})}$ in
$\mathcal{D}_{n-1}$ with $d(i^{\prime})-i^{\prime}=mc_{n-2}-wc_{n-3}$. Then we
have $w_{d(i)}=((r-1)x+(r-2)y,x+y)$, and $d(i)-i=mc_{n-1}-wc_{n-2}$.

If $w_{i}$ is not a northwest corner, then $w_{d(i)}$ is the northwest corner
immediately following $w_{i}$. In particular, $d(i)-i\leq r-1$.

###### Proof.

By 4.13, $w_{d(i)}$ must be a northwest corner. Thus 4.10 implies that
$w_{d(i)}$ must be the image of $w_{d(i^{\prime})}$ under the correspondence
between vertices of $\mathcal{D}_{n-1}$ to northwest corners of
$\mathcal{D}_{n}$. It is straightforward to check that the distances follow
the formula described under this correspondence. The second claim follows
directly from 4.13. ∎

From this, we can determine that the values $d(i)-i$ are of a particular form.

######  Lemma 4.16.

For all positive integers $i$, we have $d(i)-i=c_{m}-wc_{m-1}$ for a unique
choice of $2\leq w\leq r-1$ and $3\leq m\leq n-2$.

###### Proof.

We prove this via induction on $n$. The base case $n=3$ is straightforward to
check. Suppose the statement holds on $\mathcal{C}_{n-1}$. Whenever $w_{i}$ is
not a northwest corner of $\mathcal{C}_{n}$, then we have $d(i)-i<r$ by 4.15,
so $d(i)-i=c_{3}-wc_{2}$ for some $2\leq w\leq r-1$ or
$d(i)=r-1=c_{4}-(r-1)c_{3}$. Otherwise, if $w_{i}$ is a northwest corner of
$\mathcal{C}_{n}$, then $w_{i}=\mu(w_{i^{\prime}})$ for some
$w_{i^{\prime}}\in\mathcal{C}_{n-1}$. By assumption,
$d(i^{\prime})-i^{\prime}=c_{m}-wc_{m-1}$ for some appropriate choice of
$m,w$. Applying 4.15, we can conclude that $d(i)-i=c_{m+1}-wc_{m}$.

The fact that this representation is unique follows immediately from the fact
that, for all $m\geq 2$, we have $c_{m}-(r-1)c_{m-1}>c_{m-1}-2c_{m-2}$. ∎

We are now ready to establish that the value $t(i)-i$, as given in 2.2, can
also be represented as $c_{m}-wc_{m-1}$ for an appropriate choice of $m,w$.

###### Proof of 2.2.

We prove this via induction on $n$, with the straightforward base case $n=3$.
Fix a northwest corner $v_{i}=w_{j}\in\mathcal{C}_{n}$, and let $t(i)$ be the
minimum positive integer such that $s(v_{i},v_{t(i)})\geq s$. Since $v_{i}$ is
a northwest corner, we have $v_{i}=\mu(w_{i^{\prime}})$ for some
$w_{i^{\prime}}\in\mathcal{C}_{n-1}$. Applying 4.16, we have that
$d(i^{\prime})-i^{\prime}=c_{m}-wc_{m-1}$ for a unique choice of $2\leq w\leq
r-1$ and $3\leq m\leq n-2$. By 4.7, we have that
$t(i)-i=d(i^{\prime})-i^{\prime}$. Thus $t(i)-i$ is of the desired form. ∎

The following lemma establishes that $\chi$ preserves weights.

######  Lemma 4.17.

For all $n\geq 3$, we have $\beta\in\mathcal{F}(\mathcal{D}_{n})$ if and only
if $\chi(\beta)\in\mathcal{F}^{\prime}(\mathcal{D}_{n})$. Moreover, we have
$|\beta|_{i}=|\chi(\beta)|_{i}$ for $i=1,2$.

###### Proof of 4.17.

Fix $\beta\in\mathcal{F}(\mathcal{D}_{n})$. Suppose $\alpha(i,k)\in\beta$. By
2.2, either we have that $\alpha(i,k)$ is $(m,w)$-green, or we have that
$t(i)-i=1$. In this case, condition $(2^{*})$ of 2.3 enforces that the edge
immediately preceding $\alpha(i,k)$ is contained in $\beta_{j}$. By the non-
overlapping condition for membership in $\mathcal{F}(\mathcal{D}_{n})$, we
have $\beta_{j}\neq\alpha(i^{\prime},k^{\prime})$ for any
$i^{\prime},k^{\prime}$. Thus $\beta_{j}=\alpha_{i}^{\prime}$ for some
$i^{\prime}$. In particular, it does not contribute to $|\beta|_{1}$ and
contributes $1$ to $|\beta|_{2}$, which is the same as if we had considered
$\alpha(i,k)$ to contain this preceding edge. ∎

We can now combine the results about the map $\chi$ in order prove the
expansion formula in our setting.

###### Proof of 2.4.

This modified expansion formula follows immediately from 2.2 and the expansion
formula (4.2) given by Lee and Schiffler [12]. ∎

We now furthermore discuss a generalization of 2.4 to the setting of the
_framed $r$-Kronecker cluster algebra with principal coefficients_. While
there is a more general theory of cluster algebras with coefficients (see, for
example, [8]), we will give a brief explicit description of this cluster
algebra here. For a positive integer $r$, initial cluster variables
$X_{1},X_{2}$ and coefficient variables $Y_{1},Y_{2}$, we consider the
sequence $\\{\widetilde{X}_{n}\\}_{n\in\mathbb{Z}}$ and
$\\{\widetilde{Y}_{n}\\}_{n\in\mathbb{Z}}$ defined by

$\displaystyle\widetilde{Y}_{n+1}$
$\displaystyle=\frac{\widetilde{Y}_{n}^{r}}{\widetilde{Y}_{n-1}}\,,\text{
where }\widetilde{Y}_{1}=Y_{1}\text{ and }\widetilde{Y}_{2}=Y_{1}^{r}Y_{2},\;$
$\displaystyle\widetilde{X}_{n+1}$
$\displaystyle=\frac{\widetilde{X}_{n}^{r}+\widetilde{Y}_{n-1}}{\widetilde{X}_{n-1}}\,,\text{
where }\widetilde{X}_{1}=X_{1}\text{ and }\widetilde{X}_{2}=X_{2}\,.$

The use of tildes is to distinguish the settings with and without
coefficients. Let $\mathbb{P}$ be the tropical semifield
$\operatorname{Trop}[Y_{1},Y_{2}]$. The _framed $r$-Kronecker cluster algebra
$\widetilde{\mathcal{A}}(r,r)$ with principal coefficients_ is the
$\mathbb{Z}\mathbb{P}[\widetilde{X}_{1},\widetilde{X}_{2}]$ algebra generated
by the cluster variables $\\{\widetilde{X}_{n}\\}_{n\in\mathbb{Z}}$. Note that
when we specialize to the case $Y_{1}=Y_{2}=1$, we recover the classical
$r$-Kronecker cluster algebra.

######  Corollary 4.18.

Consider the framed $r$-Kronecker cluster algebra
$\widetilde{\mathcal{A}}(r,r)$ with principal coefficients, having initial
cluster variables $X_{1},X_{2}$ and coefficient variables $Y_{1},Y_{2}$. For
$n\geq 4$, the cluster variable $\widetilde{X}_{n}$ is given by

$\widetilde{X}_{n}=X_{1}^{-c_{n-1}}X_{2}^{-c_{n-2}}\sum_{\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{n})}X_{1}^{r|\beta|_{1}}X_{2}^{r\left(c_{n-1}-|\beta|_{2}\right)}Y_{1}^{|\beta|_{2}}Y_{2}^{|\beta|_{1}}$

and

$X_{3-n}=X_{2}^{-c_{n-1}}X_{1}^{-c_{n-2}}\sum_{\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{n})}X_{2}^{r|\beta|_{1}}X_{1}^{r\left(c_{n-1}-|\beta|_{2}\right)}Y_{2}^{-|\beta|_{2}}Y_{1}^{-|\beta|_{1}}\,.$

###### Proof.

Setting $\deg(X_{i})=e_{i}$ and $\deg(Y_{i})=(-1)^{i+1}re_{3-i}$, it is known
that the cluster variable $\widetilde{X}_{n}$ is a homogeneous Laurent
polynomial by [8, Proposition 6.1]. Moreover, this degree is readily
calculated from the recurrence relations on _g-vectors_ to be
$-c_{n-1}e_{1}+c_{n}e_{2}$ for $n\geq 2$ and $c_{-n+3}e_{1}-c_{-n+2}e_{2}$ for
$n<2$ (see, for example, [16, Subsection 4.1]). This determines the powers of
$Y_{1}$ and $Y_{2}$ that must appear in each monomial term, yielding the above
expansion formula directly from 2.4. ∎

##  5\. Bijection between Compatible Pairs and Colored Subpaths of Dyck Paths

In this section, we prove a conjecture of Feiyang Lin that the map $\Phi$,
constructed by Lin and described in 2.6, is a bijection between the
collections of colored subpaths introduced by Lee-Schiffler [12] and the
compatible pairs introduced by Lee-Li-Zelevinsky [11]. This shows the
correspondence between the objects summed over by each set of authors in their
expansion formulas for rank-$2$ cluster algebras. Specifically, we show that
Lin’s map is a bijection between collections $\beta$ of colored Dyck subpaths
in $\mathcal{F}^{\prime}(\mathcal{D}_{n})$ with a fixed $|\beta|_{1}$ and
$|\beta|_{2}$ and compatible pairs on $\mathcal{C}_{n}$ consisting of
$|\beta|_{1}$ vertical edges and $(c_{n-1}-|\beta|_{2})$ horizontal edges.

###  5.1. Compatible Pairs and Lin’s Map

The rank-2 cluster expansion formula given by Lee-Li-Zelevinsky [11] sums over
certain subsets of edges of a maximal Dyck path, known as compatible pairs,
which we discuss here. We will mainly work over $\mathcal{C}_{n}$, though
sometimes we work in more generality. Let $S_{1}$ be a subset of the vertical
edges of a maximal Dyck path $\mathcal{P}(a,b)$, and let $S_{2}$ be a subset
of the horizontal edges of $\mathcal{P}(a,b)$.

In order to study compatible pairs, Li, Lee, and Zelevinsky [11] introduced
the notion of the “shadow” of a set of edges. While they only defined shadows
of subsets of vertical edges, we extend this notion to subsets of horizontal
edges as well. These notions will be used throughout our construction of the
bijection between collections of colored Dyck subpaths and compatible pairs.

######  Definition 5.1.

For a vertical edge $\nu\in S_{2}$ with upper endpoint $w$, we define its
_local vertical shadow_ , denoted $\operatorname{sh}(\nu;S_{2})$, to be the
set of horizontal edges in the shortest subpath $\overrightarrow{tw}$ of
$\mathcal{P}(a,b)$ such that $|tw|_{1}=r|\overrightarrow{tw}\cap S_{2}|_{2}$.
Analogously, for a horizontal edge $\eta\in S_{1}$ with left endpoint $u$, we
define its _local horizontal shadow_ , denoted
$\operatorname{sh}(\eta,S_{2})$, to be the set of vertical edges in the
shortest subpath $\overrightarrow{ut}$ of $\mathcal{P}(a,b)$ such that
$|ut|_{2}=r|\overrightarrow{ut}\cap S_{1}|_{1}$. If there is no such subpath
$\overrightarrow{tw}$ or $\overrightarrow{ut}$, respectively, then we define
the local vertical (resp., horizontal) shadow to be the entire set of
horizontal (resp., vertical) edges in $\mathcal{P}(a,b)$.

For $S\subseteq S_{i}$ where $i\in\\{1,2\\}$, let
$\displaystyle\operatorname{sh}(S;S_{i})=\bigcup_{\alpha\in
S}\operatorname{sh}(\alpha;S_{i})$, and write
$\operatorname{sh}(S_{i}):=\operatorname{sh}(S_{i};S_{i})$.

######  Observation 5.2.

It is a straightforward consequence of 5.1 that for $S\subseteq S_{1}$, we
have $|\operatorname{sh}(S)|=\min(b,\,r|S|)$. Similarly, for $S\subseteq
S_{2}$, we have $|\operatorname{sh}(S)|=\min(a,\,r|S|)$

The expansion formula for cluster variables given by Lee, Li, and Zelevinsky
has monomials corresponding to compatible pairs on $\mathcal{C}_{n}$. Their
expansion formula works in the more general setting of elements of the greedy
basis, which contains the cluster variables. For further details on the greedy
basis, see [11]. We present their formula in the special case of cluster
variables.

######  Theorem 5.3.

[11, Theorem 1.11] For each $n\geq 1$, the cluster variable $X_{n}$ in
$\mathcal{A}(r,r)$ is given by

$X_{n}=x_{1}^{-c_{n-1}}x_{2}^{-c_{n-2}}\sum_{(S_{1},S_{2})}x_{1}^{r|S_{2}|}x_{2}^{r|S_{1}|}\,,$

where the sum is over all compatible pairs $(S_{1},S_{2})$ in
$\mathcal{C}_{n}$.

Figure 4. The top image depicts a collection of colored Dyck subpaths in
$\mathcal{F}^{\prime}(\mathcal{D}_{6})$, the same as in Figure 3. The bottom
image depicts the corresponding compatible pair $(S_{1},S_{2})$ in
$\mathcal{C}_{6}$, obtained by applying the map $\Phi$ to the collection of
colored Dyck subpaths. The edges that are contained in either $S_{1}$ or
$S_{2}$ are depicted by bold edges in the lower image. In particular, we have
$S_{1}=\\{\eta_{5},\eta_{15}\\}$ and
$S_{2}=\\{\nu_{1},\nu_{3},\nu_{4},\nu_{5},\nu_{7},\nu_{8}\\}$. Thus
$\operatorname{wt}((S_{1},S_{2}))=X_{1}^{18}X_{2}^{6}$.

Lin’s map from collections colored Dyck subpaths in $\mathcal{D}_{n}$ to
compatible pairs in $\mathcal{C}_{n}$ is described in 2.6. An example is shown
in Figure 4. Lin conjectured that the map $\Phi$ is a bijection between the
desired sets [16, Conjecture 3], which essentially involves showing that
$\Phi(\beta)$ is indeed a compatible pair for every
$\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{n})$. We verify this claim in the
next subsection. Lin made partial progress toward this conjecture by showing
that it was sufficient to consider only compatible pairs satisfying a certain
irreducibility condition [16, Proposition 4.8.4, Conjecture 4]. We proceed by
a different approach than Lin, so our methods do not rely on this
simplification.

In order to show the correspondence between the Lee-Schiffler and Lee-Li-
Zelevinsky expansion formulas, one needs to show not only bijectivity between
the sets summed over, but also that the resulting monomials correspond. Lin
defined a weight function for collections of colored subpaths and for
compatible pairs that keeps track of the exponents associated to these
monomials.

######  Definition 5.4.

We define the _weight_ of a compatible pair $(S_{1},S_{2})$ by

$\operatorname{wt}((S_{1},S_{2}))=X_{1}^{r|S_{2}|}X_{2}^{r|S_{1}|}$

and the _weight_ of a collection of colored subpaths
$\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{n})$ by

$\operatorname{wt}_{n}(\beta)=X_{1}^{r|\beta|_{1}}X_{2}^{r(c_{n-2}-|\beta|_{2})}\,.$

Lin showed that $\Phi$ is a weight-preserving map from a superset of
$\mathcal{F}(\mathcal{D}_{n})$ to $\mathcal{F}(\mathcal{C}_{n})$, and
conjectured that it restricted to a bijection between
$\mathcal{F}(\mathcal{D}_{n})$ and $\mathcal{F}(\mathcal{C}_{n})$. We prove
this in the next subsection. We convert Lin’s map into the setting of
$\mathcal{F}^{\prime}(\mathcal{D}_{n})$ instead of
$\mathcal{F}(\mathcal{D}_{n})$ in order to make easier reference to the
results of the previous section, though it is straightforward to show the
equivalence between these two settings.

###  5.2. Proof of Bijectivity

We now proceed to show that Lin’s map $\Phi$ indeed takes every collection of
colored subpaths in $\mathcal{F}^{\prime}(\mathcal{D}_{n})$ to a unique
compatible pair on $\mathcal{C}_{n}$. It then follows from the work of Lee-
Schiffler [12] and Lee-Li-Zelevinsky [11] that $\Phi$ is a bijection. For
$2\leq w\leq r-1$ and $m\geq 3$, we define $a_{m,w}$ to be the quantity
$c_{m}-wc_{m-1}$. We can use the quantities $a_{m,w}$ to define the size of
images of atomic colored paths under $\Phi$, as well as their shadows.

######  Observation 5.5.

It is readily deduced from the recursive definition of the sequence $c_{n}$
that for $w,m\geq 1$, we have $ra_{m,w}=a_{m+1,w}+a_{m-1,w}$.

In order to establish that the conditions for compatibility correspond to the
conditions for membership in $\mathcal{F}^{\prime}(\mathcal{D}_{n})$ via
$\Phi$, we first show that this is true for certain simple colored subpaths.

######  Definition 5.6.

We call a subpath of $\mathcal{D}_{n}$ _atomic_ if it consists of a single
edge, is blue, or is an $(m,w)$-brown path of the form $\gamma(i,i+a_{m,w})$.

######  Observation 5.7.

Any subpath of $\mathcal{D}_{n}$ can be written uniquely as a union of atomic
components meeting only at vertices such that

  1. (i)

no component is a single edge unless the entire path is a single edge, and

  2. (ii)

only the last component can be blue.

This decomposition is obtained by removing minimal brown paths from the front
until only a blue path remains.

Note that when we decompose a path into its atomic components, adjacent
components will necessarily overlap at a vertex. Thus after this
decomposition, the set of paths may no longer be non-overlapping, and hence
not in $\mathcal{F}^{\prime}(\mathcal{D}_{n})$. An example is shown in Figure
5.

Figure 5. The decomposition of the collection $\beta$ of subpaths from Figure
4 into its atomic components. Note that $(3,2)$-brown path $\gamma(2,3)$ and
blue path $\gamma(3,5)$, which meet at the vertex $v_{3}$, are the two atomic
components of the brown path $\gamma(2,5)$ from $\beta$.

We now study the structure of the $(m,w)$-brown paths as Christoffel words.
Recall the morphism $\lambda=\\{E\mapsto E^{r-1}N,N\mapsto E^{r-2}N\\}$.

######  Lemma 5.8.

The Christoffel word for an atomic $(m,w)$-brown path is
$\lambda^{(m-2)}(E^{r-w-1}N)$, where $\lambda^{0}$ is the identity map, which
has length $a_{m+1,w}$.

###### Proof.

Let $\gamma(i,k)$ be an atomic $(m,w)$-brown path, and let $\rho$ denote the
corresponding Christoffel word. Fix $i^{\prime},k^{\prime}\in\mathbb{Z}$ such
that $v_{i}=\mu(w_{i^{\prime}})$ and $v_{k}=\mu(w_{k^{\prime}})$.

First note that if $w_{i^{\prime}}$ is not a northwest corner, then by 4.13 we
have $k^{\prime}-i^{\prime}\leq r$. Thus we have $m=3$ and
$\rho=\lambda(E^{r-w-1}N)$ where $r-w-1=k^{\prime}-i^{\prime}-1$, which is of
the desired form.

Now suppose that $w_{i^{\prime}}=v_{i^{\prime\prime}}$ is a northwest corner.
We automatically have that $w_{k^{\prime}}=v_{k^{\prime\prime}}$ is a
northwest corner from 4.13. We aim to show that
$\gamma(i^{\prime\prime},k^{\prime\prime})$ is an $(m-1,w)$-brown path. Thus
the statement would follow from induction, since the Christoffel word
corresponding to $\gamma(i,k)$ is given by applying $\lambda$ to the word
corresponding to $\gamma(i^{\prime\prime},k^{\prime\prime})$.

In order to show that $\gamma(i^{\prime\prime},k^{\prime\prime})$ is an
$(m-1,w)$-brown path, we study the slopes from
$s(w_{i^{\prime}},w_{i^{\prime}+j})$ for $1\leq j\leq
k^{\prime}-i^{\prime}=k-i$. It readily follows from the recurrence for the
sequence $c_{n}$ and the formula for $\mu$ given in 4.5 that

$s(w_{i^{\prime}},w_{i^{\prime}+j})-s(v_{0},v_{c_{n}})=s(\mu(w_{i^{\prime}}),\mu(w_{i^{\prime}+j}))-s(v_{0},v_{c_{n+1}})=s(v_{i},v_{i+j})-s(v_{0},v_{c_{n+1}})\,.$

Since $s(v_{i},v_{i+j})-s(v_{0},v_{c_{n+1}})<0$ for $1\leq j\leq k-i$ and
$s(v_{i},v_{k})\geq s(v_{0},v_{c_{n+1}})$, the same holds for
$s(w_{i^{\prime}},w_{i^{\prime}+j})-s(v_{0},v_{c_{n}})$. That is, the slope
from $w_{i^{\prime}}$ to any vertex on
$\gamma(i^{\prime\prime},k^{\prime\prime})$ does not exceed that of the
diagonal, except the slope from $w_{i^{\prime}}$ to $w_{k^{\prime}}$.

We now just need to determine $k^{\prime\prime}-i^{\prime\prime}$, or
equivalently, the number of vertical edges in
$\gamma(i^{\prime\prime},k^{\prime\prime})$. Since $\gamma(i,k)$ as $a_{w,m}$
vertical edges, then we know $\gamma(i^{\prime\prime},k^{\prime\prime})$ has
$a_{w,m}$ total edges. Since the $a_{w,m}$ uniquely determine $w,m$, we can
conclude from the inductive hypothesis that
$\gamma(i^{\prime\prime},k^{\prime\prime})$ is $(m-1,w)$-brown. Moreover, by
5.5, we have

$(r-1)a_{w-1,m}+r(a_{w,m}-a_{w-1,m})=a_{w+1,m}\,,$

so we can conclude that $\gamma(i,k)$ has length $a_{w+1,m}$. ∎

We are interested in the portion of the path spanned by the vertical shadow of
the image of an $(m,w)$-brown path. We determine the structure of this portion
of the path with the following result.

######  Corollary 5.9.

The $a_{m-1,w}$ edges preceding an $(m,w)$-brown path form an $(m-2,w)$-brown
path or, when $a_{m-1,w}<r$, a path corresponding to the Christoffel word
$E^{a_{m-1,w}-1}N$.

###### Proof.

By definition, the edge preceding an $(m,w)$-brown path is vertical, so the
latter statement follows immediately.

We prove the first claim via induction on $m$. For $m\leq 5$, we note that
$a_{m-1,w}\leq r-1$, and hence the preceding path is of the form
$E^{a_{m-1,w}-1}N$. Let $\rho$ denote the Christoffel word formed by the
$a_{m-2,w}$ edges preceding an $(m-1,w)$-brown path. Then by 5.8, we can
obtain the word corresponding to the $a_{m-1,w}$ edges preceding an
$(m,w)$-brown path by applying $\lambda$ to $\rho$. By the inductive
hypothesis and 5.8, we can conclude that $\lambda(\rho)$ is an $(m-2,w)$-brown
path. ∎

######  Corollary 5.10.

Let $\gamma(i,k)$ be an $(m,w)$-brown path in $\mathcal{D}_{n}$, and let
$(S_{1},S_{2})=\Phi\left(\\{\gamma(i,k)\\}\right)$. Then the shadow of $S_{2}$
has length $a_{m+1,w}+a_{m-1,w}$.

###### Proof.

By 5.8, it follows that $S_{2}$ consists of $a_{m,w}$ consecutive vertical
edges. Thus, by 5.2, the shadow will contain $\min(ra_{m,w},c_{n-1})$
horizontal edges. Applying 5.5, we see that $ra_{m,w}=a_{m+1,w}+a_{m-1,w}$. We
then have by the bounds on $w$ and $m$ that

$ra_{m,w}=a_{m+1,w}+a_{m-1,w}\leq(c_{m+1}-2c_{m})+c_{m-1}\leq c_{m+1}\leq
c_{n-1}\,.$

So we can conclude the length of the shadow is $a_{m+1,w}+a_{m-1,w}$. ∎

We can now establish that the image under $\Phi$ of an atomic $(m,w)$-brown
path is a compatible pair in $\mathcal{C}_{n}$. As we will later see, this
encodes most of the complexity of the compatibility conditions on
$\mathcal{C}_{n}$.

######  Lemma 5.11.

Let $\gamma(i,k)$ be an atomic $(m,w)$-brown path in $\mathcal{D}_{n}$ and
$\gamma_{j}$ be one of the $a_{m-1,w}$ edges preceding $v_{i}$. Then
$\Phi(\\{\gamma(i,k),\gamma_{j}\\})$ is a compatible pair.

###### Proof.

Let $S_{2}=\Phi_{2}(\\{\gamma(i,k)\\})$. Let $\rho^{\prime}$ denote path
formed by the $a_{m-1,w}$ edges preceding $v_{i}$, and let
$\rho=\sigma(\rho^{\prime})$. It follows from 5.10 and 5.9 the shadow of
$\Phi_{2}$ spans a path of type $\rho\lambda^{2}(\rho)$.

Let $\upsilon$ be the path composed of $a_{m,w}$ north steps and $a_{m-1,w}$
west steps starting from the vertex immediately below $\Phi(v_{i})$, defined
as follows: for $i\geq 2$, the $i$-th north step of $\upsilon$ is $(i-1)r$
units to the west of the $(i-1)$-st edge in $S_{2}$. By definition, $\upsilon$
forms the eastern border of the shadow of each edge of $S_{2}$ except the
last. Moreover, the position of west steps in $\upsilon$ is determined by the
occurrence of subwords $E^{r-1}N$ and $E^{r}N$ in $\lambda^{2}(\rho)$. From
the definition of $\lambda$, it follows that Christoffel word corresponding to
the $90$ degree clockwise rotation of $\upsilon$ is precisely
$\lambda^{-1}(\lambda^{2}(\rho))=\lambda(\rho)$.

Figure 6. The yellow and black path constitute a portion of $\mathcal{D}$. The
thick black edges depict those that are contained in $\Phi_{2}(\gamma)$, where
$\gamma$ is a $(6,2)$-brown path. The shadow of $\Phi_{2}(\gamma)$ spans the
black and yellow paths. The purple path shows the left endpoint of the shadow
of the vertical edge to its right. Note that the yellow and purple paths
overlap in one edge.

We are now interested in the vertical distance from each horizontal edge
$\eta_{i}$ of $\rho$ to that of $\lambda(\rho)$. This determines the maximum
number $M$ of horizontal edges to the right of and including $\eta_{i}$ that
can be included in $S_{1}$ while satisfying the compatibility conditions.
Namely, this distance is the maximum height of the shadow at $\eta_{i}$, i.e.,
$rM$. Using the same inductive techniques as in 5.8, it is readily seen from
the base case $\rho=E^{w}N$ that this distance is $r(a_{m-1,w}-i+1)$ when
$i>1$. Hence it is possible that any combination of these edges is contained
in $S_{1}$. When $i=1$, the distance is $ra_{m-1,w}-1$. Since this is less
than $ra_{m-1,w}$ but greater than $r(a_{m-1,w}-1)$, it is not possible that
all edges of $\rho$ are contained in $S_{1}$, but it is possible that all but
one are. That is, the pair $(S_{1},S_{2})$ is compatible if and only if at
least one edge of $\rho$ is not contained in $S_{1}$. ∎

######  Example 5.12.

Figure 6 illustrates the construction in the proof of 5.11. Observe that the
purple path is a $90$ degree counterclockwise rotation of a Dyck path. Letting
$\rho=E^{2}NEN$ denote the Christoffel word for the yellow path, observe that
the Christoffel word corresponding to the (90 degree clockwise rotation of
the) purple path is given by $\lambda(\rho)$. Moreover, the Christoffel word
corresponding to the black path is given by $\lambda^{2}(\rho)$. The vertical
distance between the purple and yellow paths is precisely the maximum height
of the shadow at each horizontal yellow edge such that the compatibility
conditions are satisfied.

######  Lemma 5.13.

Let $\beta$ consist of a single atomic path and, if $\beta$ is $(m,w)$-brown,
one of the $a_{m-1,w}$ edges preceding this path. Then $\Phi(\beta)$ is a
compatible pair.

###### Proof.

We break into three cases based on the form of $\beta$. If $\beta$ consists of
a single edge, then $\Phi(\beta)$ has no vertical edges and hence is
compatible. If $\beta$ is $(m,w)$-brown, then this is precisely the result of
5.11.

Thus, the only remaining case is when $\gamma(i,k)$ is blue. Then it is the
prefix of an atomic $(m,w)$-brown path, obtained by extending $\gamma(i,k)$
until its slope exceeds that of the diagonal. Let $\gamma(i,k^{\prime})$
denote this atomic $(m,w)$-brown path, and let
$\beta^{\prime}=\\{\gamma(i,k^{\prime}),\eta_{j}\\}$ where $\eta_{j}$ is the
$(a_{m-1})$-th edge preceding $v_{i}$. In the previous case, we have shown
that $\Phi(\beta^{\prime})$ is compatible. Note now that
$\Phi_{2}(\beta)\subseteq\Phi_{2}(\beta^{\prime})$ and
$\operatorname{sh}(\Phi_{2}(\beta))\subseteq\Phi_{1}(\beta^{\prime})$.
Therefore the compatibility of $\Phi(\beta)$ follows directly from the
compatibility of $\Phi(\beta^{\prime})$. ∎

Now that we have handled the case of atomic paths, we show that we can
determine the compatibility of the image of many colored subpaths paths by
restricting to the compatibility conditions on each of its atomic components.

######  Definition 5.14.

Given a pair $(S_{1},S_{2})$ on $\mathcal{P}(a_{1},a_{2})$, and let
$(S_{1}^{\prime},S_{2}^{\prime})$ be a pair on
$\mathcal{P}(a_{1}^{\prime},a_{2}^{\prime})$. We define the _insertion of
$(S_{1}^{\prime},S_{2}^{\prime})$ into $(S_{1},S_{2})$ at position
$(j_{1},j_{2})$_ to be the pair $(S_{1}^{\prime\prime},S_{2}^{\prime\prime})$
on $\mathcal{P}(a_{1}+a_{1}^{\prime},a_{2}+a_{2}^{\prime})$ determined as
follows:

$e_{i}\in S_{k}^{\prime\prime}\iff\begin{cases}e_{i}\in S_{k}\text{ and }1\leq
i\leq j_{k}\,,\text{or}\\\ e_{i-j_{k}}\in S_{k}^{\prime}\text{ and
}j_{k}<i\leq j_{k}+a_{k}^{\prime}\,,\text{or}\\\ e_{i-a_{k}^{\prime}}\in
S_{k}\text{ and }j_{k}+a_{k}^{\prime}<i\leq
a_{k}+a_{k}^{\prime}\,,\end{cases}$

for $k\in\\{1,2\\}$ and $(j_{1},j_{2})\in V(\mathcal{P}(a_{1},b_{1}))$. Here,
each $e_{i}$ refers to the $i^{\text{th}}$ horizontal or $i^{\text{th}}$
vertical edge of the corresponding path, where the orientation of the edge is
determined by the context.

######  Lemma 5.15.

Let $(S_{1},S_{2})$ and $(S_{1}^{\prime},S_{2}^{\prime})$ be compatible pairs
on $\mathcal{P}(a_{1},a_{2})$ and
$\mathcal{P}(a_{1}^{\prime},a_{2}^{\prime})$, respectively, and suppose that
$(S_{1}^{\prime},S_{2}^{\prime})$ has non-spanning shadows. Then for any
$(j_{1},j_{2})\in V(\mathcal{P}(a_{1},a_{2}))$, the insertion
$(S_{1}^{\prime\prime},S_{2}^{\prime\prime})$ of
$(S_{1}^{\prime},S_{2}^{\prime})$ into $(S_{1},S_{2})$ at position
$(j_{1},j_{2})$ is a compatible pair on
$\mathcal{P}(a_{1}+a_{1}^{\prime},a_{2}+a_{2}^{\prime})$.

###### Proof.

Since $(S_{1}^{\prime},S_{2}^{\prime})$ is a compatible pair on
$\mathcal{P}(a_{1}^{\prime},a_{2}^{\prime})$, then either
$rS_{1}^{\prime}<a_{2}^{\prime}$ or $rS_{2}^{\prime}<a_{1}^{\prime}$. Without
loss of generality, suppose $rS_{1}^{\prime}\leq a_{2}^{\prime}$.

We can then see that for $e_{i}\in S_{1}^{\prime\prime}$ with $1\leq i\leq
j_{1}$, we have

$|\operatorname{sh}(e_{i};S_{1}^{\prime\prime})|\leq|\operatorname{sh}(e_{i};S_{1})|+\operatorname{sh}(e_{1},S_{1}^{\prime\prime})|<|\operatorname{sh}(e_{i};S_{1})|+a_{2}^{\prime}\,.$

Similarly, for $j_{2}+a_{2}^{\prime}<i\leq a_{2}+a_{2}^{\prime}$, we have

$|\operatorname{sh}(e_{i};S_{2}^{\prime\prime})|\leq|\operatorname{sh}(e_{i-a_{2}^{\prime}};S_{2})|+|\operatorname{sh}(e_{a_{2}^{\prime}},S_{2}^{\prime\prime})|\leq|\operatorname{sh}(e_{i};S_{2})|+a_{1}^{\prime}\,.$

The lengths of the shadows at the other edges, with the corresponding shift in
indices, is the same as in the original paths. Thus the horizontal and
vertical shadows will never intersect, so the pair is indeed compatible. ∎

This allows us to combine our results on atomic paths in order to handle any
collection of subpaths in $\mathcal{F}^{\prime}(\mathcal{D}_{n})$

######  Theorem 5.16.

If $\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{n})$, then $\Phi(\beta)$ is a
compatible pair.

###### Proof.

Let $(S_{1},S_{2})=\Phi(\beta)$ We proceed by induction on the number of
atomic components in $\beta$, which we denote by $t$. Note that if $t=0$, then
$S_{2}$ is empty and so $(S_{1},S_{2})$ is compatible.

If $\beta$ has one atomic component, the compatibility follows directly from
5.13.

If we add an additional singular edge to $\beta$, then the resulting pair is a
subset of the original, and hence compatible. Otherwise, suppose we add an
atomic component to $\beta$ that appears to the left of all other atomic
components. Then we can view the resulting path as the insertion of the atomic
path into the previous compatible pair. Since both paths involved in the
insertion are compatible, we can conclude using 5.15 that $\Phi(\beta)$ is
compatible. ∎

######  Lemma 5.17.

Every compatible pair in $\mathcal{C}_{n}$ is the image of some
$\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{n})$.

###### Proof.

We know from 5.16 that $\Phi(\beta)$ for
$\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{n})$ is a compatible pair in
$\mathcal{C}_{n}$. Moreover, we know from 4.2 and 5.3 that
$\mathcal{F}^{\prime}(\mathcal{D}_{n})$ and the set of compatible pairs in
$\mathcal{C}_{n}$ are equinumerous, since both are the result of the
substitution $X_{1}=X_{2}=1$. Lastly, we know from the work of Lin [16,
Proposition 4.7.3] that $\Phi$ is injective. Thus $\Phi$ is also surjective. ∎

Combining the results proven above along with work of Lee-Schiffler and Lee-
Li-Zelevinsky, we can prove the bijectivity of the map $\Phi$.

###### Proof of 2.8.

By 5.16 and 5.17, we can see that $\Phi$ is a bijection from
$\mathcal{F}^{\prime}(\mathcal{D}_{n})$ onto the set of compatible pairs in
$\mathcal{C}_{n}$. Using the work of Lin [16, Proposition 4.7.3], we
additionally see that $\Phi$ is weight-preserving. ∎

##  6\. Quantization of Colored Dyck Subpaths

In Lee and Schiffler’s work on expanding rank-$2$ cluster variables, they
showed that the coefficients of the Laurent expansion could be obtained by
taking sums over certain collections of colored Dyck subpaths. Each such
collection was taken to have weight $1$. In order to quantize this
construction, we instead weight each collection by a power of a formal
variable $q$. We then show that an analogous formula holds for rank-$2$
quantum cluster variables with an appropriate choice of $q$-weights, where
setting $q=1$ recovers Lee and Schiffler’s formula.

As discussed by Lee-Li-Rupel-Zelevinky [10, Section 3], the combinatorial
formula for greedy basis elements of a rank-$2$ cluster algebra cannot be
extended to the quantum setting by merely weighting compatible pairs by a
power of $q$, since positivity of these elements can fail in general. However,
Dylan Rupel [19, Corollary 5.4] established that the classical rank-$2$
formula for the quantum cluster variables, which are a proper subset of the
greedy basis, given by Lee-Li-Zelevinsky [11] could be extended in this way.
We proceed by applying the bijection established in the previous section to
Rupel’s expansion formula over weighted compatible pairs associated to quantum
cluster variables.

An advantage of 2.11 is that it requires fewer computations compared to
Rupel’s formula in [19, Corollary 5.4]. Our formula requires $O(|\beta|^{2})$
computations, where $|\beta|$ is the number of subpaths in
$\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{n})$. Rupel’s formula requires
$\binom{c_{n-1}+c_{n-2}}{2}$ computations, which is generally much larger.
Moreover, without knowledge of the bijection between collections of colored
subpaths and compatible pairs, it is unclear how to generate all compatible
pairs. Naïvely, one must consider all collections of edges of
$\mathcal{C}_{n}$ and check that the compatibility condition holds for each
pair of edges. It thus seems more efficient to generate all collections in
$\mathcal{F}^{\prime}(\mathcal{D}_{n})$ and compute their quantum weights
using 2.9 than to generate all compatible pairs on $\mathcal{C}_{n}$ and
compute quantum weights using [19, Corollary 5.4].

Figure 7. The top path depicts the compatible pair on $\mathcal{C}_{6}$
obtained by applying $\Phi$ to $\beta_{\emptyset}$ on $\mathcal{D}_{6}$, which
has $S_{1}=\\{\eta_{1},\eta_{2},\dots,\eta_{21}\\}$ and $S_{2}=\emptyset$. The
bottom path depicts the compatible pair obtained by applying $\Phi$ to the
collection of colored Dyck subpaths shown in Figure 4.

In our proof of 2.11, we translate each compatible pair into a finite word so
that we can refer to the language of combinatorics on words. We will work over
the alphabet $A=\\{h,v,H,V\\}$, with $A^{*}$ denoting the set of finite words
on $A$. For the purposes of this section, we represent a compatible pair by a
word in $A^{*}$. The letters $h$ and $H$ (resp., $v$ and $V$) represent
horizontal (resp., vertical) edges, with the capital letter denoting those
edges in $S_{1}$ (resp., $S_{2}$).

######  Example 6.1.

The word in $A^{*}$ corresponding to the top compatible pair in Figure 7 is

$H^{3}vH^{3}vH^{2}vH^{3}vH^{3}vH^{2}vH^{3}vH^{2}v\,.$

The word in $A^{*}$ corresponding to the bottom compatible pair in Figure 7 is

$h^{3}VhHhvh^{2}Vh^{3}Vh^{3}VHhvh^{3}Vh^{2}V\,.$

We define a morphism $f:\mathbb{Z}A^{*}\to\mathbb{Z}$, where $\mathbb{Z}A^{*}$
is the group of formal $\mathbb{Z}$-sums of words in $A^{*}$. The function
$w_{q}$ is defined on words of length $2$ in $A^{*}$ as follows:

$w_{q}(hv)=w_{q}(Hv)=w_{q}(hV)=1\,,\;\;\;w_{q}(Hh)=w_{q}(vV)=r\,,\;\;\;w_{q}(VH)=r^{2}-1\,,$

and for $x,y\in A$, $w_{q}(xy)=-w_{q}(yx)$. Note that in particular, this
implies that
${w_{q}(hh)=w_{q}(HH)=w_{q}(vv)=w_{q}(VV)=0}$. The function $w_{q}$ naturally
extends to formal $\mathbb{Z}$-sums of words of length $2$ on $A$. It is then
extended to words of larger length by taking the formal sum over all length
$2$ (not necessarily contiguous) subwords with multiplicity, and again
extended naturally to formal $\mathbb{Z}$-sums of any words on $A$. We
sometimes apply $w_{q}$ to a compatible pair; in this case, we interpret
$w_{q}$ as being applied to the corresponding word on $A$. We refer to $w_{q}$
as the _quantum weight_ of a word or compatible pair. Computing the value of
$w_{q}$ on the word associated to a compatible pair in this way is essentially
calculating Rupel’s weighting on compatible pairs associated to quantum
cluster variables [19].

######  Example 6.2.

Let $t_{1}$ (resp., $t_{2}$) denote the word in $A^{*}$ corresponding to the
top (resp., bottom) compatible pair in Figure 7, computed in 6.1. Note that
here we have $r=3$. Looking at all the length-$2$ subwords of $t_{1}$, we can
see that $t_{1}$ has $98$ instances of the subword $Hv$ and $70$ instances of
the subword $vH$. The only other length-$2$ subwords of $t_{1}$ are $HH$ and
$vv$, and we have $w_{q}(HH)=w_{q}(vv)=0$. Thus we can compute

$w_{q}(t_{1})=98w_{q}(Hv)+70w_{q}(vH)=98\cdot 1+70\cdot(-1)=28\,.$

Via a similar computation, we have

$\displaystyle w_{q}(t_{2})$
$\displaystyle=69w_{q}(hV)+45w_{q}(Vh)+21w_{q}(Hh)+17w_{q}(hH)+7w_{q}(vV)+5w_{q}(Vv)$
$\displaystyle\;\;\;+5w_{q}(VH)+7w_{q}(HV)+19w_{q}(hv)+19(vh)+3w_{q}(Hv)+w_{q}(vH)$
$\displaystyle=69-45+21\cdot 3-17\cdot 3+7\cdot 3-5\cdot 3+5\cdot 8-7\cdot
8+19-19+3-1$ $\displaystyle=28$

As we will show, there are more efficient methods for computing these weights
than looking at all length-$2$ subwords.

For a word $u\in A^{*}$ and a letter $x\in A$, let $(u)_{x}$ denote the number
of instances of $x$ in $u$. In order to find a compact formula for the weights
corresponding to collections of subpaths

######  Lemma 6.3.

Let $\beta_{\emptyset}\in\mathcal{F}^{\prime}(\mathcal{D}_{n})$ be the empty
collection of colored Dyck subpaths on $\mathcal{D}_{n}$. For all $n\geq 3$,
we have

$w_{q}\left(\Phi_{n}(\beta_{\emptyset})\right)=c_{n-1}+c_{n-2}-1\,.$

###### Proof.

We proceed by induction on $n$. For the base case $n=3$, we have

$w_{q}\left(\Phi_{3}(\beta_{\emptyset})\right)=w_{q}(H^{r}v)=rw_{q}(Hv)+\binom{r}{2}w_{q}(HH)=r=c_{3}+c_{2}-1\,.$

We now proceed to the inductive step. Let $\psi$ be the morphism $\\{H\mapsto
Hv\\}$. Observe that for a word $u\in\\{H,v\\}^{*}$ that starts with $H$, ends
with $v$, and has no consecutive instances of $v$, we have

$w_{q}(\psi(u))=w_{q}(u)+(u)_{H}\,.$

Let $u_{j}$ be the word associated to $\Phi_{j}(\beta_{\emptyset})$. Then by
3.3 and 4.5, $u_{n+1}$ can be obtained by applying the morphism
$\chi=\\{H\mapsto H^{r}v,v\mapsto H^{r-1}v\\}$ to $\psi^{-1}(u_{n})$. We can
readily calculate

$\displaystyle
w_{q}(\chi(HH))=w_{q}(H^{r}vH^{r}v)=2r=w_{q}(HH)+2w_{q}(\chi(H))\,,$
$\displaystyle
w_{q}(\chi(vv))=w_{q}(H^{r-1}vH^{r}v)=2r-2=w_{q}(vv)+2w_{q}(\chi(v))\,,$
$\displaystyle
w_{q}(\chi(Hv))=w_{q}(H^{r}vH^{r-1}v)=2r=w_{q}(Hv)+w_{q}(H^{r}v)+w_{q}(H^{r-1}v)\,,$
$\displaystyle
w_{q}(\chi(vH))=w_{q}(H^{r-1}vH^{r}v)=2r-2=w_{q}(vH)+w_{q}(H^{r-1}v)+w_{q}(H^{r}v)\,.$

Moreover, $\psi^{-1}(u_{n})$ has $c_{n-1}-c_{n-2}$ instances of $H$ and
$c_{n-2}$ instances of $v$. Therefore, we have

$\displaystyle w_{q}(\Phi_{n+1}(\beta_{\emptyset})$
$\displaystyle=w_{q}(\chi(\psi^{-1}(u_{n})$
$\displaystyle=w_{q}(\psi^{-1}(u_{n}))+(c_{n-1}-c_{n-2})w_{q}(\chi(H))+c_{n-2}w_{q}(\chi(v))$
$\displaystyle=\left(\psi(u_{n})-c_{n-2}\right)+r(c_{n-1}-c_{n-2})+(r-1)c_{n-2}$
$\displaystyle=\left(c_{n-1}+c_{n-2}-1-c_{n-2}\right)+c_{n}$
$\displaystyle=c_{n}+c_{n-1}+1\,.\qed$

Having calculated the weight of the empty collection of paths in
$\mathcal{F}^{\prime}(\mathcal{D}_{n})$, we wish to calculate the quantum
weight when we add in colored subpaths. Note that for the word corresponding
to the compatible pair, this involves swapping out certain instances of $H$
for $h$ and $v$ for $V$. We now calculate how such a substitution affects the
quantum weight of the word.

######  Lemma 6.4.

Let $t_{1},u_{1},t_{2},u_{2},\dots,t_{s},u_{s}$ be words corresponding to
compatible pairs. Furthermore, suppose that $t_{i},u_{i}\in\\{H,v\\}$. Let
$\sigma$ be the morphism $\\{H\mapsto h,v\mapsto V\\}$. Then we have

$w_{q}\left(\prod_{i=1}^{s}t_{i}\sigma(u_{i})\right)=w_{q}\left(\prod_{i=1}^{s}t_{i}u_{i}\right)+\sum_{i=1}^{s}\sum_{j=1}^{s}(-1)^{\mathbbm{1}_{i<j}}\bigg{(}r(t_{j})_{H}\cdot(u_{i})_{h}+\big{(}r(t_{j})_{v}-r^{2}(t_{j})_{H}\big{)}\cdot(u_{i})_{V}\bigg{)}$

###### Proof.

Since $w_{q}(hV)=w_{q}(Hv)$, we have
$w_{q}(u_{i})=w_{q}\left(\sigma(u_{i})\right)$ for all $i$. Hence we need only
to calculate the change its value under $w_{q}$ after applying $\sigma$ to the
length-$2$ subwords with one letter from a $u_{i}$ and the other from a
$t_{j}$.

Let $U_{i}$ denote the value under $w_{q}$ of the sum over all length-$2$
subwords of $\prod_{i=1}^{s}t_{i}\sigma(u_{i})$ with one letter in
$\sigma(u_{i})$ and the other from some $t_{j}$. We then have

$\displaystyle U_{i}$
$\displaystyle=\sum_{j=1}^{s}(-1)^{\mathbbm{1}_{i<j}}\bigg{(}(t_{j})_{H}\cdot(u_{i})_{h}\cdot\left(w_{q}(Hh)-w_{q}(HH)\right)+(t_{j})_{H}\cdot(u_{i})_{V}\cdot\left(w_{q}(HV)-w_{q}(Hv)\right)$
$\displaystyle\quad\quad\quad\quad+(t_{j})_{v}\cdot(u_{i})_{h}\cdot\left(w_{q}(vh)-w_{q}(vH)\right)+(t_{j})_{v}\cdot(u_{i})_{V}\cdot\left(w_{q}(vV)-w_{q}(vv)\right)\bigg{)}$
$\displaystyle=\sum_{j=1}^{s}(-1)^{\mathbbm{1}_{i<j}}\bigg{(}r\big{(}(t_{j})_{H}\cdot(u_{i})_{h}+(t_{j})_{v}\cdot(u_{i})_{V}\big{)}-r^{2}(t_{j})_{H}\cdot(u_{i})_{V}\bigg{)}\,.\qed$

Applying the previous general result about compatible pairs to the specific
case of $\mathcal{C}_{n}$ and using the connection between
$\mathcal{F}^{\prime}(\mathcal{D}_{n})$ and compatible pairs on
$\mathcal{C}_{n}$, we can now prove the quantum cluster variable expansion
formula.

###### Proof of 2.11.

Adding a path to $\beta\in\mathcal{F}^{\prime}(\mathcal{D}_{n})$ corresponds
to applying the morphism $\sigma$ from 6.4 to the appropriate portion of
associated compatible word. Note that $|\beta_{i}|_{2}=(\beta_{i})_{h}$ and
$|\beta_{i}|_{1}=(\beta_{i})_{V}$. Similarly,
$|\overline{\beta_{j}}|_{2}=(\overline{\beta_{j}})_{H}$ and
$|\overline{\beta_{j}}|_{1}=(\overline{\beta_{j}})_{v}$. It follows from 6.4,
6.3, and the definition of $w_{q}$ for words in $A^{*}$ that
$w_{q}(\beta)=\gamma_{\omega}+\beta_{\omega}$, where the terms on the right
hand side are those in [19, Corollary 5.4]. Thus expansion formula can be
deduced directly from [19, Corollary 5.4]. ∎

######  Example 6.5.

Let
$\beta=\\{\beta_{1},\dots,\beta_{6}\\}\in\mathcal{F}^{\prime}(\mathcal{D}_{6})$
be the collection of colored Dyck subpaths shown in Figure 4. Then we have
$\overline{\beta_{0}}=\overline{\beta_{1}}=\overline{\beta_{6}}=\emptyset$,
$\overline{\beta_{2}}=\\{\alpha_{4}\\}$, $\overline{\beta_{3}}=\\{v_{2}\\}$,
$\overline{\beta_{4}}=\\{\alpha_{14}\\}$, and
$\overline{\beta_{5}}=\\{v_{6}\\}$. Applying 2.11, we have

$\displaystyle
w_{q}(\beta)=(c_{5}+c_{4}-1)+3(15-4)-9(5-1)+3(5-1)+3(6-13)-9(2-4)+3(2-4)=28\,.$

which confirms the second calculation in 6.2.

##  7\. Further Directions

Many of our methods rely on the highly structured nature of the maximal Dyck
paths $\mathcal{C}_{n}$ and $\mathcal{D}_{n}$. We use this to better
understand the conditions for a set of edges to form a compatible pair on
$\mathcal{C}_{n}$, in particular deriving a criterion for compatibility in
terms of the sequences of consecutive vertical edges. While $\mathcal{C}_{n}$
is the relevant choice of maximal Dyck path for the cluster variables,
compatible pairs over arbitrary maximal Dyck paths were studied by Lee, Li,
and Zelevinsky [11] in their work on the greedy basis. One way to study
compatibility is in terms of forbidden edge sets, i.e., the minimal subsets of
edges which violate the compatibility condition for compatible pairs. It is
easy to verify that for the staircase Dyck path $\mathcal{P}(a,a)$, a set of
edges is compatible if and only if it does not contain a horizontal edge and
the vertical edge immediately following it. From the proof of the bijectivity
of $\Phi$, it follows that on $\mathcal{C}_{n}$ the forbidden edge sets are
the images under $\Phi$ of an $(m,w)$-brown path and the $c_{m-1}-wc_{m-2}$
edges preceding it. It could be interesting to study whether the criterion for
compatibility can also be reduced for other families of maximal Dyck paths.

While positivity fails in general for the quantum greedy basis [10, Section
3], it would be interesting to know under what conditions positivity holds.
Rupel’s formula [19, Corollary 5.4] establishes the positivity property for
the quantum cluster variables, but perhaps this is a special case of a more
general phenomenon. If so, these elements may also admit a quantum weighting
of the associated compatible pairs, similar to that given by Rupel.

##  8\. Acknowledgements

The author thanks her advisor, Lauren Williams, for introducing her to the
various interesting expansion formulas for low-rank cluster algebras and for
discussions at many stages of this research. The author is also grateful to
Kyungyong Lee, Feiyang Lin, Gregg Musiker, and Ralf Schiffler for their
helpful correspondence and comments about this work. She additionally thanks
Kyungyong Lee for suggesting the generalization of 2.4 to the case of cluster
algebras with coefficients. A portion of this work was completed while the
author was visiting l’Institut de Recherche en Informatique Fondamentale, and
the author extends her gratitude to Sylvie Corteel for hosting her. The author
was supported by NSF Graduate Research Fellowship and a Jack Kent Cooke
Foundation Graduate Fellowship.

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