Relationships between the Chicken McNugget Problem, Mutations of Brauer Conﬁguration Algebras and the Advanced Encryption Standard

: Mutations on Brauer conﬁgurations are introduced and associated with some suitable automata to solve generalizations of the Chicken McNugget problem. Additionally, based on marked order polytopes, the new Diophantine equations called Gelfand–Tsetlin equations are also solved. The approach allows algebraic descriptions of some properties of the AES key schedule via some Brauer conﬁguration algebras and suitable non-deterministic ﬁnite automata (NFA).


Introduction
Chicken McNuggets are one of the most sold products of the international fast-food restaurant chain McDonald's. Shortly after their introduction in 1979, they began to be sold in packs of 6, 9, and 20 pieces. Nowadays, several choices of this product can be ordered, which naturally generates different types of Diophantine equations. For instance, the first three types of packs give rise to the following problem [1]: Problem 1. What numbers of chicken McNuggets can be ordered using only packs with 6, 9, or 20 pieces?
Problem 1 is currently called the chicken McNugget problem (CMP), which defines the following Diophantine equation: where n is the number of ordered pieces. CMP is a version of the postage stamp problem or Frobenius coin problem or knapsack problem, which can be defined as follows: CMP deals with a more wide kind of problems based on the number of lattice points in a convex polytope P, i.e., to determine φ A (b) = |P = {x | Ax = b, x integral}|, where A = (a (i,j) ) is a m × n integral matrix and b = [b 1 b 2 . . . b m ] t is an integral m-vector. For instance, if H = [3 5 17] then φ H (58) = 9, φ H (101) = 25, φ H (1110) = 2471 [2]. Actually, The number of solutions of equations with the form n ∑ i=1 a i x i = b was studied by Sylvester [3] who called denumerant the function d(b; a 1 , a 2 , . . . , a n ), which counts the number of non-negative representations of b by a 1 , a 2 , . . . , a n . Notice that, d(b; a 1 , a 2 , . . . , a n ) is actually the number of partitions of b whose summands or parts are taken (repetitions allowed) from the sequence a 1 , a 2 , . . . , a n . Recall that a partition of a positive integer n is a finite non-increasing sequence of positive integers λ 1 , λ 2 , . . . , λ r , such that Sylvester proved that the generating function f (t) of d(b; a 1 , a 2 , . . . , a n ) is given by the identity [3,4] f (t) = 1 (1 − t a 1 )(1 − t a 2 ) . . . (1 − t a n ) .
As Ramírez-Alfonsín pointed out in [4] finding formulas for denumerants is very difficult since even the problem of determining if d(b; a 1 , a 2 , . . . , a n ) > 0 is an NP-complete problem.
It is worth noting that Euler [4][5][6] pointed out that the generating function of the number of non-negative solutions of a system of linear equations with the form Ax = b is equal to the coefficient of x .
is said to be a standard Gelfand-Tsetlin pattern (GT pattern or Gelfand-Tsetlin tableau) if its entries satisfy the following interlacing conditions [7]: Given λ, µ ∈ Z n , the Gelfand-Tsetlin polytope GT(λ, µ) is the convex polytope of GT patterns (t (i,j) ) 1≤i≤j≤n satisfying in addition that t (1,1) = µ 1 , It is worth noting that much of the theory of representation of the general linear Lie algebra gl n C (which consists of complex n × n matrices with the usual matrix commutator) is based on a result of Gelfand and Tsetlin related with the enumeration of lattice points in GT-polytopes [7,8]. Actually, it is known that if U is the subspace of an irreducible representation V of gl n C with highest weight λ and d = dim C U then d is given by the number of integral points in GT(λ, µ).

Contributions
The main contribution of this paper is the introduction of the novel notion of the mutation of a Brauer configuration and its properties. Such mutations and their specializations allow solving different types of Diophantine problems. For instance, specializations of some mutations can be used to find out the Frobenius number of variations of some Diophantine equations called Gelfand-Tsetlin equations of the form n−3 ∑ i=−1 a i x i+2 + a n−2 x n = d, with suitable positive integers a i , for each −1 ≤ i ≤ n − 2. For these particular equations, we prove that there exists a solution if the constant term d is the number of Gelfand-Tsetlin patterns of a given type. To achieve that, we define some marked posets whose points are in bijective correspondence with some classes of Gelfand-Tsetlin patterns.
Specializations of mutations and suitable automata are also used to solve Diophantine problems of type D(n 1 , n 2 , K m ), which arise from the research of generalizations of denumerants. Such a problem is defined as follows: Find out positive integers λ 1 , λ 2 , . . . , λ m such that for positive integers n 1 ≤ 16 and n 2 , it holds that where K m = {k 1 , k 2 , . . . , k m } is a fixed set of positive integers.
As an application, we note that the AES key schedule is nothing but the specialization of a mutation. Thus, the algebraic properties of the corresponding Brauer configuration algebra allow the description of some characteristics of an AES key. It is worth noting that the use of Brauer configuration algebras to analyze the structure of an AES key does not appear in the literature, neither focused on the theory of representation of algebras nor focused on cryptography.
The following diagram (7) shows a way that some of the main results presented in this paper are related. In this case, Th(i), Cl(i) and Pr(i) denote Theorem i, Corollary i, and Proposition i, respectively. This paper is organized as follows; basic notation and definitions to be used throughout the paper are given in Section 2. In particular, we introduce an algorithm to build Brauer configuration algebras. The notion of mutation associated with Brauer configurations is also introduced in Section 2.3.2. In Section 3, we prove results regarding Gelfand-Tsetlin patterns. Gelfand-Tsetlin equations, and Gelfand-Tsetlin numbers are introduced in this section as well. In Section 4, we give properties of some Diophantine equations whose solutions are given by mutations of some Brauer configurations. In this section properties of the AES key schedule are described in terms of specializations of mutations of suitable Brauer configurations. Conclusions are given in Section 5.

Preliminaries
In this section, we introduce basic definitions and notation to be used throughout the paper. Henceforth, we will use the customary symbols N, Z, and R to denote the set of natural numbers, integers, and real numbers, respectively.

On the Frobenius Number
If n 1 , n 2 , . . . , n k are positive integers then the set S of all integers which can be presented in the form [1]: is a submonoid of (N, +). We let n 1 , n 2 , . . . , n k denote the set numbers n i are said to be generators of S, which is called a numerical monoid, e.g., 6,9,20 is the chicken McNugget monoid generated by 6, 9, and 20. S is said to be a numerical semigroup if |N\S| < ∞.
The following result regards numerical semi-groups.
The following result is a version of Ramírez-Alfonsín [4] of a theorem given by Roberts [10] where x is the largest integer less than or equal to x.
In the general case, there is no known formula for the Frobenius number of the kgenerated numerical semigroup S, but for k fixed, there is an algorithm that computes the Frobenius number in polynomial time. Actually, Ramírez-Alfonsín proved that the knapsack problem can be reduced to the Frobenius problem in polynomial time [11].
We call a non-zero element x ∈ n 1 , n 2 , . . . , n k irreducible if whenever x = y + z, either y = 0 or z = 0 (hence, x is irreducible if its only proper divisors are only 0 or itself).
The following result determines which elements are irreducible in a numerical monoid.

Path Algebras
In this section, we give a brief discussion on quivers, path algebras, and their ideals based on the work of Assem et al. [12].
A quiver Q = (Q 0 , Q 1 ) is a quadruple consisting of two sets: Q 0 (whose elements are called points or vertices) and Q 1 (whose elements are called arrows), and two maps s, t : Q 1 → Q 0 which associate to each arrow α ∈ Q 1 its source s(α) ∈ Q 0 and its target t(α) ∈ Q 0 , respectively. Note that, under these circumstances, a quiver is nothing but an oriented graph without any restrictions as to the number of arrows between two points, to the existence of oriented cycles or loops.
An arrow α ∈ Q 1 of source a = s(α) and target b = t(α) is usually denoted by α : a → b. A quiver Q = (Q 0 , Q 1 , s, t) is usually denoted by Q = (Q 0 , Q 1 ) or even simply by Q.
A path of length l ≥ 1 with source a and target b (or more briefly, from a to b) is a sequence (a|α 1 , α 2 , . . . , α l |b) where α k ∈ Q 1 for all 1 ≤ k ≤ l, and t(α l ) = b. Such a path is denoted by α 1 α 2 . . . α l .
We let Length(P) (Q l ) denote the length of a path P (the set of all paths P for which Length(P) = l). We also agree to associate to each point a ∈ Q 0 a path of length 0, called the trivial or stationary path at a, and denoted by e a = a||a. If X is a set of paths in a quiver Q then Length(X) = ∑ x∈X Length(x). Example 1. The following is an example of a quiver Q with three vertices a 1 , a 2 , and a 3 and two arrows α 1 and α 2 . Note that, Q 1 = {α 1 , α 2 }, and Q 2 = {α 1 α 2 }. Thus, Length(Q 1 ) = Length(Q 2 ) = 2.
If F is an algebraically closed field then the path algebra FQ of Q is the F-algebra whose underlying F-vector space has, as its basis, the set of all paths of length l ≥ 0 in Q and such that the product of two basis vectors is given by the usual concatenation of paths. For instance, B = {e a 1 , e a 2 , e a 3 , α 1 , α 2 , α 1 α 2 } is a basis of the algebra FQ, where Q is the quiver given in (10).
Let Q be a finite and connected quiver. The two-sided ideal R Q of the path algebra FQ generated (as an ideal) by the arrows of Q is called the arrow ideal of FQ. A two-sided ideal I of FQ is said to be admissible if there exists m ≥ 2, such that R m Q ⊆ I ⊆ R 2 Q . If I is an admissible ideal of FQ, the pair (Q, I) is said to be a bound quiver. The quotient algebra FQ/I is said to be the algebra of the bound quiver (Q, I) or, simply, a bound quiver algebra.
Let Q be a quiver. A relation in Q with coefficients in F is a F-linear combination of paths of length at least two having the same source and target. Thus, a relation ρ is an element of FQ, such that where the λ i are scalars (not all zero) and the p i are paths in Q of length at least 2 such that, if i = j, then the source (or the target, respectively) of p i coincides with that of p j . If m = 1, the preceding relation is called a zero relation or a monomial relation. If it is of the form p 1 − p 2 (where p 1 and p 2 are two paths), it is called a commutativity relation.
If (ρ j ) j∈J is a set of relations for a quiver Q such that the ideal they generate ρ j | j ∈ J is admissible, we say that the quiver Q is bound by the relation (ρ j ) j∈J or by the relations ρ j = 0 [12].
Henceforth, we let rad Λ denote the radical of a path algebra Λ = FQ, which is the intersection of all maximal ideals. Actually, if I is an admissible ideal of Λ, it holds that rad(FQ/I) = R Q /I. If ≺ is an admissible well-ordering on the set of paths, i.e., ≺ is a well-ordering such that 1.
If a, b, u, v ∈ Q where uav and ubv are both not-zero a ≺ uav or a = uav; 2.
if a ≺ b then uav ≺ ubv.
Then the tip Tip(x) = w of an element x ∈ FQ is the maximal path w with respect to ≺, such that w has a non-zero coefficient in x when it is writing as a linear combination of the elements of a fixed basis of FQ. Tip(X) = {Tip(x) | x ∈ X} is the set of tips of elements of elements in X [13].
Let I be an ideal in a path algebra FQ and let G ⊂ I. If Tip(G) = Tip(I) then G is a Gröbner basis for I with respect to ≺.

Brauer Configuration Algebras
Brauer configuration algebras are multi-serial path algebras introduced recently by Green and Schroll in [14]. These algebras constitute a generalization of Brauer graph algebras, which have as one of their properties that its representation theory is encoded by some combinatorial data based on graphs.
The following is a description of the structure of Brauer configuration algebras as the first author and Espinosa and Green and Schroll present in [14,15], respectively.

4.
O is a well-ordering < defined on Γ 1 . Such that for each vertex α ∈ Γ 0 , the collection If it is assumed that V α,1 = min S α and V α,t = max S α then a new circular relation R α = {V α,t < V α,1 } is added. S α is called the successor sequence at the vertex α and C α = S α ∪ R α ; 5.
occ(α, V) denotes the number of times that a vertex α occurs in a polygon V and the sum ∑ V∈Γ 1 occ(α, V) is said to be the valency of α, denoted val(α).;
Following the Green and Schroll ideas [14], Algorithm 1 builds the Brauer quiver Q Γ and the Brauer configuration algebra Λ Γ = FQ Γ /I Γ induced by the Brauer configuration Γ, where I Γ is an admissible ideal (see Remark 2).
Algorithm 1: Construction of a Brauer configuration algebra Output The Brauer configuration algebra Construct the quiver Q Γ of the Brauer configuration Γ, Each ordered set C α defines a cycle in Q Γ called a special cycle.
Construct the ideal I Γ , which is generated by the following relations: If a is a loop associated with a vertex α with val(α) = 1 and µ(α) > 1 then a µ(α)+1 = 0. 6.
For the construction of a basis of Λ Γ follow the next steps: (a) For each V ∈ Γ 1 choose a non-truncated vertex α V and exactly one special Define:

Remark 1.
In the last few years, the computational aspects of the theory of representation of quivers have been extensively studied. For instance, Struble [13] gave results regarding the complexity of algorithms based on Gröbner bases used to construct path algebras and their dimensions. Struble proved that if I is an ideal of a path algebra FQ and G is a Gröbner basis for I under some admissible ordering ≺. Then there exist algorithms to determine whether FQ/I is finitedimensional in O(Length(Tip(G))) time, to compute dim F FQ/I in O(Length(Tip(G))) time. Additionally, Haugland [16] proved that if X is the matrix, which simultaneously encodes all the constraints imposed on any homomorphism between two representations R M and R N of a quiver Q, then there is an algorithm, which computes a basis of Hom FQ (R M , R N ) in O(r 3 + rc 2 + |Q 0 |r), where r (c) is the number of rows (columns) of the matrix X.
The complexity of algorithms used to compute the center of an associative algebra has been also studied by Strubel who described an algorithm, which finds the center Z(FQ) of a path algebra FQ with dimension d and h generators in O(d 4 ht(< Q ) + d 3 ht( * ) + t( * F ) + t(= F ) + t(reduce)). Where, d 4 ht(< Q ) means that up to d 4 h elements in Q have non-zero coefficients, the algorithm develops d 3 ht( * ) multiplications in Q, t( * F ) and t(= F ) denote the time that the algorithm takes making scalar multiplications and comparisons to determine if a given element is = 0 in the field F, t(reduce) is the running time that a specialized algorithm reduces an element of the basis of FQ to a normal form.
Several up-to-date software packages have been designed to develop basic operations and testing conjectures on path algebras and representation of quivers. For instance, the GAP package QPA [17] extends the GAP functionality for computations with finite-dimensional quotient path algebras, whereas King et al. have been working on the implementation of Sage packages [18] with the same purposes as QPA.
Henceforth, if no confusion arises, we will assume notations Q, I, and Λ instead of Q Γ , I Γ and Λ Γ , for a quiver, an admissible ideal, and the Brauer configuration algebra induced by a fixed Brauer configuration Γ.

Example 2.
As an example of the application of Algorithm 1, consider the following reduced Brauer configuration: The following is a list of special cycles: The admissible ideal I Γ is generated by the following relations: where a is the first arrow of the special cycle C k i,V j , The next result regards the dimension of a Brauer configuration algebra.

Proposition 3 ([14]
, Proposition 3.13). Let Λ be a Brauer configuration algebra associated with the Brauer configuration Γ and let C = {C 1 , . . . , C t } be a full set of equivalence class representatives of special cycles. Assume that for i = 1, . . . , t, C i is a special α i -cycle where α i is a non-truncated vertex in Γ. Then where |Q 0 | denotes the number of vertices of Q, |C i | denotes the number of arrows in the α i -cycle C i and n i = µ(α i ).

Proposition 4 ([14]
, Proposition 3.6). Let Λ be the Brauer configuration algebra associated with a connected Brauer configuration Γ. The algebra Λ has a length grading induced from the path algebra FQ if, and only if, there is an N ∈ Z >0 such that for each non-truncated vertex α ∈ Γ 0 val(α)µ(α) = N.
The following result regards the center of a Brauer configuration algebra.

Theorem 3 ([19]
, Theorem 4.9). Let Γ be a reduced and connected Brauer configuration and let Q be its induced quiver and let Λ be the induced Brauer configuration algebra such that rad 2 Λ = 0 then the dimension of the center of Λ denoted dim F Z(Λ) is given by the formula: Example 3. The dimension of the algebra Λ Γ = FQ Γ /I Γ defined by (12) and (13) is given by the following identity: Note that,

The Message of a Brauer Configuration
The notion of labeled Brauer configurations and the message of a Brauer configuration were introduced by Fernández et al. to define suitable specializations of some Brauer configuration algebras [15,20]. According to them, since polygons in a Brauer configuration Γ = (Γ 0 , Γ 1 , µ, O) are multisets, it is possible to assume that any polygon U ∈ Γ 1 is given by a word w(U) of the form The message is, in fact, an element of an algebra of words W Γ associated with a fixed Brauer configuration such that for a given field F the word algebra W Γ consists of formal sums of words with the form ∑ α i ∈F U∈Γ 1 α i w(U), 0w(U) = ε is the empty word, and 1w(U) = w(U) for any U ∈ Γ 1 . The usual word concatenation gives the product in this case. The formal product (or word product) is said to be the message of the Brauer configuration Γ.
A mutation M(Γ, X) = (Γ , X ) is given by a Brauer configuration M(Γ) = Γ = (Γ 0 , Γ 1 , µ , O ) and a vector X = (x 0 , x 1 , . . . , x l−1 ) with x i ∈ F s , 0 ≤ i ≤ l − 1, such that: Note that, if the indices of the original seed are considered then polygons in Γ 1 can be written as follows: For successor sequences, the orientation O is defined in such a way that, It turns out that according to the indices assumed for the original seed, the ith mutation M i has the form: Brauer configurations obtained from mutations are said to be Brauer clusters. Polygons are called cluster polygons.
For a fixed positive integer m 0 , the m 0 -Brauer cluster, Φ m 0 is a Brauer configuration, such that: If M(i) denotes the message associated with the ith Brauer cluster then for successor sequences, it is assumed the order M(0) < M(1) < · · · < M(m 0 ).

Example 4.
As an example, in the sequel we describe mutations of the following seed (Γ, Successor sequences at 0 and 1 are defined as follows: 0 : 1 < x, 1 : 1 < x. The first mutation M(Γ) = (Γ , X = {x, x + 1}) is defined as follows: In this case 1 is the only non-truncated vertex with a successor sequence of the form: The second mutation M 2 = (Γ , X = {x, 1}) is described as follows: In this case, the successor sequences at 0 and 1 are 0 : x < 1, 1 : x < 1.
The admissible ideal I in the Brauer configuration algebra Λ (Φ n ,X) is generated by the following relations: 1.
(l i j ) 2 , l i 0 l j 1 , for all possible values of i and j; 2.
α i 1 β r 0 , α i 1 l r 0 , l r 1 α i 1 , for all possible values of i and r; 3.
β r 0 α i 1 , β r 0 l s 1 , l s j β r 0 , for all possible values of i, r, and s.
Proof. Since each polygon M(i) ∈ Φ m 0 1 contains at least one 0 and at least one 1, we conclude that Φ m 0 does not contain truncated vertices. The result follows provided that Theorem 5. If F is a finite field then the set M Φ of m 0 -Brauer clusters obtained by mutation is finite, and the special cycle of maximal length has one of the two shapes (26) or (27). Moreover, there exists m ∈ N, such that (Γ, X) = (Γ m+1 , X m+1 ) for a given initial seed (Γ, X).
where β m β m+1 = 0.  (26), otherwise the special cycle of the quiver Q Γ m+n after m + n mutations has the form (27). We are done.

Remark 3.
For m 0 > 1 and a fixed seed, the dimension of an algebra Λ Φ m 0 and its center Z(Λ Φ m 0 ) can be estimated by using some statistical methods. For instance, we use a sample of 10 6 random seeds to obtain confidence intervals for these values. Such samples allow us to infer that if m 0 = 10, then where P r (X) denotes the probability of an event X.

Remark 4.
For the sake of applicability, the definition of mutation of a Brauer configuration has been given by taking into account that elements X constituting a seed are elements of finite products of finite fields. However, it can be extended to arbitrary rings, in such a case, if R 1 and R 2 are rings then any map τ : X ∈ R k 1 → X ∈ R s 2 , r, s ≥ 1 can be used to transform a given seed (Γ, X) into a Brauer cluster of the form (Γ , X ).

Deterministic and Non-Deterministic Automata
In this section, we recall definitions of deterministic and non-deterministic automata as Rutten et al. present in [21]. We use these definitions to interpret Brauer configuration algebras as automata with acceptance language given by relations generating suitable admissible ideals.
Given an alphabet A, a deterministic automaton is a pair (X, α) consisting of a (possibly infinite) set X of states and a transition function α : X → X A . The following is an illustration of this kind of transitions, where α(x)(a) = y = x a [21].
x a G G y If ε denotes the empty word then x ε = x, for any x ∈ X and x wa = α(x w )(a) with w ∈ A * . A deterministic automaton can be decorated by means of a coloring function c : X → 2 = {0, 1}, such that c(x) = 1 if x is an accepting (or final) state, c(x) = 0, if x is a nonaccepting state. A triple (X, c, α) is said to be a deterministic colored automaton. In the following diagram, an accepting state is denoted with a double circle, x is an accepting state, whereas y is a non-accepting state. Given a deterministic colored automaton (X, c, α) and a state x ∈ X, the set is called the language accepted or recognized by the automaton (X, c, α) starting from the state x. A deterministic automaton can also has an initial state x ∈ X represented by a function x : 1 = {0} → X. The triple (X, x, α) is said to be a deterministic pointed automaton.
A non-deterministic automaton is a pair (X, α) consisting of a set X (possibly infinite) of states and a transition function α : X → P w (X) A , that assigns to each letter and to each state a finite set of states [21]. If to each state it is assigned a single new state, the definition of a deterministic automaton is recovered. As in the deterministic case, a state x in a non-deterministic automaton can be either accepting (c(x) = 1) or non-accepting (c(x) = 0); and x ε = {x}, x w a = {y a | y ∈ x w }. A triple (X, c, α) is called a colored non-deterministic automaton.
If the set of states X is finite then the automaton is said to be a deterministic (nondeterministic) finite automaton, denoted DFA (NFA), respectively.
A regular language associated with a Brauer configuration algebra Automata associated with path algebras have been studied by Rees [22], who introduced some automata associated with some string algebras, such automata were used by her to describe indecomposable representations over these types of algebras, she points out that the set of strings defining representations of string algebras, and many other bounded path algebras, constitute a regular set. In this section, we follow Rees ideas to describe an automaton associated with a Brauer configuration algebra.
Values of the map c : Q 1 → 2 can be obtained by endowing to the successor sequences a length-lexicographic order, in this case, both Γ 0 and Γ 1 are well-ordered sets with partial orders ≺ and <, respectively. In such a way that initial states in the corresponding automaton are given by minimal successor sequences (see Algorithm 1). Note that, if S a,U denotes a successor sequence starting in a polygon U with a ∈ U, and if |S α, A Brauer configuration algebra Λ Γ induced by a Brauer configuration Γ = (Γ 0 , Γ 1 , µ, O) has associated a regular language L Γ = A * Γ / ∼, where the alphabet A Γ = {x i α | α ∈ Γ 0 , 1 ≤ i ≤ val(α)}, each letter x i α corresponds to a unique arrow in (Q Γ ) 1 . Each path P ∈ Q Γ corresponds to a word w ∈ L Γ . Two words w, w ∈ L Γ are equivalent (i.e., w ∼ w ) if their corresponding paths are equivalent as elements of the Brauer configuration algebra. In this case, if S α is a successor sequence associated with the vertex α ∈ Γ 0 , then w S α denotes the word associated with the corresponding special cycle up to equivalence. If min S α = U ∈ Γ 1 then w S α ∈ O c (U) (final vertices of special cycles are final states up to equivalence).
In the associated automaton of a Brauer configuration algebra, polygons are states. Actually, all states we represent are accepted states. The transition between states is given by the order <, in other words, if x α i is the letter associated with an arrow U i < U i+1 , that is, U i α belongs to the admissible ideal I, with Λ Γ = kQ Γ /I then it is not accepted as a word in L Γ if α = α .
Note that, according to the automaton associated with the Brauer configuration defined by (12) and (13), it holds that V 1 is the initial and final state and c(α i j β i j ) = c(β i j α i j ) = 0, for all the possible values of i, j, i , j .

Enumeration of Gelfand-Tsetlin Patterns
In this section, we recall some well-known results regarding the enumeration of GT patterns. Furthermore, we introduce the notion of the heart of a GT pattern which allows us to define posets and marked order polytopes associated with the number of some GT patterns [2,7,8,23,24].
If λ = (λ (n,1) , λ (n,2) , λ (n, 3) , . . . , λ (n,n) ) is an integer partition and V(λ) is a finitedimensional irreducible representation of gl n C with highest weight λ then a basis of V(λ) is parametrized by GT patterns T = T((n, 1), (n, n), (1, 1)) associated with λ. These are arrays of integer row vectors with the shape: such that the upper row coincides with λ and the following conditions hold: This setting together with terms of the form l (k,i) = λ (k,i) − i + 1 allow establishing the existence of a basis of V(λ) parametrized by T, such that the action of generators of gl n C are given by some Gelfand-Tsetlin formulas [7,8].
A GT pattern with first row of the form 1, 2, . . . , n is called a monotone triangle of length n. It is also known that there is a bijection between GT patterns with first row λ n ≤ λ n−1 ≤ · · · ≤ λ 1 and column-strict plane partitions of type λ = (λ 1 , λ 2 , . . . , λ n ) (λ i parts in row i) and largest part ≤ n [23,24].
The following theorem regarding monotone triangles was given by Zeilberger in 1996.

Theorem 6 ([25]
, Main Theorem). The number of monotone triangles of length n with bottom entry a (n,n) = r is equal to Note that, A n (x) = ∑ T x s(T) , where s(T) denotes the number of standard elements of T such that t (i−1,j−1) < t (i,j) < t (i−1,j) , for 2 ≤ i ≤ j ≤ n. In particular, A n (2) = 2 ( n 2 ) .
For n ≥ 4 fixed, it is possible to define an order on the set of GT-hearts over gl n C (associated with GT patterns with fixed first row) whose covers are defined as follows: If T(H n ) denotes the set of Gelfand-Tsetlin arrays with H n as heart, then two hearts H n and H n are said to be equivalent denoted H n ∼ = H n , provided that |T(H n )| = |T(H n )|. Thus, H (n,r) is a poset endowed with an equivalence relation.

Theorem 7.
The number g (n,r) of GT patterns over gl n C defined by a weight vector of the form w = (n, n − r, n − 2r, . . . , n − r(n − 1)) is given by the formula g (n,r) = (r + 1) ( n 2 ) .

Marked Posets and Marked Polytopes
In this section, we describe a special class of marked posets and marked polytopes introduced by Fourier in [26]. We also prove that some posets of type H (n,r) are marked by the number of some suitable Gelfand-Tsetlin arrays.
Let (P, ) be a finite poset and a subset A of P containing at least all maximal and all minimal elements of P, we set: and call the triple (P, A, λ) a marked poset. Then the marked chain polytope associated with λ ∈ Q A is defined: ≥0 | s x 1 + s x 2 + · · · + s x n ≤ λ b − λ a for all chains a x 1 · · · x n b} while the marked order polytope is defined as where a, b ∈ A, x i , x, y ∈ P\A (the definition of marked order polytope is also valid if in R is considered the usual order ≤ and the order in P is reversed). The marked poset (P, A, λ) is regular if for all a = b in A, λ a = λ b , and there are no obviously redundant relations.
If a, b ∈ P, then a chain c(a, b) = {x 0 = a x 1 x 2 x n−1 b = x n } ⊆ P is said to be saturated, if for each i, 0 ≤ i ≤ n − 1, the relation x i x i+1 is a cover (i.e., if there exists z ∈ P, such that x i z x i+1 , then either z = x i or z = x i+1 ).
Let (P, A, λ) be a regular marked poset then the number of facets in the marked order polytope is equal to the number of cover relations in P.
If c(a, b) denotes the number of saturated chains a x 1 x 2 · · · x p b, x i ∈ P\A, then the number of facets in the marked chain polytope is equal to |P\A|+ ∑ a b c(a, b).
The following result regards posets of type (H (n,r) , ). Proposition 6. For n = 4, and r > 1, If Q denote the set of numbers q i 's given by identities (29) and (30). Then Q ⊂ Q H (n,r) (see (31)).
Proof. Consider the following relation defined by hearts H 4 H 4 associated with suitable Gelfand-Tsetlin arrays.
x y z x y z Then, if x = x , y = y , z = z − 1 or x = x , y = y + 1, z = z and the number of Gelfand-Tsetlin arrays is given by a sequence with the form S gt (n 1 , r) = n 1 q 1 + 2r+1 ∑ j=2 q j then H 4 covers H 4 , and the associated sum to H 4 has the form S gt (n 1 , r) = n 1 q 1 + 2r+1 ∑ j=2 q j with n 1 = n 1 − 1. Note that, if H 4 and H 4 are incomparable then x = x .
The following table shows Frobenius numbers associated with some Gelfand-Tsetlin equations: We let H (4,r,y 0 ) denote the subposet of (H (4,r) , ) consisting of hearts with a fixed entry λ (3,2) = y 0 , with the shape x y 0 z , the corresponding equivalence classes have associated a unique number of the form S gt (n 1 , j). If it is assumed that n 1 = 1 for any of these numbers, then the following result holds:

Gelfand-Tsetlin Equation
Theorem 8. Numbers S gt (n 1 , r) with n 1 = 1 associated with equivalence classes of points of the subposet H (4,r,y 0 ) define a Brauer configuration algebra Λ Γ gt with length grading induced from the path algebra FQ Γ gt .
Proof. If n 1 = 1 for any number S gt (n 1 , r) then such an assignation defines the Brauer configuration Γ gt = (Γ gt0 , Γ gt1 , µ, O), such that: Γ gt0 = {q 1 , q 2 , . . . , q 2r+1 }, (see identities (29) and (30)), where w(P i ) denotes the word associated with the polygon P i (see (15)). In this case the Brauer quiver has the form: The admissible ideal is generated via equivalence of special cycles, products of the is a special cycle and α 2r+2 q s is an arrow connecting polygons P 2r+1 and P 1 . Since µ(q i )val(q i ) = 2r + 1, for any q i ∈ Γ gt0 , the result follows as a consequence of Theorem 4.

A Relationship between Brauer Configuration Algebras and Gelfand-Tsetlin Equations
For k ≥ 5, let us consider a seed (Γ k , X k ), such that X k = (y 1 , y 2 , . . . , y k−1 ) ∈ R k−1 is the ring of polynomials over C in variables x 1 , x 2 , and x 3 . Γ k = (Γ 0,k , Γ 1,k , µ k , O k ) is a Brauer configuration such that: For j ≥ 1, the word w(U j ) associated with the polygon U j is defined as follows:

4.
The successor sequences S x i at vertices x 1 , x 2 , and x 3 have the following forms: j denotes a sequence of the form U j < U j < · · · < U j i times and t i = i(i+1) 2 ;
For k ≥ 5 fixed, the ideal I k of the Brauer configuration algebra Λ Γ k = FQ Γ k /I is generated by the following relations: 1.
If a is the first arrow of a special cycle C x i then C x i a = 0; 3. (l h x s ) 2 = 0.
For k ≥ 5, the mutation M(Γ k , X k ) = (Γ , X k ) of a seed (Γ k , X k ) is defined as follows: 1.
The Brauer configuration Γ k = (Γ 0,k , Γ 1,k , µ k , O k ) is defined in such a way that: For j ≥ 1, the word w(U j ) associated with the polygon U j is given by the following identities: The successor sequences at x 1 and x 2 are defined by the following chains: (40) The following is the Brauer quiver Q Γ 5 defined by the Brauer configuration Γ 5 .
The Brauer configuration algebra Λ Γ k = FQ Γ k /I is bounded by the ideal I ⊂ I, whose relations are obtained by restricting to x 1 and x 2 the relations contained in I.
The following results regards algebras Λ Γ k and their corresponding mutations Λ Γ k for k ≥ 5 fixed.
The second identity holds provided that in Γ k , val(x 1 ) = k, and val(x 2 ) = t k−1 . The third identity is obtained by taking into account that the number of loops in We are done.

On Diophantine
Equations of Type D(n 1 , n 2 , K m ) In this section, we give criteria to solve equations of type D(n 1 , n 2 , K m ), i.e., Diophantine equations of the form: with fixed k 1 , k 2 , . . . , k m , n 1 , and n 2 .
Proof. Since the length of the message |M(Φ m 0 )| = l 2 2 n then it consists of l 2 2 n−2 lists of four bits. We let A = {α 1 , α 2 , . . . , α l 2 2 n−2 } denote this set of lists. Define a map T : A → Hex, such that T(α i ) = α i ∈ Hex, where Hex is a notation for the hexadecimal numbering system Hex = {0, 1, 2, . . . , 9, A, B, C, . . . , F}. In this case, The map T defines a new Brauer configuration T(Φ m 0 ) = (Img T, T(Φ m 0 1 ), µ , O ), which we assume reduced without loss of generality. Each polygon T(M i ) consists of elements of the form α j with α j ∈ M i . Actually, if M(M i ) = α 1 α 2 . . . α r i is the message of M i then α 1 α 2 . . . α r i ∈ A * is the message of T(M i ). Additionally, if M i < M i+1 in Φ m 0 then T(M i ) < T(M i+1 ) in T(Φ m 0 ); and µ (α i ) = 1 for any α i ∈ Img T. Thus, the message M(T(Φ m 0 )) ∈ A * is a l 2 2 n−2 word whose letters α j can be grouped according to its valency. Thus, M(T(Φ m 0 )) has the form: where A i s is a multiset with |A i s | = L i s and A i x ∩ A i y = ∅. Note that A i s consists of all letters α i , such that val(α i ) = v i s , i.e, the message M(A i s ) associated with A i s can be written as Therefore, Then terms L i h give a solution of a Diophantine equation of type Since any Brauer configuration algebra defines a regular language, whose associated automaton uses arrows of the Brauer quiver as transitions between states given by polygons, which, in this case, are obtained by mutation. Then the probability P r (M) that a given message M or Diophantine equation occurs after applying a mutation to a fixed seed is such that 0 < P r (M) < 1 (see Remark 3), we conclude that the associated automaton is non-deterministic. Example 6. Suppose that a seed (Γ, X) is defined by the polynomial p(x) = x 8 + x 4 + x 3 + 1, and that we use set Hex = {0 = 0000, 1 = 0001, 2 = 0010, . . . , A = 1010, . . . , F = 1111} to denote polynomials. Then X = (x 1 , x 2 , x 3 , x 4 ) can be denoted as follows: x 1 = (AF, C0, 13, 10), λ 1 + λ 2 + λ 3 = 15, 3λ 1 + 2λ 2 + λ 3 = 32.
For m 0 = 10, any mutation of this seed gives rise to a solution of a Diophantine equation of type (n 1 ≤ 16, 32, K m ) with K m being a set of the form: In this section, we give some properties of the AES key schedule based on the fact that such schedule arises from Brauer configurations, as described in (51) and its mutations are given by identities (52)-(54).
AES is a symmetric block cypher also considered as a substitution-permutation network, which was adopted in 2001 by the US government as the current standard cryptographic [27]. It is considered secure against different types of attacks. It requires keys of 128 bits (for 10 rounds), 192 bits (for 12 rounds), and 256 bits (for 14 rounds). Encryption and decryption algorithms are carried out via polynomials defined over suitable Galois fields. Many routers provide protocols WPA2-PSK (TKIP), WPA2-PSK (AES), and WPA2-PSK (TKIP/AES) as options to ensure Wi-Fi security. The new protocol WPA3 uses keys of 128 bits (CCMP-128), 192 bits (WPA3-Enterprise mode) to secure wireless computer networks.
In the cryptosystem AES, a plaintext, also called a state, is a sequence of 16 bytes, the encryption process also generates a 16-bytes sequence by using keys of 128, 192, or 256 bits. Such length depends on the number of rounds developed in the encryption process, 10, 12, or 14, respectively.
For the last round the function MixColumns is not executed. The next table gives all the possible outputs of the transformation SubBytes, whose operations are made modulo the polynomial p(x) = x 8 + x 4 + x 3 + 1 as described in identities (52)  The key schedule is the process for which all the keys to be used in the encryption process are generated. Such keys are called subkeys. For keys of 128 bits (or 16 bytes) of length, the process generates 11 subkeys, the initial key, the nine main rounds and the final round.
The expanded key can be seen as an array of 32-bit words numbered from 0 to 43 (0 for the initial key, which is the message of a Brauer configuration Γ associated with an initial seed (Γ, X). Γ and X are described as in Example 6. Identities (51)-(54) are used to generate the schedule of the initial seed-key in terms of mutations of Brauer configurations), words that are a multiple 4 (w 4 , w 8 , . . . , w 40 ) are calculated as follows: