Snake graphs are connected planar graphs consisting of a finite sequence of adjacent tiles (squares) $T_1,T_2,\dots ,T_n$. In this case, for $1\le j\le n-1$, two consecutive tiles $T_j$ and $T_{j+1}$ share exactly one edge, either the edge at the east (west) of $T_j$ ($T_{j+1}$) or the edge at the north (south) of $T_j$ ($T_{j+1}$). Finding the number of perfect matchings associated with a given snake graph is one of the most remarkable problems regarding these graphs. It is worth noting that such a number of perfect matchings allows a bijection between the set of snake graphs and the positive continued fractions. Furthermore, perfect matchings of snake graphs have also been used to find closed formulas for cluster variables of some cluster algebras and solutions of the Markov equation, which is a well-known Diophantine equation. Recent results prove that snake graphs give rise to some string modules over some path algebras, connecting snake graph research with the theory of representation of algebras. This paper uses this interaction to define Brauer configuration algebras induced by schemes associated with some multisets called polygons. Such schemes are named Brauer configurations. In this work, polygons are given by some admissible words, which, after appropriate transformations, permit us to define sets of binary trees called groves. Admissible words generate codes whose energy values are given by snake graphs. Such energy values can be estimated by using Catalan numbers. We include in this paper Python routines to compute admissible words (i.e., codewords), energy values of the generated codes, Catalan numbers and dimensions of the obtained Brauer configuration algebras.

Recently, Postnikov introduced Bert Kostant’s game to build the maximal positive root associated with the quadratic form of a simple graph. This result, and some other games based on Cartan matrices, give a new version of Gabriel’s theorem regarding algebras classification. In this paper, as a variation of Bert Kostant’s game, we introduce a wargame based on a missile defense system (MDS). In this case, missile trajectories are interpreted as suitable paths of a quiver (directed graph). The MDS protects a region of the Euclidean plane by firing missiles from a ground-based interceptor (GBI) located at the point $(0,0)$. In this case, a missile success interception occurs if a suitable positive number associated with the launches of the enemy army can be written as a mixed sum of triangular and square numbers.

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.