# Dyck Paths Categories And Its Relationships With Cluster Algebras

aa r X i v : . [ m a t h . R T ] F e b Dyck Paths Categories And Its Relationships With Cluster Algebras

Agust´ın Moreno Ca˜[email protected] Bravo R´ı[email protected] of MathematicsUniversidad Nacional de Colombia

Abstract

Dyck paths categories are introduced as a combinatorial model of the category ofrepresentations of quivers of Dynkin type A n . In particular, it is proved that there is abijection between some Dyck paths and perfect matchings of some snake graphs. Theapproach allows us to give formulas for cluster variables in cluster algebras of Dynkintype A n in terms of Dyck paths. Keywords and phrases : Auslander-Reiten quiver, cluster algebras, Dyck paths, perfectmatchings, quiver representation, snake graph.Mathematics Subject Classiﬁcation 2010 : 16G20; 16G30; 16G60.

In the last few years, researches regarding connections between cluster algebras anddiﬀerent ﬁelds of mathematics have been growing. For instance, relationships betweencluster algebras, quiver representations, combinatorics and number theory have beenreported by Fomin et al., Shiﬄer et al., K. Baur et al., Assem et al. amongst a greatnumber of mathematicians [1, 3, 6–8, 10–12].Perhaps the Catalan combinatorics (which consists of all the enumeration problemswhose solutions are Catalan numbers) is the most appropriated environment for theinvestigation of cluster algebras of Dynkin type A n . Among all these kinds of problems,it is possible to prove (for example) that the Catalan numbers count [16]:1. The number of plane binary trees with n + 1 endpoints (or 2 n + 1) vertices,2. The number of ways to parenthesize a string of length n + 1 subject to a nonassociative binary operation,3. The number of paths P in the ( x, y )-plane from (0 ,

0) to (2 n,

0) with steps (1 , , −

1) that never pass below the x -axis. Such paths are called Dyck paths ,4. The number of triangulations of an ( n + 3) polygon,5. The number of clusters of a cluster algebra of Dynkin type A n . Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos

Regarding integer friezes, we point out that Propp in [15] reminds that Conway andCoxeter completely classiﬁed the frieze patterns whose entries are positive integers, andshowed that these frieze patterns constitute a manifestation of the Catalan numbers.Speciﬁcally, that there is a natural association between positive integer frieze patternsand triangulations of regular polygons with labelled vertices. According to Baur andMarsh [3], a connection between cluster algebras and frieze patterns was establishedby Caldero and Chapoton [5], which showed that frieze patterns can be obtained fromcluster algebras of Dynkin type A n .Another example of the use of the Catalan combinatorics as a tool to describe thestructure of cluster algebras, was given by Schiﬄer et al. [6, 7, 14], who found outformulas for cluster variables based on its relations with some triangulated surfacesand perfect matchings of snake graphs. They also proved that there is a way ofobtaining the number of perfect matchings of a given snake graph by associating asuitable continued fraction deﬁned by the sign function of the graph.Given a non-negative integer n and a triangulation T of a regular polygon with ( n + 3)vertices. Caldero, Chapoton and Schiﬄer [4] gave a realization of the category C C of representations of a quiver Q C associated to a cluster C of a cluster algebra interms of the diagonals of the ( n + 3) polygon. They proved that there is a categoricalequivalence between the categories C T and Mod Q T , where C T is the category whoseobjects are positive integral linear combinations of positive roots (i.e., diagonals thatdoes not belong to the triangulation T ), whereas Mod Q T denotes the category ofmodules over the quiver Q T with triangular relations induced by the triangulation T .Follow the ideas of Caldero, Chapoton and Schiﬄer, in this paper, a combinatorialmodel of the category of representations of Dynkin quivers with relations is developedby using Dyck paths. To do that, Dyck paths categories are introduced and it is provedthat these categories are equivalent to categories of representations of Dynkin quiversof type A n . This approach allows us to realize perfect matchings of snake graphsas objects of suitable Dyck paths categories, and with this machinery a formula forcluster variables based on Dyck paths is obtained.This paper is distributed as follows; In Section 2, notation and basic deﬁnitions tobe used throughout the paper are introduced. In Section 3, we deﬁne Dyck pathscategories and some of its main categorical properties are given in Section 4. In section5, relationships between objects of the categories of Dyck paths, perfect matchings andcluster algebras are given. Fomin and Zelevinsky introduced the term cluster algebra in [10], as a subalgebra ofa ﬁeld of rational functions generated by a set of n cluster variables [8, 11, 12]. Thecluster algebras are in connection with diﬀerent topics in mathematics, as algebraiccombinatorics, Lie theory, discrete dynamical systems, tropical geometry, and others.Afterwards, Fomin, Schiﬄer et al introduced cluster algebras associated to surfaces[7, 8, 14]. yck Paths... For the sake of clarity, we remind here the deﬁnition given by Fomin et al. [11] of acluster algebra.Let T n the n -regular tree whose edges are labeled by the numbers 1 , . . . , n , so that the n -edges incident to each vertex receive diﬀerent labels. The symbol t k − t ′ is used todenote that vertices t, t ′ ∈ T n are joined by an edge labeled by k .If F is a ﬁeld isomorphic to the ﬁeld of rational functions over C (alternatively over Q ) in m independent variables, then a labeled seed of geometric type over F is a pair( e x, e B ) where;1. e x = ( x , x , . . . , x m ) is an m -tuple of elements of F forming a free generating set;that is; x , x , . . . , x m are algebraically independent, and F = C ( x , . . . , x m ),2. e B = ( b ij ) is an m × n extended skew-symmetrizable integer matrix. e B is saidto be the extended exchange matrix of the seed. Its top n × n submatrix B isthe exchange matrix.Let ( e X, e B ) be a labeled seed as above. Take an index k ∈ { , , . . . , n } . The seedmutation in direction k transforms ( e x, e B ) into the new labeled seed µ k ( e x, e B ) = e x ′ , e B ′ deﬁned as follows; e B ′ = µ k ( e B ) = ( b ′ ij ) , (1)where b ′ ij = − b ij , if i = k or j = k,b ij + b ik b kj , if b ik > b kj > ,b ij − b ik b kj , if b ik < b kj < ,b ij , otherwise . The extended cluster e x ′ = ( x ′ , . . . , x ′ m ) is given by the identiﬁcations x ′ j = x j for j = k , whereas x ′ k ∈ F is determined by the exchange rule. x k x ′ k = Y b ik > x b ik i + Y b ik < x − b ik i . (2)A seed pattern is deﬁned by assigning a labeled seed ( e x ( t ) , e B ( t )) to every vertex, t ∈ T n ,so that the seeds assigned to the end points of any edge t k − t ′ are obtained from eachother by the seed mutation in direction k . A seed pattern is uniquely determined byone of its seeds.Let ( e x ( t ) , e B ( t )) t ∈ T n be a seed pattern as above, and let X = S t ∈ T n x ( t ) be the set of allcluster variables appearing in its seeds. We let the ground ring be R = C [ x n +1 , . . . , x m ]the polynomial ring generated by the frozen variables .The cluster algebra A (of geometric type over R ) associated with the given seed patternis the R -subalgebra of the ambient ﬁeld F generated by all cluster variables A = R [ X ]. Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos

For quivers, cluster algebras are deﬁned as follows:Fix an integer n ≥

1. In this case, a seed (

Q, u ) consists of a ﬁnite quiver Q with-out loops or 2-cycles with vertex set { , . . . , n } , whereas u is a free-generating set { u , . . . , u n } of the ﬁeld Q ( x , . . . , x n ).Let ( Q, u ) be a seed and k a vertex of Q . The mutation µ k ( Q, u ) of (

Q, u ) at k is theseed ( Q ′ , u ′ ), where;(a) Q ′ is obtained from Q as follows;(1) reverse all arrows incident with k ,(2) for all vertices i = j distinct from k , modify the number of arrows between i and j , in such a way that a system of arrows of the form ( i r −→ j, i s −→ k, k t −→ j ) is transformed into the system ( i r + st −→ j, k s −→ i, j t −→ k ). Andthe system ( i r −→ j, j t −→ k, k s −→ i ) is transformed into the system ( i r − st −→ j, i s −→ k, k t −→ j ). Where, r , s and t are non-negative integers, an arrow i l −→ j , with l ≥ l arrows go form i to j and an arrow i l −→ j ,with l ≤ − l arrows go from j to i .(b) u ′ is obtained form u by replacing the element u k with u k = 1 u k Y arrows i → k u i + Y arrows k → j u j . (3)If there are no arrows from i with target k , the product is taken over the empty setand equals 1. It is not hard to see that µ k ( µ k ( Q, u )) = (

Q, u ). In this case the matrixmutation B ′ has the form b ′ ij = ( − b ij , if i = k or j = k,b ij + sgn ( b ik )[ b ik b kj ] + , else , where [ x ] + = max(x , Q is a ﬁnite quiver without loops or 2-cycles withvertex set { , . . . , n } , the following interpretations have place:1. the clusters with respect to Q are the sets u appearing in seeds, ( Q, u ) obtainedfrom a initial seed (

Q, x ) by iterated mutation,2. the cluster variables for Q are the elements of all clusters,3. the cluster algebra A ( Q ) is the Q -subalgebra of the ﬁeld Q ( x , . . . , x n ) generatedby all the cluster variables.As example, the cluster variables associated to the quiver Q = 1 −→ { x , x , x x , x + x x x , x x } . yck Paths... Let S be a connected oriented 2 − dimensional Riemann surface with nonempty bound-ary, and let M be a nonempty ﬁnite subset of the boundary of S , such that eachboundary component of S contains at least one point of M . The elements of M arecalled marked points . The pair ( S, M ) is called a bordered surface with marked points .Marked points in the interior of S are called punctures (For technical reasons, we re-quire that ( S, M ) is not a disk with 1,2 or 3 marked points) [7].An arc γ in ( S, M ) is a curve in S , considered up to isotopy, such that:(i) the endpoints of γ are in M ,(ii) γ does not cross itself, except that its endpoints, may coincide,(iii) except for the endpoints, γ is disjoint from the boundary of S ,(iv) γ does not cut out a monogon or a bigon.Curves that connect two marked points and lie entirely on the boundary of S withoutpassing through a third marked point are boundary segments. Note that boundarysegments are not arcs. For any two arcs γ , γ ′ in S , let e ( γ, γ ′ ) be the minimal numberof crossings of arcs α and α ′ , where α and α ′ range over all arcs isotopic to γ and γ ′ ,respectively. We say that arcs γ and γ ′ are compatible if e ( γ, γ ′ ) = 0.An ideal triangulation is a maximal collection of pairwise compatible arcs (togetherwith all boundary segments). Triangulations are connected to each other by sequencesof ﬂips. Each ﬂip replaces a single arc γ in a triangulation T by a (unique) arc γ ′ = γ that, together with the remaining arcs in T , forms a new triangulation.According to Schiﬄer and Canakci [7] , Fomin, Shapiro and Thurston [10] associateda cluster algebra A ( S, M ) to any bordered surface with marked points (

S, M ), and thecluster variables of A ( S, M ) are in bijection with the (tagged) arcs of (

S, M ).The following theorem regarding relationships between cluster algebras and surfacetriangulations was obtained Fomin, Shapiro, and Thurston [8, 9]

Theorem 1. [14]

Fix a bordered surface ( S, M ) and let A be the cluster algebra asso-ciated to the signed adjacency matrix of a tagged triangulation. Then the (unlabeled)seed Σ T of A are in bijection with tagged triangulations T of ( S, M ) , and the clustervariables are in bijection with the tagged arcs of ( S, M ) (so we can denote each by x γ , where γ is a tagged arc). Moreover, each seed in A is uniquely determined byits cluster. Furthermore, if a tagged triangulation T ′ is obtained from another taggedtriangulation T by ﬂipping a tagged arc γ ∈ T and obtaining γ ′ , then Σ T ′ is obtainedfrom Σ T by the seed mutation replacing x γ by x γ ′ . In this section, we recall the deﬁnition of a snake graph, the number of perfect match-ings associated to these graphs, and the way that these concepts can be used to ﬁndout a formula for the cluster variables of a cluster algebra associated to a surface[6, 7, 14].A tile G is a square of ﬁxed side-length in the plane whose sides are parallel or orthog-onal to the ﬁxed basis. Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos G West East.NorthSouthWe consider a tile G as a graph with four vertices and four edges in the obvious way.A snake graph G is a connected graph consisting of a ﬁnite sequence of tiles G , . . . , G d with d ≥

1, such that for each i = 1 , . . . , d − G i and G i +1 share exactly one edge e i and this edge is either the north edge of G i and the south edge of G i +1 or the east edge of G i and the west edge of G i +1 .(ii) G i and G j have on edge in common whenever | i − j | ≥ G i and G j are disjoint whenever | i − j | ≥ G [ i, i + t ] = ( G i , . . . , G i + t ) is the subgraph of G = ( G , . . . , G n ), the d − e , . . . , e d − which are contained in two tiles are called interior edges of G andthe other edges are called boundary edges . A perfect matching P of a graph G is asubset of the set of edges of G such that each vertex of G is incident to exactly oneedge in P . We let Match( G ) denote the set of all perfect matchings of the graph G .Let T be a triangulation of a surface ( S, M ) and let γ be an arc in ( S, M ) which isnot in T . Choose an orientation on γ , let s ∈ M be its starting point, and let t ∈ M be its endpoint. Denote by s = p , p , . . . , p d +1 = t the ordered points of intersectionof γ and T . For j = 1 , , . . . , d , let τ i j be the arc of T containing p j , and let ∆ j − and∆ j be the two triangles in T on either side of τ i j . Then, for j = 1 , . . . , d −

1, arcs τ i j and τ i j +1 form two sides of the triangle ∆ j in T and we deﬁne e j to be the third arcin this triangle.Let G j be the quadrilateral in T that contains τ i j as a diagonal (a tile) whose edges arearcs in T , thus, they are labeled edges. Deﬁne a sign function f of the edges e , . . . , e d by f ( e j ) = ( +1 , if e j lies on the right of γ when passing through ∆ j , − , otherwise. (4)The labeled snake graph G γ = ( G , . . . , G d ) with tiles G i and sign function f is calledthe snake graph associated to the arc γ . Each edge e of G γ is labeled by an arc τ ( e )of the triangulation T . Such an arc deﬁnes the weight x ( e ) of the edge e as the clustervariable associated to the arc τ ( e ). Thus x ( e ) = x τ ( e ) .In [14] Musiker, Schiﬄer, and Williams showed a combinatorial formula for clustervariables of a cluster algebra of surface type A ( S, M ) with principal coeﬃcients Σ T =( x T , y T , B T ). In such a case, if γ is a generalized arc in a triangulation T which hasno self-folded triangles, and G γ is its snake graph. Then the corresponding clustervariable x γ is given by the identity x γ = 1cross( γ, T ) X P ∈ Match( G γ ) x ( P ) . (5) yck Paths... where the sum runs over all perfect matchings of G γ , the summand x ( P ) = Q e ∈ P x ( e )is the weight of the perfect matching P , and cross( T, γ ) = Q dj =1 x τ ij is the product ofall initial cluster variables whose arcs cross γ .A relationship between cluster variables and continued fractions is described by Schif-ﬂer and Canakci in [7], who claimed that, the numerator of a continued fraction isequal to the number of perfect matchings of the corresponding abstract snake graph,and that it can therefore be interpreted as the number of terms in the numerator of theLaurent expansion of an associated cluster variable. Thus, the Laurent polynomialsof the cluster variable can be recovered from the continued fraction. In 2006 [4], Caldero, Chapoton, and Schiﬄer introduced the category of diagonals ofa polygon with n + 3 sides associated to a triangulation T , in this case, the diagonalsare called roots which can be classiﬁed as negative or positive, negative roots are thoseroots belonging to the triangulation T .The combinatorial C -linear additive category C T is described as follows. The objectsare positive integral linear combinations of positive roots, and the space of morphismsfrom a positive root α to a positive root α ′ is a quotient of the vector space over C spanned by pivoting paths from α to α ′ . The subspace which deﬁnes the quotient isspanned by the so-called mesh relations . For any couple α, α ′ of positive roots suchthat α is related to α ′ by two consecutive pivoting elementary moves with distinctpivots, the mesh relations are given by the identity P v ′ P v = P v ′ P v , where v , v (resp. v ′ v ′ ) are the vertices of α (resp. α ′ ) such that P v ′ P v = α ′ .Let T be a triangulation, then one can deﬁne a planar tree t T as follows. Its verticesare the triangles of T and the edges connect adjacent triangles. In the same way, wecan deﬁne a graph Q T whose vertices are the inner edges of T and are related to eachother by an edge, if they bound the same triangle. An orientation can be deﬁned byusing graph Q T , in such a way that a vertex i connects a vertex j (denoted i → j ),if − α j can be obtained from the diagonal − α i by rotating anticlockwise about theircommon vertex.The triangulation T deﬁnes a C − linear abelian category Mod Q T , that is, the categoryof modules over the quiver Q T , such that in any triangle, the composition of twosuccessive maps is zero. These relations are named triangle relations .The following result regarding the category of diagonals was given by Caldero, Chapo-ton, and Schiﬄer in [4]. Theorem 2. If T is a triangulation of a polygon with n + 3 sides then there is a cat-egorical equivalence between the category of diagonals C T and the category of modulesover the quiver Q T . In this section, we introduce the category of Dyck paths of length 2 n . Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos

A Dyck path is a lattice path in Z from (0 ,

0) to ( n, n ) with steps (1 ,

0) and (0 , y = x . The number of Dyck paths oflength 2 n is equal to the nth Catalan number [16]. Henceforth, Dyck words as deﬁnedin the following Remark 3 are used to denote Dyck paths. Remark 3.

The set of

Dyck words is the set of words w ∈ X ∗ = { U, D } ∗ characterizedby the following two conditions [2]: • for any left factor u of w , | u | U ≥ | u | D , • | w | U = | w | D .where | w | a is the number of occurrences of the letter a ∈ X in the word w and theword u is a left factor of the word w = uv .Let D n be the set of all Dyck paths of length 2 n , let UW D = Uw . . . w n − D be aDyck path in D n with A = { UD, DU, UU, DD } the set of choices in W .The support of UW D (denoted by Supp

UW D ⊆ { , , . . . , n − } = n-1 ) is a set ofindices such thatSupp UW D = { q ∈ n-1 | w q = UD or w q = UU , 1 ≤ q ≤ n − } . A map f : A −→ A such that for any w ∈ A , it holds that f ( w ) = f ( ab ) = w − = ba , a, b ∈ { U, D } is said to be a shift . An unitary shift is a map f i : D n −→ D n suchthat f i ( Uw . . . w i − w i w i +1 . . . w n − D ) = Uw . . . w i − f ( w i ) w i +1 . . . w n − D. We will denote a unitary shift by a vector of maps from D n to itself of the form(1 , . . . , i − , f i , i +1 , . . . , n − ), where 1 k is the identity map associated to the i -thcoordinate.An elementary shift is a composition of unitary shifts . A shift path of length mUW D −→ UW D −→ · · · −→ UW m D −→ UV D from

UW D to UV D is a com-position of elementary shifts. The set of all Dyck paths in a shift path between

UW D and

UV D will be denoted by J . For notation, we introduce the identity shift as theelementary shift (1 , . . . , n − ). Irreversibility condition . Suppose that a map R : D n → D n is deﬁned by theapplication of successive elementary shifts to a given Dyck path. Then R is said tobe an irreversible relation over D n if and only if elementary shifts transforming Dyckpaths (from one to the other) are not reversible. In other words, if an elementary shift F = f p ◦ · · · ◦ f p q transforms a Dyck path UW D into a Dyck path

UV D then thereis not an elementary shift F = f p ◦ · · · ◦ f p q transforming UV D into

UW D , for some p, q ∈ Z + . Shift Relation . If there exist two paths G ◦ F and G ′ ◦ F ′ of irreversible relations(of length 2) transforming a Dyck path UW D into the Dyck path

UV D over R in thefollowing form: yck Paths... UW D UW ′ DUW ′′ D UV D , ✏✏✏✏✏✶PPPPPq ✏✏✏✏✏✶PPPPPq FF ′ GG ′ with W ′ = W ′′ . Then G ◦ F is said to be related with G ′ ◦ F ′ (Denoted G ◦ F ∼ R G ′ ◦ F ′ )whenever G ′ = F and G = F ′ . Category of Dyck paths of length n . As for the case of diagonals [4], we can alsodeﬁne a k -linear additive category ( D n , R ) based on Dyck paths, in this case, objects are k -linear combinations of Dyck paths in D n with space of morphisms from a Dyckpath UW D to a Dyck path

UV D over R being the setHom ( D n ,R ) ( UW D, UV D ) = h{ g | g is a shift path over R }i / h∼ R i . The vector space

Hom ( D n ,R ) ( UW D, UV D ) = 0 if and only if there are shift pathsfrom UW D to UV D and \ i ∈ J Supp UW i D = ∅ , (6)for each shift path, with UW D and

UV D in D n .Figure 1 shows the elementary shifts over ( D , R ) associated to an irreversible relation R deﬁned over the set of all Dyck paths of length 6. And such that, R ( UW D ) = ( f ( UW D ) , if w = UD,f ( UW D ) , if w = UD. (7) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)✻ ✻✛✛

Figure 1: Elementary shifts in ( D , R ). R i ...i k j ...j m If n = { , , . . . , n } is an n -point chain then C (1,n) stands for all admissible subchains C of n with min C = 1 and max C = n . For instance, { , , } and { , , } are three-pointsubchains contained in C (1,8) . The admissible subchain C = { j , . . . , j m , i , . . . , i k } ⊆ n must satisfy the following constraints for 1 ≤ r k and 1 ≤ s m : Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos • If i = 1 and k = m then i < j < · · · < i k < j m = n . • If i = 1 and k = m + 1 then i < j < · · · < i k < j m < i k = n . • If j = 1 and k = m then j < i < · · · < j m < i k = n . • If j = 1 and m = k + 1 then j < i < · · · < j m < i k < j m = n .Let σ : { i , j } → { , } be a map such that σ ( i ) = 1 and σ ( j ) = 0. For a ∈ { i , j } ,we assume i r ( i r + σ ( a ) ) ∈ { i , . . . , i k } and j r +1 − σ ( a ) ( j r ) ∈ { j , . . . , j m } . The orientationbetween the interval [ i r , j r +1 − σ ( a ) ] ([ j r , i r + σ ( a ) ]) over Z + is from left to right, denoted −−→ [ a, b ] (right to left, denoted ←−− [ a, b ]). The following words are deﬁned by using intervals: • w t = min { w s | w i r ≤ w s ≤ w j r +1 − σ ( a ) , w s = UD } (cid:0) w t = max { w s | w j r ≤ w s ≤ w i r + σ ( a ) , w s = UD } (cid:1) , • w p = min { w s | w t < w s ≤ w j r +1 − σ ( a ) , w s = DU } (cid:0) w p = max { w s | w j s ≤ w s < w t , w s = DU } (cid:1) .We introduce the following elementary shifts:ES1. If w s = UD for all w s ∈ [ i r , j r +1 − σ ( a ) ] ([ j r , i r + σ ( a ) ]), −−−−−−−→ [ j r − σ ( a ) , i r ] ←−−−−−−−−− [ i r , j r +1 − σ ( a ) ] −−−−−−−−−−−→ [ j r +1 − σ ( a ) , i r +1 ] , ( ←−−−−−−−−− [ i r + σ ( a ) − , j r ] −−−−−−−→ [ j r , i r + σ ( a ) ] ←−−−−−−−−− [ i r + σ ( a ) , j r +1 ]) , then g ( UW D ) = f j r +1 − σ ( a ) ◦ · · · ◦ f i r ( UW D ) , if there exists s ∈ Z + such that j r − σ ( a ) ≤ s ≤ i r , | s − j r | > w x = UD if s ≤ x ≤ i r over [ j r − σ ( a ) , i r ] and w y = ( UD, if y = j r +1 − σ ( a ) , DU, otherwise, (8)over [ j r +1 − σ ( a ) , i r +1 ] for j r +1 − σ ( a ) = n − j r − σ ( a ) , i r ]for j r +1 − σ ( a ) = n − g ( UW D ) = f i r + σ ( a ) ◦ · · · ◦ f j r ( UW D ) , if there exists s ∈ Z + such that i r + σ ( a ) ≤ s ≤ j r +1 , | s − i r + σ ( a ) | > w x = UD if i r + σ ( a ) ≤ x ≤ s over [ i r + σ ( a ) , j r +1 ] and w y = ( UD, if y = j r , DU, otherwise, (9)over [ i r + σ ( a ) − , j r ] for j r = 1 or the ﬁrst condition over [ i r + σ ( a ) , j r +1 ] for j r = 1 ! ,with i r = 1 ( i r + σ ( a ) = n − yck Paths... ES2. If t = 1 or n − g ( UW D ) = f t ( UW D ). Then there are not elementaryshifts associated to elements of a subchain diﬀerent from 1 and n − w i r < w t < w j r +1 − σ ( a ) ( w j r < w t < w i r + σ ( a ) ) then g ( UW D ) = f t ( UW D ).ES4. If w p = w j r +1 − σ ( a ) ( w j r ) then g ( UW D ) = ( f i r +1 ◦ · · · ◦ f j r +1 − σ ( a ) ( UW D ) if j r +1 − σ ( a ) = n − f j r +1 − σ ( a ) ( UW D ) if j r +1 − σ ( a ) = n − g ( UW D ) = ( f i r + σ ( a ) − ◦ · · · ◦ f j r ( UW D ) if j r = 1, f j r ( UW D ) if j r = 1 . ! ES5. If w t < w p < w j r +1 − σ ( a ) ( w j r < w p < w t ) then g ( UW D ) = f p ( UW D ).For a given subchain C = { j , . . . , j m , i , . . . , i k } ⊆ n-1 , two Dyck paths D and D ′ oflength 2 n are said to be related by a relation of type R i ...i k j ...j m if there is an elementaryshift ES i , 1 ≤ i ≤ D into D ′ or D ′ into D . Proposition 4.

The relation R i ...i k j ...j m is irreversible. Proof.

Suppose that there is an elementary shift f r ◦ · · · ◦ f r t from a Dyck path UW D to a Dyck path

UV D and that there is an elementary shift f r ◦ · · · ◦ f r t from UV D to a

UW D , then we have ﬁve cases:(i) If f r ◦ · · · ◦ f r t arises from ES1 over [ i r , j r +1 − σ ( a ) ]. Shifts ES2, ES3 and ES5allow to conclude that from UV D to a

UW D , f t = f j r +1 − σ ( a ) ◦ · · · ◦ f i r or f p = f j r +1 − σ ( a ) ◦ · · · ◦ f i r and this is a contradiction. If ES1 is an elementaryshift from UV D to UW D , then two cases arise: If j r +1 − σ ( a ) = n −

1, thus

UV D equals Uv . . . v jσ ( a ) . . . v js . . . v ir − | {z } UD v ir . . . v jr +1 − σ ( a ) | {z } UD ( v jr +1 − σ ( a )+1 ) . . . v ir +1 | {z } DU v ri +1+1 . . . v n − D, it turns out that f j r +1 − σ ( a ) ◦ · · · ◦ f i r ( UV D ) has the form Uw . . . w jσ ( a ) . . . w js . . . ( w ir − | {z } UD w ir . . . w jr +1 − σ ( a ) | {z } DU ( w jr +1 − σ ( a )+1 ) . . . w ir +1 | {z } DU w ri +1+1 . . . v n − D, which is a contradiction. If j r +1 − σ ( a ) = n − UV D is equal to Uv . . . v jσ ( a ) . . . v js . . . v ir − | {z } UD v r . . . v jr +1 − σ ( a ) | {z } UD D, and f j r +1 − σ ( a ) ◦ · · · ◦ f i r ( UV D ) has the shape Uw . . . w jσ ( a ) . . . w js . . . w ir − | {z } UD w r . . . w jr +1 − σ ( a ) | {z } DU D, again a contradiction. We also get a contradiction if a elementary shift is doneby using ES4 from UV D to a

UW D , indeed, in these cases it holds that, if j r +1 − σ ( a ) = n −

1, there are t and p such that p = j r +1 − σ ( a ) < t ≤ i r +1 and UV D is equal to Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos Uv . . . v ir − v ir . . . v jr +1 − σ ( a ) | {z } DU v jr +1 − σ ( a )+1 . . . v t | {z } UD v t +1 . . . v n − D, and f j r +1 − σ ( a ) ◦ · · · ◦ f i r ( UV D ) is Uw . . . w ir − w ir . . . w jr +1 − σ ( a ) | {z } UD w jr +1 − σ ( a )+1 . . . w t | {z } UD w t +1 . . . w n − D. If v j r +1 − σ ( a ) = n − f r +1 − σ ( a ) = f r +1 − σ ( a ) ◦ · · · ◦ f i r but this is a contradiction.(ii) If f r ◦ · · · ◦ f r t arises from ES2 over [ i , j ] then we cannot use elementary shiftsdeﬁned in cases ES1, ES4, ES5 or ES3, provided that, i = 1, t = p or 1 < t < j .Therefore, ES2 guarantees the existence of a walk from UV D to UW D such that;

U v |{z} UD . . . v j . . . v n − D, and f ( UW D ) has the form

U w |{z} DU . . . w j . . . w n − D, which is a contradiction (if t = n −

1, the proof is dual).(iii) If f r ◦ · · · ◦ f r t arises from ES3 over [ i r , j r +1 − σ ( a ) ], provided that, i r < t < p

UW D , UV D equals Uv . . . v i r . . . v t − | {z } DU v t |{z} UD . . . v j r +1 − σ ( a ) . . . v n − D, and f t ( UV D ) has the shape Uw . . . w i r . . . w t | {z } DU w t +1 . . . w j r +1 − σ ( a ) . . . w n − D, but this is a contradiction.(iv) If f r ◦ · · · ◦ f r t arises from ES4 over [ i r , j r +1 − σ ( a ) ], provided that t < p , we donot use ES2, ES3 nor ES5. If j + 1 − σ ( a ) = n −

1, we cannot use ES1. If j + 1 − σ ( a ) = n − UV D to a

UW D (Note that, it is notnecessary with v m = UD for all s ∈ [ j r +1 − σ ( a ) + 1 , i r +1 ]) UV D is equal to Uv . . . v ir − v ir . . . v t . . . v p − | {z } DU v p v jr +1 − σ ( a )+1 . . . v ir +1 | {z } UD v ir +1+1 . . . v n − D, it turns out that g ( UV D ) has the form Uw . . . w ir − w ir . . . w t . . . w p − w p w jr +1 − σ ( a )+1 . . . w ir +1 | {z } DU w ir +1+1 . . . w n − D, which is a contradiction. Using ES5 from UV D to UW D , if j r +1 − σ ( a ) = n − UV D is equal to Uv . . . v i r . . . v t . . . v p − | {z } UD v p |{z} DU v j r +1 − σ ( a ) +1 . . . v i r +1 v i r +1 +1 . . . v n − D, yck Paths... and UW D has the shape Uw . . . w i r . . . w t . . . w p | {z } UD w j r +1 − σ ( a ) +1 . . . w i r +1 | {z } f ( ab ) w i r +1 +1 . . . w n − D, again a contradiction. If j r +1 − σ ( a ) = n − UV D is equal to Uv . . . v i r . . . v t − v t . . . v p − | {z } UD v p |{z} DU D, it turns out that UW D has the shape Uw . . . w i r . . . w t − w t . . . v p | {z } UD D, this is a contradiction.(v) If f r ◦ · · · ◦ f r t arises from ES5 over [ i r , j r +1 − σ ( a ) ]. Then we cannot use ES1,ES2, ES3 nor ES4, because f p = f j r +1 − σ ( a ) ◦ · · · ◦ f i r and t < p . Using ES5 from UV D to a

UW D , we observe that

UV D is equal to Uv . . . v i r . . . v t − v t . . . v p − | {z } UD v p |{z} DU . . . v j r +1 − σ ( a ) . . . v n − D, and f p ( UW D ) has the form Uw . . . w i r . . . w t − w t . . . w p | {z } UD v p +1 . . . v j r +1 − σ ( a ) . . . v n − D, again this is a contradiction.Taking into account that if f r ◦· · ·◦ f r t arises from ES ES ES ES ES i r , j r + σ ( a ) ] then same arguments as described above applied dually allow to concludethe proposition. We are done. (cid:3) A n − -Dyck Paths Categories For n ≥ A n − -Dyck paths category is a category of Dyck paths ( D n , R )where R is a relation of type R i ...i k j ...j m as described before. As an example we let( D , R ) denote the A -Dyck paths category with the admissible subchain 1 < D , R ).We let S denote the set of all Dyck paths with exactly n − S . Proposition 5.

Let

UW D be a Dyck path of length n , then UW D ∈ S if and onlyif there is a unique sequence w l w l +1 . . . w r − w r such that w i = ( UD, if l ≤ i ≤ r,DU, otherwise. (10) Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)✲ ✛❍❍❍❍❍❍❍❍❍❥ ✟✟✟✟✟✟✟✙❄(cid:0)(cid:0)✒ (cid:0)(cid:0)✒(cid:0)(cid:0)✒❅❅❅❅❘ ❅❅❅❅❘❅❅❅❅❘✛ ❄ ❄

Figure 2: Elementary shifts in an A -Dyck paths category. Proof.

Firstly, let δ be a map δ : { ) , ( } → { U, D } where left bracket is associated tothe letter U and right bracket is associated to the letter D , suppose UW D ∈ S , thenthere exist bracket-subchains such that UW D can be written in the following form( )( |{z} )( |{z} . . . )( |{z} l − )( |{z} l − ( ( ) |{z} l . . . ( ) |{z} r ) )( |{z} r +1 . . . )( |{z} n − )( |{z} n − ) , therefore w i = UD if l ≤ i ≤ r and w i = DU . On the other hand, suppose UW D hasa unique subsequence w l w l +1 . . . w r − w r that satisﬁes identity (10), then if we apply δ − to UW D , the sequence( ) |{z} ( ) |{z} . . . ( ) |{z} l − ( ( ) |{z} l . . . ( ) |{z} r ) ( ) |{z} r +1 . . . ( ) |{z} n − ( ) |{z} n − , is obtained, therefore UW D ∈ S . We are done. (cid:3) Lemma 6.

Let

UW D be a Dyck path in S , and integers r, l deﬁned as in Proposition 5 with | r − l | > , then there exists an elementary shift from UW D to another Dyck pathwith exactly n − peaks. Proof

Let

UW D be a Dyck path in S , let l and r be positive integers such that w m = UD for l ≤ m ≤ r . Let l ∈ [ i r , j r +1 − σ ( a ) ], we have the following cases: yck Paths... (1) If l = i r = 1, then g ( UW D ) =

U f ( w ) | {z } DU w . . . w r | {z } UD w r +1 . . . w n − D ∈ S. (2) If l = i r = 1, then there is p = l − j r − σ ( a ) , i r ] such that g ( UW D ) = Uw . . . f ( w p ) w l . . . w m | {z } UD . . . w n − D ∈ S. (3) If i r < l < j r +1 − σ ( a ) , then g ( UW D ) = Uw . . . f ( w l ) | {z } DU w l +1 . . . w r | {z } UD w r +1 . . . w n − D ∈ S. (4) If l = j r +1 − σ ( a ) and | l − r | >

0, then r ∈ [ i r , j r +1 − σ ( a ) ] with | r − r | > i r ≤ r < j r +1 − σ ( a ) , there is p = r + 1 such that, if p = j r +1 − σ ( a ) then g ( UW D ) = uw . . . w l . . . w r f ( w p ) | {z } UD . . . w n − D ∈ S, if p = j r +1 − σ ( a ) = n −

1, then g ( UW D ) = Uw . . . w l . . . w r f ( w p ) | {z } UD D ∈ S, or if p = j r +1 − σ ( a ) = n − g ( UW D ) = Uw . . . w l . . . w r f ( w p ) . . . f ( w i r ) | {z } UD . . . w n − D ∈ S. (4.2) If r = j r +1 − σ ( a ) g ( UW D ) = Uw . . . w l . . . w i r − | {z } UD f ( w i r ) . . . f ( w r ) | {z } DU . . . D ∈ S. (4.3) Now, if | r − r | > r = r + 1 and r > i r +1 + 2 then g ( UW D ) = Uw . . . f ( w l ) . . . f ( w i r +1 ) | {z } DU w i r +1 +1 . . . w r | {z } UD . . . D ∈ S. For r ∈ [ j r +1 − σ ( a ) , i r +1 ] with | r − r | ≥ s = t = i r +1 = n −

1, then g ( UW D ) = Uw . . . w l . . . w r − | {z } UD f ( w r ) | {z } DU D ∈ S. On the other hand, if s = t = i r +1 = n −

1, then there is p ∈ [ i r +1 , j r +2 − σ ( a ) ]satisfying ﬁrst condition of (4.1). Thus, if j r +1 − σ ( a ) < s < i r +1 , it holdsthat g ( UW D ) = Uw . . . w l . . . w r − | {z } UD f ( w r ) | {z } DU . . . w n − D ∈ S. Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos (4.5) If s = j r +1 − σ ( a ) then | r − r | > | r − r | = 0, | l − f | = 0 which is acontradiction) g ( UW D ) = Uw . . . w l . . . w i r − | {z } UD f ( w i r ) . . . f ( w r ) | {z } DU w r +1 . . . w n − D ∈ S. (4.6) Now, suppose that in UW D | r − r | >

0, then it satisﬁes the ﬁrst conditionin (4.3).In case that l ∈ [ j r , i r + σ ( a ) ], we have the following cases:(5) If jr < l ≤ i r + σ ( a ), then there exists p = l + 1 such that, if p = j r then g ( UW D ) = Uw . . . w j r . . . f ( w p ) w l . . . w r | {z } UD . . . w n − D ∈ S. Note that, if p = j r = 1 then g ( UW D ) =

U f ( w p ) w l . . . w r | {z } UD . . . w n − D ∈ S, or if p = j r = 1, then g ( UW D ) = Uw . . . f ( w i r − σ ( a ) ) . . . f ( w p ) w l . . . w r | {z } UD . . . w n − D ∈ S. (6) If l = j r and | l − r | >

0, then r ∈ [ j r , i r + σ ( a ) ] with | r − r | ≥

0, then the followingcases hold:(6.1) If j r + 2 ≤ r ≤ i r + σ ( a ) , then there exists p satisfying (4.4).(6.2) If j r ≤ r < j r + 2, then | r − r | > r = j r +1 satisﬁes (6.1), or if r = j r then UW D satisﬁes (4.5).(6.3) Now, if | r − r | > g ( UW D ) =

U . . . f ( w l ) . . . f ( w i r + σ ( a ) ) | {z } DU w i r + σ ( a )+1 . . . w s | {z } UD . . . D ∈ S, or r ∈ [ i r + σ ( a ) , j r +1 ] with | r − | ≥ i r + σ ( a ) ≤ r ≤ j r +1 .Same arguments are used for the cases r ∈ [ i r , j r +1 − σ ( a ) ]([ j r , i r + σ ( a ) ]) to conclude thelemma. We are done. (cid:3) Lemma 7.

Suppose that

UW D is a Dyck path in S and that integers l and r asdeﬁned in Proposition 5 are such that l = r , then the following statements hold :(a) If l / ∈ { j s } then there is an elementary shift to a Dyck path with exactly n − peaks. (b) If l ∈ { j s } then there is an elementary shift from a Dyck path with exactly n − peaks to UW D . Proof.

Let

UW D be a Dyck path in S , and positive integers l and r with l = r . yck Paths... (a) Suppose l / ∈ { j s } and l ∈ [ i r , j r +1 − σ ( a ) ]. If i r ≤ l < j r +1 − σ ( a ) , then UW D satisﬁes (4.1) and (4.2) of Lemma 6. In particular, if l = i r = 1 there is p ′ = l − j r − σ ( a ) , i r ] that satisﬁes the ﬁrst condition of (5) of Lemma 6. Thecase l ∈ [ j r , i r + σ ( a ) ] is dual.(b) Suppose l = j r +1 − σ ( a ) , we have the following cases:(i) If | i r − j r +1 − σ ( a ) | = 1 (or | i r +1 − j r +1 − σ ( a ) | = 1) and i r = 1 (or i r +1 = n − UV D which is equal to

U w |{z} UD w l . . . D ∈ S (or U . . . w l w n − | {z } UD D ∈ S ) , and Uf ( w ) w l . . . D = UW D (or

U . . . w l f ( w n − ) D = UW D )(ii) If | i r − j r +1 − σ ( a ) | = 1 (or | i r +1 − j r +1 − σ ( a ) | = 1) and i r = 1 (or i r +1 = n −

1) then there is l ′ = j r − σ ( a ) and r ′ = j r +1 − σ ( a ) (or l ′ = j r +1 − σ ( a ) and r ′ = j r +2 − σ ( a ) ) such that UV D is equal to

U . . . w l ′ . . . w r ′ − w l | {z } UD . . . D (or U . . . w l w l ′ +1 . . . w r ′ | {z } UD . . . D ) ∈ S and U . . . f ( w l ′ ) . . . f ( w r ′ − ) w l . . . D (or U . . . w l f ( w l ′ +1 ) . . . f ( w r ′ ) . . . D ) = UW D. (iii) If | i r − j r +1 − σ ( a ) | > | i r +1 − j r +1 − σ ( a ) | >

1) then there is

UV D whichis equal to

U . . . w l − | {z } UD w l . . . D (or U . . . w l w l +1 | {z } UD . . . D ∈ S )and U . . . f ( w l − ) w l . . . D = UW D (or

U . . . w l f ( w l +1 ) . . . D = UW D ) . Similar arguments dually applied can be used to obtain the lemma in the case l = j r .We are done. (cid:3) Remark 8.

Note that, in general there is an elementary leftshift and an elementaryrightshift over S , and these elementary shifts are disjoint, i.e. if f p ◦ · · · ◦ f p q and f p ′ ◦ · · · ◦ f p ′ q ′ are elementary left and right shifts, respectively. Then { p , . . . , p q } ∩ { p ′ , . . . , p ′ q ′ } = ∅ , these elementary shifts are unique according to Lemma 6 and Lemma 7. If F p = f p ◦ · · · ◦ f p q is an elementary leftshift (rightshift) we write F pl ( F pr ). Proposition 9.

Let C = { i , . . . i k , j , . . . j m } be an admissible subchain, then all Dyckpaths of S constitute a connected quiver Q . Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos

Proof.

It suﬃces to prove that Q is connected, to do that, consider Dyck paths UW D and

UV D of S . Then if there is a shift path between UW D and

UV D theyare connected. Otherwise, Lemmas 6 and 7 allow to deﬁne a Dyck path UW (1) D anda shift path F (1) = F (1) p ◦ · · · ◦ F (1)1 with F (1) m = f (1) m ◦ · · · ◦ f (1) m q such that UW D F (1)1 −−−→ . . . F (1) p −−−→ UW (1) D, and if there is a shift path from UV D to a UW (1) D then they are connected. If thereis not a shift path from UV D to UW (1) D , then there is a Dyck path UW (2) D and ashift path F (2) = F (2) p ◦ · · · ◦ F (2)1 with F (2) m = f (1) m ◦ · · · ◦ f (2) m q such that UW (2) D F (2)1 −−−→ . . . F (2) p −−−→ UW (1) D F (1) p ←−−− . . . F (1)1 ←−−− UW D, again, if there is a shift path from UW (2) D to a UV D then they are connected. Since S is ﬁnite, the procedure ends in such a way that UW D and

UV D are connected andwith this argument we are done. (cid:3)

Henceforth, we let C denote the subcategory of ( D n , R i ...i k j ...j m ) whose objects are k − linear combinations of Dyck paths of S . Lemma 10 and Proposition 11 give someproperties of the Hom-spaces of this category. Lemma 10.

Let

UW D , UW ′ D , UW ′′ D and UV D be Dyck paths in C and let F = F l ◦ F r (resp. F r ◦ F l ) be a shift path UW D F r −−→ UW ′ D F l −−→ UV D (resp.

UW D F l −−→ UW ′ D F r −−→ UV D ), if there is other shift path G = G ◦ G such that UW D G −−→ UW ′′ D G −−→ UW ′′ D with UW ′ D = UW ′′ D then G = F l and G = F r (resp. G = F r and G = F l ). Proof.

Let F = F l ◦ F r be a shift path such that U . . . w l . . . w r . . . D F r −−→ U . . . w ′ l . . . w ′ r . . . D F l −−→ U . . . v l . . . v r . . . D, with l = l and r = r and suppose that there is other shift path G = G ◦ G suchthat U . . . w l . . . w r . . . D G −−→ U . . . w ′′ l . . . w ′′ r . . . D G −−→ U . . . v l . . . v r . . . D, with UW ′ D = UW ′′ D . Given the elementary rightshift F r , then since G = F r ,it holds that UW D satisﬁes the conditions of UW ′ D in order to apply the sameelementary leftshift F l , i.e., F l = G and l = l . Since r = r , UW D and UW ′′ D satisfy the conditions to apply the same elementary rightshift, i.e., F r = G . Case F r ◦ F l is obtained via a dual argument. (cid:3) Proposition 11. If Hom C ( UW D, UV D ) = 0 then dim k Hom C ( UW D, UV D ) =1 . Proof.

Suppose that Hom C ( UW D, UV D ) = 0, then there is a shift path F of theform UW D F x −−→ . . . F i − xi − −−−−→ UW i − D F i − xi − −−−−→ UW i D F ixi −−→ UW i +1 D F i +1 xi +1 −−−−→ . . . F mxm −−−→ UV D, yck Paths... with x i ∈ { l, r } and for some m ∈ Z + . Now, for each pair F ix i ◦ F i − x i − with x i − = l and x i = r ( x i − = r and x i = l ) that satisﬁes conditions described in Lemma 10there is another shift path F ′ of the form UW D F x −−→ . . . F i − xi − −−−−→ UW i − D F ixi −−→ UW i ′ D F i − xi − −−−−→ UW i +1 D F i +1 xi +1 −−−−→ . . . F mxm −−−→ UV D, transforming

UW D and

UV D . Thus F ∼ R i ...ikj ...jm F ′ . (cid:3) In this section, we establish an equivalence between the category C and the categoryof representations of a quiver of Dynkin type A n . Θ Functor

Given an admissible subchain C = { j , . . . , j m , i , . . . , i k } , C n the subcategory of( D n , R i ...i k j ...j m ) and Q a quiver of type A n − with { i , . . . , i k } and { j , . . . , j m } be-ing the sets of sinks and sources, respectively. Then the k -linear additive functorΘ : C n −→ rep Q is deﬁned in such a way that, for an object UW D ∈ C n , it holdsthat, Θ( UW D ) = (Θ( w i ) , ϕ Θ( w i ,w i +1 ) ) , where Θ( w i ) = ( k, if w i = UD ,0 , if w i = DU . (11)If w i , w i +1 ∈ [ i r , j r +1 − σ ( a ) ] (cid:0) [ j r , i r + σ ( a ) ] (cid:1) then s (Θ( w i , w i +1 )) = i + 1, is the startingpoint of the corresponding arrow, whereas t (Θ( w i , w i +1 )) = i is the ending vertex ofthe corresponding arrow (cid:0) s (Θ( w i , w i +1 )) = i , t (Θ( w i , w i +1 )) = i + 1 (cid:1) and, ϕ Θ( w i ,w i +1 ) : Θ( w s (Θ( w i ,w i +1 )) ) −→ Θ( w t (Θ( w i ,w i +1 )) ) ,ϕ Θ( w i ,w i +1 ) = ( k , if w i = UD = w i +1 ,0 , if w i = DU or w i +1 = DU . (12)Functor Θ acts on morphisms as follows;Let f q ◦ · · · ◦ f q = (1 , . . . , q − , f q , . . . , f q , q +1 , . . . n − ) , be a elementary shift between UW D and

UV D , then:Θ((1 , . . . , q − , f q , . . . , f q , q +1 , . . . n − )) , (Θ(1 ) , . . . , Θ(1 q − ) , Θ( f q ) , . . . , Θ( f q ) , Θ(1 q +1 ) , . . . , Θ(1 n − )) , where Θ( f m ) = 0 and, Θ(1 m ) = ( k , if w m = UD = v m ,0 , otherwise, (13)for 1 ≤ m ≤ q − q ≤ m ≤ q and q + 1 ≤ m ≤ n − Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos

Remark 12.

Note that, it is easy to see that Θ is an additive covariant functor.

Lemma 13.

Let

UW D and

UV D be Dyck paths of C . If Hom C ( UW D, UV D ) = 0 then Hom rep Q (Θ( UW D ) , Θ( UV D )) = 0. Proof.

Suppose Hom C ( UW D, UV D ) = 0, and let F be a shift path UW D F −−→ UW D F −−→ . . . F m − −−−−→ UW m − D F m − −−−−→ UW m D from UW D = UW D to UV D = UW m D for some m ∈ Z + , then there exist q and q such that { q , q + 1 , . . . , q − , q } = \ i ∈ J Supp UW i D, applying Θ we obtain the following diagram: . . . Θ( w q − ) a q − c q − (cid:15) (cid:15) k (cid:15) (cid:15) . . . k (cid:15) (cid:15) Θ( w q +1 ) a q d q (cid:15) (cid:15) . . .. . . Θ( w q − ) a q − c q − (cid:15) (cid:15) k (cid:15) (cid:15) . . . k (cid:15) (cid:15) Θ( w q +1 ) a q d q (cid:15) (cid:15) . . . ... c m − q − (cid:15) (cid:15) ... (cid:15) (cid:15) ... (cid:15) (cid:15) ... d m − q (cid:15) (cid:15) . . . Θ( w m − q − ) a m − q − c m − q − (cid:15) (cid:15) k (cid:15) (cid:15) . . . k (cid:15) (cid:15) Θ( w m − q +1 ) a m − q d m − q (cid:15) (cid:15) . . .. . . Θ( w mq − ) a m +1 q − k . . . k Θ( w mq +1 ) a mq . . . Diagram 1.where c iq − a iq − , a iq , d iq +1 ∈ { , k } , squares in the diagram are commutativebetween q and q (independently of the chosen orientation). For the sub-shift path F ( x,y ) to F with 0 ≤ x ≤ y ≤ m − q ( x,y )1 and q ( x,y )2 suchthat S ( x,y ) = { q ( x,y )1 , q ( x,y )1 + 1 , . . . , q ( x,y )2 − , q ( x,y )2 } = \ i ∈ J ( x,y ) Supp UW i D, and for the diagramsΘ( w xq ( x,y )1 − ) a xq ( x,y )1 − (cid:15) (cid:15) k (cid:15) (cid:15) Θ( w yq ( x,y )1 − ) a yq ( x,y )1 − k and k (cid:15) (cid:15) Θ( w xq ( x,y )2 +1 ) a xq ( x,y )2 (cid:15) (cid:15) k Θ( w yq ( x,y )2 +1 ) a yq ( x,y )2 Diagram 2. Diagram 3.we have the following cases: yck Paths... (1) If q ( x,y )1 ∈ [ i r , j r +1 − σ ( a ) ] ( i r < q ( x,y )1 ≤ j r +1 − σ ( a ) ) four cases must be considered.(1.1) If Θ( w xq ( x,y )1 − ) = k and Θ( w yq ( x,y )1 − ) = k , q ( x,y )1 belong to S ( x,y ) , which is acontradiction.(1.2) If Θ( w xq ( x,y )1 − ) = k and Θ( w yq ( x,y )1 − ) = 0, then the Diagram 2 commutes.(1.3) If Θ( w xq ( x,y )1 − ) = 0 and Θ( w yq ( x,y )1 − ) = k , then there is an elementary shift f q ( x,y )1 − on the interval and this is again a contradiction.(1.4) If Θ( w xq ( x,y )1 − ) = 0 and Θ( w yq ( x,y )1 − ) = 0, then the Diagram 2 commutes.(2) If q ( x,y )1 ∈ [ j r +1 − σ ( a ) , i r +1 ] ( j r +1 − σ ( a ) < q ( x,y )1 ≤ i r +1 ), the conditions (1.1)-(1.4)are satisﬁed on the interval.(2.1) If Θ( w xq ( x,y )1 − ) = k and Θ( w yq ( x,y )1 − ) = 0, then they satisfy condition (1.3).(2.2) If Θ( w xq ( x,y )1 − ) = 0 and Θ( w yq ( x,y )1 − ) = k , then they satisfy condition (1.2).(3) Case q ( x,y )2 ∈ [ i r , j r +1 − σ ( a ) ] is similar to case (2) for the Diagram 3.(4) Case q ( x,y )2 ∈ [ j r +1 − σ ( a ) , i r +1 ] is similar to case (1) for the Diagram 3.therefore the Diagram 1 commutes. Since the cases over [ j r , i r + σ ( a ) ] can be showed byusing dual arguments. We are done. (cid:3) Lemma 14.

Functor Θ is faithful and full. Proof.

Let φ be the map φ : Hom C ( UW D, UV D ) → Hom rep Q (Θ( UW D ) , Θ( UV D )) , such that φ ( λF ) = λ Θ( F ) with F = (1 , . . . , q − , f q , . . . f q , q +1 , . . . , n − ), forsome 1 ≤ q , q ≤ n − λ ∈ k . Note, φ is well deﬁned and Lemma 13 allows usto observe that the image of a non-zero morphism in C is a non-zero morphism inrep Q . Thus, φ is surjective and injective. (cid:3) Theorem 15.

Functor Θ is a categorical equivalence between the categories C n and rep Q . Proof.

Lemma 14 implies that functor Θ is faithful and full. Now, let ( M i , ϕ α ) i ∈ Q ,α ∈ Q be an indecomposable representation in rep Q of the form0 · · · k k · · · k k · · · q z}|{ q z}|{ with { i , . . . , i k } and { j , . . . , j m } the sets of sinks and sources respectively. Let ϕ : { , k } → { DU, UD } be a map such that ϕ ( k ) = UD and ϕ (0) = DU . Deﬁne theDyck path UW D such that

UW D = U DU z }| { w . . . w q − UD z }| { w q . . . w q DU z }| { w q +1 . . . w n − D. Proposition 5 allows us to observe that

UW D has n − { j , . . . , j m , i , . . . , i k } and Θ( UW D ) = ( M i , ϕ α ) i ∈ Q ,α ∈ Q . Thus, Θ is essentially surjective. (cid:3) Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos

Corollary 16.

There exists a bijection ϕ between the set of representatives of inde-composable representations of rep Q (denoted Ind(rep Q ) ) and the set of Dyck pathsof length n with exactly n − peaks. Proof.

The Narayana number with exactly n − n is the triangular number T n − = ( n − n )2 , which is equal to the number of inde-composable representations of rep Q , then we deﬁne ϕ : S → Ind (rep Q ) such that ϕ ( UW D ) = Θ(

UW D ). (cid:3) Corollary 17.

The category C n is an abelian category. C n In this section, we introduce some properties of C n regarding simple, projective andinjective indecomposable objects, also we construct the Auslander-Reiten quiver foralgebras of Dynkin type A n − . Some conditions for morphisms between objects of thecategory are introduced as well. Theorem 18.

Let C = { j , . . . , j m , i , . . . , i k } be an admissible subchain, and let C n be the corresponding category, then (i) Indecomposable simple objects of C n are objects of the form S ( x ) = US ( w x ) . . . S ( w xn ) D where S ( w xy ) = ( UD, if x = y , DU, otherwise. (14)(ii)

Indecomposable projective objects of C n have the form P ( x ) = UP ( w x ) . . . P ( w xn ) D where P ( w yx ) = ( UD, if x, y ∈ [ i r , j r +1 − σ ( a ) ] ([ j r , i r + σ ( a ) ) and y ≤ x ( x ≤ y ) ,DU, otherwise. (15)(iii) Indecomposable injective objects of C n have the form I ( i ) = UI ( w x ) . . . I ( w xn ) D where I ( w yx ) = ( UD, if x, y ∈ [ i r , j r +1 − σ ( a ) ] ([ j r , i r + σ ( a ) ]) and x ≤ y ( y ≤ x ) ,DU, otherwise. (16) Proof. (i) Let S ( x ) = ( S ( x ) y , ϕ α ) be an indecomposable simple object of rep Q suchthat S ( x ) y = k if x = y and S ( x ) y = 0 if x = y . Functor Θ allows us to observe that,there is UW D ∈ C n satisfying the required conditions.(ii) Let P ( x ) = ( P ( x ) y , ϕ α ) be an indecomposable projective object of rep Q , if P ( x ) y = k then there is a path from x to y , as well as, a source j r +1 − σ ( a ) ( j r )and a sink i r ( i r + σ ( a ) ) such that i r ≤ y ≤ x ≤ j r +1 − σ ( a ) ( j r ≤ x ≤ y ≤ i r + σ ( a ) ), and P ( x ) y = 0. Thus, there is not a path between x and y , then functor Θ determines anobject UW D of C n with i , . . . i k , j , . . . j m being an admissible subchain satisfyingthe required conditions. Case (iii) follows by dually applying the arguments used inthe case (ii). (cid:3) yck Paths... Corollary 19.

The indecomposable simple objects of C n have exactly a subsequence UUDD . Proof.

Let S ( x ) be an indecomposable simple object of C n , then the identity S ( x ) = U . . . S ( w xx − ) S ( w xx ) S ( w xx +1 ) . . . D = U . . . DU . . . DU |{z} x − UD |{z} x DU |{z} x +1 . . . DU . . . D has place as a consequence of Theorem 18. (cid:3) Remark 20.

The Auslander-Reiten translate can be obtained by using the Coxetertransformation and the dimension vector associated to the support of a Dyck path in C n .Morphisms in C n also have the following properties.Let UW D be a Dyck path of C n , then • p UWD = w t and b UWD = max { w s | w i r ≤ w s ≤ w j r +1 − σ ( a ) , w s = UD } over[ i r , j r +1 − σ ( a ) ], • p UWD = min { w s | w j r ≤ w s ≤ w i r + σ ( a ) , w s = UD } and b UWD = w t over[ j r , i r + σ ( a ) ] . Theorem 21.

The vector space

Hom C n ( UW D, UV D ) = 0 if and only (i) Supp( UW D ) ∩ Supp(

UV D ) = ∅ ,(ii) p UWD ≤ p UV D and b UWD ≤ b UV D over [ i r , j r +1 − σ ( a ) ],(iii) p UWD ≥ p UV D and b UWD ≥ b UV D over [ j r , i r + σ ( a ) ], for all [ i r , j r +1 − σ ( a ) ] , [ j r , i r + σ ( a ) ] such that i r ≤ q ≤ j r +1 − σ ( a ) and j r ≤ q ≤ i r + σ ( a ) with q ∈ Supp ( UW D ) ∩ Supp ( UV D ) . Proof.

The result follows as a consequence of the deﬁnition of the functor Θ and theconstruction of Lemma 3.1. (cid:3)

Figure 3 describes a quiver Q of type A and the Auslander-Reiten quiver of rep Q . In [13] Marczinzik, Rubey and Stump presented a connection between the Auslander-Reiten quiver of Nakayama algebras and Dyck paths. In such a work for a Nakayamaalgebra A , they associated the vector space dimension of the indecomposable projectivemodules e i A to a Dyck path, this vector is called the Kupisch series. If we takea Nakayama algebra A = kQ/I , with I = h x x , x x x i , then the Kupisch series of kQ/I is [3 , , , , kQ/I has the shape describedin Figure 5.Let C n +1) be the category with the admissible subchain 1 < n , j = 1 and i = n ,and let D i be the sets Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos Q = ❞ ❞ ❞ ❞ ❞ ✲✛ ✲✛ (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✒ (cid:0)✒ (cid:0)✒(cid:0)✒ (cid:0)✒(cid:0)✒ (cid:0)✒ (cid:0)✒(cid:0)✒ (cid:0)✒❅❅❅❘ ❅❅❅❘❅❅❅❘ ❅❅❅❘ ❅❅❅❘❅❅❅❘ ❅❅❅❘❅❅❅❘ ❅❅❅❘ ❅❅❅❘ Figure 3: Quiver Q and the Auslander-Reiten quiver of rep Q . ❞ / / ❞ / / ❞ / / ❞ / / ❞ x x x x Figure 4: Quiver Q of type A . (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Figure 5: Dyck path associated to kQ/I . D = { X ∈ Ob ( C n +1) ) | w = UD } ,D i = { X ∈ Ob ( C n +1) ) | w m = DU, ≤ m ≤ i − } , (17) yck Paths... for 1 < i ≤ n . Then, we take the subset D i,j ⊆ D i , D i,j i = { Y ∈ D i | i ≤ r Y ≤ m ( i, j i ) + i − } , (18)such that the vector v = ( n − ( m ( i, j i ) + i − ni =1 constitutes an integer partition with n parts. Now, let N v be the full subcategory of C n +1) whose objects are k − linearcombinations of the Dyck paths in the following set L = n [ i =1 D i,j i , (19)and morphisms deﬁned by the category C n ( n +1) .We assume the following numbering and orientation for a quiver Q associated to aNakayama algebra ❞ / / ❞ / / · · · / / ❞ / / ❞ n − nx x x n − x n − Figure 6: Quiver Q of type A n . The functor Θ ′ between the category N v and the category of representations of kQ/I where kQ/I is a Nakayama algebra with Kupisch series [ m (1 , j ) , . . . , m ( n, j n )] is de-ﬁned in such a way that, Θ ′ ( UW D ) = Θ(

UW D ) and Θ ′ ( F ) = Θ( F ) for UW D ∈ L and F being an elementary shift in N v . Corollary 22.

The functor Θ ′ is an equivalence of categories. Proof.

It is a direct consequence of Theorem 15. (cid:3)

As an example, Figure 7 shows the Auslander-Reiten quiver of the Nakayama algebra A = kQ/I associated to the quiver Q shown in Figure 4 with I = h x x , x x x i . (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)❅❅❘ ❅❅❘ ❅❅❘ ❅❅❘❅❅❘ ❅❅❘ ր ր ր րր ր Figure 7: Auslander-Reiten quiver of rep kQ/I .6 Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos

In this section, we construct an alphabet associated to Dyck paths. And it is given aformula for cluster variables of cluster algebras associated to Dynkin diagrams of type A n . For n >

2, let U i = u . . . u n and U i = u ′ . . . u ′ n be Dyck paths in D n with thefollowing form: u j = ( U, if 1 ≤ j ≤ i + 1 or j = 2( i + 1) + k ≤ n,D, if i + 2 ≤ j ≤ i + 1) or j = 2( i + 1 + k ) ≤ n, (20)and u ′ j = ( U, if 2 i < j ≤ i + n or j = 1 + 2 k ≤ i,D, if i + n < j ≤ n or j = 2 k ≤ n, (21)for k > i ≤ n −

2. The alphabet H n is the union of the set { U jr | r = 1 , ≤ i ≤ n − } and the Dyck path with exactly one peak in D n (denoted by E n ). Figure8 shows the alphabet H . (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (a) U (b) U (c) E Figure 8: Alphabet H . Let C = { i , . . . , i k , j , . . . , j m } be an admissible subchain of n-1 . We ﬁx two diﬀerentrelations of concatenation f and f over H n such that f ( V i ) = E n , if V i = E n or V i = U i ,U i +12 , if V i = E n or V i = U i ,U i +11 , if V i = U i , (22)and f ( V i ) = E n , if V i = U i ,U i +11 , if V i = E n or V i = U i ,U i +12 , if V i = U i . (23)Then, we take the set of words V = V . . . V n − in H ∗ n such that V i = ( f ( V i − ) , if i / ∈ C ,f ( V i − ) , if i ∈ C − { , n − } , (24)for 1 < i ≤ n − n ≥

4. This set is denoted by X C , in particular case X { , } = H . yck Paths... Let G = ( G , . . . , G n − ) be a snake graph, then we can associate to G an admissiblesubchain C of n-1 in the following way:If G i − , G i and G i +1 denote tiles of the following snake graph G i − G i G i +1 then, i ∈ C for 1 < i < n −

1. For example, for the snake graph G shown in Figure 9 G G G G G Figure 9: Snake graph G . it holds that the corresponding admissible subchain is given by the identity { , , } = { i , j , i } = { j , i , j } . By notation, G can be written as G C .The following result establishes a relationship between the alphabet X C and perfectmatchings of snake graphs. Lemma 23.

Let C = { i , . . . , i k , j , . . . , j m } be an admissible subchain of n-1 . Then,there is a bijective correspondence between the set X C and the perfect matchings of G C . Proof.

Let C be an admissible subchain of n-1 , X C be a set of words, and G C be asnake graph associated to C . Assume a numbering over the edges of G C in the followingway:For boundary edges of G i , we have the following four possibilities G i − G i − G i G i G i +1 G i +1 U i − U i − U i − U i − U i U i G i − G i − G i G i G i +1 G i +1 U i − U i − U i − U i − U i U i Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos with 1 < i < n − E n . Now, a perfect matching P of G C can be written as a vector v = ( v , . . . , v n ), where each v i corresponds to an edge of G C (this vector is uniqueup to permutation). Deﬁne a map f : X C → Match( G C ) such that f ( V . . . V n − ) =( E n , V , . . . , V n − , E n ). Firstly, we will prove that f is well deﬁned by induction over n . To start note that for n = 3, we have the following three cases:(I) If V = E , it turns out that F ( V ) = ( E , E , E ), which is given by G G (II) If V = U , it holds that f ( U ) = ( E , U , E ), which is equal to G G (III) If V = U , then f ( U ) = ( E , U , E ), which is of the form G G Suppose that the result holds for n = k . Let n = k +1, by hypothesis ( E k +1 , V , . . . V k )are disjoint sets containing all the previous tiles in G C , then there are two possibilitiesfor k .(I) for k ∈ C − { , k + 1 } , we have the following conditions:(1.1) If V k − = E k +1 , then f ( V . . . E k +1 E k +1 ) = ( E k +1 , V , . . . , E k +1 , E k +1 , E k +1 )and f ( V . . . E k +1 U k ) = ( E k +1 , V , . . . , E k +1 , U k , E k +1 ), which are givenby G k − G k − G k G k G k +1 G k +1 and rr rr (1.2) If V k − = U k − , then f ( V . . . U k − E k +1 ) = ( E k +1 , V , . . . , U k − , E k +1 , E k +1 )and f ( V . . . U k − U k ) = ( E k +1 , V , . . . , U k − , U k , E k +1 ), which are equalto G k − G k − G k G k G k +1 G k +1 and(1.3) If V k − = U k − , then f ( V . . . U k − U k ) = ( E k +1 , V , . . . , U k − , U k , E k +1 )which is of the form G k − G k G k +1 rr (II) for k / ∈ C , there are the following cases: yck Paths... (2.1) If V k − = E k +1 , then f ( V . . . E k +1 U k ) = ( E k +1 , V , . . . , E k +1 , U k , E k +1 ),which is given by G i − G i G i +1 rr (2.2) If V k − = U k − , then f ( V . . . U k − U k ) = ( E k +1 , V , . . . , U k − , U k , E k +1 ),which is equal to G i − G i G i +1 rr (2.3) If V k − = U k − , then f ( V . . . U k − E k +1 ) = ( E k +1 , V , . . . , U k − , E k +1 , E k +1 )and f ( V . . . U k − U k ) = ( E k +1 , V , . . . , U k − , U k , E k +1 ), which are of theform G i − G i − G i G i G i +1 G i +1 and rr rr Dual arguments prove the result for the other labelings. We also note that by deﬁni-tion map f is injective and surjective. (cid:3) Remark 24.

Each perfect matching of G C is in correspondence with just only oneobject of the A n − − Dyck paths category associated to the admissible subchain C = { i , . . . , i k , j , . . . , j m } .For each Dyck path Y = y . . . y n with n − Y ∩ X C ∈ H ∗ n such that; Y ∩ X C = { Y ∩ V z | V z ∈ X C } , (25)where Y ∩ V z = ( V z if there exists j such that y j = v zj for 1 < j < n,E n otherwise, (26)with V z = v z . . . v z n in X C . For the set Y ∩ X C , it can be deﬁned a relation ∽ suchthat Y ∩ V z ∽ Y ∩ V z if and only if Y ∩ V z and Y ∩ V z are the same word. (27)In this case, ∽ is an equivalence relation and ( Y ∩ X C ) / ∽ is denoted by [ Y ∩ X C ]. Also,we remind that a Dyck path Y can be written as the word UW D = Uw , . . . w n − D ,where y = U , y n = D and, w i = y i y i +1 . Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos

Lemma 25.

Let C = { i , . . . , i k , j , . . . , j m } be an admissible subchain of n-1 andlet Y a Dyck path of length n with exactly n − peaks. Then, there is a bijectivecorrespondence between the set [ Y ∩ X C ] and the set of perfect matchings of the snakegraph belonging to G C and induced by the words w t = UD in Y . Proof.

Let C be an admissible subchain of n-1 and Y = UW D be a Dyck path in S ,then by Proposition 5 there are l, r ∈ Z > with 1 ≤ l ≤ r ≤ n − w t = UD for l ≤ t ≤ r and w t = DU otherwise. Now, let G C l,r = G [ l, d ] be a snake graphbelonging to G C induced by Y . Deﬁne a map g : [ Y ∩ X C ] → Match( G C l,r ) such that:(I) If 1 < l ≤ r < n −

1, then g ([ Y ∩ V i ]) = g ( E n . . . E n V il − . . . V ir E n . . . E n ) =( V il − , . . . , V ir ).(II) If l = 1 and 1 = l ≤ r < n −

1, then g ([ Y ∩ V i ]) = g ( V i . . . V ir E n . . . E n ) =( E n , V il , . . . , V ir ).(III) If r = n − < l ≤ r = n −

1, then g ([ Y ∩ V i ]) = g ( E n . . . E n V il − . . . V in − ) =( V il − , . . . , V in − , E n ).(IV) If l = 1 and r = n −

1, then g = f .Since in the four cases g is a restriction of f . It follows that g is a bijection as aconsequence of Lemma 23. (cid:3) In this section, Dyck paths categories are used to give a formula for cluster variablesof cluster algebras of Dynkin type A n , to do that, we use the category of Dyck pathsassociated to an admissible subchain. We also present a connection between clustervariables of algebras of type A n − and Dyck paths with n − C = { i , . . . i k , j , . . . j m } be an admissible subchain of n-1 and let Y = UW D bea Dyck path in S , then we deﬁne the monomials η Y = Y UD = w i ∈ Y x i , (28)and X V = Y m ∈ M V x m , (29)with M V being the set of indices m such that m = i + 1 , if U i ∈ V,i, if U i ∈ V, , if E n ∈ V, (30) V ∈ [ Y ∩ X C ]. For this case x = 1.The following theorem gives the cluster variable associated to a Dyck path in the set S and its connection with cluster algebras of type A n − . yck Paths... Theorem 26.

Let C = { i , . . . , i k , j , . . . , j m } be an admissible subchain of n-1 , Y = UW D a Dyck path with n − peaks and M the set of all cluster variables of a clusteralgebra of type A n − with { i , . . . , i k } and { j , . . . , j m } the sets of sinks and sources,respectively. Then :(i) The cluster variable associated to Y in the category C n is given by X Y = ( η Y ) − X V ∈ [ Y ∩ X C ] X V ! . (31)(ii) There exists a bijective correspondence between Dyck paths with n − peaks andthe set M \ x with x the initial seed. Proof.

Let C = { i , . . . , i k , j , . . . , j m } be an admissible subchain of n-1 , and let T C be the triangulation of the polygon with n + 2 vertices given by C ✏✏✏✏✏✏✏✏PPPP PPPP✏✏✏✏ ✏✏✏✏PPPP PPPP PPPP PPPP✏✏✏✏ ✏✏✏✏PPPP PPPP PPPP PPPP✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏PPPP PPPP✏✏✏✏✏✏✏✏ ...... ...... ...... ...... ...... ...... ...... ......... ... ... ... i i j j i i j k − i k j k i k − j k − i k j j i i j j i m − j m i m j m − i m − j m Let α l,r be a diagonal that is not in T C that cuts the diagonals α l , . . . α r ∈ T C . Wedeﬁne a functor χ : C T C → C n such that χ ( α l,r ) = UW l,r D , where w j = ( UD, if l ≤ j ≤ r,DU, otherwise, (32)and for any pivoting elementary move E : α r,l → α ′ r ′ ,l ′ , χ ( E ) is the elementary shift F = f t ◦ · · · ◦ f t k from UW l,r D to UW l ′ ,r ′ D . Theorems 2 and 15 allow us to establishthe following sequence of equivalences: C T C ⋍ Mod Q T C ⋍ C n , (33)therefore χ is a categorical equivalence. Thus,(i) Functor χ and Lemma 25, allow to establish that x γ = X Y .(ii) The map ψ : S → M \ x such that ψ ( Y ) = X Y is a bijection as a consequenceof Theorem 1 and the deﬁnition of functor χ . We are done. (cid:3) For instance, let C = { j = 1 , i = 2 , j = 4 } be an admissible subchain of , the set X C is in correspondence with the objects of C shown in Figure 10.Then, for Y = UDUUDUDDUD , we deﬁne the set Y ∩ X C such that[ Y ∩ X C ] = { E E U , E U E , U U U } . (34) Agust´ın Moreno Ca˜nadas and Gabriel Bravo R´ıos (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) ⊕ ⊕ (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) ⊕ ⊕ ⊕ ⊕⊕ ⊕ ⊕ ⊕⊕ ⊕ ⊕ ⊕ (a) E ⊕ E ⊕ U (b) E ⊕ U ⊕ E (c) E ⊕ U ⊕ U (d) U ⊕ E ⊕ U (e) U ⊕ U ⊕ E (f) U ⊕ U ⊕ U (h) U ⊕ U ⊕ U Figure 10: Objects in C . Thus, identities (28), (29) and (30) deﬁne the polynomials η Y = x x , X E E U = x x x , X E U E = x x x , X U U U = x x x , (35)therefore, the cluster variable associated to the Dyck path Y is given by the expression X Y = x + x + x x x x x . (36) References [1] I. Assem, C. Reutenauer, and D. Smith,

Friezes , Adv. Math (2010), 3134-3165.[2] E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani,

Nondecreasing Dyck paths andq-Fibonacci numbers , Discrete Mathematics (1999), 211-217.[3] K. Baur and R.J. Marsh,

Categoriﬁcation of a frieze pattern determinant , J. Combin.Theory Ser. A (2012), 1110-1122.[4] P. Caldero, F. Chapoton, and R. Schiﬄer,

Quivers with relations arising from clusters( A n case) , Trans. Am. Math. Soc. (2006), no. 3, 1347-1364.[5] P. Caldero and F. Chapoton, Cluster algebras as Hall algebras of quiver representations ,Commentarii Mathematici Helvetici (2006), 595-616. yck Paths... [6] I. Canakci and R. Schiﬄer, Snake graph calculus and cluster algebras from surfaces , J.Algebra (2013), 240-281.[7] ,

Cluster algebras and continued fractions , Compositio Mathematica (2018),no. 3, 565-593.[8] S. Fomin, M. Shapiro, and D. Thurston,

Cluster algebras and triangulated surfaces.PartI: Cluster complexes. , Acta Math. (2008), 83-146.[9] S. Fomin and D. Thurston,

Cluster algebras and triangulated surfaces.Part II: LambdaLengths. , Vol. 255, Memoirs of the American Mathematical Society., 2018.[10] S. Fomin and A. Zelevinsky,

Cluster algebras. I: Foundations. , J. Amer. Math. Soc. (2002), 497-529.[11] , Cluster algebras. II: Finite type classiﬁcation. , Invent. Math. (2003), no. 1,63-121.[12] ,

Cluster algebras. IV: Coeﬃcients. , Compositio Mathematica (2007), 112-164.[13] R. Marczinzik, M. Rubey, and C. Stump,

A combinatorial classiﬁcation of 2-regularsimple modules for Nakayama algebras , Journal of Pure and Applied Algebra (2020),no. 3.[14] G. Musiker, R. Schiﬄer, and L. Williams,

Positivity for cluster algebras from surfaces ,Adv. Math. (2011), 2241-2308.[15] J. Propp,