Solutions to the T-systems with Principal Coefficients

The $A_\infty$ T-system, also called the octahedron recurrence, is a dynamical recurrence relation. It can be realized as mutation in a coefficient-free cluster algebra (Kedem 2008, Di Francesco and Kedem 2009). We define T-systems with principal coefficients from cluster algebra aspect, and give combinatorial solutions with respect to any valid initial condition in terms of partition functions of perfect matchings, non-intersecting paths and networks. This also provides a solution to other systems with various choices of coefficients on T-systems including Speyer's octahedron recurrence (Speyer 2007), generalized lambda-determinants (Di Francesco 2013) and (higher) pentagram maps (Schwartz 1992, Ovsienko et al. 2010, Glick 2011, Gekhtman et al. 2014).


Introduction
The A ∞ T-system [DFK13], also called the octahedron recurrence, is a discrete dynamical system of formal variables T i,j,k for i, j, k ∈ Z satisfying T i,j,k−1 T i,j,k+1 = T i−1,j,k T i+1,j,k + T i,j−1,k T i,j+1,k .
This recurrence relation preserves the parity of i + j + k and therefore there are two independent systems depending on the parity of i + j + k. We will assume throughout the paper that i+j +k ≡ 1 mod 2. Several combinatorial solutions have been considered including solutions in terms of alternating sign matrices [RR86], domino tilings [RR86,EKLP92], perfect matchings [Spe07] and networks [DFK13].
The A ∞ T-system can also be interpreted [DFK09] as mutation in an infiniterank cluster algebra [FZ02] of geometric type without coefficients. Its corresponding quiver is called the octahedron quiver [DFK13].
One can also consider cluster algebras with coefficients [FZ07]. For cluster algebras of geometric type, this is equivalent to adding frozen vertices to the quiver. A quite general type of coefficients are principal coefficients. It corresponds to adding one frozen vertex for each quiver vertex and an arrow pointing from it to the quiver vertex. In some literature, this new quiver is called the coframed quiver associated with the octahedron quiver. The reason why the principal coefficients are very important is due to the separation formula [FZ07,Theorem 3.7], stating that a cluster algebras with any coefficients can be written in terms of one with principal coefficients.
Some generalizations of T-systems with coefficients have been suggested by Speyer in his work on Speyer's octahedron recurrence [Spe07] and by Di Francesco in his work on the generalized lambda determinant [DF13]. In this paper, we consider A ∞ T-systems with principal coefficients using cluster algebras definition. We then give combinatorial solutions in terms of perfect matchings, non-intersecting paths and networks.
The paper is organized as follows. In Section 2, we review some basic definitions and results in cluster algebras from [FZ02,FZ07]. In Section 3, we define the Tsystem with principal coefficients, whose initial condition is in the form of initial data on a stepped surface.
The goal is to find, for an arbitrary point (i 0 , j 0 , k 0 ) and a stepped surface k, an expression of T i0,j0,k0 in terms of initial data on k. Laurent phenomenon for cluster variables [FZ02,Theorem 3.1] guarantees that the expression is indeed a Laurent polynomial in the initial data and coefficients. In this paper, we give explicit combinatorial expressions of T i0,j0,k0 in terms of initial data when the point (i 0 , j 0 , k 0 ) is above k and k is above the fundamental stepped surface fund : (i, j) → (i + j mod 2) − 1. Some other cases will be discussed in Section 9.
Section 4 is devoted to a perfect matching solution. Following the construction in [Spe07], we first construct a finite bipartite graph with open faces G depending on both (i 0 , j 0 , k 0 ) and k, then construct face-weight w f and pairing-weight w p on perfect matchings of G. This leads to the perfect-matching solution (Theorem 4.9): where the sum runs over all perfect matchings of G. The weight w p (M ) is a monomial in the cluster coefficients, while w f (M ) is a Laurent monomial in the initial data (cluster variables).
In Section 5, we define the closure G of the graph G and transform our previous two weights to the edge-weight w e . This gives another form of the perfect-matching solution (Theorem 5.4): T i0,j0,k0 = M w e (M ) w e (M 0 ) ci,j =1 .
The sum runs over all perfect matchings of G with a certain boundary condition, see Definition 5.2.
In Section 6, we orient all the edges of G and G and give an explicit bijection between perfect matchings of G and non-intersecting paths on G with certain sources and sinks. This bijection can also be extended to G. Using the modified edge-weight w e obtaining from w e together with the bijection, the perfect-matching solution for G gives the nonintersecting-path solution (Theorem 6.18): T i0,j0,k0 = P w e (P ) where the sum runs over all non-intersecting paths on G with certain sources and sinks, see Theorem 6.18. In Section 7, we first consider the network N , studied in [DF10,DFK13] associated with (i 0 , j 0 , k 0 ) and k. It is obtained from the shadow of the point (i 0 , j 0 , k 0 ) on the lozenge covering on k. We point out that it can also be obtained from the graph G by tilting all the diagonal edges of G so that they become horizontal. This allows us to pass the modified edge-weight w e on G to a weight on the network N . Paths on G then become paths on N . From Theorem 6.18, we get the network solution (Theorem 7.3) as a partition function of weighted non-intersecting paths on the network N , which can also be written as a certain minor of the network matrices (Theorem 7.8).
Acknowledgements. The author would like to thank his advisors R. Kedem and P. Di Francesco for their helpful advice and comments. The author also thank G. Musiker and M. Glick for discussions. This work was partially supported in part by a gift to the Mathematics Department at the University of Illinois from Gene H. Golub, the NSF grant DMS-1100929, NSF grant DMS-1301636 and the Morris and Gertrude Fine endowment.

Cluster algebras
In this section, we quote some basic definitions and important results in the theory of cluster algebras mainly from [FZ02,FZ07].
2.1. Finite rank cluster algebras. Let (P, ⊕, ·) be a semifield, i.e., (P, ·) is an abelian group, and ⊕ is an auxiliary addition: commutative, associative and distributive with respect to the multiplication. The following are two important examples of semifields.
Definition 2.1 (Universal semifield). For a set of labels J, the universal semifield on the set of variables {y j | j ∈ J} denoted by Q sf (y j : j ∈ J) := (Q sf (y j : j ∈ J), +, ·) is the set of all subtraction-free expressions of rational functions in independent variables {y j | j ∈ J} over Q with the usual addition and multiplication.
Definition 2.2 (Tropical semifield). For a set of labels J, the tropical semifield on the set of variables {y j | j ∈ J} denoted by Trop(y j : j ∈ J) is the free multiplicative abelian group generated by {y j | j ∈ J} with the auxiliary addition defined by: Here are some notations that will be used throughout the paper: Let n ∈ N, we now state the main definitions of cluster algebras of rank n. Let P be a semifield, F = QP(x 1 , . . . , x n ) an ambient field, T n the n−regular tree whose n edges incident to each vertex have different labels from 1 to n. Definition 2.3 (Cluster patterns and Y-patterns). A cluster pattern (resp. Ypattern) is an assignment t → (x t , y t , B t ) (resp. t → (y t , B t )) of any vertex t ∈ T n to a labeled seed (resp. labeled Y-seed) such that: • The cluster tuple x t = (x 1;t , . . . , x n;t ) is an n−tuple of elements of F forming a free generating set. • The coefficient tuple y t = (y 1;t , . . . , y n;t ) is an n−tuple in P.
• The exchange matrix B t = (b (t) ij ) ∈ M n×n (Z) is a skew-symmetrizable matrix.
• t k − − t in T n if and only if (x t , y t , B t ) µ k ←→ (x t , y t , B t ), where µ k is the seed mutation in direction k defined by: kj ] + , otherwise.
-y t = (y 1,t , . . . , y n,t ) where Definition 2.4 (Cluster algebra). The cluster algebra A associated with a cluster pattern t → (x t , y t , B t ) for t ∈ T n is defined to be the ZP-algebra generated by all cluster variables, i.e.
In this paper, we will consider only cluster algebras with skew-symmetric exchange matrix. In this case, we can think of the cluster mutation as a combinatorial rule performing on a quiver.
Definition 2.5 (The quiver associated with a skew-symmetric B). When the exchange matrix B = (b ij ) i,j∈[1,n] is skew-symmetric, we define Q B , the quiver associated with B, to be a directed graph with vertices 1, . . . , n. There are b ij arrows from i to j if and only if b ij > 0.
All the information of the exchange matrix B, skew-symmetric, is encoded in the quiver Q B . The mutation µ k will then act on Q B by the following process: (1) For every pair of arrows i → k and k → j, add an arrow i → j.
(2) Reverse all the arrows incident to k.
One can easily check that µ k (Q B ) = Q µ k (B) . The following is an example of the quiver associated with an exchange matrix and the quiver mutation.
Fixing t 0 ∈ T n , we can express variables x 1;t , . . . , x n;t , y 1,t , . . . , y n;t of the labeled seed at arbitrary t ∈ T n in terms of variables x 1;t0 , . . . , x n;t0 , y 1,t0 , . . . , y n;t0 at t 0 . We will then call the labeled seed (resp. labeled Y-seed) at t 0 an initial labeled seed (resp. initial labeled Y -seed) and assign a simpler notations: . Unless stated otherwise, the initial labeled seed (x, y, B) is always at t 0 and we denote A(x, y, B) the cluster algebra with an initial labeled seed (x, y, B). Clearly, a cluster pattern (resp. Y-pattern) is completely determined by its initial labeled seed (resp. initial labeled Y-seed). In addition, a cluster variable is a Laurent polynomial in the initial variables, as stated in the following theorem.
Theorem 2.6 (Laurent phenomenon [FZ07, Theorem 3.7]). The algebra A(x, y, B) is contained in the Laurent polynomial ring ZP[x ±1 ], i.e. every cluster variable is a Laurent polynomial over ZP in the initial cluster seed x 1 , . . . , x n .
If y i = 1 for all i ∈ [1, n], we have y i,t = 1 for all i ∈ [1, n], t ∈ T n . We call it a coefficient-free cluster algebra, and write just (x t , B t ) for for its labeled seeds.
Definition 2.7 (Frozen variables). For a cluster algebra (resp. cluster pattern) of rank m with initial seed (x, y, B), we consider a subpattern t ∈ T n ⊆ T m → (x t , y t , B t ). It is the pattern obtained by µ 1 , . . . , µ n . That means the directions n + 1, . . . , m are not mutated. We call it a cluster algebra (resp. cluster pattern) of rank n with frozen variables x n+1 , . . . , x m .
Remark 2.8. In the cluster algebra of rank n with frozen variables x n+1 , . . . , x m , the cluster seeds are not mutated in directions n + 1, . . . , m. So, the necessary information in the m × m matrix B for mutations are only the columns 1 to n. Hence we often use a reduced exchange matrixB instead of the full exchange matrix B, whereB is the m × n submatrix of B obtained by deleting columns n + 1 to m. Definition 2.9 (Geometric type). A cluster algebra (or cluster pattern, or Ypattern) is of geometric type if P is a tropical semifield.
Remark 2.10 (Geometric type and Frozen variables). For a cluster algebra or cluster pattern of geometric type, the notion of coefficients and frozen variables are interchangeable. Let t ∈ T n → (x t , y t , B t ) be a cluster pattern of geometric type of rank n where P = Trop(x n+1 , . . . , x m ). Since x n+1 , . . . , x m generate P, we can choose the initial seed coefficients to be Then the pattern is equivalent to a coefficient-free cluster pattern: t ∈ T m → ((x 1;t , . . . , x m;t ),B t ) with frozen variables x n+1 , . . . , x m , whereB = (b ij ) m×n .
Example 2.11. Consider a semifield P = Trop(x 5 , x 6 ) and a rank 4 cluster algebra of geometric type with an initial seed (x, y, B) where We have y + 1 = x 6 and y − 1 = x 5 . After the seed mutation µ 1 (Definition 2.3), we have On the other hand, by Remark 2.10, we can think of x 5 , x 6 as frozen variables and transform our cluster algebra with coefficients to the coefficient-free cluster algebra of rank 6 with the following cluster variables and reduced exchange matrix.
The mutation µ 1 gives the new cluster tuple x = x 4 x 6 + x 2 x 5 x 1 , x 2 , x 3 , . . . , x 6 and the following quiver mutation. We see that the mutated quiver encodes the information of y = x 5 x 2 , 1, 1, 1 x 5 .
Also notice that we can omit arrows between frozen variables because they will not effect any mutations at non-frozen variables.
Definition 2.12 (Principal coefficient). A cluster algebra (or cluster pattern, or Y-pattern) has a principal coefficients at t 0 ∈ T n if P = Trop(y 1 , . . . , y n ) where the initial coefficient tuple is y t0 = (y 1 , . . . , y n ). We denote A • (B) for the cluster algebra with principal coefficients.
The quiver QB is obtained from Q B by adding one vertex i and an arrow i → i for any vertex i in the quiver Q B . The new QB is called the coframed quiver associated with Q B .
Definition 2.14 (The functions X l;t ). Given an exchange matrix B, we consider the unique (up to isomorphism) cluster pattern t → (X t , Y t , B t ) with principal coefficients at t 0 and an initial seed (X, Y, B). For l ∈ [1, n] and t ∈ T n , we let X l;t ∈ Q sf (X 1 , . . . , X n ; Y 1 , . . . , Y n ) be the l-th component of the cluster tuple at t, and In short, X (B) l;t is a cluster variable in the cluster algebra with principal coefficients. For a fixed B, we often view it as a function on the initial variables X i and Y i for i ∈ [1, n]. The function F The next theorem states that cluster variables of any cluster pattern can be written in terms of the functions X l;t with some restriction.
The notation F l;t | P (y 1 , . . . , y n ) means that we compute F l;t (y 1 , . . . , y n ) in P by changing + to ⊕.
Example 2.16. Consider the cluster algebra with principal coefficients with the same exchange matrix as in Example 2.11. Let P = Trop(Y 1 , Y 2 , Y 3 , Y 4 ), we can write an initial seed as (X, Y, B) where By Remark 2.13, we think of Y i 's as frozen variables and get a coefficient-free cluster algebra of rank 8 with the following quiver and exchange matrix, where Y i is the cluster variable on the vertex i + 4.
Then the mutation µ 1 gives Let us try to compute x 1 in Example 2.11 using the separation formula. From the formula, we think of X 1 as a function (X 1 , . . . , X 4 ; Y 1 , . . . , Y 4 ) → Y 1 X 4 + X 2 X 1 . Then l,t ). Given an exchange matrix B, we consider the unique (up to isomorphism) Y-pattern t → (Y t , B t ) having an initial seed Again, we think of Y where Y (B) j;t | P (y 1 , . . . , y n ) can be interpreted as a cluster coefficient in the Y-pattern with principal coefficients with an initial coefficient tuple (y 1 , . . . , y n ).
In order to apply Theorem 2.18, we also need to compute the cluster variables of the cluster algebra with principal coefficients of the same quiver. Let (X 1 , X 2 , X 3 , X 4 ) be the initial cluster tuple, we then get the following.
This is the same result as we computed directly in (1).

2.2.
Infinite rank cluster algebras. We define infinite rank cluster algebras in a similar way. The cluster tuple, coefficient tuple and the exchange matrix now are infinite dimensional. For the mutation to make sense, we only need the condition: For each j, b ij = 0 for all but finitely many i. If B is also skew-symmetric, this condition is equivalent to saying that an infinite quiver Q B has only finitely many arrows incident to each of its non-frozen vertex. For the cluster pattern, although we think of it as an assignment from the infinite tree T, we usually restrict the study to only those seeds obtainable from the initial seed by finitely many mutations. In the next section, we will review the fact that the T-system can be realized as an infinite-rank coefficient-free cluster pattern, and then define the T-system with principal coefficients in the same way as we have already discussed cluster algebras with principal coefficients in this section. We pick a specific cluster seed to put principal coefficients. This choice generates a new recurrence relation which we will call the octahedron recurrence with principal coefficients.

T-systems
In this paper, we consider the A ∞ T-system [DFK13], which is also known as the octahedron recurrence. It is the infinite-rank version of A r T-system. We will refer to A ∞ T-system as just the T-system when there is no ambiguity.
The T-system can be realized as mutation in an infinite-rank coefficient-free cluster algebra of geometric type [DFK09,DFK13]. Its exchange matrix is skewsymmetric, so we can express it as a quiver, the octahedron quiver. In this section, we review this connection and define T-systems with generic coefficients by inserting principal coefficient into the relation, as in Definition 2.12, in the initial quiver. We will also show that it is equivalent to the recurrence relation (4).
3.1. T-systems without coefficients. Let Z 3 odd = {(i, j, k) ∈ Z 3 | i + j + k ≡ 1 mod 2}. The T-system, so called the octahedron recurrence, is a recurrence relation on the set of formal variables A stepped surface is a subset {(i, j, k(i, j)) | i, j ∈ Z} ⊂ Z 3 odd defined by a function k : Z × Z → Z satisfying: We will also denote this surface by the function k. The condition |i−i |+|j −j | = 1 is referred to as (i, j) and (i , j ) are lattice-adjacent, and k(i, j) is called the height of (i, j) with respect to k. There are three important stepped surfaces which we will use throughout the paper. We define fund : (i, j) → (i + j mod 2) − 1, and call them the fundamental stepped surface, the stepped surface projected from a point (i , j , k ) and the adjusted stepped surface associated with a surface k and a point p, respectively. See Figure 1 for examples. To each k, we can attach an initial condition X k (t) : {T i,j,k(i,j) = t i,j | i, j ∈ Z} for some formal variables t = {t i,j | i, j ∈ Z}, to which we refer as initial data/values along the stepped surface k.
It is worth pointing out that for a point (i 0 , j 0 , k 0 ) ∈ Z 3 odd , not every initial data gives a finite solution to T i0,j0,k0 . In other words, an expression of T i0,j0,k0 in terms of t i,j 's may not be finite. We call an initial data on k that gives a finite expression for T i0,j0,k0 an admissible initial data with respect to (i 0 , j 0 , k 0 ).
Example 3.1. The fundamental stepped surface is always admissible with respect to any point in Z 3 odd . The stepped surface proj (0,0,m) is not admissible with respect to a point (0, 0, n) when n > m.
The T-system can also be interpreted as an infinite-rank coefficient-free cluster algebra [DFK09]. Using Z 2 as the index set, the initial seed is (x, B) = (x i,j ) (i,j)∈Z 2 , (b (i ,j ),(i,j) ) where The quiver Q B associated with B is shown in Figure 2. We call it the octahedron quiver. We embed the vertices of the quiver into the 3-dimensional lattice Z 3 odd so that they lie on the fundamental stepped surface, i.e. the vertex (i, j) of the octahedron quiver lies at the point (i, j, fund(i, j)) ∈ Z 3 odd . The reason for picking this choice is to associate it to the index of the initial cluster variables We then allow mutations only on vertices (i, j, k) having the property that there are exactly two incoming and two outgoing arrows incident to (i, j, k). This property is equivalent to saying that all four neighbors of (i, j, k) have the same third coordinate in Z 3 odd , i.e., the four neighbors are all either (i ± 1, j ± 1, k − 1) or (i ± 1, j ± 1, k + 1).
If the neighbors are (i ± 1, j ± 1, k − 1), after the mutation at (i, j, k), we move the vertex that used to be at (i, j, k) to (i, j, k − 2) and call the new cluster variable obtained by the mutation T i,j,k−2 . We call this mutation a downward mutation. On the other hand, when the neighbors are (i ± 1, j ± 1, k + 1), (i, j, k) is moved to (i, j, k + 2) and the new cluster variable is called T i,j,k+2 . We call it an upward mutation.
The set of vertices of a quiver Q obtained from the octahedron quiver by allowed mutations forms a stepped surface, denoted by k Q . On the other hand, we can j i Figure 2. The octahedron quiver. The red dots correspond to indices (i, j) where i + j is even, and the blue to the odd i + j. The quiver is infinite in both i and j directions. create a quiver from a stepped surface by reading the arrangement of the quiver arrows from Table 1, and call this quiver Q k . We notice that the quiver mutation at (i, j) corresponds to moving (i, j, k(i, j)) to (i, j, k(i, j) ± 2), depending on the height of its neighbors as discussed above. We say that k is obtained from k by a 3.2. T-systems with principal coefficients. We define the T-systems with principal coefficients from the cluster algebra setting. Instead of the coefficient-free cluster algebra with the octahedron quiver, we consider the cluster algebra with principal coefficients (Definition 2.12) on the same quiver, where the initial coefficient at (i, j) is c i,j . Due to Remark 2.13, it is the same as the coefficient-free cluster algebras on the coframed octahedron quiver, where the variables c i,j on the added vertices are frozen, see Figure 3. We show that the cluster variables satisfy the recurrence relation (4) on {T i,j,k | (i, j, k) ∈ Z 3 odd } with an extra set of coefficients {c i,j | (i, j) ∈ Z 2 }. We will use this recursion as an alternative definition of the T-system with principal coefficients.
odd } be the set of cluster variables obtained from the T-system with principal coefficients. Then We call the relation (4) the octahedron recurrence with principal coefficients. The pictorial representation of I and J are shown in Figure 4.
In order to prove Theorem 3.2, we first compute the coefficients at the vertices in any quiver obtained by the octahedron quiver. We note that unlike cluster variables, a coefficient at (i, j) on a stepped surface k depends on the height of (i, j) and its neighbors (i ± 1, j ± 1).
Proposition 3.3. Consider the T-system with principal coefficients. Let k be a stepped surface obtained from fund by a finite number of allowed mutations. Let , as described as follows: Example 3.4. Consider a stepped surface k having height as the following.
The coefficient at the vertex (i, j), y (i,j),k , computed by Proposition 3.3 is Proof of Proposition 3.3. We fist show the second equality in (6). Notice that which can be easily derived from the definition of I and J in (5).
We will then prove the proposition by induction on the number of mutations from the fundamental stepped surface. On fund, the vertices are in the forms (i, j, −1) or (i, j, 0) depending on the parity of i + j, and y (i,j),fund = c i,j for all (i, j). When i + j ≡ 0 mod 2, fund(i, j) = −1 and the neighbors of (i, j, −1) are (i ± 1, j ± 1, 0). So = 1 for all at (i, j). We also have When i + j ≡ 1 mod 2, fund(i, j) = 0 and the neighbors of (i, j, 0) are (i ± 1, j ± 1, −1). So = −1 for all . We have Hence the proposition holds for the fundamental stepped surface. Next we assume that the proposition holds for a stepped surface k. Consider a stepped surface k obtained from k by a mutation at (i, j). Then k = k on every point except at (i, j). Also y (a,b),k = y (a,b),k for all (a, b) but at most five points: (i, j), (i ± 1, j ± 1, ). So we only need to consider the coefficients at these five points.
Let us assume that k(i, j) = k. Since k is mutable at (i, j), we have two cases: k(i ± 1, j ± 1) = k − 1 or k(i ± 1, j ± 1) = k + 1, as shown in the following pictures.
Case 1: We know that y (i,j),k = I i,j,k−1 /J i,j,k−1 by the induction hypothesis.
After the mutation at (i, j), the point (i, j, k) becomes (i, j, k − 2). So on k , i = 1 for all i. We also get Hence the expression of y (i,j),k agrees with the proposition. At (i, j + 1, k − 1), the induction hypothesis gives We know that 1 = 1 since k(i, j) = k = k(i, j + 1) + 1. Then the mutation at (i, j) gives which agrees to the proposition. By the similar argument, we can show that all four of the y (i±1,j±1),k agree to the proposition.
Case 2: We know that y (i,j),k = J i,j,k+1 /I i,j,k+1 by the induction hypothesis. After the mutation at (i, j), the point (i, j, k) becomes (i, j, k + 2). So on k , i = −1 for all i. We also get Hence the expression of y (i,j),k agrees with the proposition. At (i, j + 1, k − 1), the induction hypothesis gives which agrees to the proposition. By the similar argument, we can show that all four of the y (i±1,j±1),k agree to the proposition. By both cases, we proved the proposition.
Proof of Theorem 3.2. To show (4), it is enough to show that it is the mutation rule at (i, j, k − 1) on k such that k(i ± 1, j ± 1) = k and k(i, j) = k − 1. The quiver at (i, j) will look like the following: So it is equivalent to show that y (i,j),k = J i,j,k /I i,j,k , which comes from Proposition 3.3. Due to Theorem 3.2, we view the T-system with principal coefficients as a recurrence relation on odd and an admissible initial data on a stepped surface k, Theorem 2.6 guarantees that the expression of T i0,j0,k0 is a Laurent polynomial in the initial data The goal is to give combinatorial interpretation for this expression. In this paper, we study the case when p is above the k and k is above fund, i.e. k 0 ≥ k(i 0 , j 0 ) and k(i, j) ≥ fund(i, j) = (i + j mod 2) − 1. In this case, we have explicit combinatorial solutions in terms of perfect matchings in Sections 4 and 5, non-intersecting paths in Section 6 and networks in Section 7.

Perfect-matching solution
The goal of this section is to give an expression of T i0,j0,k0 in terms of a partition function of weighted perfect matchings of a certain graph. There are previous works [Spe07, MS10, JMZ13] on expressing cluster variables by using perfect matchings of certain weighted graphs. Regarding only cluster variables, the weight studied in [Spe07] coincides with the "face-weight" in Definition 4.5, while the weight in [MS10,JMZ13] coincides with the "edge-weight" in Definition 5.1. 4.1. Graphs from stepped surfaces. We fix a point p = (i 0 , j 0 , k 0 ) ∈ Z 3 odd , an admissible stepped surface k and an initial data X k (t) : From the stepped surface k, we follow the construction in [Spe07] and define, using Table 1, an infinite bipartite graph G k associated with k. This graph can also be realized as the dual of the quiver Q k associated with k with vertex bi-coloring, see the end of Section 3.1. Faces of Q k become vertices of G k . Since all faces of Q k are always oriented, we color a vertex of the graph in white if the arrows around its corresponding face of the quiver are oriented counter-clockwise and black if they are oriented clockwise. Vertices of Q k become faces of G k . Since the vertices of the quiver are indexed by Z 2 , we will use (i, j) ∈ Z 2 to represent a face of the graph. Arrows of Q k gives edges of G k . There are three types of edges in the graphs: horizontal, vertical and diagonal, which came from vertical, horizontal and diagonal arrows of the quiver, respectively. See Figure 5 for an example.
If k is obtained from k by a mutation at (i, j), then we can see from Table  1 that the face (i, j) in G k must be a square. In addition, G k can be obtained [Ciu03,Spe07] from G k by the following steps.
(1) Apply urban renewal at the face (i, j), see Figure 6.
(2) Collapse any degree-2 vertices created by the previous step, see Figure 7. Table 1. All six local pictures of Q k and G k for four points A = (i, j), B = (i + 1, j), C = (i + 1, j + 1), and D = (i, j + 1) on a stepped surface k.   We use the notations F (G), V (G) and E(G) for the set of faces, vertices and edges of a graph G, respectively. We then define two subsetsF =F (p, G k ) and ∂F = ∂F (p, G k ) of F (G k ) = Z 2 depending on p and k as follows.
We also assume that ∂F = {(i 0 , j 0 )} when k 0 = k(i 0 , j 0 ). The setF can be illustrated as the set of points inside (excluding boundary) the shadow projecting from p onto k, while ∂F is the boundary of the projection. The following example shows elements ofF in blue and elements of ∂F in red when p = (0, 0, 3) and The picture on the left shows the faces on (i, j)-plane, discarding the k-direction.
The picture on the right shows the projection in the section j = 0 of the whole 3-dimensional space.
We will see later from the solution to the T-system (Theorem 4.9) that the expression of T i0,j0,k0 depends only on t i,j 's where (i, j) ∈F ∪ ∂F . Due to this reason, we will work on a finite subgraph G p,k of G k generated by the faces inF , while considering faces in ∂F as "open faces" as in the following definition. Since we can always determine ∂F (G) from F (G), we will omit ∂F (G) by writing just G instead of (G, ∂F (G)). The faces in F (G) =F are called closed faces, while the faces in ∂F (G) = ∂F are called open faces.
Later in the paper, some other solutions to the T-systems with principal coefficients will look nicer if written in terms of the "closure" of G instead of G. This will be a graph with no open faces.
Definition 4.2 (The closure G of G). For a point p and a surface k, let k p be the adjusted stepped surface associated with k and p defined in (3) and G ∞ := G kp be the graph associated to k p . We define the closure G of G to be the finite subgraph of G ∞ generated byF ∪ ∂F , and we think of it as a graph with no open face.
We note that k(i, j) = k p (i, j) for all (i, j) ∈ F (G) ∪ ∂F (G). So the graphs with open faces G p,k and G p,kp are exactly the same except for the shape of the open faces. Due to the following proposition, we can obtain G directly from G by closing all the open faces of G in a certain way.   We define the face-weight w f and the pairing-weight w p on G, which contribute cluster variables/initial data t i,j 's and coefficients c i,j 's, respectively, to the expression of T i0,j0,k0 . Definition 4.5. For a face (i, j) ∈F ∪ ∂F, we define the face-weight depending on a perfect matching M of G as: where a contribution of a face to the product is defined as: where a is the number of sides of (i, j) in the matching M and b the number of sides in E(G) \ M.
The pairing-weight will be defined on pairs of horizontal edges in M. We first note that there are exactly two types of horizontal edges in G as follows.
• a white-black horizontal edge, an edge joining a white vertex on the left and a black vertex on the right. We will call it N (i, j), indexing by the face (i, j) below it (the north side of the face (i, j)). • a black-white horizontal edge, an edge joining a black vertex on the left and a white vertex on the right. We will call it S(i, j), indexing by the face (i, j) above it (the south side of the face (i, j)).
Let an allowed pair be a pair of S(i, j 1 ) and N (i, j 2 ) when j 1 ≤ j 2 in the same column of the graph. In the other words, an allowed pair is a pair of a whiteblack horizontal edge above a black-white horizontal edge in the same column. We denote N (i,j2) S(i,j1) for an allowed pair. Since F (G) ⊂ Z 2 , for each i we can consider a subgraph of G generated by the faces in F (G) ∩ ({i} × Z). In this column subgraph, we read from the bottom to the top and get a sequence of horizontal edges in the matching M . We then pair these edges into allowed pairs by the following steps.
(1) If S(i, j 1 ) and N (i, j 2 ) where j 1 ≤ j 2 are consecutive in the sequence, we pair the two.
(2) Remove both S(i, j 1 ) and N (i, j 2 ) from the sequence, and repeat the first step until the sequence is empty.
We do this to all of the columns of G. The set P of all allowed pairs obtained by this process is called the perfect pairing of M . Proposition 4.8 will guarantee that the process works and the perfect pairing always exists. Now the pairing-weight is defined in the following definition.
Definition 4.6. Let P be the perfect pairing of a perfect matching M of G. The pairing-weight on M is defined to be: where a contribution of an allowed pair in the product is defined as: A contribution of an allowed pair in the perfect pairing to the pairing-weight can be illustrated by Figure 10.
The perfect pairing is Also, the face-weight of M is w f (M ) = t −2,0 t −1 0,0 t 2,0 . Proposition 4.8. Let M be a perfect matching of G. Then the following holds.
(1) If all four adjacent faces of a face (i, j) ∈F have the same height, then the face (i, j) is a square. Also, the coloring around the face depends on the height of (i, j) and its neighbors as shown in Figure 11.
That means the number of black-white horizontal edges in M and the number of white-black horizontal edges in M in the same column are equal. proj p by a finite number of downward mutations. We will show (2) and (3) using induction on the number of downward mutations from proj p . When k is away from proj p by only one downward mutation, G is a square of type (S1) in Figure 11. There are only two perfect matchings of the graph, which both satisfy (2) and (3).
Next, we assume that the claims hold for any surfaces which are away from proj p by less than n mutations. Let k be a surface away from proj p by n downward mutations. There must be an intermediate surface k such that k is obtained from proj p by n − 1 downward mutations and k is obtained from k by one downward mutation, says at (i, j). We have two cases: is a closed face of G p,k , then (i, j) is also a closed face of G p,k . G p,k is obtained from G p,k by applying the urban renewal action at the face (i, j) then collapsing all degree-2 vertices created by the urban renewal. For any perfect matching M of G p,k , there exists a perfect matching M of G p,k differing from M only at the face (i, j), see Figure 2. Since M satisfies (2) and (3) by the induction hypothesis, we see from Figure 2 that M also satisfy (2) and (3). So they hold for any matchings of G p,k .
Case 2. If (i, j) is an open face of G p,k , then (i, j) becomes a closed face of G p,k . We first consider the case when i > i 0 and j > j 0 . G p,k is obtained from G p,k by applying the urban renewal action at the face (i, j) and collapsing all degree-2 vertices created by the urban renewal. This yields the correspondence of the matchings of G p,k and G p,k via Figure 3. With the same argument as for a closed face, (2) and (3) hold for any perfect matchings of G p,k . Similarly, if i > i 0 and j = j 0 , the correspondence is shown in Figure 4, which implies (2) and (3) for any perfect matchings of G p,k . The other cases can be treated similarly.
From both cases, the statements (2) and (3) hold for every perfect matching of G p,k . By induction, we proved (2) and (3).
We have defined face-weight and pairing-weight for perfect matchings of G. The previous proposition ensures that the pairing-weight is well-defined. We are now ready to state the main theorem. Table 2. The list of all correspondences between matchings before and after a single downward mutation at a closed face. Theorem 4.9 (Perfect-matching solution). Let p = (i 0 , j 0 , k 0 ) and k be an admissible initial data stepped surface with respect to p where k 0 ≥ k(i 0 , j 0 ) and k ≥ fund. Then where M is the set of all the perfect matchings of G = G p,k . This solution specializes to the solution in [Spe07] for the coefficient-free Tsystem [Spe07, The Aztec Diamonds theorem] when c i,j = 1 for all (i, j) ∈ Z 2 .
The proof of the theorem follows from the proof in [Spe07] using the "infinite completion" of G = G p,k , which is the same thing as G ∞ = G kp in our setup. To do so, we need to make sense of perfect matchings of G ∞ and weight on them.
Definition 4.10 (Acceptable perfect matching of G ∞ ). We call a perfect matching We then extend the definition of the face-weight and the pairing-weight to acceptable perfect matchings of G ∞ . Notice that G ∞ has no open faces. Also the weight of M and M ∞ are equal, i.e.
The following proposition gives a bijection between the perfect matchings of G and the acceptable perfect matching of G ∞ . Example 4.12. Figure 12 shows an example of a perfect matching M of G and its corresponding acceptable perfect matching M ∞ of G ∞ from the bijection in Proposition 4.11. An edge in M ∞ is either an edge in M (described in red) or a diagonal edge in E(G ∞ ) \ E(G) (described in blue). From (10) and Proposition 4.11, we can see that Theorem 4.9 is equivalent to the following theorem.  Proof. Since T i0,j0,k0 depends only on G p,k = G p,kp , we can assume without loss of generality that k = k p . We will prove the theorem by using induction on the number of downward mutations from the top-most stepped surface proj p to k. The base case is when k = proj p . The graph G proj p is shown in Figure 13. There is only one acceptale perfect matching and its weight is t i0,j0 = T i0,j0,k0 . So the theorem holds for the base case.
Assuming that the theorem holds for any stepped surfaces away from proj p by less than n downward mutations, we let k be a surface obtained from proj p by n downward mutations. Then we can find an intermediate surface k such that it is obtained from proj p by n − 1 downward mutations and k is obtained from k by one downward mutation at (i, j). We also assume that k(i, j) = k − 1 and k (i, j) = k + 1 for some k ∈ Z. By the induction hypothesis we have Let M be any acceptable perfect matching of G k . By Proposition 4.8, the face (i, j) of G k is a square of type (S2) in Figure 11. Then the matching M at the face (i, j) must be one of the 7 cases in the first column of Figure 2.
If M is of type (A) at (i, j), there are two matchings M A1 and M A2 of G k of type (A1 ) and (A2 ), respectively, such that the matchings are the same except locally at the face (i, j). We then have The term J i,j,k in the second equation came from an extra pair N (i,j) S(i,j) in M A2 which gives an extra term J i,j,k to the pairing-weight. By (4), we have If M is of type (B), there exists a unique corresponding matching M B of G k of type (B ). We see that the weight of M and M B are equal. That is If M is of type (C1) (resp. (C2)), let M be another matching of G k of type (C2) (resp. (C1)) which is the same as M except for the two edges at the face (i, j). Without loss of generality, we assume that M is of type (C1) and M is of type (C2). Then there exists a corresponding perfect matching M C of G k of type (C ). We then have To write w p (M ) in terms of w p (M C ), we first notice that there must be two other edges S(i, j 1 ), N (i, j 2 ) ∈ M where j 1 < j < j 2 such that both pairs N (i,j−1)

S(i,j1)
and N (i,j2) S(i,j+1) are in the perfect pairing of M , while in its corresponding M C there is only N (i,j2) S(i,j1) . Thus we have and so Hence, By (11), (12), (13) and the induction hypothesis, we can conclude that So the statement holds for k. By the induction, we proved the theorem. Now we have proved Theorem 4.9 and Theorem 4.13. In the proof of Theorem 4.13, we notice that for any acceptable perfect matching M ∞ of G ∞ , any face outsideF ∪ ∂F always gives 1 for its face-weight. Also edges in M ∞ \ E(G) are all diagonal, so they will not contribute any weight to the partition function. We then have the following perfect-matching solution for the closure G of G.  We now have a combinatorial expression of T i0,j0,k0 as a partition function of face-weight and pairing-weight over all perfect matchings of a graph. In the next section, we will combine the two weights together and construct an edge-weight. This will be the first step toward our next aim to construct a solution in terms of networks, analog to [DF14] for the coefficient-free T-system.

Perfect-matching solution via edge-weight
Now that we have multiple versions of the perfect matching solution to the T-system with principal coefficients in Theorem 4.9, Theorem 4.13 and Theorem 4.14, our next goal is to find a network solution analog to the network solution for coefficient-free T-systems studied in [DFK13,DF14]. One big advantage of the network solution is its explicit solution via the network matrices. In the perfect matching solution, we need to enumerate all the perfect matchings of the graph G in order to compute the solution. For the network solution, we associate an initial data stepped surface with a product of network matrices. Then the solution is just a certain minor of the product.
In order to get the network solution, we first transform the face-weight and the pairing-weight studied in the last section to the edge-weight w e (Definition 5.1) on edges of the closure G of G, which also gives us a new perfect-matching solution but with the edge-weight w e (Theorem 5.4). This solution will be used to construct a nonintersecting-path solution in the next section. We also note that our edgeweight coincide with the weight studied in [MS10,JMZ13] in the case when all c i,j = 1.
Definition 5.1 (Edge-weight w e ). Let k be an admissible initial stepped surface with respect to p, G be the closure of the graph G = G p,k . For each edge of G we assign the edge-weight w e as follows: where p a and p a are the following formal products when a = (i, j) and k = k(i, j). We also assume that t a = 1 when a / ∈ F (G) and p a = p a = 1 when a / ∈ F (G). We note that M and M 0 are not necessary perfect matchings of G. Also for j 1 ≤ j 2 , By Proposition 4.8, the product in (14) is indeed a finite product of pairing-weights, hence a finite product of c i,j 's. The following lemma interprets the face-weight of a matching M as a function of M and our special matching M 0 .
Since x must be one of the cases in Figure 14 Theorem 5.4 (Perfect-matching solution for G with the edge-weight w e ). Let k be an admissible initial data stepped surface with respect to p = (i 0 , j 0 , k 0 ), M and M 0 be defined as in Definition 5.2. Then where w e (M 0 ) ci,j =1 denotes the substitution c i,j = 1 for any (i, j) ∈ Z 2 .
x x x x Figure 14. All possible faces of G up to π 2 rotation and i-axis reflexion.
Proof. Let M ∈ M. By Lemma 5.3, we get By Definition 5.1, it equals to By the definition of w e , we consider From (15) and the fact that M \ M contains only diagonal edges, we then get we conclude that w f (M )w p (M ) = w e (M ) w e (M 0 ) c=1 for any M ∈ M. By Theorem 4.14, we have T i0,j0,k0 = M ∈M w e (M ) w e (M 0 )| c=1 . Now we have a combinatorial expression of T i0,j0,k0 in terms of a partition function of edge-weight over all matchings G. In the next section, we give an explicit bijection between perfect matchings and non-intersecting paths (with certain sources and sinks) in both G and G. Using this bijection, we are able to transform the perfect-matching solutions to a solution in terms of non-intersecting paths.

Non-intersecting path solution
In this section, we provide an explicit bijection (Proposition 6.13) between the perfect matchings of G and the non-intersecting paths in the oriented graph G with certain sources and sinks. It can be extended to a bijection between the matchings in M of G and the non-intersecting paths in the oriented graph G with certain sources and sinks (Proposition 6.15). Using this bijection and a new weight w e modified from the edge-weight w e , we can write the solution to the T-system in terms of non-intersecting paths in G (Theorem 6.18). 6.1. Some setup. We first show some properties of the graph G and G.
Proposition 6.1. G and G are bipartite and connected.
Proof. It was proved in [Spe07, Section 3.5] that G is bipartite and connected. The extended result to G follows from Proposition 4.3.
Definition 6.2. For two vertices v, v of a graph, we say that v (resp. v ) is on the left (resp. right) of v (resp. v) if there is a sequence of vertices of the graph v = v 0 , v 1 , v 2 , . . . , v n = v such that any two consecutive vertices are connected by one of the following edges.
We center the graph at the face (i 0 , j 0 ). Then the notions of the North-West, North-East, South-West and South-East of the center are well-defined. Definition 6.3. We denote V NW (G), V NE (G), V SW (G) and V SE (G) to be the set of all the vertices of a graph G in the North-West, North-East, South-West and South-East of the center face (i 0 , j 0 ), respectively. Also let V leftNW (G), V rightNE (G), V leftSW (G), V rightSE (G) be the set of all left-most NW vertices, right-most NE, leftmost SW and right-most SE vertices of G, respectively. See Figure 15 for an example.
Proposition 6.4. G has the following properties. ( Proof. To show (1), we use the same analysis as in the proof of Proposition 5 in [Spe07]. It is clear that a left-most vertex must be on the boundary of G. We then consider an open face (i, j) ∈ ∂F on the South-West of (i 0 , j 0 ) with height k ∈ Z. All the eight possibilities are shown in Figure 16, where the face of height k in the circle is (i, j). We see that all the left-most South-West vertices must be black. A similar argument can be used to show the other case of (1) and (2). In order to show (3), we consider a maximal sequence v 0 , v 1 , . . . , v n of vertices of G such that v i is This gives a bijection between V leftSW (G) and V rightSE (G). So we proved (3). The similar argument also shows (4).
6.2. Non-intersecting paths and perfect matchings. We will give an explicit bijection between the perfect matchings of G and the non-intersecting paths from V leftSW (G) to V rightSE (G) in G (Proposition 6.13). This bijection can be extended to G in Proposition 6.15. In order to define a path in G and G, we need an orientation of the graphs.
Definition 6.5 (Edge orientation of G and G). Let the orientation of the edges of G and G be such that it goes from left to right for diagonal and horizontal edges and from a black vertex to a white vertex for the vertical as follows.
Definition 6.6. Let G = G p,k and G be as in Definition 4.2. With the notations from Definition 6.3, we define The map Φ (resp. Φ) in the following definition is indeed the bijection between M and P (resp. between M and P). They will be key ingredients to construct our nonintersecting-path solution.
is the symmetric difference of A and B. The notation represents the disjoint union.
Remark 6.8. It is worth mentioning that if we consider M ∈ M and P ∈ P as sets of dimers on edges of G, then the action of Φ can be interpreted as superposing with M 0 and counted the number of dimers on each edge modulo 2. Similarly, Φ is the superposition with M 0 with number of dimers being counted modulo 2.
Example 6.9. This example shows an interpretation of the map Φ on M as the superposition with M 0 modulo 2.
We notice that the image of the map is indeed an element in P.
Using the following lemmas, Proposition 6.13 shows that Φ is a well-defined bijection from M to P. (1) For any vertex in V leftSW (G) ∪ V rightSE (G), exactly one of its incident edges is in Φ(M ).
(2) For the other vertices, either none or two of their incident edges are in Φ(M ).
Proof. To show (1), we first consider a vertex in V leftSW (G). By the fact that M is a perfect matching, exactly one of its incident edges is in M . By Proposition 6.4, the vertex is black. Since it is the left-most, none of its incident edges is in M 0 . This is because there cannot be a white vertex on its left. So, exactly one of its incident edges is in Φ(M ). Similarly, exactly one incident edge of a vertex in V rightSE (G) is in M . There is none of its edges is in M 0 because the vertex is white and the right-most. Hence (1) holds. For (2), if a vertex is not in V leftSW (G) ∪ V rightSE (G), it must be either in V leftNW (G) ∪ V rightNE (G) or it has both left and right adjacent vertices. For a vertex in V leftNW (G) ∪ V rightNE (G), there is one of its incident edges in M, and also one in M 0 . They can be the same or different. So, either none or two of its incident edges are in Φ(M ). For a vertex having left and right adjacent vertices, one of its incident edges must be in M 0 . This is because the vertex is either black and is on the right of a white vertex, or white and on the left of a black vertex. Also its one of the incident edges must be in M because M is a perfect matching. So either none or two of its incident edges are in Φ(M ). Hence (2) holds. Figure 17. All configurations of two incident edges of a black vertex of G. The horizontal edges in the picture represent horizontal/diagonal edges of G. Figure 18. All configurations of two incident edges of a white vertex of G. The horizontal edges in the picture represent horizontal/diagonal edges of G.
Lemma 6.11. For a black vertex of G having two incident edges in Φ(M ), the two edges must be of the form A, B or C in Figure 17 where the horizontal edges in the figure represent horizontal or diagonal edges of G.
Proof. It is easy to see that the six cases in Figure 17 are all configurations of two incident edges of a vertex of G. Since M is a perfect matching, one of the two edges must come from M 0 . Since the cases (D), (E) and (F) contain no edge from M 0 , they cannot happen.
Lemma 6.12. For a white vertex of G having two incident edges in Φ(M ), the two edges must be of the form A, B or C in Figure 18 where the horizontal edges in the figure represent horizontal or diagonal edges of G.
Proof. Similar to Lemma 6.11.
Proof. To show that Φ(M ) is well-defined, we need to show that Φ(M ) is a collection of non-intersecting paths from V leftSW (G) to V rightSE (G) in G, where G is oriented as in Definition 6.5. From Lemma 6.11 and Lemma 6.12, only the case (A), (B) or (C) can happen at any vertex. From this, all the vertices incident to two edges in Φ(M ) are neither source nor sink. So Φ(M ) is a set of non-intersecting paths in the oriented graph G, where a source or a sink is incident to exactly one edge in Φ(M ). From Lemma 6.10, the sources and the sinks must in V leftSW (G) ∪ V rightSE (G). But (1) and (2) of Proposition 6.4 guarantees that V leftSW (G) must be the source and V rightSE (G) must be the sink. It remains to show that there is no loop in Φ(M ). Since every vertex in a loop are neither source nor sink, there must be an horizontal/diagonal edge oriented from right to left. This contradicts the orientation of G. So Φ(M ) contains no loop. To show that Φ is a bijection, we construct another map Ψ : P → M by letting Ψ(P ) be the superposition of the path P with M 0 and counting dimers on each edge modulo 2. In other words Ψ(P ) = (P ∪ M 0 ) \ (P ∩ M 0 ). It is obvious that Ψ • Φ = id M and Φ • Ψ = id P . So, Ψ = Φ −1 provided that Ψ is well-defined. To show that Ψ(P ) is a perfect matching of G, it suffices to show that any vertex has exactly one incident edge in Ψ(P ).
We first consider a black vertex.
• If it is in V leftSW (G), it has one incident edge in P because it is a source. Also, it has no edge in M 0 since it is the left-most. So, it still has one incident edge in Ψ(P ) after the superposition. • If it is not in V leftSW (G), it has either none or two edges in P. In both cases, since the vertex is not the left-most, there must be a white vertex on its left. So it has an incident edge in M 0 .
-If it has no edge in P, it will receive an edge from M 0 after mapping by Ψ.
-If it has two edges in P, it is either of the case A, B or C in Figure 17.
So exactly one of the edges gets removed after superposing with M 0 . From both cases, the vertex is incident to exactly one edge in Ψ(P ).
A similar argument holds for a white vertex. So we conclude that Ψ(P ) ∈ M. Now we have analogs to Proposition 6.13 and Proposition 6.14 for G.
Proposition 6.15. Let Φ be defined as in Definition 6.7. Then we have the following: (1) Φ : M → P is a bijection, (2) For e ∈ M 0 , e ∈ M if and only e / ∈ Φ(M ).
Proof. We recall that the symmetric difference is commutative and associative. Consider P ∈ P, we can see from Proposition 4.3 that the paths from V leftSW (G) must go to the right via horizontal or diagonal edges in E(G) \ E(G). Similarly, the paths will arrive V rightSE (G) via horizontal or diagonal edges in E(G) \ E(G). Since there is a unique choice of these edges, we have P = P ∪P 0 where P 0 ⊂ E(G)\E(G) is the set of all horizontal and diagonal edges in the South-West, South or South-East. Note that P 0 is also equal to the set of white-black horizontal and black-white diagonal edges in E(G)\E(G). Since P ∩P 0 = ∅ for P ∈ P, P and P are in bijection via the map: g : P → P ∪ P 0 = P P 0 with the inverse map: g −1 : P → P P 0 .
If we can show that Φ = g • Φ • f −1 , (1) will automatically follows. To show this, we first show that that M 0 = D M 0 P 0 . Consider Hence we proved (2).
6.3. Modified edge-weight and nonintersecting-path solution. We recall the edge-weight w e in Definition 5.1. It is compatible with the perfect-matching solution (Theorem 5.4). In order to construct a nonintersecting-path solution from the bijection Φ, the edge-weight w e requires some modification. Due to Proposition 6.15, we only need to inverse the weight of all edges in M 0 .
Definition 6.16 (Modified edge-weight w e ). For x ∈ E(G) we define the modified edge-weight as follows: where w e is the edge-weight defined in Definition 5.1.
Definition 6.17 (Modified edge-weight for paths in G). For a path p = x 1 x 2 . . . x n in G, its modified edge-weight is defined to be the following product Then the weight for a non-intersecting path is defined by w e (P ) := p∈P w e (p) for P ∈ P.
Now we are ready for a nonintersecting-path solution for G. Note that we can also consider the solution for G, but it turns out to be more complicated due to the present of open faces of G, which needs to be treated separately.
Theorem 6.18 (Nonintersecting-path solution for G). Let k be an admissible initial data stepped surface with respect to p = (i 0 , j 0 , k 0 ), M 0 be defined as in Definition 6.6. Then where P is the set of all non-intersecting paths in G from V leftSW (G) to V rightSE (G).
Hence we proved the theorem.
The nonintersecting-path solution obtained in this section gives us a hint that it is possible to have a network solution analog to [DFK13,DF14]. In the next section, we will transform the oriented graph G to a network. The modified edgeweight will be used on the network as well. This leads to a network solution and a network-matrix solution.

Network solution
In this section, we will construct a weighted directed network N associated with the oriented graph G and the modified edge-weight w e . We then decompose N into network chips and their associated elementary network matrices (adjacency matrices). The product of all the elementary matrices associated with the network chips, according to an order of the chips, is then called the network matrix associated to N . The nonintersecting-path solution for G (Theorem 6.18) can then be interpreted as a path solution on this network, and can also be computed from a certain minor of the network matrix. We also show that our network and elementary network matrices coincide with the objects studied in [DFK13] in the case of coefficient-free T-systems (c i,j = 1 for all (i, j) ∈ Z 2 ). 7.1. Network associated with a graph. We construct the directed network N associated with the oriented graph G by tilting all the diagonal edges so that they become horizontal, and tilting all the vertical edges so that the vertex on the left is black as shown in Figure 19. Also, the network is directed from left to right. So a path in the oriented graph G (Definition 6.5) corresponds to a path in N .
We then introduce the notion of "row" for vertices of N . Two vertices are said to be in the same row if they are joined by a connected path of horizontal or Figure 19. An example of G and its associated network N .
diagonal edges. In other words, they are on the left and right of each other in G, see Definition 6.2. We number the rows so that they increase by one from bottom to top and the center face (i 0 , j 0 ) of N lies between the row −1 and 0. See Figure  19. The precise definition is as the following.
Definition 7.1. The vertex v ∈ V (G) is in row r if two of its incident faces are (i, j 0 + r) and (i, j 0 + r + 1) for some i ∈ Z.
Let N be the directed network associated with G, where r min and r max are the smallest and the largest row numbers of N . We then put weight on the network locally around each black vertex as shown in Figure 5. The weight comes directly from the modified edge-weight w e on G (Definition 6.16), so we will carry the notation w e for the weight on N .
Remark 7.2. When a black vertex in Figure 5 is on the boundary, we will assume the following.    Figure 20. A network is decomposed into elementary network chips 7.3. Network matrix. We will now decompose N into elementary network chips, which are small pieces of N around black vertices illustrated in Figure 5. This can be done by breaking the network at all of its white vertices. See Figure 20 for an example. We notice that we will have the following choice when there are two incident vertical edges at a white vertex.
Since the weight of a path on N is independent of this choice, the nonintersectingpath solution is independent of this ambiguity. Figure 21. A totally-order from a decomposition in Figure 20 A decomposition gives a partially-order on the network chips, we then pick a "finer" totally-order which does not contradict the partially-order. This totallyorder can be thought as the order in which to pull chips out of the network from the left.
Example 7.4. From the example in Figure 19, we can pick a network decomposition as in Figure 20, and then pick a totally-orders as in Figure 21.
The next step is to associate each chip with an elementary network matrix shown in Figure 5. The matrices W r , V r , V r , U r and U r are defined in the following definition.
Then the modified network matrix associated with N is defined as the product of all modified elementary network matrices according to a totally-order of the network chips as before. Theorem 7.8 becomes the following. The main reason that the lozenge covering has a rich connection to our story is because it is indeed a dual of our bipartite graph. This can be locally described as follows: A choice to triangulate a white square corresponds to a choice of decomposing a white vertex, and a choice to triangulate a black square corresponds to a choice of collapsing a degree-2 white vertex in Remark 7.6. These can be illustrated by the following pictures: We have provided various solutions to the T-system with principal coefficients. These solutions give combinatorial expressions of T i0,j0,k0 in terms of coefficients c i,j 's and initial data t i,j 's on k under the conditions: We will discuss some other cases in Section 9.

Other coefficients
In this section, we discuss a few examples of other choices of coefficients on Tsystems: Speyer's octahedron recurrence [Spe07], generalized lambda-determinants [DF13] and (higher) pentagram maps [Gli11,GSTV14]. Almost all of them have their own explicit combinatorial solutions and are treated with different techniques. Applying Theorem 2.15 and Theorem 2.18 to our solutions, we get a partial solution to each of them when the initial data stepped surface is fund.
In [Spe07], the author provides a perfect matching solution to the Speyer's octahedron recurrence, which is a partition function of perfect matchings of G. In fact, our perfect matching solution is developed from the method used in the paper. The author uses face-weight (same as Definition 4.5) which gives cluster variables and edge-weight (instead of our pairing-weight) which gives cluster coefficients. So the main difference is that the edge-weight is specific to a choice of coefficients.
For the generalized lambda-determinant in [DF13], the author provides a network solution which is a partition function of non-intersecting paths in a weighted directed network. This network is the same as the network discussed in Section 7. However, the weight used in [DF13] is different to our weight due to the choice of coefficients.
The (higher) pentagram maps [Sch92,OST10,Gli11,GSTV14] can be realized as a Y-pattern, i.e. a dynamic on cluster coefficients, not cluster variables. So it is not directly a T-system, but is instead a Y-pattern on the octahedron quiver. In [Gli11], the author gives a combinatorial solution to the pentagram map using alternating sign matrices. i,j,k | (i, j, k) ∈ Z 3 odd } together with a set of extra variables, called coefficients, {A i,j , B i,j , C i,j , D i,j | (i, j) ∈ Z 2 even } satisfying We note that in [Spe07], the condition on the index (i, j, k) of T (s) i,j,k is i + j + k ≡ 0 mod 2, and the coefficients are defined on Z 2 odd . With a shift in the indices, we make it coherent with our construction.
Speyer's octahedron recurrence can also be interpreted [Spe07] as a cluster algebra with coefficients. Its initial quiver is the octahedron quiver with initial cluster variables T (s) i,j,fund(i,j) similar to what we discussed in Section 3.1. The only difference is the initial coefficients: in the semifield P = Trop(A i,j , B i,j , C i,j , D i,j : (i, j) ∈ Z 2 even ). By Remark 2.10, it can be interpreted as a coefficient-free cluster algebra with frozen variables {A i,j , B i,j , C i,j , D i,j | (i, j) ∈ Z 2 even } with the quiver illustrated in Figure 22. Since we will only consider the initial stepped surface fund, we let T (s) i0,j0,k0 denote its expression in terms of the initial data t i,j := T (s) i,j,fund(i,j) and the coefficients A i,j , B i,j , C i,j , D i,j for (i, j) ∈ Z 2 . For the T-system with principal coefficients, T i0,j0,k0 denotes its expression in terms of t i,j := T i,j,fund(i,j) and c i,j for (i, j) ∈ Z 2 .
In order to get T (s) i0,j0,k0 , we will have to specialize values of t i,j and c i,j in T i0,j0,k0 according to Theorem 2.15. Let T i0,j0,k0 (c i,j = y i,j ) denote the expression of T i0,j0,k0 where t i,j stayed untouched but c i,j is substituted by y i,j . T i0,j0,k0 | P (t i,j = 1; c i,j = y i,j ) denotes the expression of T i0,j0,k0 where t i,j is set to 1, c i,j is set to y i,j , and then the whole expression is finally computed in P.
By the separation formula (Theorem 2.15), we get a solution to the Speyer's octahedron recurrence from the solution to the T-system with principal coefficients: where P = Trop(A i,j , B i,j , C i,j , D i,j : (i, j) ∈ Z 2 even ) and We now compare our result to the solution in [Spe07]. From our perfect matching solution (Theorem 4.9), we then have where M is the set of perfect matchings of the graph G p,fund . The denominator is a sum in P = Trop(A i,j , B i,j , C i,j , D i,j : (i, j) ∈ Z 2 even ), hence a monomial in where the sum runs over all the perfect matchings of G = G p,k . The weight w s is defined by w s (M ) := x∈M w s (x) and the weight w s (x) is defined for x ∈ E(G) as follows: Comparing (21) to Theorem 8.1, we can write w s on a perfect matching on G p,fund in terms of the pairing weight as follows: for M ∈ M, where the sum in the denominator is computed in the semifield P = Trop(A i,j , B i,j , C i,j , D i,j : (i, j) ∈ Z 2 even ). Example 8.2. Let p = (0, 0, 1), then we get the graph G and its two matchings as follows: i,j,k | (i, j, k) ∈ Z 3 odd } together with a set of coefficients {λ i , µ i | i ∈ Z} satisfying for all (i, j, k) ∈ Z 3 odd . It can also be realized [DF13] as a cluster algebra with coefficients. The quiver is the octahedron quiver, the initial cluster variables are T i,j,fund(i,j) and the initial coefficients are y i,j = λ i /µ j , i + j ≡ 0 mod 2, µ j /λ i , i + j ≡ 1 mod 2, in P = Trop(λ i , µ i : i ∈ Z). By Remark 2.10, we can also interpret it as a coefficientfree cluster algebra with frozen variables {λ i , µ i | i ∈ Z} with the quiver illustrated in Figure 23.
8.3. Pentagram maps. The pentagram map [Sch92,OST10] is a discrete evolution on points in RP 2 . It maps on a twisted polygon with n vertices to give another twisted n-gon whose vertices are the intersections of the shortest diagonals of the original polygon. In [Gli11] the pentagram map evolution is interpreted as the mutation in a Y-pattern (cluster mutation on cluster coefficients). Using this interpretation, the authors of [GSTV14] give a generalization of the pentagram map called higher pentagram maps. For a given integer 3 ≤ κ ≤ n − 1, the higher pentagram map on a twisted n−gon produces a new polygon using (κ − 1) th -diagonals (connecting vertex i to vertex i+κ−1) instead of the shortest diagonals (connecting vertex i to vertex i + 2) in the case when κ = 3. The following is an example of one evolution of the higher pentagram map, κ = 4, on a closed 9-gon.
For a twisted n−gon, one can define 2n variables p 1 , . . . , p n , q 1 , . . . , q n ∈ R. Then the evolution [Gli11,GSTV14] of the higher pentagram map on these variables are as follows (1 + p i−r )(1 + p i+r ) (1 + p −1 i−r−1 )(1 + p −1 i+r +1 ) , where r = κ−2 2 and r = κ−2 2 and p 1 , . . . , p n , q 1 , . . . , q n are the new variables associated with the new polygon produced by the higher pentagram map. We note that the variables originally defined in [GSTV14] differ from the ones considered here by a change of variables and a shift in indices. See [GSTV14] and [KV15] for more details.
The evolution of the variables p i , q i can be also realized [Gli11,GSTV14] as the Y-pattern of a cluster algebra of rank 2n in the universal semifield P = Q sf (p 1 , . . . , p n , q 1 , . . . , q n ) with the initial coefficient tuple (p 1 , . . . , p n , q 1 , . . . , q n ). The exchange matrix is where C = (c ij ) is an n × n matrix defined by c ij = δ i,j−1 − δ i,j − δ i,j+1 + δ i,j+1 (the indices are read modulo n). The quiver corresponding to B, the generalized Glick's quiver, is a bipartite graph with 2n vertices labeled by p 1 , . . . , p n , q 1 , . . . , q n with four arrows adjacent to each q i as the following. By Theorem 2.18 and Proposition 3.3, we then have q (k) = y (i,j),k T i,j−1,k−1 T i,j+1,k−1 T i−1,j,k−1 T i+1,j,k−1 T i,j,fund(i,j) =1 ci,j =π(i,j) = I i,j,k−1 J i,j,k−1 T i,j−1,k−1 T i,j+1,k−1 T i−1,j,k−1 T i+1,j,k−1 T i,j,fund(i,j) =1 ci,j =π(i,j) where y (i,j),k is defined as in Proposition 3.3. This gives an expression of all the pentagram variables in terms of the solution to the T-system with principal coefficients.

Conclusion and Discussion
In this paper, we have defined the T-system with principal coefficients from cluster algebra aspect. We obtain the octahedron recurrence with principal coefficients, which is a recurrence relation governing the T-system with principal coefficients. Various explicit combinatorial solutions and their connection have been established. This is for a special case when the point p and the initial data stepped surface k satisfy the following conditions These solutions to the T-system with principal coefficients allow us to solve any other systems having other choices of coefficients on the T-system as we seen in the previous section. In particular, we are able to give a solution to the higher pentagram maps as a product of T-system variables and coefficients, see (27).
We notice a symmetry {i ↔ j, k ↔ −k − 1} of the T-system with principal coefficients (4). This symmetry basically switches the roles between i and j and reflects the system upside down. So if we have a point p and an initial data stepped surface k such that k 0 ≤ k(i 0 , j 0 ) and k(i, j) ≤ fund(i, j) for (i, j) ∈ Z 2 , after applying the symmetry the system will satisfy the conditions (28) and (29). Furthermore, the condition (29) can be relaxed a little more. Since the expression of T i0,j0,k0 depends only on the values of k(i, j) when (i, j) ∈F ∪ ∂F (see (8)), Condition (29) can be relaxed to k(i, j) ≥ fund(i, j) for (i, j) ∈F ∪ ∂F.
Nevertheless, an explicit combinatorial solution for arbitrary p and k is still unknown.
Our general solution for the T-systems with principal coefficients may be applied to various problems related to the octahedron recurrence. For instance, there are known connections between the T-systems and Bessenrodt-Stanley polynomials discussed in [DF15]. We expect the solutions to the T-systems with principal coefficients to provide generalizations of this family of polynomials. It would also be interesting to apply our solutions to study the arctic curves of the octahedron equation with principal coefficients in the same spirit as [DFSG14]. We expect the coefficients to act as additional probability for dimer configurations, which may give other shapes to the arctic curves. Lastly, it would be interesting to investigate the quantum version of the T-systems with principal coefficients analog to [DF11,DFK12].