New Combinatorial Formulas for Cluster Monomials of Type A Quivers

Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type A cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial formulas for the cluster monomials in terms of the so-called globally compatible collections. We give bijective proofs of these formulas by comparing with the well-known combinatorial models of the T-paths and of the perfect matchings in a snake diagram. For cluster variables of a type A cluster algebra, we give a bijection that relates our new formula with the theta functions constructed by Gross, Hacking, Keel and Kontsevich.


Introduction
Cluster algebras were first introduced by S. Fomin and A. Zelevinsky in [7] to design an algebraic framework for understanding total positivity and canonical bases for quantum groups. A cluster algebra is a subring of a rational function field generated by a distinguished set of Laurent polynomials called cluster variables. The long-standing Positivity Conjecture, now proved in [14] and [9], asserts that the coefficients in any cluster variable are positive integers. From the combinatorial point of view, the Positivity Conjecture suggests that these coefficients should count some combinatorial objects. Lots of research focuses on building such combinatorial models. We give a brief summary of the pros and cons of four such models.
• T -paths: In [18], Schiffler (independently Carroll-Price and Fomin-Zelevinsky in their unpublished work) obtained a formula for the cluster variables of a cluster algebra of finite type A (see §2 for the definition) in terms of T -paths. This formula has been modified and generalized to cluster algebras coming from surfaces [15,16,19,20,10].
The first author was supported by the Korea Institute for Advanced Study (KIAS), the AMS Centennial Fellowship, NSA grant H98230-14-1-0323, and the University of Nebraska-Lincoln.
The second author was partially supported by the Oakland University URC Faculty Research Fellowship Award.
• Perfect matchings of a snake diagram: A description that is similar to the T -path model but has a more graph-theoretic flavor [15]. Interesting combinatorics, for example the snake graph calculus [4,5], arises in the study of this model. This formula is also restricted to cluster algebras coming from surfaces.
• Compatible pairs in a Dyck path: In [13], the cluster variables of rank 2 quivers, which do not necessarily come from surfaces, are described in terms of Dyck paths. A more general construction of the so-called compatible pairs is used in the study of greedy bases in [12], and another generalization called GCC is used in [1,11].
• Broken lines and Theta functions: Discovered in [9], they are the most general combinatorial models so far. But the model is mysterious that even the finiteness of the number of broken lines is not immediate from the definition.
Our ultimate goal is to find a combinatorial model that is both general and effective in computation. Even though this goal appears out of reach for now, we feel that the model of maximal Dyck paths and compatible pairs has the potential to be generalized. This motivates the main goal of this paper: For a type A quiver, give a new formula for the cluster monomials using a combinatorial model similar to compatible pairs, and find the bijections to other known models.
We reach this goal by proving three equivalent formulas.
-In Theorem 4.1, we give a formula using a sequence of 0-1 sequences called GCS (globally compatible sequence), where each vertex of the quiver is assigned a 0-1 sequence satisfying a certain compatibility condition.
-In Theorem 4.5, we give a formula using globally compatible collections (GCCs) in Dyck paths. This formula has a similar flavor to the combinatorial formula for greedy bases in [12].
-In §3, we first use a combinatorial gadget called pipelines to decompose the d-vector of a cluster monomial into the ones of cluster variables, then give a formula for cluster variables using GCCs in Theorem 4.6.
Moreover, we construct a bijection between GCSs (which is equivalent to GCCs) and broken lines in Theorem 7.10 (the even rank case) and 7.13 (the general case), which relates our new formula with the theta functions constructed in [9]. The simplicity of this bijection came as a surprise for us: namely, under our setting, the i-th number in a GCS (which is a 0-1 sequence) is 0 if and only if the corresponding broken line bends at the i-th coordinate hyperplane e ⊥ i . This suggests that there could be further connections between our new combinatorial formulas and theta functions, and thus could provide a new approach to understanding broken lines (which are difficult to describe explicitly in general).
The paper is organized as follows. In §2 we recall the definition of cluster algebra and some facts about type A quivers. In §3 we define the d-vector of a cluster monomial and introduce its decomposition using pipelines. §4 consists of the statements of the main results of the paper. In §5 we prove the GCC formula for cluster variables (Theorem 4.6) by establishing a bijection from GCCs to perfect matchings. Then in §6 we give the proof of the other main results of §4. In §7 we give the bijection between GCSs and broken lines. Then we give some examples in §8. In the appendix, we give another proof of Theorem 4.6 using T -paths.

Background on cluster algebras and type A quivers
In this section, we recall some definitions and fix notations about quivers and skewsymmetric cluster algebras ( §2.1) and some special type A quivers ( §2.2).
2.1. Quivers and skew-symmetric cluster algebras. Recall that a finite oriented graph is a quadruple Q = (Q 0 , Q 1 , h, t) formed by a finite set of vertices Q 0 , a finite set of arrows Q 1 and two maps h and t from Q 1 to Q 0 which send an arrow α respectively to its head h(α) and its tail t(α). An arrow α whose head and tail coincide is a loop; a 2-cycle is a pair of distinct arrows β and γ such that h(β) = t(γ) and t(β) = h(γ). Similarly, it is clear how to define n-cycles for n ≥ 3. A vertex is a source (respectively a sink ) if it is not the head (resp. the tail) of any arrow.
In this paper, a quiver is a finite oriented graph without loops or 2-cycles.
Given a quiver Q and a vertex v ∈ Q 0 , the mutation µ v (Q) is the new quiver Q ′ obtained as follows: (1) For every path of the form u → v → w, add a new arrow from u to w.
(2) Reverse all arrows incident to v.
Let Q = (Q 0 , Q 1 , h, t) be a quiver. Let Q 0 = {v 1 , . . . , v n } = {1, . . . , n} (for simplicity, we denote v i by i in this paper if no confusion arises; and later we also use notation I = I uf = Q 0 to denote the same set). Let F = Q(x 1 , . . . , x n ) be the field of rational functions in x 1 , x 2 , . . . , x n with rational coefficients. A seed is a pair (u, Q) where u = {u 1 , u 2 , . . . , u n } is a set of elements of F which freely generate the field F . For any vertex i ∈ Q 0 , we denote The mutation µ i (u, Q) is the seed (u ′ , Q ′ ) where Q ′ = µ i (Q) and u ′ is obtained from u by replacing u i by Let ({x 1 . . . , x n }, Q) be the initial seed. A cluster is a set u ′ which appears in a seed (u ′ , Q ′ ) obtained from the initial seed by iterated mutations. An element in a cluster is called a cluster variable. A cluster monomial is a product of cluster variables in the same cluster. The (coefficient-free) cluster algebra A(Q) associated with Q is the subring of F generated by all cluster variables.
Next we recall the definition of the cluster algebra A prin with principal coefficient corresponding to the coefficient-free cluster algebra A = A(Q). Let I uf = Q 0 = {1, . . . , n}, I = {1, . . . , 2n}. Define a quiverQ with the vertex setQ 0 := I and the edge set In other words,Q is obtained from Q by adding an arrow from n+i to i for each i = 1, . . . , n. We call i ∈ I uf unfrozen vertices and i ∈ I \ I uf frozen vertices. Starting with the initial seed ({x 1 , . . . , x 2n },Q), we mutate it iteratively similar as above, with the restriction that we only use the mutations µ i for 1 ≤ i ≤ n (that is, only mutate at the unfrozen vertices). For each new seed ({x ′ 1 , . . . , x ′ 2n },Q ′ ), the first n rational functions x ′ 1 , . . . , x ′ n form a cluster. The union of all clusters gives the set of cluster variables. The cluster algebra A prin is the subring of Q(x 1 , . . . , x 2n ) generated over ZP := Z[x ±1 n+1 , . . . , x ±1 2n ] by all cluster variables.
and h(e) = h ′ (e) and t(e) = t ′ (e) for any arrow e ∈ Q 1 . We say that Q is a full subquiver of Q ′ if Q can be obtained from Q ′ by removing vertices Q ′ 0 \ Q 0 and their incident arrows. A linear quiver is a quiver with n vertices {v 1 , v 2 , . . . , v n } and n − 1 arrows in which any two consecutive vertices v i and v i+1 (i ∈ [1, n − 1]) are connected by a single arrow in either direction and there are no others arrows.

2.2.3.
Completely extended linear quiver. We define a completely extended linear quiver Q ′ as obtained from a linear quiver Q by attaching a 3-cycle to every edge, a 3-cycle to v 1 , and a 3-cycle to v n . (So Q ′ has 2n + 3 vertices.) By abuse of terminology, we also call the pair (Q, Q ′ ) a completely extended linear quiver, whenever we need to specify the linear quiver Q.
For convenience, if v j,k = v i , then we denote the variable x j,k = x i .

Extended linear quivers.
An extended linear quiver (Q, P ) is obtained from a completely extended linear quiver (Q, Q ′ ) by removing some vertices (or none) in Q ′ 0 \ Q 0 and  Figure 1. A completely extended linear quiver the arrows incident with them. Equivalently, we can characterize P as a quiver obtained from Q by adding some (or none) of the following: • a 3-cycle or an edge hung on v 1 , or • a 3-cycle or an edge hung on v n , or • 3-cycles attached to some edges of Q.
There is an obvious way to obtain a completely extended linear quiver (Q, Q ′ ) from an extended linear quiver (Q, P ) (up to relabeling vertices in Q ′ 0 \ P 0 ). An example is shown in  3.1. Cluster monomials and d-vectors. It is well known that any cluster algebra associated to a type A quiver with n vertices can be constructed from a triangulation on the (n + 3)-gon. A diagonal on the (n + 3)-gon is a line segment connecting two non-adjacent vertices. A connected curve on the polygon is called a pseudo-diagonal if it is isotopic to a diagonal (and its endpoints are the same as those of the diagonal) and if its interior is in the interior of the polygon. Two pseudo-diagonals are said to be crossing if they intersect in the interior of the polygon. Let Q be a type A quiver with n vertices, and let {T 1 , ..., T n } be the diagonals given by the corresponding triangulation on the (n + 3)-gon and {T n+1 , ..., T 2n+3 } the boundary edges. It is also known that the cluster variables of A(Q) are in natural bijection with all the diagonals of the polygon. Using this bijection, a cluster monomial of A(Q) can be identified with a finite set of pairwise non-crossing pseudo-diagonals, or equivalently with where m is a non-negative integer, D 1 , ..., D m are pairwise non-crossing diagonals, and d 1 , ..., d m are positive integers.
It is easy to see that the d-vector (a 1 , ..., a n ) of any cluster monomial satisfies the following: Property A. For any 3-cycle i → j → k → i in Q such that a i , a j , a k are positive and satisfy the triangle inequalities (i.e., the sum of any two numbers is strictly greater than the third), the sum a i + a j + a k is even.

3.2.
Construction of pipelines. Let W be the set of all integer vectors (a 1 , ..., a n ) satisfying Property A. In this subsection we prove that the cluster monomials of A(Q) are in bijection with W. We will define a map from W to the cluster monomials, which would then induce the immediate bijection. Let [x] + = max(x, 0) for any real number x. Let a = (a 1 , ..., a n ) ∈ W. We define a function σ : R 3 ≥0 → R ≥0 as follows: For convenience, we denote σ a ijk = σ(a i , a j , a k ). (If no confusion shall arise, we denote σ ijk = σ a ijk .) We are ready to construct the so-called pipelines associated to a.
Step 1: If a i > 0 then draw a i marking points on the diagonal T i to separate it into a i + 1 segments. If a i < 0 then draw −a i pipes so that these pipes are pairwise non-crossing pseudo-diagonals isotopic to T i and that they do not cross any other T j (j = i).
Step 2: If T i and T j are two sides of a triangle with the third side T k , then for 1 ≤ r ≤ σ ijk , we join the two r-th marking points on T i and T j (ordering in the increasing distance from the common endpoint of T i and T j ) by a pipe inside the triangle. Draw these pipes so that they are disjoint from each other and from the pipes constructed in Step 1.
Step 3: Suppose that T i , T j , T k form a triangle. Then for each marking point on T i that is not connected by a pipe to any marking point on T j or T k , we draw a pipe from the marking point to the common endpoint of T j and T k . Draw these pipes inside the triangle in such a way that they are non-crossing with each other and with the pipes constructed in Step 1,2.
A pipeline is a union of pipes connected consecutively through the marking points (but not through the vertices of the (n + 3)-gon). Then the pipelines are pairwise non-crossing. Since the endpoints of each pipeline are non-adjacent vertices of the polygon, every pipeline is a pseudo-diagonal. Hence the union of these pipelines corresponds to a cluster monomial. Clearly the d-vector of this cluster monomial is equal to a.
Using the above construction of pipelines, it is straightforward to prove the following:  (2) Two distinct cluster monomials have different d-vectors.
This allows us to denote the (unique) cluster monomial with d-vector (a 1 , . . . , a n ) by x[a 1 , . . . , a n ] or x[a].
Each pipeline Λ corresponds to a 0- Note that Λ corresponds to a linear full subquiver of Q. Let S be the multiset of all such sequences. Then Example 3.3. The first two pictures in Figure 3 are a type A quiver with 7 vertices and its corresponding triangulation on the 10-gon. For clearer illustration, the 10-gon is drawn as a concave polygon. The bottom illustrates the construction of pipelines for a = (3, 3, 3, 2, 4, 3, 1).
and that x[b] and x i (a i < 0) are in the same cluster.
Proof. If a i < 0, then there is no b satisfying b i > 0. Thus no pseudo-diagonals corresponding to pipelines for [a] + will cross the diagonal T i . Therefore, after adding T i we still get a set of non-crossing pseudo-diagonals, which means that x[b] and x i (a i < 0) are in the same cluster. It then follows that Remark 3.5. Assume that Q is a full subquiver of Q ′ . We compare cluster variables in various cluster algebras.
For simplicity denote the vertex sets Q 0 = {1, . . . , n} and Q ′ 0 = {1, . . . , n ′ }. By definition, the cluster algebra with coefficients, denoted A(Q, Q ′ ), is generated by only those cluster variables in A(Q ′ ) obtained from iteratively mutating the initial cluster variables There is also a natural bijection sending a cluster variable x[d 1 , . . . , d n ] ∈ A(Q, Q ′ ) to the cluster variable x[d 1 , . . . , d n ] ∈ A(Q), given by substituting x i by 1 for i ∈ [n + 1, n ′ ]. More generally, if Q is a full subquiver of P , and P is a full subquiver of Q ′ , then there is a natural bijection sending a cluster variable If Q is a full subquiver of Q ′′ , and Q ′ is the vertex-induced subquiver of Q ′′ whose vertex set consists of vertices in Q 0 and those adjacent to Q 0 , then the cluster variables in A(Q, Q ′ ) have the same expressions as those in A(Q, Q ′′ ).

Globally Compatible Collections
In this section, we give three formulas for the cluster monomial x[a] for a ∈ W. All results here will be proved in §6.
First we reduce to a special case. We can replace a by [a] + , thus can assume a i ≥ 0, because of Lemma 3.4. Moreover, for any edge i → j of Q that is not in a 3-cycle, we can add a vertex k and two arrows j → k and k → i (the vertex k is a frozen vertex). Indeed, assume the modified quiver is Q ′ . Then by Remark 3.5, once we have a formula for cluster monomials for Q ′ , we can set x i = 1 for all i ∈ Q ′ 0 \ Q 0 and obtain a formula for cluster monomials for Q. In the rest of the paper, we assume that a = [a] + and (4.1) Q is of type A with more than one vertex, and every edge of Q is in a 3-cycle.
For every 3-cycle i → j → k → i, σ ijk is a nonnegative integer by Proposition 3.2, and it is not hard to verify that σ kij + σ ijk ≤ a i .

4.1.
A formula using 0-1 sequences. Fix a deg-2 vertex i 0 of Q (which exists because of (4.1)). For i ∈ Q 0 , denote by d(i) be the distance (i.e., the length of the shortest directed path) from i 0 to i. Let s i = (s i,1 , s i,2 , . . . , s i,a i ) ∈ {0, 1} a i be a 0-1 sequence, and define We say that a sequence of 0-1 sequences s := (s 1 , . . . , s n ) is a globally compatible sequence (abbreviated GCS) if the following holds for any 3-cycle i → j → k → i: Theorem 4.1. For any d-vector a = (a 1 , . . . , a n ) ∈ Z n ≥0 (i.e., a ∈ W ∩ Z n ≥0 ), we have the following formula for the corresponding cluster monomial: here s runs through all GCSs.
This formula specializes to the formula given in [11] for a linear quiver.

4.2.
A formula using Dyck paths. We recall the following definition from [11].
Let (a 1 , a 2 ) be a pair of nonnegative integers. Let c = min(a 1 , a 2 ). The maximal Dyck path of type a 1 × a 2 , denoted by D = D a 1 ×a 2 , is a lattice path from (0, 0) to (a 1 , a 2 ) that is as close as possible to the diagonal joining (0, 0) and (a 1 , a 2 ), but never goes above it. A corner is a subpath consisting of a horizontal edge followed by a vertical edge.
Definition 4.2. Let D 1 (resp. D 2 ) be the set of horizontal (resp. vertical) edges of a maximal Dyck path D = D a 1 ×a 2 . We label D with the corner-first index in the following sense: (a) edges in D 1 are indexed as u 1 , . . . , u a 1 such that u i is the horizontal edge of the i-th corner for i ∈ [1, c] and u c+i is the i-th of the remaining horizontal ones for i ∈ [1, a 1 −c], and v c+i is the i-th of the remaining vertical ones for i ∈ [1, a 2 − c].
(Here we count corners from bottom left to top right, count vertical edges from bottom to top, and count horizontal edges from left to right.) . We say that S 1 and S 2 are s-compatible if for every 1 ≤ r ≤ s, either u r / ∈ S 1 or v r / ∈ S 2 . In other words, neither of the first s corners are in the subpath S 1 ∪ S 2 .

Figure 4. A maximal Dyck path
Note that the following definition of global compatibility is different from [11].
a j . We say that the collection is a globally compatible collection (abbreviated GCC) if and only if for any k → i → j in Q: Theorem 4.5. Assume n > 1. For any d-vector a = (a 1 , . . . , a n ) ∈ Z n ≥0 , we have the following formula for the corresponding cluster monomial: where the sum runs over all GCCs. (Note that because of the rotational symmetry, each 3-cycle contributes three terms to the last product.)

4.3.
A more explicit formula for cluster variables. The third method of computing the cluster monomial with given d-vector is to first decompose the cluster monomial into a product of cluster variables using pipelines and then use a formula for cluster variables using GCCs.
For given d-vector, we construct pipelines as in the subsection 3.2. Then each pipeline corresponds to a cluster variable. We give the GCC formula of the cluster variable x[a] in a completely extended linear quiver (Q, Q ′ ), where a = (a 1 , . . . , a n ′ ) such that a i = 1 if i ∈ Q, and a i = 0 otherwise. For any arrow (i + δ i ) → (i + 1 − δ i ) of Q we attach a Dyck path D (i) = D 1×1 , which consists of one horizontal edge and one vertical edge. (Recall that δ i is defined in §2.2.2.) Then Definition 4.4 specializes to the following: , and the following holds for i ∈ [2, n − 1]: Theorem 4.6. Let (Q, Q ′ ) be a completely extended linear quiver, and n = |Q 0 | > 1, where the sum runs over all GCCs {S i,r }, , and x n,1 , otherwise.
Remark 4.7. Note that the above theorem induces a formula for the cluster variables of any type A quiver. Indeed, letQ be any type A quiver and non-initial x[a] be a cluster variable with d-vector a = (a 1 , . . . , a n ′ ) (here n ′ = |Q 0 |; the formulas for initial cluster variables are trivial). Then the subset of vertices {i | a i = 1} is equal to the set of vertices Q 0 of a linear full subquiver Q. By relabeling vertices if necessary, we assume Q 0 = {1, . . . , n} (thus a 1 = · · · = a n = 1 and a n+1 = · · · = a n ′ = 0 and for convenience we denote a = 1 n 0 n ′ −n . By removing vertices inQ 0 but not in or adjacent to Q 0 , we get an extended linear quiver (Q, P ); from this extended linear quiver we can obtain a completely extended linear quiver Q ′ . Define n ′ = |Q ′ 0 |, m = |P 0 |. Thanks to Remark 3.5, a formula for cluster variable Remark 4.8. We also give a formula for cluster variables of any type A quiver in terms of GCS as follows. LetQ be a type A quiver and Q be a linear full subquiver ofQ, as in Remark 4.7. Define Relabelling vertices if necessary, we assume that Q is 1 ←→ 2 ←→ · · · ←→ n where each arrow can go either direction. We will give a formula of the cluster variable x[a Q ].
Because s i = 0 for i > n, by abuse of notation we also view s as an element in {0, 1} n .
Let deg − Q (i) be the number of arrows inQ 1 whose tails are in Q 0 and heads are the vertex i, let deg + Q (i) be the number of arrows inQ 1 whose heads are in Q 0 and tails are the vertex We have the following formula, where arrows i → j are inQ: r∈Q 0 ∪K x r and s runs through all GCSs.

A bijection between perfect matchings and GCCs
In this section, we first recall the construction of snake diagram and the formula of cluster variables using perfect matching as in [15], then give a bijective proof of Theorem 4.6 via perfect matching.
Associated to a completely extended linear quiver (Q, Q ′ ), we recursively construct the snake diagram by gluing n-tiles together as follows: we first put the 2 nd -tile to the right side of the 1 st -tile; suppose the i th -tile is placed, we add the (i + 1) th -tile to the right side or on top of the i th -tile such that the (i − 1) th , (i) th and (i + 1) th -tiles are in the same row or column if and only if δ i−1 = δ i .
Next, we label the edges as follows.
• The common edge of the i th -tile and the (i + 1) th -tile is labeled T i,i+1 .
• Denote by P l (i) the parallelogram bounded by the main diagonals of the i th -tile and the (i+1) th -tile and two boundary edges. Any edge forming an angle of 135 • with the main diagonal of the i th -tile will be labeled T (j) i (where j indicates the parallelogram to which the edge belongs).
• For convenience, we let P l (0) be the right triangle with legs T 1,0 and T 1,1 , and let P l (n) the right triangle with legs T n,0 and T n,1 .
• The edges of the first and the last tiles are labeled as in Figure 6. Figure 5. Labels of edges in (i − 1) th , i th and (i + 1) th -tiles in two cases Figure 6. Labels of edges in the first and last tiles A perfect matching of the snake diagram is a set of edges such that each vertex is incident to exactly one edge in the set.
For the fixed completely extended linear quiver (Q, Q ′ ), let M be the set of all perfect matchings in the associated snake diagram, and G be the set of all GCCs. We shall construction a bijective map ψ 1 : M → G and its inverse ψ 2 . First we prove a simple lemma. Proof. The statement is obviously true for i = 0 and i = n. Suppose the statement is false for some i ∈ [1, n − 1]. Since γ is a perfect matching, we are in one of the following two cases.
i+1 are in γ. If we remove P l (i) (4 vertices and 3 edges), then the rest graph has two components which have odd number of vertices and have perfect matchings. This is a contradiction.
Case 2: none of the three edges T Then we remove the three edges (but do not remove the vertices) and apply the same argument as in Case 1.
(By Lemma 5.1, exactly one of the three cases occurs.) (ii) We define a map ψ G,M : G → M by sending {S i,r } ∈ G to the set of edges γ = {γ 0 , γ 1 , . . . , γ n } such that We assign a weight w(u) for each edge u of the snake diagram as follows for all i, j: For a perfect matching γ = (γ 0 , . . . , γ n ), define its weight w(γ) = n i=0 w(γ i ). In [15] it is proved that the cluster variable with d-vector a is For (a), we suppose (δ i−1 , δ i ) = (0, 0) and need to show that T This is true because the two edges T These two edges are opposite edges of a tile which is the middle of three tiles in a row or a column. Deleting these two edges will separate the snake diagram into two graphs with even number of vertices each. Thus the two edges must be both in γ or not in γ. (See the right diagram in Figure 5.) (ii) We show that ψ G,M is well-defined, that is, ψ G,M ({S i,r }) = γ is a perfect matching. Since the snake diagram has 2n + 2 vertices and γ has n + 1 edges, it suffices to show that all edges in γ are disjoint. We assume the contrary that γ c shares a vertex with γ d for some 0 ≤ c < d ≤ n. Since γ c ∈ P l (c) and γ d ∈ P l (d) , P l (c) and P l (d) much be consecutive, thus d = c + 1.
) or (T c,c+1 , T c+1,c+2 ). We get the expected contradiction by observing Figure 8.  Figure 9. We have explained in §4.3 that, in order to compute cluster variables, it suffices to have the formula for a completely extended linear quiver, namely Theorem 4.6. This theorem follows from Theorem 5.4. In this section, we show how to derive Theorem 4.1 and Theorem 4.5 from Theorem 4.6.   1, 0). In this case, Λ intersects edges i and j at the r-th marking points for some r ≤ σ a ijk . Then either u . In the former case, |S Since the left hand side is equal to |, and Λ σ b jki = σ a jki . The first two are clear. To show the last equality: first note that if Λ is disjoint from the edge j, then b j = 0 and thus σ b jki = 0. So we only need to consider those Λ that intersect j. Let Λ r (1 ≤ r ≤ a j ) be the pipeline that intersects j at the r-th marking point (ordered in the increasing distance to the common endpoint of j and k). If r ≤ σ a jki , . By Remark 3.5, it suffices to show that, in the setting of Theorem 4.6, the right hand side of (4.5) is equal to x ′ [a]. It breaks down to show that, for i ′ ∈ [0, n], the following equality holds for the i ′ -th 3- We shall only prove the case when i ′ ∈ [1, n − 1] and δ i ′ = 0, because other cases can be proved in a similar way. In this case, the i ′ -th 3-cycle is , and the left hand side of (6.1) is equal to So (6.1) holds.

6.2.
Proof that Theorem 4.5 implies Theorem 4.1. We give a bijection between GCSs and GCCs. Let s be a GCS. Consider a 3-cycle i → j → k → i, labeled in the way that d(k) < d(i) < d(j). Then we define It is then easy to check that the conditions of GCSs and GCCs, as well as the two theorems, are equivalent under this bijection.

A bijection between GCSs and broken lines
This section is devoted to the construction of a bijection between GCSs and broken lines in the even rank case (Theorem 7.10) and the general case (Theorem 7.13). In §7.1, we give a description of the g-vector of a cluster variable, which will determine the initial direction of the broken lines. In §7.2, we recall the necessary facts on scattering diagrams, broken lines, and theta functions. The rest of the section is to state and prove Theorem 7.10 and 7.13.
In this section, we letQ be a type A quiver and Q be a linear full subquiver ofQ. Relabelling vertices if necessary, we assume that Q is (7.1) Q: 1 ←→ 2 ←→ · · · ←→ n (each arrow can go either direction).

g-vectors.
It is shown in [9, Theorem 7.5 (4)] that, if m is the g-vector of a cluster variable, then the cluster variable is exactly ϑ m . So we first study g-vectors.
Lemma 7.1. Let a Q be defined as in (4.6). Then the g-vector m 0 = (g r ) of the cluster variable x[a Q ] is given by Proof. We use the description of g-vectors in [16, §13.1]: It is easy to verify that P − corresponds to the GCS s = (s i ) where s i = 1 if and only if i ∈ Q (this is the unique GCS satisfyings i = 0). Thus . So we can compute the g-vector case by case: If r ∈ Q: the power of x r in (7.2) is equal to the number of arrows i → r with s r = 1, which is deg − Q (r). Since a r = 1, g r = deg − Q (r). If r / ∈ Q and (deg + Q (r), deg − Q (r)) = (0, 1): then r / ∈ K, thus the power of x r in (7.2) is deg − Q (r) = 1. If r / ∈ Q and (deg + Q (r), deg − Q (r)) / ∈ (0, 1): then (deg + Q (r), deg − Q (r)) = (1, 1), (1, 0) or (0, 0). If it is (1, 1), then r ∈ K, thus the power of x r in (7.2) is deg − Q (r) − 1 = 0; if it is (1, 0) or (0, 0), then deg − Q = 0 and r / ∈ K, thus x r does not appear in (7.2). In all cases, we get g r = 0.
Remark 7.2. The above is equivalent to the following description of the g-vector of x[a Q ]: is not the head of any arrow in Q; 1, if "r ∈ Q is the head of two arrows in Q", or "r / ∈ Q is adjacent to only one vertex in Q, either 1 → r or n → r"; 0, otherwise.

Scattering diagrams and broken lines.
We only recall the necessary facts needed in our paper, specialized in our setting. For more reference, see [9].
Recall that in §2.1 we defined I = I uf = {1, . . . , n} for a coefficient free cluster algebra A of rank n, and I = {1, . . . , 2n} and I uf = {1, . . . , n} for a principal coefficient cluster algebra A prin of rank n.
Let M ∼ = Z n ′ , N = Hom(M, Z) with a basis (e i ) i∈I , M R := M ⊗ R with dual basis (f i ) i∈I . N is equipped with a skew-symmetric form {·, ·}. Let N uf be a sublattice of N with basis (e i ) i∈I uf . Define We assume the Fundamental Assumption: the map p * 1 : N uf → M given by n → {n, ·} is injective.
Choose a strictly convex top-dimensional cone σ ⊆ M R , with associated monoid P := σ ∩ M, such that p * 1 (e i ) ∈ P \ {0} for all i ∈ I uf . Let Z[P ] be the completion of Z[P ] at the maximal monomial ideal m generated by {x m |m ∈ P \ {0}}.
A wall is a pair (d, f d ) where σ ∈ M R is a (rankN −1)-dimensional convex rational polyhedral cone contained in n ⊥ 0 for some primitive vector n 0 ∈ N + , and f d = 1 + k>0 c k z kp * 1 (n 0 ) . A scattering diagram D is a collection of walls such that there are only finite many walls (d, f d ) ∈ D satisfying f d ≡ 1 mod m k for each k > 0.
For a smooth path γ : [0, 1] → M R \ Sing(D) whose endpoints are not in Supp(D) and that it is transversal to each wall that it crosses, we define an automorphism θ γ,D of Z[P ] as follows: for each k > 0, we can find numbers 0 < t 1 ≤ t 2 ≤ · · · ≤ t s < 1 such that Note that all walls in D in are incoming.  Remark 7.4. The broken lines that we shall construct in this paper will not bend on walls that are not in the initial scattering diagram D in defined in (7.3). A priori, there may exist broken lines that do bend on walls in D \ D in (i.e., outgoing walls); but we shall argue that there are no such broken lines in our setting (where we only consider those appear in the theta function corresponding to cluster variables (see §7.4.2). In general, broken lines can bend on outgoing walls if we consider those appear in the theta function corresponding to cluster monomials.

A bijection between GCSs and broken lines (in the even rank case).
In this subsection, we give a bijection between GCSs and broken lines for the cluster variable x[a Q ] when n ′ (the rank of the cluster algebra A) is even. This is exactly the case when the exchange matrix B is of full rank, which guarantees that the Fundamental Assumption in [9] is satisfied. (Indeed, since [2, Lemma 3.2] asserts that the rank is invariant under mutation, it is suffices to consider the linear quiver 1 → 2 → · · · → n ′ , in which case the determinant of B is 1 if n ′ is even, 0 if n ′ is odd.) where deg 1→ Q,s (i) is the number of arrows j → i in Q ′ 1 such that j ∈ Q 0 and s j = 1, and deg 0← Q,s (i) is the number of arrows j ← i in Q ′ 1 such that j ∈ Q 0 and s j = 0.
Remark 7.9. (1) Note that s (0) = [1, . . . , 1] ∈ {0, 1} n . We claim that m 0 = (g r ) coincides with the definition given in Lemma 7.1. Indeed, note that deg 0← Q,s (r) = 0. If r ∈ Q 0 , nothing needs to be proved. We assume r / ∈ Q 0 in the rest of the paragraph. If (deg + Q (r), deg − Q (r)) = (0, 1), then we are in the second case of (7.4), g r = deg 1→ Q,s (r) = deg − Q (r) = 1; if (deg + Q (r), deg − Q (r)) = (0, 0) or (1, 0), then we are in the second case of (7.4), thus g r = deg 1→ Q,s (r) = deg − Q (r) = 0; if (deg + Q (r), deg − Q (r)) = (1, 1), then we are in the first case of (7.4), thus g r = deg 1→ Q,s (r) − 1 = deg − Q (r) − 1 = 0. (2) It is easy to check that every coordinate of m i = (g r ) satisfies −1 ≤ g r ≤ 1. Indeed, g r ≥ −1 is obvious; to check g r ≤ 1, note that every vertex r is adjacent to at most two vertices in Q 0 , so we only need to show that in the second case of (7.4), it is impossible to have deg 1→ Q,s (r) + deg 0← Q,s (r) ≥ 2. But this equality implies that r is adjacent to at least two vertices in Q 0 ; it follows from the description of type A quivers that r and two vertices in Q 0 form a 3-cycle, which contradicts the condition of the second case.
The following main theorem gives a bijective construction of ϑ Q,m 0 = x[a Q ].
Let m 0 be defined as in Lemma 7.1. Then there is a bijection ϕ between the set of GCS and the set of broken lines for m 0 with endpoint Q, such that each GCS s is sent to a broken line γ = ϕ(s) satisfying Mono(γ) = z s (defined in (4.7)): using notation in Definition 7.7, the broken line γ can be explicitly described as follows: (i) it has ℓ + 1 domains of linearity L 0 , L 1 , . . . , L ℓ (where L 0 is unbounded); (ii) it bends from the domain of linearity L i−1 to L i at a point on the wall d w i ; (iv) the monomial attached to L i is z m L .

7.4.1.
We show that (i)-(iv) determine a valid broken line. The setting of [9] in the type A setting is specialized as follows: the lattice N uf = N = Z n ′ with a basis e 1 , . . . , e n ′ and with a skew-symmetric bilinear form {·, ·} : N × N → Q satisfying {e i , e j } = ǫ ij = −b ij . (Since type A is skew-symmetric, all the multipliers d 1 = · · · = d n ′ = 1 as noted in [9, p114]). The dual lattice M has a basis f 1 , . . . , f n ′ dual to e 1 , . . . , e n ′ . The vector v j defined in [9, p29] is (Here n 0 is the primitive vector annihilating the tangent space to d w i and that n 0 , m i−1 is positive.) Since n 0 = ±e w i , and by Remark 7.9(2), all coordinates of m i−1 take value in {−1, 0, 1}, we must have n 0 , m i−1 = 1. Meanwhile, by the initial scattering diagram described [9, p31], and we are left to show that v w i = m i − m i−1 , or equivalently b rw i = m i,r − m i−1,r , or equivalently . Since s (i) and s (i−1) only differ in the w i -th coordinate (where the former has coordinate 0 and the latter has coordinate 1), we have Step 2: denoting by Q i ∈ d w i the point where γ bends from the domain of linearity Note that Q i = (q (i) j ) are determined by the following conditions: We see that the coordinates of Q i can be expressed as linear combinations of q 1 , . . . , q n ′ (by treating q 1 , . . . , q n ′ as variables). For f in span(q 1 , . . . , q n ′ ) and 1 ≤ j ≤ n, we denote f ≈ q j ⇐⇒ f − q j ∈ span(q 1 , . . . , q j−1 , q n+1 , . . . , q n ′ ).
The purpose of introducing this definition is: if f ≈ q j , then f > 0 under the condition (7.5). So Step 2 is done once we prove the following lemma.
it is equivalent to show that the above degrees are both 0.
Since s (ii) Assume that Q i − Q i+1 = λ m i . To determine λ, it suffices to consider the w i -th coordinate on both sides: We prove it by fixing i and using downward induction on i ′ . For i ′ = ℓ + 1, q (i ′ ) w i = q w i , hence the statement holds. For i ′ < ℓ + 1, using (ii) we have We argue in two cases: It suffices to show that the two degrees in the following expression are 0 and 1, respectively: But then w i is also a adjustable position in s (i ′ ) , which contradicts the property of being "smallest" in the definition of w i ′ (Definition 7.7).

If deg 0←
Q,s (i ′ ) (w i ) ≥ 2, then by our assumption on Q (see (7.1)), there must be two arrows w i → (w i − 1) and w i → (w i + 1) in Q satisfying s w i +1 = 0. Consider the longest path starting from w i and going left: j ← (j + 1) ← · · · ← (w i − 1) ← w i in Q s (i ′ ) (defined in Remark 7.6). Then j is a sink in Q s (i ′ ) , hence adjustable and satisfying j < w i < w i ′ , again contradicts the property of being "smallest" in the definition of w i ′ .

7.4.2.
The map ϕ is bijective. It is easy to see that ϕ injective, since different GCS determine different sets of w i ( i.e., the sets of walls where the broken line bend), thus determine different broken lines. Thus to show the bijectivity of ϕ, it suffices to show that the number of GCC is not less than the number of broken lines. To show the latter, we use [9,Theorem 7.5 (4)] which asserts that the cluster variable x[a Q ] is equal to the theta function ϑ Q,m 0 . Since the number of GCS is equal to x[a Q ]| x 1 =···=x n ′ =1 , on the other hand each broken line contributes at least 1 to x[a Q ]| x 1 =···=x n ′ =1 , we conclude that the number of GCS is not less than the number of broken lines. Therefore ϕ is bijective. This completes the proof of Theorem 7.10.
Note that the above argument implies that each broken line that contributes to ϑ Q,m 0 will not bend on any outgoing walls. 7.5. A bijection between GCSs and broken lines (in the general case). To extend Theorem 7.10 to odd ranks, we need to consider principal coefficients. We denote byx[a] the cluster variable in A prin that corresponds to the cluster variable x[a] in A.
Lemma 7.12. Let A be a (coefficient-free) cluster algebra of rank n, and A prin the corresponding cluster algebra with principal coefficients. Let x[a] be a cluster variable in A with g-vector g ∈ Z n . Then the corresponding cluster variablex[a] in A prin has g-vector Proof. Recall that the g-vector of x[a] ∈ A is the multidegree of the corresponding cluster variable in principal coefficients. More precisely, start with matrixB t 0 = B I , and for each mutation t k --t ′ ,B t andB t ′ are related by the rule the new cluster variable is determined by Each cluster variable is homogeneous with the assignment that, This multidegree is the g-vector of the cluster variable.
The g-vector ofx[a] ∈ A prin is defined similarly as above, with B andB t 0 being replaced byB , respectively. Using (7.7), it can be proved by a simple induction that, (In fact, D t = 0 if the sign-coherence conjecture is true [17,Conjecture 8.8]; in particular, it is true if B is skew-symmetric, but we do not need this fact in this paper.) Using (7.8), it can be proved by a simple induction thatx[a] can be obtained from x[a] by the substitution x i →x i and x n+i →x n+ix2n+i , for 1 ≤ i ≤ n. Since the multidegree (i.e., the g-vector) ofx[a] must be g 0 where g is the multidegree (i.e., the In the rest we show that Theorem 7.10 can be adapted to A prin . Theorem 7.13. AssumeQ = (q 1 , q 2 , . . . , q 2n ′ ) such that 0 < q i ≪ q 1 ≪ q 2 ≪ · · · ≪ q n , for each i = n + 1, . . . n ′ .
(There is no condition on q n ′ +1 , . . . , q 2n ′ .) Let where m 0 is defined in Lemma 7.1. Then there is a bijection ϕ between the set of GCS and the set of broken lines form 0 with endpointsQ, such that each GCS s is sent to a broken line γ = ϕ(s) satisfying Mono(γ) x n ′ +1 =···=x 2n ′ =1 = z s (defined in (4.7)): using notation in Definition 7.7, the broken line γ can be explicitly described as follows: (i) it has ℓ + 1 domains of linearity L 0 , L 1 , . . . , L ℓ (L 0 is unbounded); (ii) it bends from the domain of linearity L i−1 to L i at a point on the wall d w i ; (iv) the monomial attached to L i is zm L .
Proof. SinceB is full rank, the Fundamental Assumption in [9] is satisfied, thus the proof in §7.4 works with little change; we point out the nontrivial revision change.
In Step 1, n 0 is replaced byñ 0 = n 0 ∈ R 2n ′ . Since the first n ′ coordinates ofm i−1 form the vector m i−1 , we have ñ 0 ,m i = n 0 , m i = 1. The function attached to the wall d i is Thus > 0 as in Lemma 7.11.

Examples
In this section, we give several examples to illustrate the computation of cluster variables and cluster monomials using methods introduced in previous sections.
(But at least it is easy to see that there are 27 terms; indeed, since each pair has 3 choices (0, 0), (1, 1), (0, 1), the total number of GCSs is 3 × 3 = 27.) -Using formula (4.3): there are three Dyck paths of size 2 × 2 corresponding to the three arrows 1 → 2, 2 → 3, 3 → 1: such that (i) we do not choose both u 1 and v 1 in each Dyck path, and (ii) we choose v r in the i-th Dyck path if and only if we do not choose u 3−r in the (i + 1)-th Dyck path for r = 1, 2 (by convention, the 4th Dyck path is the 1st one). In the example of GCC in Figure 10 (Right), the corresponding product x 1 x 2 x 1 x 2 x 1 x 3 Figure 10. Left: the 2 × 2 maximal Dyck path Right: An example of GCC.
For example, we consider the d-vector a = (1, 1, 1, 0, 0, 0, 0). Then the subset of vertices {i|a i = 1} is equal to the set of vertices Q 0 of the full linear subquiver Q = 1 → 2 ← 3. It is a subquiver of an extended linear subquiver P and after completing P , we get a completely extended linear quiver Q ′ as shown in Figure 12.

Appendix: a bijection between T -paths and GCCs
In this section, we first recall the construction of T -paths and the formula of cluster variables using T -paths as in [18], then give a bijective proof of Theorem 4.6 via T -paths.
Let P be a convex polygon with n+3 vertices. A diagonal of P is a line segment connecting two non-adjacent vertices. Two diagonals are said to be crossing if they intersect in the interior of P . A triangulation T of P is a maximal set of non-crossing diagonals together with the boundary edges of P . Any triangulation has n diagonals and n + 3 boundary edges.
Our initial triangulation of P will consist of the set T = {T 1 , . . . , T n }∪{T 1,0 , T 1,1 , T n,0 , T n,1 }∪ {T i,i+1 : i ∈ [1, n − 1]}, where the first set is the set of diagonals and the last two set is the set of boundary edges. Figure 13. The initial triangulation of the quiver (Q, Q ′ ) in Example 2.1.
The process of constructing the initial triangulation starts with placing the diagonal T 1 . We obtain T 1,0 by rotating T 1 in the counterclockwise direction. The edge T 1,1 is obtained by rotating T 1 in the clockwise direction. Let v be the common vertex of T 1,0 and T 1,1 .
The boundary edge between T i and T i+1 is labeled T i,i+1 .
When i = n then T n,1 is the boundary edge clockwise from T n and T n,0 is the boundary edge counterclockwise from T n . Denote the common vertex of T n,0 and T n,1 by w.
v Figure 14. Boundary edges and diagonals of an initial triangulation We can view both the snake diagram and the triangulation T as graphs. Then there is a natural graph homomorphism p between them that sends an edge of the snake diagram to an edge of the triangulation as follows: In [8], Fomin and Zelevinsky showed that the cluster variables of A(Q) are in bijection with the diagonals of the polygon P where the initial cluster variables {x 1 , . . . , x n } corresponds to {T 1 , . . . , T n }.
Let M v,w be the diagonal connecting v and w, thus crossing the diagonals T 1 , . . . , T n . For i ∈ [1, n], let p i be the intersection of M v,w and T i . Definition 9.1. [18] A T -path α from v to w is the sequence such that 1) v = w 0 , w 1 , . . . , w l(α) = w are vertices of P .
4) l(α) is odd. 5) T i k crosses M v,w if k is even. 6) If j < k and both T i j and T i k cross M then p i j is closer to v than p i k is to v.
Let P be the set of all T -path from v to w. For any α ∈ P, let (9.2) x(α) = k odd Let a = (a 1 , . . . , a n ′ ) ∈ {0, 1} n ′ such that a i = 1 if and only if i ∈ Q. The following formula of the cluster variable x[a] is proved in [18]: Definition 9.2. We define a map ψ G,P : G → P by sending {S i,r } ∈ G to the T -path α obtained by first constructing a path α ′ 1 α ′ 2 . . . α ′ 2n+1 from v to w where α ′ 2i = T i for i ∈ [1, n], T n,0 if |S n−1,2−δ n−1 | = δ n−1 , T n,1 otherwise, then define α to be the path obtained from α ′ by canceling duplicate pairs.
Proof. In order to prove Theorem 9.3, we shall show that all maps below are bijective, and that their composition is ψ G,P : (i) We first define L, which is exactly the folding map in [15, §4.3]. As defined in [15,20], a complete T -path α from v to w is similar to a T -path from v to w defined in Definition 9.1, in the sense that we require 1), 2), and 5') the 2j-th edge T i 2j = T j (i.e., i 2j = j), where we use the order (9.4) T 1,0 < T 1,1 < T 1 < T 1,2 < T 2 < T 2,3 < · · · < T n < T n,0 < T n,1 .
Note that we do not require edges in α to be distinct. It is easy to see that a complete T -path has length 2n + 1. For simplicity, we denote α using its edge sequence. For γ ∈ M, we define (recall that p is defined in (9.1)): (9.5) L(γ) = L 1 L 2 · · · L 2n+1 , where L 2j = T j for j ∈ [1, n], L 2j+1 = p(γ j ) for j ∈ [0, n].
Note that the starting point of each L i is determined by L 1 · · · L i−1 . The union of a perfect matching γ with all the diagonals of tiles form a path α ′ γ in the snake diagram. If we consider the quotient map from the snake diagram to the triangulation of P , by identifying the diagonal edge i with T (i) and identifying diagonal edge i + 1 with T (i) i+1 , then the image of α ′ γ is the complete T -path L(γ). (ii) We show that L has a well-defined inverse map L −1 (which is the unfolding map in [15, §4.5]), thus L is bijective. Indeed, L −1 sends a complete T -path θ = L 1 · · · L 2n+1 to γ = {γ 1 , γ 2 , . . . , γ n }, where γ j is the unique edge in P l (j) ∩ p −1 (L 2j+1 ), that is, γ 1 = L 1 , γ n = L 2n+1 , and γ j = T (j) j (resp. T (j) j+1 , T j,j+1 ) if L 2j+1 is T j (resp. T j+1 , T j,j+1 ) for j ∈ [1, n − 1]. Next we show that γ is indeed a perfect matching, it suffices to prove that the edges in γ are disjoint, because it has the correct number (= n + 1) of edges.
By looking at Figure 15, we see that γ j and γ j+1 are disjoint in each case.
(iii) We show that π is bijective by giving its inverse π −1 . Suppose that α = T i 1 T i 2 · · · T i l(α) is a T -path from v to w. If n = 1, then α is already a complete T -path, so we define π −1 (α) = α. Now assume n > 1. The sequence π −1 (α) = L = L 1 L 2 · · · L 2n+1 is obtained as a result of the following algorithm.
(2) Let j run from 1 to n: if L 2j = T j , then insert T j T j to L so that the resulting L is nondecreasing with the order given in (9.4).
We claim that L is a complete T -path. Conditions 1) 2) 6') are obviously satisfied, and 5') can be proved by induction.
Combining (i)(ii)(iii) and Theorem 5.4, we have proved that ψ G,P is bijective.
Finally, we show that n i=1 x −1 i n i=0 y i = x(α). By the construction of π −1 (α) in (iii), x(α) remains unchanged if we replace α by the complete T -path π −1 (α) = T i 1 T i 2 · · · T i 2n+1 ; this is because each time we insert the pair T j T j , the extra contribution to the product (9.2) is x j x −1 j = 1. So it suffices to show By 5'), k even x −1 i k = n i=1 x −1 i , so it suffices to show n i=0 y i = k odd x i k , or to show that y j = x i 2j+1 for j ∈ [0, n]. Indeed, T i 2j+1 = L 2j+1 = p(γ j ) by (9.5), thus x i 2j+1 = w(γ j ) by the definition of the weight w in (5.1). Moreover, Theorem 5.4 asserts that w(γ j ) = y j . Thus y j = x i 2j+1 .