Brauer Conﬁguration Algebras Arising from Dyck Paths

: The enumeration of Dyck paths is one of the most remarkable problems in Catalan combinatorics. Recently introduced categories of Dyck paths have allowed interactions between the theory of representation of algebras and cluster algebras theory. As another application of Dyck paths theory, we present Brauer conﬁgurations, whose polygons are deﬁned by these types of paths. It is also proved that dimensions of the induced Brauer conﬁguration algebras and the corresponding centers are given via some integer sequences related to Catalan triangle entries.


Introduction
In the last few years, several combinatorial objects have allowed the research development of the theory of representation of algebras. For instance, the number of triangulations of a polygon with n + 3 sides, or the number of Dyck paths of length 2n in the plane connecting the origin with a point P = (2n, 0). Edges in these paths are either rises (linking points (x, y) and (x + 1, y + 1) in N 2 ) or falls (connecting points (x, y) and (x + 1, y − 1) in N 2 ) [1]. Caldero, Chapoton, and Schiffler [2] proved that any triangulation T of an (n + 3)-polygon defines a category mod F Q T of finitely generated modules over a path algebra FQ T induced by a quiver Q T arisen from the triangulation T. They also proved that mod F Q T can be identified with the category of diagonals defining the triangulation.
Following the ideas by Caldero et al., Cañadas et al. [3] introduced a categorical equivalence between a category C 2n of Dyck paths and a category of representations rep A n−1 of a quiver of Dynkin type A n−1 . These approaches allow obtaining formulas for cluster variables of type A and frieze patterns in terms of Dyck paths and perfect matchings of some snake graphs. It is worth pointing out that finding formulas of these types is a significant problem in the cluster algebras theory.
On the other hand, Brauer configuration algebras (BCAs) are bound quiver algebras introduced by Green and Schroll in [4]. The structure of the indecomposable projective modules over these types of algebras is given by some combinatorial data, which are also used to determine their dimension. BCAs were used by Espinosa [5] to categorify integer sequences in the sense of Ringel and Fahr, i.e., the numbers in these sequences can be considered as invariants of objects in a category. They are also helpful in several fields of applied mathematics (cryptography, graph energy theory, algebraic combinatorics, etc. [6][7][8][9]).

Contributions
This paper establishes a connection between Dyck paths and Brauer configuration algebras researches by proving that Dyck paths define suitable words associated with polygons in a Brauer configuration. We introduce new Catalan-Brauer configurations (CBCs) obtaining formulas for the dimensions of the corresponding BCAs and their centers. To do that, we introduce an integer sequence S = {h i x,j } whose properties allow obtaining formulas relating its elements with entries of the Catalan triangle. Thus, the approach allows giving another manifestation of Catalan numbers via BCAs. Figure 1 shows how Brauer configuration algebras and Dyck paths theories are related to the main results presented in this paper.  (7) and (8) C C Main results presented in this paper (targets of blue and red arrows) allow establishing a connection between Brauer configuration algebras and Dyck paths theories. Propositions 1-3 give properties of a suitable integer sequence related to the Catalan triangle entries via Lemma 1 and Proposition 4. We introduce a Brauer configuration Γ x , whose vertices occur in polygons according to entries of suitable matrices whose properties are given in Lemma 2. Proposition 5 proves that the Catalan triangle gives entries of such matrices. Theorem 2 gives a formula for the dimension of a Brauer configuration algebra (and its corresponding center) defined by Dyck paths.
The organization of this paper goes as follows; the main definitions and notation are given in Section 2, we recall the definitions and notation used throughout the document. In particular, we recall notions of Dyck paths, Catalan triangle, and Brauer configuration algebra (Section 2.2). In Section 3, we give our main results. We introduce an integer sequence whose elements are related to the entries of the Catalan triangle. The numbers in this sequence allow giving a formula to compute the dimension of some Brauer configuration algebras, whose polygons are defined by Dyck paths. The concluding remarks are given in Section 4.

Background and Related Work
This section is devoted to reminding the basic notation and results concerning Dyck paths, the Catalan triangle, and Brauer configuration algebras, which are helpful for a better understanding of the paper.

Dyck Paths and Catalan Triangle
In this section, we make a brief review on Dyck paths and the Catalan triangle [1,10]. Dyck paths, as defined in the introduction, were enumerated by Stanley [1], who proved that there are C n = 1 n+1 ( 2n n ) Dyck paths of length 2n, where C n denotes the nth Catalan number. p 2n = {P | P is a Dyck path, |P| = 2n}.
For u ≥ 0 and 0 ≤ v ≤ u. The Catalan's triangle is an integer array whose entries C(u, v) are given by the formula [10].
There are many ways of finding Catalan triangle entries in the specialized literature. For instance, the array (3) gives such numbers according to the recurrence (2). Such a recurrence was found by Cañadas et al. [3] via some seed vectors associated with positive integral diamonds of type A n arising from the theory of integer friezes. (2)

Brauer Configuration Algebras
Green and Schroll introduced the notion of a BCA (Brauer Configuration Algebra). The authors refer the interested reader to [4,11] for a more detailed study of BGAs (Brauer Graph Algebras) and BCAs. In the sequel, we make a brief description of the structure of these algebras.
A BCA Λ Γ is a bound quiver algebra induced by a Brauer configuration Γ = (Γ 0 , Γ 1 , µ, O) of sets, with the functions and orders satisfying the following conditions: occ(h, W) is said to be the valency of the vertex h, which is said to be non-truncated if val(h)µ(h) = 1 (otherwise, it is non-truncated). We let S h denote the maximal set of polygons containing a non-truncated vertex. If S h = {U i 1 , U i 2 , . . . , U i m }. Then S h is endowed with a well-order <, which allows writing S h in the following form: The symbol U x i j is used to denote that occ(h, U i j ) = x. In successor sequences U x denotes a subsequence of length x with the form: The set (S h , <) is said to be the successor sequence associated with the vertex h. Note that if U i < U j is a covering in S h and h ∈ U i ∩ U j with h = h then the relation U i < U j also appears in the sequence S h .
In this work, it is assumed that each polygon U i ∈ Γ 1 is given by a word w U i of the form where for each i, 1 ≤ i ≤ t, α i is a vertex, and s i = occ(α i , U i ).

4.
Define the Brauer quiver To each ordered successor sequence S α define a special cycle C α associated with a vertex α by adding a relation r α of the form Define the path algebra FQ Γ . 6.
Define an admissible ideal I Γ generated by the following relations: , for any pair of special cycles associated with vertices α, α ∈ U, U ∈ Γ 1 . 7.
Define the Brauer configuration algebra The union of classes of proper prefixes of special cycles with classes of special cycles provide an F basis B of Λ Γ .
Later on, if there is not possibility of confusion, we will assume notations Q (for a quiver), I (for an admissible ideal) and Λ (for a Brauer configuration algebra).

1.
There is a bijection between the set of indecomposable projective modules over Λ and Γ 1 ; 2.
If P V is an indecomposable projective module over a BCA Λ defined by a polygon V in Γ 1 .
and r is the number of (non-truncated) vertices of V; 3.
I is admissible, whereas Λ is a multiserial symmetric algebra. Moreover, if Γ is connected, then Λ is indecomposable as an algebra; 4.
If rad P (soc P) denotes the radical (socle) of an indecomposable projective module P, and rad 2 P = 0. Then, the number of summands in the heart rad P/soc P of P equals the number of non-truncated vertices of the polygons in Γ corresponding to P counting repetitions; 5.
If Λ Γ and Λ Γ are BCAs, induced by Brauer configurations Γ = (Γ 0 , Γ 1 , µ, O), and Green and Schroll in [4] proved that the dimension of a Brauer configuration algebra is given by the following formula.
Sierra [12] obtained the next formula for the dimension of the center of a connected Brauer configuration algebra Λ Γ with radical square different from zero.
As an example, we use compositions of the number 3 to define a Brauer configuration ∆ = (∆ 0 , ∆ 1 , µ, O) for which; The following Figure 2 shows the Brauer quiver Q ∆ . Figure 2. Brauer quiver associated Q ∆ induced by the Brauer configuration ∆.
The admissible ideal I ∆ is generated by the following relations (a and a denote first arrows of special cycles C α associated with a vertex α).
, for all possible special cycles associated with vertices 1 and 2.

Main Results
The results in this section allow establishing interactions between Catalan combinatorics via Dyck paths and Brauer configuration algebras. It is proved that indecomposable projective modules over some Brauer configuration algebras define Dyck paths. We compute the dimension of these algebras and their corresponding centers.

Dyck Paths Arising from Brauer Configuration Algebras
For a fixed integer x, and sets of letters It is defined a Brauer configuration Γ If a product or concatenation c is defined on the set Γ x 0 in the following fashion: for suitable integers p and q.
Then the word w V associated with a polygon V ∈ Γ x 1 has the form δ h , where δ 1 = a 1 0 and δ q = c(δ q−1 ). The orientation O x is defined by an order < satisfying the following conditions: Successor sequences associated with vertices can be defined by adopting the following relation V < V if and only if there exists a positive integer r such that where N r g V is the number of a words appearing before the first occurrence of an r g -b word in V.
If no confusion arises, henceforth, we will write polygons in terms of their corresponding words. In successor sequences polygons V i ∈ Γ x 1 are (linearly) ordered in the form Remark 1. It is worth pointing out that under these circumstances, it is easy to prove that there is a bijection between words of type V i and Dyck paths of type p 2n .
The multiplicity function µ x : Γ x 0 → N + is defined in such a way that Brauer configurations of type Γ x are called Catalan-Brauer configurations. We note that, if V ∈ Γ x 1 , then occ(a x 2 x 1 , V) is given by an entry of the ( Figure 3 (see identities (2)). On the other hand, the same row in a matrix B x = (b x i,j ) of the same size gives occ(b x 2 x 1 , V). Note that, ) are given by the following identities: are given by the same formulas for −1 ≤ p ≤ x − 1. Paths in FQ associated with the Brauer quiver induced by Γ x are given by words (a p q ) 2 and (b p q ) 2 denote loops associated with vertices δ for which val(δ) = 1. Special cycles C p associated with a given non-truncated vertex δ p are given by the following identities: The admissible ideal I associated with the Brauer configuration Γ x is generated by the following set of relations.

•
Relations of type I.
for all the possible products in Λ Γ x .
We let Λ Γ x = FQ Γ x /I x denote the Catalan-Brauer configuration algebra induced by the Brauer configuration Γ x .
As an example, we define the Catalan-Brauer configuration Γ 2 = (Γ 2 0 , Γ 2 1 , µ 2 , O 2 ), for which The following are the matrices A 2 and B 2 . Figure 4 shows the Brauer quiver Q Γ 2 . • C u i ∼ C v i , for all special cycles associated with vertices u i , v i ∈ V i , i = 1, 2; • C u i a, for any special cycle associated with a vertex u i ∈ Γ 0 , a is the first arrow of C u i .
The Catalan-Brauer configuration algebra Λ Γ 2 is defined in such a way that Λ Γ 2 = FQ Γ 2 /I Γ 2 , where I Γ 2 = ρ Γ 2 is an admissible ideal generated by relations ρ Γ 2 . Figure 5 shows the indecomposable projective Λ Γ 2 -modules. Figure 5. Indecomposable projective Λ Γ 2 -modules. Note that the number of composition series equals the number of non-truncated vertices in the corresponding polygon.

Dimension of a Catalan-Brauer Configuration Algebra and Its Corresponding Center
The dimensions of the Catalan-Brauer configuration algebras are given in this section based on a new family of integer sequences, whose elements are related to Catalan triangle entries.
Let h p x,q be integer numbers, such that p ≥ x − 1 q ≤ p + 1. Figure 6 shows integer sequences h p x,q for x = 2, . . . , 5. Arithmetic properties of numbers h p x,q are given by the following Propositions 1-3.
We suppose that the statement holds true for p = m and 1 < x < m + 1. Then, for p = m + 1 and x = 2, if it is assumed that the proposition holds for The result follows by induction. We are done.
Proof. If p = 1 then h 1 2,2 = 1 = h 1 2,1 . If the proposition is valid for p = ι and 1 < x < ι + 1, then the following identities hold for p = ι + 1 and If the validness of the proposition holds for Thus, the statement holds by induction. We are done.

Proof.
To proceed by induction, we note that for any q ≥ 1, h 3 1,q = 1, and h 4 4,q = q. Since h . We are done. Lemma 1 and Proposition 4 give the relationships between integer numbers h p x,q and entries of Catalan triangle p x,y (see identities (2)).
Each m × n-matrix M = (m pq ) 1≤p≤m, 1≤q≤n with entries in a commutative ring F has associated a list of column vectors where, Remark 2. Henceforth, we will assume the notation φ 0 (A The following result holds for maps φ ν : Lemma 2. Let A x and B x be the matrices given in Figure 3. Then × p x,p+1 matrix, whose entries satisfy identities (15).
and a x,p i = 0, otherwise.
The item 2 can be proved by using similar arguments as in the case 1. We are done.
The following notation is assumed in the proof of Proposition 5.

Proposition 5.
Let A x and B x be the matrices shown in Figure 3. Then, the following identities hold. 1.
The proof of case 2 requires similar arguments as those exposed in case 1. We are done.
The following map δ : N → {1, 2} such that is used to give dimension formulas for Catalan-Brauer algebras Γ x and their centers.
where, x > 0 and t i denotes the ith triangular number.

Proof. 1.
Firstly, we note that the number of vertices in the Brauer quiver Q Γ x is given by the xth Catalan number C x = 1 x+1 ( 2x x ). Secondly, we note that val(a x 2 x 1 ) (resp. val(b n 2 n 1 )) is given by a x,0 m a , m a = xx 1 + . As a consequence of Proposition 5, we have that a x,0 x = 1 = b x,0 x for any x. In particular, for x = 2, it holds that a 0,1 2 = b 0,1 2 = 1.

Concluding Remarks
Catalan-Brauer configuration algebras (CBCAs) is a way to relate Catalan combinatorics with the BCAs theory. Dimension formulas of such CBCAs and their centers can be obtained by using entries of the Catalan triangle. The procedure interprets such entries as numbers of some novel integer sequences h i x,j dealing with binomial numbers. It is an interesting task for the future to investigate additional relationships between sequences h i x,j and different Catalan objects.

Abbreviations
The following abbreviations are used in this manuscript:

BCA
Brauer Configuration Algebra C n nth Catalan number C(n, k) Catalan triangle entry dim F Λ Γ Dimension of a Brauer configuration algebra dim F Z(Λ Γ ) Dimension of the center of a Brauer configuration algebra CBCA Catalan-Brauer Configuration Algebra Set of vertices of a Brauer configuration Γ occ(α, V) Number of occurrences of a vertex α in a polygon V t n nth triangular number Ordered sequence of polygons val(α) Valency of a vertex α w V The word associated with a polygon V