Twenty Vertex model and domino tilings of the Aztec triangle
aa r X i v : . [ m a t h . C O ] F e b TWENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTECTRIANGLE
PHILIPPE DI FRANCESCO
Abstract.
We show that the number of configurations of the 20 Vertex model on cer-tain domains with domain wall type boundary conditions is equal to the number ofdomino tilings of Aztec-like triangles, proving a conjecture from [DFG20]. The resultis based on the integrability of the 20 Vertex model and uses a connection to the U-turnboundary 6 Vertex model to re-express the number of 20 Vertex configurations as a sim-ple determinant, which is then related to a Lindstr¨om-Gessel-Viennot determinant forthe domino tiling problem. The common number of configurations is conjectured to be2 n ( n − / Q n − j =0 (4 j +2)!( n +2 j +1)! = 1 , , , , ... The enumeration result is extended toinclude refinements of both numbers.
Contents
1. Introduction 21.1. Triangular Ice model combinatorics 21.2. Outline of the paper and main results 52. Partition function of the 20V model 82.1. The 20V model: integrable weights 82.2. Boundary conditions 112.3. Transformation to a mixed 6V model 122.4. Transformation to a U-turn boundary 6V model 142.5. Inhomogeneous determinant formula 162.6. Homogeneous limit 172.7. The 20V model partition function 213. Domino tilings of the Aztec triangle 223.1. Path model 223.2. Generating functions and determinant formula 234. 20V-Domino Tiling correspondence 244.1. Determinant relations 244.2. Proof of the equivalence theorem 255. Refined partition functions of the 20V model 265.1. Setting and weights 265.2. Refined partition functions 26
Figure 1.
The local vertex configurations of the 20V model (top) and theirreformulation in terms of osculating Schr¨oder paths (bottom). k = 2 407.2. Proof of the 20V-DT conjecture for k = 3 418. Discussion and Conclusion 448.1. An exact (conjectured) formula 448.2. Alternative expressions for Z Vn Introduction
Triangular Ice model combinatorics.
Two-dimensional integrable lattice modelsof statistical physics have exhibited an extremely rich mathematical content, ranging from
WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 3
Figure 2.
DWBC1 boundary conditions for the 20V model and a sample con-figuration in the osculating Schr¨oder path formulation. representation theory to probability and combinatorics. The present paper concentrateson the two-dimensional triangular lattice version of the ice-type models, in the form ofthe so-called Twenty Vertex (20V) model [Kel74, Bax89]. The model is defined on afinite domain of the triangular lattice, by considering all possible orientations of edgesof the elementary triangles, obeying the ice rule : “there are equal numbers of incom-ing and outgoing edges at each vertex of the domain”. This gives rise to the (cid:0) (cid:1) = 20local vertex configurations displayed in the top two rows of Fig. 1. The 20V model isreally the triangular lattice version of the celebrated Six Vertex (6V) model [Lie67] de-scribing ice on a two-dimensional square lattice. The latter was at the core of the sagaof Alternating Sign Matrices (ASM) [Bre99], a remarkable sequence of connections be-tween purely combinatorial objects such as Descending Plane Partitions (DPP) [And80]in bijection with cyclically symmetric rhombus tilings of an hexagon with a central tri-angular hole [Kra06], Totally Symmetric Self-Complementary Plane Partitions (TSSCPP)[MRR86, Zei96] and two-dimensional models of statistical physics such as the 6V model[ICK92, Kup96, Tsu98, Kup02], the O(1) loop model [MNdGB04] and finally the Fully-Packed Loop model (FPL), at the center of the Razumov-Stroganov conjecture [RS04a],later proved by Cantini and Sportiello [CS11]. A striking feature of all the correspondencesis the apparent absence of natural bijections between different classes of objects with thesame cardinality A n = Q n − j =0 (3 j +1)!( n + j )! = 1 , , , , , ... The cases studied in this paper canbe seen as other remarkable examples of coincidences of cardinalities of sets, between whichno natural bijection is known. Throughout the paper we view the two-dimensional triangular lattice in a sheared fashion: the verticesform a square lattice, and edges are those of the square lattice plus the second diagonal within each square.This allows for easier drawings, and is more adapted to our choices of domains.
PHILIPPE DI FRANCESCO
In its physics literature debut, the 20V model was shown to be integrable provided somesuitable choice of Boltzmann weights is made [Kel74, Bax89], governed by 3 independentparameters, for which the thermodynamic free energy was computed. This used an explicitconnection between the integrable 20V model on the triangular lattice and the integrable6V model on the square lattice. Recently the model was revisited from a combinatorialperspective [DFG20], by considering domains with so-called “domain-wall boundary con-ditions” (DWBC), i.e. by imposing orientations on the boundaries of the domain thatforce the existence of non-local fault lines within the domain, across which orientationsare reversed (domain walls). In [DFG20], several possible DWBC were considered (labeled1,2,3,4) on a square n × n grid. Each type of DWBC gives rise to generalizations of ASM,coined Alternating Phase Matrices with entries among 0 and the sixth roots of unity andconservation conditions along rows, columns and diagonal lines.The 20V model can be reformulated in terms of lattice paths, by picking a preferredorientation of all edges of the lattice (say right, down and diagonal down), and decidingthat in any given orientation configuration, only edges with the preferred orientation arepath steps. There are 3 possibilities for these steps: (1 , , (0 , − , (1 , −
1) namely right,down and diagonal down. They form non-intersecting but possibly “kissing” paths (calledosculating Schr¨oder paths) that are allowed to share vertices, where up to three pathsbounce against each-other. The 20 local configurations of osculating Schr¨oder paths aredisplayed in the two bottom rows of Fig. 1.In the case of DWBC1,2 illustrated in Fig. 2 , it was shown in [DFG20] that the partitionfunction of the corresponding 20V model with uniform weights counts also the number of2 × q = e i π , the 20V DWBC1,2 APM are enumerated by thepartition function of a 6V DWBC model with q = e i π . The connection between 20V and6V being still valid in the presence of some non-trivial spectral parameters, the result wasgeneralized to include refinements as well.In the same paper, other combinatorial conjectures were made regarding the 20V DWBC3model on a more general pentagonal grid P n,k , 0 ≤ k ≤ n −
1, illustrated in Fig. 3. It wasconjectured in [DFG20] that the 20V DWBC3 configurations on P n,k are equinumerous tothe 2 × T n,k of the triangular domain T n, of the Ref. [DFG20] considers two choices labelled 1,2 for this first DWBC, which turn out to be equivalentmodulo a 180 ◦ rotation, and we show here a DWBC1 example. WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 5 n−k−1n−1k
Figure 3.
Extended DWBC3 boundary conditions on the pentagon P n,k for the20V model and a sample configuration in the osculating Schr¨oder path formulation,with trivial forced vertices removed. square lattice introduced by Pachter [Pac97] (see Fig. 4 for an illustration). In particular,for k = 0 and n = 1 , , ... the number of configurations of the 20V DWBC3 model on thesquare grid P n, forms the sequence 1 , , , , ... which matches conjecturally thatof domino tilings of the Pachter triangle T n, .The aim of the present paper is to shed some light on these latter conjectures and toprove some of them. More precisely, we will concentrate on the 20V DWBC3 model on themaximal domains Q n := P n,n − (as in Fig. 4 top right), whose configuration numbers giverise to the sequence 1 , , , , ... and prove that they are counted by the numbersof 2 × T n := T n,n − (as in Fig. 4bottom right) .1.2. Outline of the paper and main results.
The paper is organized as follows.The partition function of the 20V model is computed in Section 2. Using integrability,we transform the 20V model into a 6V model on a rectangular grid with different boundaryconditions, the so-called U-turn boundary conditions [Kup02], and with quantum param-eter q = e i π . The fully inhomogeneous version of the 20V model involves a large number ofarbitrary “spectral” parameters, which can be specialized to the combinatorial point whereall the Boltzmann weights are 1. On the other hand, the 6V U-turn boundary conditionpartition function is expressed as an explicit determinant involving the spectral parameters Remarkably, the same model but with the quantum parameter q = e i π was shown to enumerateVertically Symmetric ASM (VSASM) [Kup02]. PHILIPPE DI FRANCESCO T T T TP P P P
Figure 4.
Extended Pachter’s triangular domains T n,k for n = 4 (bottom row),and the corresponding 20 V pentagonal domains P n,k (top row). The domain T n,k must be tiled by means of 2 × P n,k must carry configurationsof the 20 V model with the specified boundaries. of the model (Section 2.5). The main difficulty in computing the 20V model partitionfunction is to derive the homogeneous limit of this determinant, when all spectral param-eters tend to their combinatorial point values. The result takes the form of a compactdeterminant expression for the total number Z Vn of configurations of the 20V model onthe domain Q n (Theorem 2.7): Theorem 1.1.
The total number Z Vn of configurations of the 20V DWBC3 model on thedomain Q n reads: Z Vn = det ≤ i,j ≤ n − (1 + u )(1 + 2 u − u )(1 − u v ) (cid:16) (1 − u ) − v (1 + u ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u i v j The enumeration of domino tilings Z DTn of the Aztec triangle T n is performed in Section3 by a bijective formulation in terms of non-intersecting Schr¨oder lattice paths, leading toa compact determinant formula (Theorem 3.4): WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 7
Theorem 1.2.
The total number Z DTn of domino tilings of the Aztec triangle T n reads: Z DTn = det ≤ i,j ≤ n − (cid:18) u − v − uv − u v + u v (cid:12)(cid:12)(cid:12)(cid:12) u i v j (cid:19) The identity between 20V and domino tiling enumerations is proved in Section 4 bynoting that both enumeration results of Theorems 2.7 and 3.4 express the numbers asthe determinant of a finite truncation of infinite matrices P and M respectively, and byexhibiting an explicit infinite lower triangular matrix L with entries 1 on the diagonal suchthat L M = P (Theorem 4.3): Theorem 1.3.
The partition function of the 20V DWBC3 model on the quadrangle Q n andthat of the domino tilings of the Aztec triangle T n coincide for all n ≥ . The second part of the paper is devoted to a refinement of the enumeration result,in which we keep track of one particular statistic on the set of configurations. For 20Vconfigurations, we record the position of the first visit to the rightmost vertical column.For domino tilings, we record the first visit to the line of slope − w in the last column, all other parametersbeing sent to their combinatorial point values. This gives rise to a compact determinantformula for the generating polynomial of refined 20V partition functions in the variable τ = q − − q wq − q − w (Theorems 5.3 and 5.4).In Section 6, we use again the non-intersecting Schr¨oder lattice path formulation of thedomino tiling problem to derive a compact determinant formula for a generating polynomialof the refined partition functions (Theorem 6.1).In Section 6.2, the results of Theorems 5.3 and 6.1 are shown to be equivalent (Theorem6.2). The proof, inspired by techniques employed in [BDFZJ12, DFL18] relies on theobservation that both results are perturbations of the uniform case, in which both infinitematrices M, P are modified into infinite matrices ¯ M ( n ) ( τ ) , P ( n ) ( τ ) by an additional termaffecting only their n -th and higher columns, and for which the relation L ¯ M ( n ) ( τ ) = P ( n ) ( τ )still holds.Section 7 is devoted to an investigation of the other conjectures regarding the 20V modelon P n,n − k and the domino tilings of T n,n − k for k >
1. We present direct proofs of theconjectures for k = 2 and k = 3 based on recursion relations involving the k = 1 result andits refinements.A few concluding remarks are gathered in Section 8. In Section 8.1, we present thefollowing: PHILIPPE DI FRANCESCO a a b b c c
Figure 5.
The local vertex configurations of the 6V model (top row) and thecorresponding osculating path configurations (bottom row). We have indicatedthe weights a, b, c of the respective configurations.
Conjecture 1.4.
The total numbers Z Vn = Z DTn of configurations of the 20V DWBC3model on the quadrangle Q n and of domino tilings of the Aztec triangle T n read: (1.1) Z Vn = Z DTn = 2 n ( n − / n − Y i =0 (4 i + 2)!( n + 2 i + 1)!for which we have no proof at this time. In Section 8.2, we present alternative formulasfor the numbers Z Vn = Z DTn , including a determinant involving binomials (Theorem 8.2)and a constant term formula (Theorem 8.4), in the same spirit as the TSSCP enumerationof Zeilberger [Zei96]. In Section 8.3, we show how a simple modification of the results ofthis paper allows to include an extra weight γ per horizontal step in the non-intersectingSchr¨oder lattice path formulation of the domino tiling problem. Section 8.4 is the conclusion per se . Acknowledgments.
We acknowledge partial support from the Morris and Gertrude Fineendowment and the NSF grant DMS18-02044.2.
Partition function of the 20V model
The 20V model: integrable weights.
Given a domain D of the two-dimensionaltriangular lattice, we consider the set of orientations of all lattice edges within D that satisfythe ice rule, namely that each vertex is incident to exactly 3 incoming and 3 outgoing edges .Boundary edge orientations are fixed, as part of the definition of the model. As mentionedbefore, the 20V model configurations are alternatively described by osculating Schr¨oderpaths on edges of the domain D that travel right and down. The boundary edges of D arefixed to be occupied/empty as part of the definition of the model. The occupied boundaryedges split into equal numbers of starting and endpoints of the paths. WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 9
311 11 11 11 1111113 3 3 32 2 2 2 2 2111 3 a a b b b b a a a a b b c c c c c c Figure 6.
The local vertex configurations of the three 6V models after resolvingall triple intersections in the 20V model.
Let us now describe the weighting of the configurations. Each vertex receives a localweight according to its configuration, and each configuration on D receives a weight equalto the product of the local weights of its inner vertices. The weights are chosen in a specialmanner, which we describe now. We attach complex “spectral parameters” to the linesof the lattice that intersect D . If a vertex is at the intersection of three such lines, saywith parameters z, t, w , we attach a weight W i ( z, t, w ), i = 1 , ...,
20 according to the list inFig. 1. In [Kel74, Bax89, DFG20], the weights are built out of elementary pieces as follows.Consider a triple intersection between a horizontal, vertical and diagonal line, and trans-late slightly the diagonal, say to the right: t 1 23wz . This amounts to resolv-ing the triple intersection into three simple intersections, labeled 1 , ,
3. Assume the edgeswere originally oriented and obeying the ice rule. The resolution above keeps the six outeredge orientations and creates 3 new inner edges, which may be oriented in such a way thatthe ice rule is now satisfied at each simple intersection, which now has 4 adjacent edges.At each simple intersection, this gives rise to (cid:0) (cid:1) = 6 possible local configurations, thoseof the so-called six vertex (6V) model (see Fig. 5 top for a list of the vertex environmentsand their 3 customary weights a, b, c that are invariant under reversal of all orientations).Here we are really dealing with three different 6V models, attached to the three simpleintersections. The 6V weights are parameterized by the two spectral parameters of theirtwo lines, and an additional index 1,2,3 corresponding to respectively a horizontal-vertical,horizontal-diagonal, and diagonal-vertical intersection. The three sets of 6V weights aredenoted by a i , b i , c i and are functions of their two spectral parameters (see Fig. 6 for the listof all 18 possible configurations and their weights). The 20V weights can now be definedin terms of the 6V weights of the resolution as follows: W V ( z, t, w ) := X inner edge configs . w V ( z, w ) w V ( z, t ) w V ( t, w ) a a a b a b b a c c a a b c a b b a a b b c c c + ! ! ! ! ! ! ! Figure 7.
Integrable weights of the 20V model and their expression in terms ofweights of the three sublattice 6V models. where the sum extends over all allowed inner edge orientations, and w Vi ∈ { a i , b i , c i } standsfor the 6V weight of the corresponding local configuration. For instance, the resolution:(2.1) = + corresponds a 20V weight ω = a b c + c c b .The weights are further constrained by the integrability condition which amounts torequiring that these values be independent of the resolution. For instance, we could havemoved the diagonal slightly to the left instead of the right, giving rise to an a priori different definition of the W V s. The requirement that the two definitions coincide resultsin a system of cubic relations for the 6V weights (the celebrated Yang-Baxter equations).In the case (2.1), the Yang-Baxter equation reads ω = a b c + c c b = b a c . We use thesolution of these equations from [DFG20], equivalent to that of [Kel74, Bax89]. Define thefollowing three functions of the two spectral parameters z, w :(2.2) A ( z, w ) = z − w, B ( z, w ) = q − z − q w, C ( z, w ) = ( q − q − ) √ zw, and pick the following 6V weights: (2.3) a = A ( z, w ) = z − w, b = B ( z, w ) = q − z − q w, c = C ( z, w ) = ( q − q − ) √ zwa = A ( qz, q − t ) = qz − q − t, b = B ( qz, q − t ) = q − z − qt, c = C ( qz, q − t ) = ( q − q − ) √ zta = A ( qt, q − w ) = qt − q − w, b = B ( qt, q − w ) = q − t − qw, c = C ( qt, q − w ) = ( q − q − ) √ tw WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 11
It is easy to show that the Yang-Baxter equations are satisfied by these, and lead to thefollowing expressions for the 20V integrable weights: ω = ( z − w )( qz − q − t )( qt − q − w ) ω = ( q − z − q w )( qz − q − t )( q − t − qw ) ω = ( q − z − q w )( qz − q − t )( q − q − ) √ twω = ztw ( q − q − ) + ( z − w )( q − z − qt )( q − t − qw ) ω = ( q − q − ) √ zw ( qz − q − t )( qt − q − w ) ω = ( q − z − q w )( q − q − ) √ zt ( qt − q − w ) ω = ( q − z − q w )( q − z − qt )( qt − q − w )(2.4)where the index refers to the vertex configurations of Fig. 7 (our running example (2.1)corresponds to ω = ω ).We will be first interested in the pure enumeration of the configurations of the 20V modelon specific domains with prescribed boundary conditions, which requires uniform weightsfor all vertices. As it turns out, there exists a choice of the spectral parameters and of theparameter q which ensures that all the weights (2.4) are equal to 1. Definition 2.1.
The “combinatorial point” where all ω i = 1 corresponds to the values (2.5) q = e iπ/ and ( z, t, w ) = α ( q , , q − ) , α = 2 − / q − The integrability property of the weights (2.4) will be crucially used in this paper totransform partition functions. The property can be used to deform/move any line of thelattice without changing the value of quantities such as the partition function, i.e. theweighted sum over configurations. For that, the presence of arbitrary spectral parametersis essential. Combinatorial (enumeration) results will be obtained by eventually sending allparameters to their combinatorial point values (2.5).2.2.
Boundary conditions.
We now consider the configurations of the 20V model on thequadrangular domain Q n = P n,n − depicted in Fig. 8, with a horizontal north boundary(N), vertical west and east (W,E) boundaries and diagonal south-west (SW) boundaries.We assign spectral parameters z = ( z , z , ...z n − ) to horizontal lines (top to bottom), w = ( w , w , ..., w n ) to vertical lines (left to right) and t = ( t , t , ..., t n − ) to diagonal lines(bottom to top). The corresponding fully inhomogeneous partition function is denoted by Z Vn [ z ; t ; w ]: it is the sum over all configurations of the 20V model compatible with theimposed boundary conditions, of the product of corresponding local weights ω m ( z i , t j , w k )at the intersection of the three horizontal, diagonal and vertical lines carrying spectralparameters z i , t j , w k respectively.We also consider a semi-homogeneous version of this partition function Z Vn [ z, t ; w ]obtained by setting z i = z and t i = t for all i , while keeping the w i arbitrary. . . . w w w n z z n z t t t t n z n+1 z . . . . . .. . . Figure 8.
Quadrangular domain Q n for the partition function of the 20V model.We have indicated the horizontal spectral parameters z , z , ..., z n − , the verticalones w , w , ..., w n and the diagonal ones t , t , ..., t n − . We impose the followingboundary conditions: arrows pointing towards the domain on the E boundary, outof the domain on the N and SW boundaries, and horizontal arrows point inwardand diagonal arrows point outward along the W boundary. The total number Z Vn of configurations is obtained by considering the abovementionedcombinatorial point, namely further restricting w i = w for all i , and using the values (2.5).2.3. Transformation to a mixed 6V model.
The semi-homogeneous partition function Z Vn [ z, t ; w ] can be identified with that of a special mixed 6V model on a rectangular grid,by use of the Yang-Baxter equation, which allows to displace spectral lines while keepingtrack of their intersection pattern (see Fig. 9 (a-d)).Starting from (a), we first move up the n uppermost diagonal lines, and down the n − fixed arrow configurations (all withidentical arrow orientations along lines), due to the propagation of the boundary conditionsvia the ice rule. The two top domains have n ( n + 1) / n ( n − / WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 13 (c)(a) (d)(b) w w w n . . . w w w n . . . tttttzzzzzzzzzzzz z z z z z ttttttttttt Figure 9.
Transformation of the original 20V model partition function Z Vn [ z, t ; w ] (a) into the alternating boundary mixed 6V model partition function Z Vn [ z, t ; w ] (d) (represented here for n = 6), by moving the diagonal lines, usingthe Yang-Baxter equation. The marking of vertices corresponds to the weights ofthe three 6V models on the three sublattices 1 (empty square), 2 (empty circle)and 3 (filled circle). In (a), spectral parameters have uniform values z (horizon-tal), t (diagonal), and arbitrary values w i (vertical). As a consequence, in (d) thehorizontal spectral parameters alternate between z, t and the vertical ones are w , w , ..., w n . Erasing them all, we get a new partition function. The original one is the product of thetrivial weights of all the erased vertices, namely: a ( z, t ) n Q ni =1 b ( z, w i ) i − a ( t, w i ) i timesthe new partition function with all the trivial vertices erased. (Here a i , b i , c i denote theuniform weights of the three sublattices i = 1 , , n ( n − / Z m Vn [ z, t ; w ],up to an extra factor of a ( z, t ) n ( n − / . The acronym m V stands for mixed 6V model z n z z −11 z −12 z −1n z −1 z −1 w w w n ... w w w n ...(a) (b) (c) ... z zz ... z Figure 10.
The U-turn boundary 6V model inhomogeneous partition function(a) reduces when all z i = z = p − q − to the configurations of (b) where all thearrow orientations at the U-turns point down, and where the top vertices are allfrozen in a b-type configuration. Erasing the latter and cutting the U-turns leadsto the partition function (c), in which horizontal spectral parameters alternatebetween z and z − from bottom to top, while the vertical ones are w n , ..., w , w from left to right. partition function: it is defined on a rectangular grid of size (2 n − × n with 6V weightsof sublattices 1 and 3 alternating on horizontal lines, respectively with uniform horizontalspectral parameters z and t , and vertical spectral parameters w , w , ..., w n , and with arrowspointing alternatively in and out along the W border.Recording all the weights of the erased vertices, we get the relation:(2.6) Z Vn [ z, t ; w ] = a ( z, t ) n (3 n − / n Y i =1 b ( z, w i ) i − a ( t, w i ) i ! Z m Vn [ z, t ; w ]2.4. Transformation to a U-turn boundary 6V model.
The U-turn boundary 6Vmodel was independently studied by Tsuchiya [Tsu98] and Kuperberg [Kup02]. The latteruses the 6V weights (
A, B, C )( qz, q − w ) / √ zw (and up to a renaming of variables x = p z/w , a = q , b = p ). Here we drop the overall factor of 1 / √ zw in the definition of theweights, which amounts to using the 6V weights (2.3) of the sublattices 2 or 3.Let Z V − Un [ z , z , ..., z n ; w , w , ..., w n ; p ] ≡ Z V − Un [ z ; w ; p ] denote the partition function ofthe inhomogeneous 6V model on a rectangular grid of size 2 n × n with alternating horizontal WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 15 spectral parameters z , z − , ..., z n , z − n (from bottom to top), vertical spectral parameters w n , w n − , ..., w (from left to right), with entering arrows along the W boundary, exitingarrows along the N and S boundaries, and U-turn boundaries on the E, as described inFig. 10 (a). The marked dot on the U-turn indicates that the spectral parameter changesfrom z i (bottom) to z − i (top) and that the configuration of the corresponding U-turnreceives a weight Y ( z i ), where(2.7) Y ( z ) = pq − − p − qzpqz − p − q − These weights obey Sklyanin’s Reflection Yang-Baxter equation [Skl88]. We now relate thepartition function of this model to that of the mixed 6V model of previous section.Setting all z i = z = p − q − forces each U-turn to have an arrow pointing down, as indi-cated in Fig. 10 (b). However this imposes no constraint on the vertical spectral parameters w i , which we keep arbitrary. In the resulting semi-homogeneous partition function, we maythen cut out all the marked dots on the U-turns, and moreover use the ice rule to dispose ofthe top-most horizontal line, which gives rise to n vertices with trivial b-type configurations.Erasing these while keeping track of their weight leaves us with the semi-homogeneous par-tition function Z Vn [ z ; w ] corresponding to Fig. 10 (c), with alternating uniform horizontalspectral parameters z, z − , and arbitrary vertical spectral parameters w = ( w , w , ..., w n ):(2.8) Z Vn [ z ; w ] := Z V − Un [ z ; w ; p ]( p − p − ) n Q ni =1 ( p − p − w i ) , p = z − / q − , where we have divided by the top row b-type weights q − z − − qw i = qp ( p − p − w i ) and bythe U-turn weights pq − − p − qz = p − q − ( p − p − ), using the relation z = p − q − .The partition function Z Vn [ z ; w ] (2.8) may now be identified with Z m Vn [ z, t ; w ] of theprevious section as follows. Note first that a rotation of 180 ◦ maps the configurations ofFig. 10 (c) onto those of Fig. 9 (d) for general size n , and then that this rotation flips allthe local arrow orientations of the vertices, which therefore keep the same weights (2.2).However a discrepancy subsists between the weights in Z m Vn [ z, t ; w ] and those in Z Vn [ z ; w ].Indeed, the 6V weights for Z m Vn [ z, t ; w ] alternate between those of the sublattice 1 on oddrows and 3 on even rows. The weights on the even rows are those of the sublattice 3 in bothmodels, and we must therefore identify their spectral parameters by setting t = z − . On oddrows, using the relation ( a , b , c )( z, w i ) = ( a , b , c )( q − z, qw i ) = q ( a , b , c )( q − z, w i ),we may identify these weights with ( a , b , c )( z, w i ) up to the overall factor q , provided weredefine odd row parameters z → q z (while keeping t = z − on even rows). This results in the following relation between the semi-homogeneous partition functions: Z m Vn [ q z, z − ; w ] = q n Z Vn [ z ; w ]with z = p − q − . Together with (2.6), this leads to the relation:(2.9) Z Vn [ q z, z − ; w ] = q n a ( q z, z − ) n (3 n − / Q ni =1 b ( q z, w i ) i − a ( z − , w i ) i ( p − p − ) n Q ni =1 ( p − p − w i ) Z V − Un [ z ; w ; p ]with p = z − / q − .To reach the combinatorial point (2.5), let us pick uniform 20V spectral parameters sothat ( q z, z − , w i ) = δ ( q , , q − ), namely z = q , δ = q − , p = q − and w i = q − for all i ,and finally set q = e iπ/ . Using the homogeneity of the weights, this allows to relate: Z Vn = (cid:16) αδ (cid:17) n (3 n − Z Vn [ q z, z − , w ] (cid:12)(cid:12) z = q ; w = q − = b ( q , q − ) n ( n − a ( q , q − ) n (3 n − a ( q − , q − ) n ( n +1)2 q n (2 / q ) n (3 n − ( q − − q ) n ( q − − q − ) n Z V − Un [ q , q − ; q − ]= 2 − n (5 n +3)4 q − n Z V − Un [ q , − q − ](2.10)with α as in (2.5), and where the latter partition function Z V − Un [ q , − q − ] correspondsto taking uniform values z i = q and w i = w = q − = − i , and p = q − in theU-turn boundary partition function Z V − Un [ z , w ; p ]. Remark 2.2.
We note that neither Z V − Un [ q , − q − ] nor Z Vn [ q , − are “combinatorial”in the sense that they count objects. Indeed, they are weighted countings of configurationsof the corresponding models, including irrational weights (such as √ ). For instance, wemay choose to normalize the weights of the V model by an overall factor ρ o = q √ onodd rows such that ρ o ( A, B, C )( qz, q − w ) = (1 , √ , and ρ e = q − √ on even rows, suchthat ρ e ( A, B, C )( qz − , q − w ) = ( √ , , for z = q and w = − . In that case, the V partition function ¯ Z Vn with these new weights reads ¯ Z Vn = ρ n o ρ n ( n − e Z Vn [ q , − . Yet,using the above relations at the combinatorial point, we find that ¯ Z Vn = 2 n ( n − / Z Vn . Sowe should think of the V partition functions at hand as combinatorial tools rather thancounting combinatorial objects. Inhomogeneous determinant formula.
The result for the partition function of theU-turn boundary 6V model [Tsu98, Kup02] reads as follows for our choice of weights. Forshort, we denote by ( a, b, c )( z, w ) := ( a , b , c )( z, w ) = ( A, B, C )( qz, q − w ). Let M U ( n ) bethe n × n matrix with entries(2.11) [ M U ( n )] i,j = 1 a ( z i , w j ) b ( z i , w j ) − a (1 , z i w j ) b (1 , z i w j ) WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 17
Then the inhomogeneous partition function is given by: Z V − Un [ z , z , ..., z n ; w , w , ..., w n ; p ]= det (cid:0) M U ( n ) (cid:1) × { Q ni =1( p − p − wi ) a ( qzi,q − z − i ) c ( zi,wi ) }{ Q ni,j =1 a ( zi,wj ) b ( zi,wj ) a (1 ,ziwj ) b (1 ,ziwj ) }{ Q ni =1 zn − i }{ Q ≤ i Let us denote by ∆ n [ z , ..., z n ; w , ..., w n ] the quantity:(2.13)∆ n [ z , ..., z n ; w , ..., w n ] := det (cid:0) M U ( n ) (cid:1)nQ ≤ i Throughout the paper, for any function f ( u, v ) with a power series expansionaround (0 , f ( u, v ) | u i v j the coefficient of u i v j in the series expansion of f ,in other words f ( u, v ) | u i v j := ∂ iu i ! ∂ jv j ! f (0 , Theorem 2.3. We have: (2.14) ∆ n [ z, − 1] = ( − n ( n − / n (1 − z ) n ( n +1) / det ≤ i,j ≤ n − (cid:16) f U ( z ; u, v ) (cid:12)(cid:12)(cid:12) u i v j +1 (cid:17) , where f U ( z ; u, v ) denotes the function: f U ( z ; u, v ) := 1 a ( z + u, v − b ( z + u, v − − a (1 , ( z + u )( v − b (1 , ( z + u )( v − q − − q z + u ) (cid:26) q − a ( z + u,v − − qb ( z + u,v − − q − a ( 1 z + u ,v − + qb ( 1 z + u ,v − (cid:27) (2.15) Proof. The theorem is proved in two steps. We first keep z i generic, and compute for w = − n [ z , ..., z n ; − 1] := lim wi →− i =1 , ,...,n det (cid:0) M U ( n ) (cid:1)nQ ≤ i 1] := lim vi → i =1 , ,...,n det ≤ i,j ≤ n (cid:0) f [ z i ; v j ] (cid:1)nQ ≤ i Let c n , n ∈ Z + be the coefficients in the series expansion: (2.17) 21 + e − x = X n ≥ c n x n n ! namely c = 1 , c = , c = 0 , c = − , c = 0 , c = , c = 0 , c = , ... Then for all n ≥ we have: (2.18) F n = n − X i =0 c i +1 (cid:18) n + 12 i + 1 (cid:19) F n − i − Proof. For any function f ( α ) (say polynomial), introduce the shift operator S : f Sf ,such that ( Sf )( α ) = f ( α − α = 0, we may express S as exp( − ddα ). In particular, the identity (1 + exp( − ddα )) f ( α ) = f ( α ) + f ( α − 1) is easilyinverted using the series expansion (2.17) in which x is substituted with the differentialoperator ddα : f ( α ) = 12 X n ≥ c n n ! d n dα n (cid:0) f ( α ) + f ( α − (cid:1) Let us apply this to the polynomial f ( α ) = α n +1 for n ≥ 0. We get: α n +1 = 12 ( α n +1 + ( α − n +1 + n X i =0 c i +1 (cid:18) n + 12 i + 1 (cid:19) ( α n − i + ( α − n − i ) ) which implies: α n +1 + (1 − α ) n +1 = n X i =0 c i +1 (cid:18) n + 12 i + 1 (cid:19) ( α n − i + (1 − α ) n − i ) . WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 19 Up to the overall prefactor 1 / (( q − − q ) z i ), the i -th row of the desired identity (2.18)follows immediately from this, as it takes precisely the form: α n +1 + (1 − α ) n +1 − ( β n +1 + (1 − β ) n +1 )= n − X i =0 c i +1 (cid:18) n + 12 i + 1 (cid:19) (( α n − i + ( α − n − i ) − ( β n − i + ( β − n − i ))where α = q z i +1 and β = q − z i +1 . (cid:3) Note that as F = 0 we have the series expansion F [ v ] = P n ≥ F n v n =: v G [ v ]. Thematrix with entries f [ z i ; v j ] is made of successive columns v G [ v ] , ..., v n G [ v n ], therefore thedeterminant of this matrix is v v · · · v n det( G [ v ] G [ v ] · · · G [ v n ]). Consequently we rewrite:det ≤ i,j ≤ n (cid:0) f [ z i ; v j ] (cid:1)nQ ≤ i 0, ... v n → 0. The first limit v → G [0] = F as first column in the determinant and set v = 0 in the denominator.To perform the second, we must expand G [ v ] = F + F v + F v + O ( v ) up to order 2,as both constant and linear terms are columns proportional to F by Lemma 2.4. Whenwe reach v k → 0, we must expand G [ v k ] up to order 2 k , as all previous terms are linearcombinations of the previous columns in the determinant, as a consequence of Lemma 2.4.This leads successively to:lim v → D n [ z ; v ] = 12 n det( F G [ v ] · · · G [ v n ]) Q ni =2 v i Q ≤ i 1] = lim zi → zi =1 , ,...,n ∆ n [ z , ..., z n ; − Q ≤ i 1. Asbefore, we write the series expansion R [ u ] = P n ≥ R n u n , and perform the successive limits u → u → 0, ... u n → D n [ u ] := 12 n (1 − z ) n ( n +1) / det( R [ u ] R [ u ] · · · R [ u n ]) t Q ≤ i 1, and we end up with the formulas:lim u → D n [ u ] = ( − n − n (1 − z ) n ( n +1) / det( R R [ u ] · · · R [ u n ]) t Q ni =2 u i Q ≤ i 1) ofthe matrix ( R R · · · R n ) t with the coefficient of u i v j +1 of the series expansion of f U ( u, v )around (0 , F j +1 ) i +1 corresponds to the coefficient of v j +1 in the series expan-sion of F [ v ] around v = 0, and ( R i ) j +1 to the coefficient of u i in the expansion of ( F j +1 ) i +1 around u = 0, after setting z i = z + u . This completes the proof of the theorem. (cid:3) To further the computation of ∆ n [ z ; − Lemma 2.5. Let ˜ f U ( u, v ) := γ ( u ) δ ( v ) f U (cid:16) auα ( u ) , bvβ ( v ) (cid:17) , with a, b ∈ C ∗ , and where α ( x ) , β ( x ) , γ ( x ) , δ ( x ) are power series that converge for small enough x , and take the value at x = 0 . Then we have: ∆ n [ z, − 1] = ( − n ( n − / n (1 − z ) n ( n +1) / a n ( n − / b n det ≤ i,j ≤ n − (cid:16) ˜ f U ( u, v ) (cid:12)(cid:12)(cid:12) u i v j +1 (cid:17) Proof. Repeating the first step of the proof of Theorem 2.3 with ˜ f U substituted for f U , wesee that when we send v j → 0, the column F j +1 gets multiplied by b j +1 , but otherwisethe substitution v j → bv j β ( v j ), as well as the overall factor δ ( v j ) only affect higher orderterms of the expansion in v j , hence the limiting process yields the same result as before, upto an overall factor b ··· +(2 n − = b n . The same happens for the second step, where eachvector R i gets multiplied by a i leading to an overall factor a n ( n − / , while the substitution u i → au i α ( u i ), as well as the overall factor γ ( u i ) again only affect higher order terms inthe expansion in u i . The Lemma follows. (cid:3) We now compute the quantity ∆ n [ q , − z = q . WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 21 Theorem 2.6. We have ∆ n [ q ; − 1] = q n ( n +1) n (7 n +5)4 det ≤ i,j ≤ n − (cid:16) g U ( u, v ) (cid:12)(cid:12)(cid:12) u i v j (cid:17) (2.19) g U ( u, v ) = (1 + u )(1 + 2 u − u )(1 − u v ) (cid:16) (1 − u ) − v (1 + u ) (cid:17) (2.20) Proof. We apply Lemma 2.5 with z = q , a = 2 q , b = 2 q − , α ( u ) = − q u , β ( v ) = q − v , γ ( u ) = q u − q u , δ ( v ) = β ( v ) , with the result:˜ f U ( u, v ) = 1 + q u (1 − q u )(1 + q − v ) f U (cid:18) q u − q u , q − v q − v (cid:19) = v q − q (1 + u )(1 + 2 u − u )(1 − u v ) (cid:16) (1 − u ) − v (1 + u ) (cid:17) = − q − √ v g U ( u, v )Noting that the coefficient of v j +1 in ˜ f U ( u, v ) corresponds to the coefficient of v j in g U ( u, v ),the theorem follows from assembling all the factors and using the fact that q = e iπ/ . (cid:3) The 20V model partition function. We now complete the calculation of the par-tition function for the 20 V model on the quadrangle Q n . Theorem 2.7. The partition function of the 20V model with uniform weights on the quad-rangle Q n reads: Z Vn = det ≤ i,j ≤ n − (cid:16) g V ( u, v ) (cid:12)(cid:12)(cid:12) u i v j (cid:17) (2.21) g V ( u, v ) = (1 + u )(1 + 2 u − u )(1 − u v ) (cid:16) (1 − u ) − v (1 + u ) (cid:17) = ( u − u ) − v ( u − u ) − u − vu (2.22) Proof. We use the result of Theorem 2.6 to compute the homogeneous limit of the deter-minant formula (2.12) for Z V − Un [ z , w ; p ], with z i → q , w i → − p = q − . First weassemble all the remaining factors in Z V − Un [ z , w ; p ] to find that: Z V − Un [ q , − 1] = 1 z n ( n − (cid:0) ( q − q − ) √ zw (cid:0) p − − p )( p − p − w )) n × (cid:0) ( qz − q − w )( q − z − qw )( q − q − zw )( q − − qzw ) (cid:1) n ∆ n [ z, w ] | z → q ,w →− = q − n ( n − ( q − q − ) n ( q − q − ) n ( q + q − ) n (8 q ) n ∆ n [ q ; − q n (2 n − n (3 n +2) ( − n ∆ n [ q ; − (b) (1,1) (3,3)(0,2)(0,0)(0,2n−2) (2n−1,2n−1) (a) Figure 11. The Aztec triangle T n and a typical domino tiling configuration (a),together with its bijectively associated family of non-intersecting Schr¨oder paths.In (b) we show the starting points of the n paths: (0 , i ), i = 0 , , ..., n − j + 1 , j + 1), j = 0 , , .., n − Substituting the result of Theorem 2.6 for ∆ n [ q ; − Z V − Un = q n n (5 n +3)4 det ≤ i,j ≤ n − (cid:16) g U ( u, v ) (cid:12)(cid:12)(cid:12) u i v j (cid:17) The theorem follows from combining this with (2.10), and defining g V ( u, v ) := g U ( u, v ). (cid:3) Remark 2.8. We may interpret this result as follows. Let P be the infinite matrix withentries P i,j , i, j ∈ Z + generated by the series expansion g V ( u, v ) = P i,j ≥ u i v j P i,j of (2.22) . Then Z Vn = det( P n ) where P n is the truncation of the matrix P to it n first rowsand columns. Domino tilings of the Aztec triangle Path model. The Aztec triangle T n is defined as the planar domain of Figs. 11 (a,b),where the length of the vertical West boundary is 2 n . We study and count the tilingconfigurations of this domain by means of 2 × n non-intersecting lattice paths from the West boundary (startingpoints (0 , i ), i = 1 , , ..., n − 1) to the staircase East boundary (endpoints (2 j + 1 , j + 1), WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 23 j = 0 , , ..., n − Z : up (1 , , − 1) and horizontal (2 , Z DTn denote the total number of domino tiling configurations of the Aztec triangle T n . The latter is easily computed by direct application of the Lindstr¨om-Gessel-Viennotdeterminant formula [Lin73, GV85]. Denoting by M i,j the number of configurations of asingle path from the starting point (0 , i ) to the endpoint (2 j + 1 , j + 1), we have: Lemma 3.1. The partition function for domino tilings of the Aztec triangle T n reads: Z DTn = det ≤ i,j ≤ n − ( M i,j ) Remark 3.2. Note that M i,j is independent of n , as the shape of the Aztec triangle domainimposes no further constraint on the paths apart from their fixed starting and endpoints.The Lemma 3.1 states that Z DTn = det( M n ) , where the n × n matrix M n may be viewed asthe finite truncation to the first n rows and columns of the infinite matrix M with entries M i,j , i, j ∈ Z + . Generating functions and determinant formula. The generating function f M ( u, v ) := P ∞ i,j =0 v i u j M i,j is given by the following. Theorem 3.3. We have f M ( u, v ) = ∞ X i,j =0 v i u j M i,j = 1 + u − v − uv − u v + u v Proof. We start from the generating series:(3.1) σ ( u, v ) = 11 − u − v − uv We may interpret σ ( u, v ) as the partition function for arbitrary finite length Schr¨oder pathsfrom the origin, with weight u per up step, v per down step, and uv per horizontal step.The position of the endpoint of the path reads ( U + D + 2 H, U − D ) where U, D, H denoterespectively the total numbers of up, down, horizontal steps in the path. Such a pathreceives a weight u U v D ( uv ) H = u U + H v D + H . The number Σ i,j of paths from the origin withfixed endpoint ( i, j ), with i − j even, therefore reads: Σ i,j = σ ( u, v ) | u i + j v i − j . Equivalently:(3.2) Σ i,j = I dx iπx dy iπy x i + j y i − j − x − y − xy where the contour integral picks the residues at 0. By a trivial translation of the startingpoint, we may now interpret M i,j as the total number of paths from the origin to (2 j +1 , j + 1) − (0 , i ) = (2 j + 1 , j + 1 − i ), hence M i,j = Σ j +1 , j +1 − i , or: M i,j = I dx iπx dy iπy x j +1 − i y i − x − y − xy Changing integration variables to u = y/x, v = x , we get M i,j = I du iπu dv iπv v j +1 u i − v − uv − uv = 11 − v − uv − uv (cid:12)(cid:12)(cid:12) u i v j +1 Finally, we remark that for any series f ( v ), and any j ∈ Z + , we have:(3.3) f ( v ) | v j +1 = 12 √ v (cid:0) f ( √ v ) − f ( −√ v ) (cid:1) (cid:12)(cid:12)(cid:12) v j and the theorem follows from the identity12 √ v (cid:18) − (1 + u ) √ v − uv − 11 + (1 + u ) √ v − uv (cid:19) = 1 + u − v − uv − u v + u v (cid:3) This allows to translate the result of Lemma 3.1 into the following determinant formula. Theorem 3.4. The partition function for domino tilings of the Aztec triangle T n reads: Z DTn = det ≤ i,j ≤ n − (cid:16) f DT ( u, v ) (cid:12)(cid:12)(cid:12) u i v j (cid:17) (3.4) f DT ( u, v ) = 1 + u − v − uv − u v + u v (3.5) 4. Determinant relations. We wish to compare the results of Theorems 2.7 and 3.4.Using Remarks 2.8 and 3.2, we see that both enumeration results are expressed as thedeterminant of a finite truncation of an infinite matrix. We have the following [DFL18]. Lemma 4.1. Let L (resp. U ) be arbitrary infinite lower (resp. upper) triangular matrices,with diagonal entries , and A any infinite matrix. Then we have the following properties: (1) For all n ≥ , the truncation ( LAU ) n of the infinite matrix LAU to its first n rowsand columns is equal to L n A n U n , the product of the corresponding truncations of L, A, U respectively. (2) det( A n ) = det (cid:0) ( LAU ) n (cid:1) Proof. We have for all i, j ∈ [0 , n − LAU ) n ) i,j = P ℓ,m ≥ L i,ℓ A ℓ,m U m,j . But L i,ℓ = 0for ℓ > i hence in particular for all ℓ ≥ n , as i ≤ n − 1. Similarly, U m,j = 0 for m ≥ n ,and therefore we may rewrite (( LAU ) n ) i,j = P n − ℓ,m =0 L i,ℓ A ℓ,m U m,j = (( L n A n U n ) i,j , and (1)follows. The statement (2) follows from det( L n A n U n ) = det( A n ) as det( L n ) = det( U n ) =1. (cid:3) WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 25 Lemma 4.2. Let A, B be two infinite matrices with entries generated by the series f A ( u, v ) = P i,j ≥ u i v j A i,j and f B ( u, v ) = P i,j ≥ u i v j B i,j . Then the product AB has for generating se-ries the convolution of f A and f B , namely: f AB ( u, v ) = f A ∗ f B ( u, v ) = I dt iπt f A ( u, t − ) f B ( t, v ) where the contour integral picks up the residue at t = 0 .Proof. By direct calculation. (cid:3) Proof of the equivalence theorem.Theorem 4.3. The partition function of the 20V DWBC3 model on the quadrangle Q n andthat of the domino tilings of the Aztec triangle T n coincide for all n ≥ .Proof. By Remarks 2.8 and 3.2, we have Z Vn = det( P n ), Z DTn = det( M n ) in terms oftruncations of infinite square matrices P, M respectively generated by f P = g V (2.22)and f M = f DT (3.5). Let L be the infinite matrix with generating series(4.1) f L ( u, v ) := 1 + 2 u − u − u (1 + v + uv ) . Then we have the following properties: (1) L is lower triangular, with diagonal elements 1and (2) LM = P . The property (1) follows from noticing that we have a series identity ofthe form f L ( u, v ) = h ( u, uv ) where h ( u, v ) = 1 + 2 u − u − u − v − uv , which shows that non-zero coefficients of u i v j always have j ≥ i . Moreover the terms with i = j correspond to the series expansion of h (0 , uv ) = 1 / (1 − uv ) and are all equal to 1.Property (2) follows from Lemma 4.2: f LM ( u, v ) = I dt iπ u − u t (1 − u ) − u (1 + u ) f M ( t, v )= 1 + 2 u − u − u f M (cid:18) u u − u , v (cid:19) = f P ( u, v )easily checked by substituting the values f P = g V (2.22) and f M = f DT (3.5). Finally,by Lemma 4.1 we have det( P n ) = det(( LM ) n ) = det( L n M n ) = det( M n ), and the theoremfollows. (cid:3) Refined partition functions of the 20V model Setting and weights. In this section, we consider the particular inhomogeneouspartition function Z Vn [ w ] for the 20V model on the quadrangle of Fig. 8, correspondingto a choice of uniform spectral parameters z i = q z = q , i = 1 , , ..., n , t i = z − = q − , i = 1 , , ..., n − w i = q = − i = 1 , , ..., n − 1, while w n = w is kept arbitrary. Thecorresponding weights (2.4) are all equal to 1, except those of the last column, which read:¯ ω = (1 − w )( q − q − w )2 √ ω = (1 − w )( q − − q w )2 √ ω = ¯ ω = 1 − w √− w ¯ ω = ¯ ω = ¯ ω = (cid:18) − w (cid:19) (5.1)The quantity Z Vn [ w ] gives access to refined 20V model partition functions as explained inthe next section.5.2. Refined partition functions. Let us now relate the partition function Z Vn [ w ] torefined configuration numbers of the 20V model on the quadrangle Q n . The osculatingSchr¨oder path configurations of the 20V model on Q n may be classified according to theconfiguration of the vertices in the last column (see Fig. 12 for an illustration). Let k denotethe position (labeled 1 , , ..., n − k -th vertex corresponds to the point of entry of the topmost path intothe rightmost vertical line, below which the path goes down vertically until its endpoint.Note that all vertices strictly above k are in the empty configuration (weights ω ), andall those strictly below in the configuration with exactly one vertical line passing through(weights ω ). The vertex k itself may be in either of two configurations, correspondingto whether the path enters the last column horizontally ( − , weight ω ) or diagonally ( \ ,weight ω ). Let Z V − n,k and Z V \ n,k denote the corresponding refined partition functions, withthe obvious sum rule at w = − n − X k =1 Z V − n,k + Z V \ n,k = Z Vn In the presence of the spectral parameter w in the last column, this becomes:(5.2) n − X k =1 (cid:16) ¯ ω Z V − n,k + ¯ ω Z V \ n,k (cid:17) ¯ ω n − k − ¯ ω k − = Z Vn [ w ] WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 27 k2n−1 ω ω ωω w Figure 12. A typical configuration of osculating Schr¨oder paths on the quadran-gle Q n . The vertex labeled k ∈ [1 , n − 1] corresponds to the first visit of any pathto the last column. All vertices strictly above it are in the empty configuration(weight ω ), while those strictly below have a single vertical path going through(weight ω ). In general configurations, the vertex k itself could be in either of thetwo indicated states (cameos on the right, with weights ω or ω ). with ¯ ω i as in (5.1). Let us introduce a parameter:(5.3) τ := ¯ ω ¯ ω = q − − q wq − q − w Further noting that ¯ ω = ¯ ω , we get the following expression for the generating function Z Vn ( τ ) of refined 20V partition functions:(5.4) Z Vn ( τ ) := n − X k =1 Z Vn,k τ k − = q − q − w ¯ ω n − √− w Z Vn [ w ] where we used the notation Z Vn,k = Z V − n,k + Z V \ n,k for the refined 20V partition functioncorresponding to the leftmost path entering the rightmost vertical line at position k . Itis customary to also introduce the generating function h Vn ( τ ) for normalized “one-pointfunctions” Z Vn,k Z Vn :(5.5) h Vn ( τ ) := n − X k =1 Z Vn,k Z Vn τ k − = Z Vn ( τ ) Z Vn = q − q − w ¯ ω n − √− w Z Vn [ w ] Z Vn Transformations to the 6V model. Repeating the operations of Sections 2.3 and2.4, and recording all the weights of trivial vertices, we arrive at:(5.6) Z Vn [ w ] = q n b ( q ,w ) n − a ( q − ,w ) n a ( q ,q − ) n (3 n − b ( q , − ( n − n − a ( q − , − n ( n − Z Vn [ w ]where Z Vn [ w ] denotes the partition function function of the 6V model with boundaryconditions as in Fig. 10 (c), with the vertex weights ( A, B, C )( qz i , q − w j ) identical to thoseof the sublattices 2 and 3, with vertical spectral parameters w = · · · = w n − = − w n = w , and horizontal spectral parameters alternating between the value z = q on oddrows and z − = q − on even rows. Eq. 5.6 boils down to:(5.7) Z Vn [ w ] Z Vn = (cid:18) − w (cid:19) n − Z Vn [ w ] Z Vn trivially satisfied for w = − 1. Next, (2.8) for p = q − implies:(5.8) Z Vn [ w ] = Z V − Un [ w ]( p − p − ) n ( p + p − ) n − ( p − p − w )and equivalently:(5.9) Z Vn [ w ] Z Vn = √ q − − q w (cid:18) − w (cid:19) n − Z V − Un [ w ] Z V − Un Refined partition functions of the 6V model. We now turn to the partitionfunction Z Vn [ w ] of the 6V model. To easier compare it to that of the 6V model withU-turn boundaries, let us rotate the picture of Fig. 10 (c) by 180 ◦ . The rows are nowlabeled 1 , , ..., n − w isattached to the rightmost vertical line. We consider the osculating path configurations ofthe model (see Fig. 5 bottom), with n paths starting on odd rows along the left boundaryand ending on the S boundary (see Fig. 13 for an illustration). We note that the topmostpath must first visit the rightmost vertical line at a vertex corresponding to some row k , and then continue along the vertical until its endpoint. The configurations along therightmost vertical line have therefore b-type weights at all positions strictly under k , anda-type weights at all positions strictly above k , while the vertex at position k is of typec. When collecting the weights however, we must distinguish the cases according to the WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 29 k −1 z −1 z −1 a e a o b e b o c e c o z −1 z −1 zzz −1 zzw zzzz Figure 13. A typical configuration of osculating paths for the 6 V model ona rectangular grid of size (2 n − × n . The boundary conditions match thoseof Fig. 10 (c) upon global rotation by 180 ◦ . The horizontal spectral parametersalternates between the values z = q on odd lines and z − = q − on even lines. Thevertical parameters are w i = q = − w n = w .The vertex labeled k ∈ [1 , n − 1] corresponds to the first visit of any path to thelast column. All vertices strictly above it are in the empty configuration (weight a e , a o on even/odd rows), while those strictly below have a single vertical pathgoing through (weight b e , b o on even/odd rows). In general configurations, thevertex k itself could be in either of the two indicated states (cameos on the right,with weights c e if k is even or c o if k is odd). parity of k , as odd and even horizontal spectral parameters are different. Using the oddrow weights: ( A, B, C )( q , q − w ) and even row weights ( A, B, C )( q − , q − w ), we get thefollowing relative weights (ratios of the weights at generic w to those at w = − 1) in thelast column: ¯ a o = q − q − w √ ¯ b o = − w ¯ c o = √− w ¯ a e = − w ¯ b e = q − − q w √ ¯ c e = √− w where the subscript e/o stands for even/odd row. Let Z Vn,k denote the refined partition function corresponding to paths first visiting therightmost vertical at position k . Apart from the trivial sum rule at w = − n − X k =1 Z Vn,k = Z Vn we have in the presence of a non-trivial w : n X k =1 Z Vn, k − (¯ b o ¯ b e ) k − ¯ c o (¯ a e ¯ a o ) n − k + n − X k =1 Z Vn, k (¯ b o ¯ b e ) k − ¯ b o ¯ c e ¯ a o (¯ a e ¯ a o ) n − k − = Z Vn [ w ]Noting that ¯ c e = ¯ c o , ¯ b o ¯ a o = ¯ a e ¯ a o = ¯ ω and ¯ b o ¯ b e = ¯ ω , and using the parameter τ of (5.3),we finally get an expression for the generating function of the sums of refined partitionfunctions Z Vn, k − + Z Vn, k :(5.10) Z Vn ( τ ) := n X k =1 ( Z Vn, k − + Z Vn, k ) τ k − = 1¯ ω n − √− w Z Vn [ w ]or equivalently the generating function h Vn ( τ ) for sums of one-point functions Z Vn, k − + Z Vn, k Z Vn :(5.11) h Vn ( τ ) := n X k =1 Z Vn, k − + Z Vn, k Z Vn τ k − = Z Vn ( τ ) Z Vn = 1¯ ω n − √− w Z Vn [ w ] Z Vn where we set Z Vn, n = 0 by convention.Combining (5.5), (5.11) and the relation (5.7), we finally get the following relation be-tween one-point generating functions of the 20 V and 6 V models:(5.12) h Vn ( τ ) = ¯ ω n − q − q − w √ (cid:18) − w ω (cid:19) n − h Vn ( τ ) = (cid:18) τ (cid:19) n − h Vn ( τ )where we have identified the quantity − w √ q − q − w ) = τ .5.5. Relation to the inhomogeneous U-turn boundary partition function. Let usfinally relate the above quantities to the U-turn boundary partition function Z V − Un [ w ].First we note that (5.8) implies: Z Vn [ w ] Z Vn = √ q − − q w Z V − Un [ w ] Z V − Un As already mentioned in Remark 2.2, the quantities Z Vn,k are not integers as the weighting of the 6 V model is not uniform and involves √ z = q , w = − 1] indicating that we are at the combinatorial point, and we use the notation [ w ] to indicatea non-trivial value w for the n -th vertical spectral parameter, while all other spectral parameters have thevalues of the combinatorial point. WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 31 and moreover that (2.12) implies the relation: Z V − Un [ w ] Z V − Un = √− w (cid:18) − w (cid:19) n (cid:18) q − − q w √ (cid:19) n +1 (cid:18) q − q − w √ (cid:19) n ∆ n [ w ]∆ n where we denote for short ∆ n [ w ] the quantity ∆ n [ z , ..., z n ; w , ..., w n ] (2.13) with all z i = q and all w i = − w n = w , and ∆ n := ∆ n [ − h Vn ( τ ) = 1¯ ω n − √− w √ q − − q w Z V − Un [ w ] Z V − U n × n = (cid:18) − w (cid:19) n +1 (cid:18) q − q − w √ (cid:19) (cid:18) q − − q w √ (cid:19) n ∆ n [ w ]∆ n (5.13)Similarly:(5.14) h Vn ( τ ) = (cid:18) − w (cid:19) n (cid:18) q − q − w √ (cid:19) (cid:18) q − − q wq − q − w (cid:19) n ∆ n [ w ]∆ n Refined U-turn boundary. We now turn to the calculation of the refined determi-nant ∆ n [ w ]. We start with a refinement of Theorem 2.3. We consider the semi-homogeneouslimit ∆ ′ n [ z, w ] := lim z ,...,zn → zw ,...,wn − →− ,wn → w ∆ n [ z , ..., z n ; w , ..., w n ]As in the proof of Theorem 2.3, we proceed in two steps. First, we take the successivelimits w , w , ..., w n − → − 1. This step is rigorously identical, as only the limit w n → w isdifferent. Let us consider:∆ ′ n [ z , w ] := lim w ,...,wn − →− wn → w ∆ n [ z , ..., z n ; w , ..., w n ]= lim v ,v ,...,v n → det (cid:16) F [ v ] F [ v ] · · · F [ v n − ] ˜ F [ v n ] (cid:17) n − ( w + 1) n − (1 − w ) Q ≤ i 0, we end up with∆ ′ n [ z , w ] = det ( F F · · · F n − F [ w + 1])2 n − ( w + 1) n − (1 − w ) The second step consists in computing:∆ ′ n [ z, w ] = lim z ,...,z n → z ∆ ′ n [ z , w ]= lim u ,...,u n → det ( F F · · · F n − F [ w + 1])2 n − ( w + 1) n − (1 − w )(1 − z ) n ( n +1) / Q ≤ i 0, we arriveat: ∆ ′ n [ z, w ] = ( − n ( n − / det (cid:16) ( ˜ R ˜ R · · · ˜ R n ) t (cid:17) n − ( w + 1) n − (1 − w )(1 − z ) n ( n +1) / The entries ( ˜ R m ) i of the vector ˜ R m are easily identified as f U ( u, v ) | u i v m − for m = 1 , , ..., n − f U ( u, w + 1) | u i for m = n . This result may be recast into the following. Lemma 5.1. We have ∆ ′ n [ z, w ] = ( − n ( n − / det ≤ i,j ≤ n − ( f U ( z, w ; u, v ) | u i v j +1 )2 n − ( w + 1) n − (1 − w )(1 − z ) n ( n +1) / in terms of the function: (5.15) f U ( z, w ; u, v ) := f U ( z ; u, v ) + v n − ( f U ( z ; u, w + 1) − f U ( z ; u, v ) | v n − )(Note the subtraction of the contribution of order v n − of f U ( z ; u, v ), to avoid over-counting.). We are now ready to compute the quantity ∆ n [ w ] = ∆ ′ n [ q , w ] correspondingto z = q . Theorem 5.2. We have the following determinant formula: ∆ n [ w ] = q n ( n +1) n (7 n +5)4 (cid:18) − w (cid:19) n det ≤ i,j ≤ n − ( g ′ U ( u, v ) | u i v j )(5.16) g ′ U ( u, v ) = g U ( u, v ) + v n − (cid:18) u − u (cid:19) n (cid:26) − u ) (1 − u ) (1 − w ) + (1 + u ) (1 + w ) − (cid:27) (5.17) with g U ( u, v ) as in (2.20) .Proof. Like in Sect. 2.6, we apply Lemma 2.5 to the function f U ( q , w ; u, v ) (5.15), withthe same choices of a, b, α, β, γ, δ . More precisely we apply the substitutions:(5.18) u → q u − q u , v → q − v q − v , w = 2 q − x q − x − WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 33 and we multiply it by an overall factor of q u (1 − q u )(1+ q − v ) . The last substitution in (5.18) issimply a change of variable w → x , which allows to treat v and w + 1 similarly. We firstcompute the effect of this transformation on the term v n − f U ( z ; u, w + 1), up to higherorders in v : 1 + q u (1 − q u )(1 + q − v ) (2 q − v ) n − f U (cid:18) q ; 2 q u − q u , q − x q − x (cid:19) = (2 v ) n − q ( − n q − q (1 + q − x ) x g U ( u, x ) + O ( v n )with g U ( u, v ) as in (2.20). The complete function f U ( q , w ; u, v ) is therefore transformedinto:˜ f U ( q , x ; u, v ) = v q − q (cid:8) g U ( u, v )+ v n − (cid:0) n − q ( − n (1 + q − x ) x g U ( u, x ) − g U ( u, v ) | v n − (cid:1)(cid:9) In this last expression we may replace xg U ( u, x ) = P m ≥ x m +1 n(cid:0) u − u (cid:1) m +2 − u m +2 o with the truncated expansion x ¯ g ( u, x ) = X m ≥ n − x m +1 ((cid:18) u − u (cid:19) m +2 − u m +2 ) = x n − (cid:18) u − u (cid:19) n − x (cid:0) u − u (cid:1) + O ( u n )Indeed, the first n − n − w by inverting the change of variables in (5.18)into x = q w − w . The corresponding function finally reads:˜ f ′ U ( q , w ; u, v ) = − vq − √ (cid:8) g U ( u, v )+ v n − (cid:18) u − u (cid:19) n (cid:18) w )1 − w (cid:19) n − − u ) (1 − u ) (1 − w ) + (1 + u ) (1 + w ) − !) =: − vq − √ g ′ U ( u, v )By Lemma 2.5 we deduce:∆ n [ w ] = q n ( n +1) n (7 n +5)4 − w (1 + w ) n − det ≤ i,j ≤ n − (˜ g ′ U ( u, v ) | u i v j )= q n ( n +1) n (7 n +5)4 (cid:18) − w (cid:19) n det ≤ i,j ≤ n − ( g ′ U ( u, v ) | u i v j ) in which we have divided explicitly the last column by (cid:16) w )1 − w (cid:17) n − , thus allowing toreplace ˜ g ′ U ( u, v ) with g ′ U ( u, v ). The theorem follows. (cid:3) Refined partition functions for 20V and 6V models. Theorem 5.2 provides thefollowing formula for the ratio ∆ n [ w ] / ∆ n :∆ n [ w ]∆ n = (cid:16) − w (cid:17) n det ≤ i,j ≤ n − ( g ′ U ( u, v ) | u i v j )det ≤ i,j ≤ n − ( g U ( u, v ) | u i v j )in terms of the functions g U ( u, v ) (2.20) and g ′ U ( u, v ) (5.17).Using the relations (5.13) and (5.14) and the result of Theorem 5.2, we immediately getthe following expressions for the one-point function generating functions of the 6V and 20Vmodels, in terms of the variable τ = − q wq − w (i.e. w = − q τq − τ ), and of the function g V = g U : Theorem 5.3. The one-point generating functions for the V and V models read for all n ≥ : h Vn ( τ ) = det ≤ i,j ≤ n − (cid:0) g ref6 V ( u, v ) | u i v j (cid:1) det ≤ i,j ≤ n − ( g V ( u, v ) | u i v j )(5.19) h Vn ( τ ) = det ≤ i,j ≤ n − (cid:0) g ref20 V ( u, v ) | u i v j (cid:1) det ≤ i,j ≤ n − ( g V ( u, v ) | u i v j )(5.20) in terms of the functions: g ref6 V ( u, v ) = g V ( u, v ) + v n − (cid:18) u − u (cid:19) n ((cid:18) 21 + τ (cid:19) n − τ n (1 − u ) ( τ − u )(1 − τ u ) − ) (5.21) g ref20 V ( u, v ) = g V ( u, v ) + v n − (cid:18) u − u (cid:19) n (cid:26) τ n (1 − u ) ( τ − u )(1 − τ u ) − (cid:27) (5.22) with g V ( u, v ) as in (2.22) . In particular, using Theorem 2.7 to identify the denominator of (5.20) as Z Vn , thisimplies immediately the following: Theorem 5.4. The refined partition function generating function for the V model reads: Z Vn ( τ ) = det ≤ i,j ≤ n − (cid:0) g ref20 V ( u, v ) | u i v j (cid:1) with g ref20 V as in (5.22) . Remark 5.5. We may interpret again the result as: (5.23) Z Vn ( τ ) = det (cid:0) ( P ( n ) ( τ )) n (cid:1) WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 35 the determinant of a finite truncation of an infinite matrix P ( n ) ( τ ) , which differs fromthe matrix P of Remark 2.8 only in its n -th column, and is generated by f P ( n ) ( τ ) ( u, v ) = g ref20 V ( u, v ) (5.22) . Direct calculation using Theorem 5.4 gives the following first few values for n = 1 , ..., Z Vn ( τ ) for the 20V model on thequadrangle Q n : Z V ( τ ) = 1 Z V ( τ ) = 1 + 2 τ + τ Z V ( τ ) = 4 + 15 τ + 22 τ + 15 τ + 4 τ Z V ( τ ) = 60 + 328 τ + 772 τ + 1008 τ + 772 τ + 328 τ + 60 τ Z V ( τ ) = 3328 + 23868 τ + 76856 τ + 145860 τ + 179088 τ + 145860 τ +76856 τ + 23868 τ + 3328 τ Remark 5.6. Note that Z Vn ( τ ) is a palindromic polynomial of degree n − , namelythat Z Vn, n − k = Z Vn,k for all k = 1 , , ..., n − . Indeed, from (5.22) we see that theonly dependence on τ comes from the n -th column of P ( n ) ( τ ) , generated by ϕ n ( τ ) := (cid:0) u − u (cid:1) n τ n (1 − u ) ( τ − u )(1 − τu ) . The property follows from the fact that ϕ n ( τ − ) = τ − n − ϕ n ( τ ) .Observe also that (5.24) Z Vn (0) = Z Vn, = Z Vn, n − = Z Vn − Indeed, the contributions to Z Vn, n − all have a trivial topmost path made of n horizontalsteps in the top row, followed by n − down steps along the rightmost vertical until itsendpoint. This effectively cuts out the top row and right column from the domain Q n ,and the remaining paths form a configuration on Q n − . Similarly, the contributions to Z Vn, = Z Vn (0) all have the bottom-most diagonal entirely occupied by paths, while therightmost vertical is empty except for the bottom-most edge. This effectively cuts out thebottom diagonal and right vertical lines of vertices, leaving us with paths on Q n − as well. Finally, Theorem 5.3 gives the corresponding 6 V one-point functions: h V ( τ ) = 1 h V ( τ ) = 1 + τ h V ( τ ) = 4 + 7 τ + 4 τ h V ( τ ) = 15 + 37 τ + 37 τ + 15 τ h V ( τ ) = 64 + 203 τ + 282 τ + 203 τ + 64 τ y+x=4n−2 (2n−1,2n−1)(1,1) (3,3)(0,0)(0,2) k(0,2n−2) Figure 14. A typical configuration of non-intersecting Schr¨oder paths for thedomino tiling model of the Aztec triangle T n , contributing to the refined partitionfunction Z DTn,k , i.e. such that the first entry point of any path into the line x + y =4 n − k of the endpoint (2 n − , n − We note again that these polynomials are palindromic, a property inherited from the 20Vone-point functions via the relation (5.12). In particular, comparing the top and constantcoefficients we get a non-trivial relation: Z Vn, + Z Vn, = Z Vn, n − .6. Refined domino tilings of the Aztec triangle and connection to 20Vrefined partition functions Refined domino tiling partition functions. Consider again the path model forthe domino tilings of the Aztec triangle T n . In any non-intersecting path configuration,the path ending at the topmost point (2 n − , n − 1) must first hit the diagonal line x + y = 4 n − n − − k, n − k ) for some k ∈ [0 , n − Z DTn,k the corresponding contribution to the total partition function (see Fig. 14 for anillustration).Alternatively, let us denote by W DTn,k the partition function for non-intersecting Schr¨oderpaths with starting points (0 , i ), i = 0 , , ..., n − j + 1 , j + 1), j =0 , , ..., n − n − − k, n − k ). The condition of first hitting WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 37 the line y + x = 4 n − n − − k, n − k ) is equivalent to imposing that the path visitsthe point (2 n − − k, n − k ) but does not visit the point (2 n − − k − , n − k + 1).Note that once a point is visited on the line x + y = 4 n − 2, the rest of the path descendsalong this line until it reaches the endpoint (2 n − , n − k ∈ [0 , n − Z DTn,k = W DTn,k − W DTn,k +1 For convenience, we define the generating function: Z DT ( t ) := n − X k =0 Z DTn,k t k for refined domino tiling partition functions of the Aztec triangle T n . We have the following: Theorem 6.1. The generating function Z DT ( t ) reads: Z DT ( t ) = det ≤ i,j ≤ n − (cid:16) f DT ( t ; u, v ) (cid:12)(cid:12)(cid:12) u i v j (cid:17) (6.1) f ref DT ( u, v ) = f DT ( u, v ) + v n − α + ( u ) n √ u + u ( t − uα + ( u ) − tu (6.2) where α + ( u ) = (cid:0) u + √ u + u (cid:1) and f DT ( u, v ) as in (3.5) .Proof. Let us first compute W DTn,k . By direct application of the Lindstr¨om-Gessel-Viennottheorem [Lin73, GV85], we have W DTn,k = det( M ( k ) n ), where the n × n matrix M ( k ) n dif-fers from the matrix M n (of Lemma 3.1, Remark 3.2 and Theorem 3.3) only in its lastcolumn, where the entry M i,n − = Σ n − , n − − i (3.2) must be replaced by ( M ( k ) n ) i,n − =Σ n − − k, n − − i + k as the corresponding endpoint has been shifted by ( − k, k ). Recall thatΣ i,j = σ ( u, v ) | u i + j v i − j , with σ ( u, v ) as in (3.1). We deduce thatΣ n − − k, n − − i + k = I dx iπx dy iπy x n − − i y i − k − x − y − xy = I du iπu dv iπv v n − − k u i − k − v (1 + u ) − uv by changing again to variables u = y/x , v = x . Noting that:11 − v (1 + u ) − uv = 1 v √ u + u (cid:18) − vα + ( u ) − − vα − ( u ) (cid:19) where α ± ( u ) = 12 (cid:0) u ± √ u + u (cid:1) we get 11 − v (1 + u ) − uv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v n − − k = α + ( u ) n − k − α − ( u ) n − k √ u + u Note that α − ( u ) = − u + O ( u ) hence the term α − ( u ) n − k = O ( u n − k ) does not contributeto the coefficient of u i − k . The generating function for the n -th column of M ( k ) n reads: v n − n − X i =0 Σ n − − k, n − − i + k u i = v n − u k α + ( u ) n − k √ u + u + O ( u n )The matrix M ( k ) n is the finite truncation to the first n rows and columns of the infinitematrix M ( n,k ) , with generating function f M ( n,k ) ( u, v ) = f M ( u, v ) + v n − u k α + ( u ) n − k − α + ( u ) n √ u + u Using the linearity of the determinant w.r.t. its last column, we immediately deducethat Z DTn,k = W DTn,k − W DTn,k +1 = det( ˜ M ( k ) n ) where the n × n matrix ˜ M ( k ) n := ( ˜ M ( n,k ) ) n is thefinite truncation of the infinite matrix ˜ M ( n,k ) with generating function: f ˜ M ( n,k ) ( u, v ) = f M ( u, v ) + v n − u k α + ( u ) n − k − u k +1 α + ( u ) n − k − − α + ( u ) n √ u + u Similarly, the generating function Z DT ( t ) = det (cid:16) ( ˜ M ( n ) ( t )) n (cid:17) where the n × n matrix( ˜ M ( n ) ( t )) n is the finite truncation of the infinite matrix ˜ M ( n ) ( t ) generated by: f ˜ M ( n ) ( t ) ( u, v ) = f M ( u, v ) + v n − P k ≥ t k ( u k α + ( u ) n − k − u k +1 α + ( u ) n − k − ) − α + ( u ) n √ u + u = f M ( u, v ) + v n − α + ( u ) n √ u + u ( t − uα + ( u ) − tu = f ref DT ( u, v )(Here we have considered the sum over all k ≥ 0, as the terms of order u n and above donot affect the truncation.) This completes the proof of Theorem 6.1. (cid:3) Direct calculation using Theorem 6.1 leads to the following first few values of the refinedpartition function generating functions Z DTn ( t ), up to n = 5: Z DT ( t ) = 1 Z DT ( t ) = 3 + tZ DT ( t ) = 37 + 19 t + 4 t Z DT ( t ) = 1780 + 1100 t + 388 t + 60 t Z DT ( t ) = 324948 + 222716 t + 100724 t + 27196 t + 3328 t WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 39 We note the obvious sum rule Z DTn (1) = Z DTn . Observe also that the leading coefficient of t n − in Z DTn ( t ) is nothing but(6.3) Z DTn − , = Z DTn − Indeed, imposing that the topmost path (from (0 , n − 2) to (2 n − , n − x + y = 4 n − n up steps, followed by n − T n to T n − for the n − Relation between the refined tiling and refined 20V partition functions.Theorem 6.2. We have the following relation between refined partition functions of the V model on the quadrangle Q n and of the domino tilings of the Aztec triangle T n : (6.4) Z Vn ( τ ) = τ n Z DT ( τ ) + τ − Z DT ( τ − )1 + τ Proof. We apply the same technique as in the case of ordinary partition functions. Let L be the infinite matrix (4.1) of Sect. 4.2 and ˜ M ( n ) ( t ) generated by f ˜ M ( n ) ( t ) ( u, v ) = f ref DT ( u, v )of Theorem 6.1. We compute: f L ˜ M ( n ) ( t ) ( u, v ) = 1 + 2 u − u − u f ˜ M ( n ) ( t ) (cid:18) u u − u , v (cid:19) = f P ( u, v ) + v n − (cid:18) u − u (cid:19) n ( t − u − tu = g V ( u, v ) + v n − (cid:18) u − u (cid:19) n (cid:26) − u − tu − (cid:27) (6.5)by use of the result LM = P of Section 4.2. Let us now denote by C ( t ) the n -th col-umn of the matrix L ˜ M ( n ) ( t ). We have Z DT ( t ) = det (cid:16) ( L ˜ M ( n ) ( t )) n (cid:17) . By linearity of thedeterminant w.r.t. the n -th column, the quantity t n t (cid:0) Z DT ( t ) + t − Z DT ( t − ) (cid:1) is the de-terminant of the finite truncation of an infinite matrix ¯ M ( n ) ( t ) defined as follows: it isidentical to L ˜ M ( n ) ( t ) except for the n -th column, where C ( t ) is replaced with the combi-nation t n t { C ( t ) + t − C ( t − ) } . Consequently: t n Z DT ( t ) + t − Z DT ( t − ) t + 1 = det (cid:0) ( ¯ M ( n ) ( t )) n (cid:1) f ¯ M ( n ) ( t ) ( u, v ) = g V ( u, v ) + v n − (cid:18) u − u (cid:19) n ( t n − u − tu + − ut − u t − ) = g ref20 V ( u, v )where we have recognized the generating function g ref20 V ( u, v ) (at τ = t ) of Theorem 5.3.The theorem follows by setting t = τ , and using the identification ¯ M ( n ) ( τ ) = P ( n ) ( τ ) givenby (5.23). (cid:3) (b)(a) (d)(c) Figure 15. The proof of the 20V-DT conjecture for k = 2. The 20V partitionfunction on Q n is decomposed according to the two possible configurations of thebottom vertex (a) and (b). The domino tiling partition function on T n is similarlydecomposed according to whether the topmost vertex is visited (c) or not (d). Remark 6.3. Note that the result above gives an independent confirmation that the poly-nomial Z Vn ( τ ) is palindromic, namely that τ n − Z Vn ( τ − ) = Z Vn ( τ ) as noted in Remark5.6. Finally Eqn.(6.4) implies the following relations between the refined partition functionsof the 20 V model and the domino tiling problem. Corollary 6.4. The refined partition functions for the V model and for the dominotilings of the Aztec triangle are related as follows: (6.6) Z DTn,k = Z Vn,n + k +1 + Z Vn,n + k = Z Vn,n − k − + Z Vn,n − k for k = 0 , , ..., n − with the convention that Z Vn, = Z Vn, n = 0 . Pentagonal 20V and truncated Aztec triangle domino tilings In this section, we prove the correspondence (20V DWBC3-DT conjecture of [DFG20])between 20V configurations on the domains P n,n − k (see Fig. 4) and the domino tilingconfigurations of the domains T n,n − k for k = 2 and 3. The method of proof is purelycombinatorial and does not rely on integrability, except for its use of the k = 1 results.7.1. Proof of the 20V-DT conjecture for k = 2 .Theorem 7.1. The number of configurations of the 20V DWBC3 model on the pentagon P n,n − is identical to the number of domino tilings of the truncated Aztec triangle T n,n − , WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 41 namely: Z V ( P n,n − ) = Z DT ( T n,n − ) Proof. The correspondence is proved by expressing both relevant partition function in termsof known objects. For the 20V model on P n,n − , let us start from the partition function Z Vn on the larger domain Q n = P n,n − . This quantity splits into two parts, accordingto the two possible local configuration of the bottom-most vertex displayed in Figs.15 (a)and (b) (circled vertex). In the case (a), the configuration of the bottom vertex inducesa chain of forced edge occupations, due to the SW boundary condition. The contributionof case (a) is readily seen to reduce to Z Vn, = Z Vn − . The case (b) contributes exactly Z V ( P n,n − ). We deduce the relation: Z V ( P n,n − ) = Z Vn − Z Vn − For the domino tiling of T n,n − let us also start from the partition function Z DTn on thelarger domain T n = T n,n − in the lattice path formulation. Analogously, this quantity splitsinto two parts according to whether the topmost vertex is visited or not, as displayed inFigs.15 (c) and (d). The case (c) corresponds to a unique configuration of the topmost path,and the contribution reduces to Z DTn − . The case (d) contributes precisely Z DT ( T n,n − ). Thisgives the relation Z DT ( T n,n − ) = Z DTn − Z DTn − The theorem follows from the result of Theorem 4.3, i.e. the identities Z Vm = Z DTm for all m . (cid:3) Proof of the 20V-DT conjecture for k = 3 .Theorem 7.2. The number of configurations of the 20V DWBC3 model on the pentagon P n,n − is identical to the number of domino tilings of the truncated Aztec triangle T n,n − ,namely: Z V ( P n,n − ) = Z DT ( T n,n − ) Proof. We proceed like in the case k = 2. For the 20V model on P n,n − , let us start from thepartition function Z V ( P n,n − ). It splits into four parts according to the local configurationof the two bottom vertices, as shown in Figs.16 (a), (c), (d) and (e).The first contribution (a) can be further split according to the configuration of therightmost column: there are two ways in which the path can leave the second rightmostvertical line at an exit point say at height k (by a horizontal or diagonal step, followed byvertical steps until the endpoint). Removing the last column, and inserting vertical stepsup to the point of exit at height k into the second vertical, we get a 2 to 1 mapping torefined configurations of the 20V on Q n − that reach the last column at or above height k ,resulting in a contribution P j ≥ k Z Vn − ,j for each k , hence a total of 2 P n − k =1 P j ≥ k Z Vn − ,j = P n − k =1 k Z Vn − ,k . (d) (e)(b)(a) (c) 2 1k Figure 16. Decomposition of Z V ( P n,n − ) into four parts (a), (c),(d), (e), ac-cording to the bottom vertex configurations (circled). Cases (a) are further de-composed according to the point of exit of the rightmost path at height k (emptycircle) from the second rightmost vertical line. There are 2 distinct configurationsof the rightmost path after this exit point (paths using edges among the dashedones), in 2-1 correspondence with the partition function (b) on Q n − , equal to thesum of the refined partition function P j ≥ k Z Vn,j . Cases (c) and (d) sum up to Z Vn, − Z Vn, , while (e) contributes Z V ( P n,n − ). (c)(b)(a) n−2n−1 Figure 17. Decomposition of Z DT ( P n,n − ) into three pieces (a), (b), (c) accord-ing to the configurations of the two top vertices (empty circles). Case (a) givesrise to 2( n − 1) times the domino problem on T n − , as there are 2( n − 1) choicesfor how the top path ends, using only edges among the dashed ones. Case (b)receives contributions from the refined partition function where paths first enterthe x + y = 4 n − n − Z DT ( T n,n − ). WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 43 The case (c) corresponds to the contributions to the refined partition function Z V \ n, such that a path goes vertically through the left vertex. We must therefore subtract from Z V \ n, the contributions with the two rightmost paths ending diagonally before their lastvertical step, in bijection with 20V configurations on Q n − . This gives a net contribution of Z V \ n, − Z Vn − . The case (d) contributes Z V − n, . Finally the case (e) contributes Z V ( P n,n − ).Collecting all the terms and using Z V \ n, + Z V − n, = Z Vn, , we get: Z V ( P n,n − ) = n − X k =1 k Z Vn − ,k + Z Vn, − Z Vn − + Z V ( P n,n − )Using Remarks 5.6 and 6.3, which imply the symmetry Z Vn − , n − − k = Z Vn − ,k , we furthernote that n − X k =1 k Z Vn − ,k = 12 n − X k =1 k ( Z Vn − ,k + Z Vn − , n − − k ) = 2( n − n − X k =1 Z Vn − ,k = 2( n − Z Vn − leading to the identity:(7.1) Z V ( P n,n − ) = Z Vn − Z Vn, − n − Z Vn − For the domino tiling model, we start from Z DT ( T n,n − ). It splits again according to theconfigurations of it two topmost vertices, as shown in Figs.17 (a) and (b). The case (a)corresponds to a path visiting the leftmost vertex, which must be made of n − n − , n − n − 1) possibilities, while the rest of the configuration is arbitrary on Q n − .The corresponding contribution is 2( n − Z DTn − . In the case (b) a path visits the rightvertex, while the left vertex is empty. The corresponding configurations are those pathscontributing to the refined partition function Z DTn,n − that do not pass through the leftvertex. We must therefore subtract from Z DTn,n − the two contributions of the topmost paththat visit both vertices in T n,n − (with either a horizontal step between the two vertices,or a down followed by an up step). In both cases, the rest of the paths correspond to anarbitrary tiling of T n − . The total contribution is therefore Z DTn,n − − Z DTn,n − . The case (c)with the two empty vertices contributes Z DT ( T n,n − ). Assembling all the terms, we get Z DT ( T n,n − ) = 2( n − Z DTn − + Z DTn,n − − Z DTn,n − + Z DT ( T n,n − )We deduce that(7.2) Z DT ( T n,n − ) = Z DTn − Z DTn,n − − (2 n − Z DTn − Finally we use eq.(6.6) of Corollary 6.4 for k = n − Z DTn,n − = Z Vn, + Z Vn, .Together with the identities Z Vm = Z DTm of Theorem 4.3 and the fact that Z Vn, = Z Vn − (5.24), this implies the equality between the r.h.s. of (7.1) and (7.2), and the theoremfollows. (cid:3) Discussion and Conclusion In this paper we have investigated the 20V model with DWBC3 boundary conditions onthe family of pentagons P n,n − k and their relation to the domino tilings of the family of Aztectriangles T n,n − k , and proved the conjectured identity between their partition functions forthe cases k = 1, 2, 3. In the case k = 1 of the quadrangle Q n = P n,n − we have extendedthe result to refined partition functions. Our proofs for this case have relied mainly onthe choice of integrable weights for the 20V model, allowing for transforming the originalproblem into a 6V model with U-turn boundaries. The further truncations of the domain Q n for k = 2 , k = 1 case and its refinements.We now discuss a conjecture for the exact number Z Vn = Z DTn and alternative formulasfor these quantities. We also show how to introduce some extra weight in the domino tilingproblems, before making some concluding remarks.8.1. An exact (conjectured) formula. Based on numerical data, we were able to for-mulate the following conjecture about the total number of configurations of the 20V modelon the quadrangle Q n : Conjecture 8.1. The total numbers Z Vn = Z DTn of configurations of the 20V model onthe quadrangle Q n and of domino tilings of the Aztec triangle T n read: (8.1) Z Vn = Z DTn = 2 n ( n − / n − Y i =0 (4 i + 2)!( n + 2 i + 1)!This formula was checked up to n = 30. It is reminiscent of the famous formula for thenumber A n of n × n Alternating Sign Matrices (ASM), which reads A n = Q n − i =0 (3 i +1)!( n + i )! orthe related formula for the dimension δ n = 3 n ( n − / A n of the irreducible representation of GL n indexed by the Young diagram Y n = ( n − , n − , n − , n − , ..., , , , z i = q s i , w j = q − s j + n for some parameter s to be sent to 1eventually.Another approach would try to reproduce the connection between the 6V model withDWBC conditions and the GL n character with partition Y n or that between U-turn 6Vmodel and characters of SP n [Str06, RS04b]. Looking for candidates, and denoting by B n the r.h.s. of (8.1), we have found that the irreducible representation of SP n withdiagram Y ′ n = ( n − , n − , n − , n − , n − , n − , ..., , , , , , 0) has the dimension All these results are product formulas obtained by specializing the corresponding characters, given byvarious determinant formulas [Miz03], when all arguments are 1. WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 45 δ ′ n = B n / n . Analogously, we found that the irreducible representation of SO n +1 withdiagram Y ′′ n = ( n − , n − , n − , ..., , , , , , ) has dimension δ ′′ n = 2 n B n . Finally, wefound that the irreducible representation of SO n − with diagram Y ′′′ n = ( n − , n − , n − , n − , n − , n − , ..., , , , , 0) has dimension δ ′′′ n = 2 n − B n − . Recalling that the in 6Vcase [Str06, RS04b] the argument involved some non-trivial Z symmetry of the partitionfunction at the combinatorial point, we expect a Z symmetry to play an analogous rolehere. We intend to return to this question in a future publication.8.2. Alternative expressions for Z Vn . The results of this paper allow us to deriveseveral alternative determinant formulas for the number Z Vn . Using Theorem 2.7, we mayexpress Z Vn as follows. Theorem 8.2. The number Z Vn of configurations of the V model on the quadrangle Q n reads: (8.2) Z Vn = det ≤ i ≤ j ≤ n − (cid:18) i (cid:18) i + 2 j + 12 j + 1 (cid:19) − (cid:18) i − j + 1 (cid:19)(cid:19) with the convention (cid:0) mp (cid:1) = 0 for all − ≤ m < p .Proof. We start from the determinant formula Z Vn = det ≤ i,j ≤ n − ( f P ( u, v ) | u i v j ) = det( P n )of Theorem 2.7, where: f P ( u, v ) = g V ( u, v ) = (cid:0) u − u (cid:1) − v (cid:0) u − u (cid:1) − u − v u and P n is the finite truncation of the infinite matrix P generated by f P . Using the sametechnique as in Sections 4.1-4.2, let us consider the infinite matrix Λ with generating func-tion: f Λ ( u, v ) = 1 − u − u − vu Then (1) Λ is lower triangular and (2) the diagonal matrix elements of Λ are all 1. Wededuce the truncation (Λ P ) n = Λ n P n , and therefore det((Λ P ) n ) = det( P n ). Explicitlycomputing: f Λ P ( u, v ) = ( f Λ ∗ f P )( u, v ) = f P (cid:18) u − u , v (cid:19) = (cid:0) − u (cid:1) − v (cid:0) − u (cid:1) − (cid:0) u − u (cid:1) − v (cid:0) u − u (cid:1) we identify the entries(Λ P ) i,j = ((cid:18) − u (cid:19) j +2 − (cid:18) u − u (cid:19) j +2 ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u i = 2 i (cid:18) i + 2 j + 12 j + 1 (cid:19) − (cid:18) i − j + 1 (cid:19) and the theorem follows. (cid:3) A slight variant of the formula consists of using the analytic continuation of binomialcoefficients. Corollary 8.3. The number Z Vn of configurations of the V model on the quadrangle Q n reads: (8.3) Z Vn = 2 n ( n − / ≤ i,j ≤ n − (cid:16) θ j +1 ( i ) + θ j +1 ( − i ) (cid:17) , θ m ( x ) = 2 x/ m ! ( x + 1)( x + 2) ... ( x + m )This latter, more symmetric form allows to derive the following constant term identity. Theorem 8.4. The number Z Vn of configurations of the V model on the quadrangle Q n reads: Z Vn = CT x ,...,x n (cid:26) Q ≤ i We also need the antisymmetrization mapASym : f ( x ) ASym( f )( x ) := 1 n ! X σ ∈ S n sgn( σ ) f ( x σ (1) , ..., x σ ( n ) )with the main propertySym Y ≤ i The constant term identity of Theorem 8.4 is reminiscent of constant term expressionsfor TSSCPP derived by Zeilberger [Zei96] in his famous proof of the ASM conjecture.8.3. Introducing step weights in the domino tiling problem. In this paper we havefound compact determinant formulas for both the number of tilings of the Aztec triangle T n and its refinements, involving the truncation of infinite matrices generated respectively bythe series f DT (3.5) and f ref DT (6.2). We note that a very simple decoration of the generatingfunction f DT ( u, v ) allows to include a non-trivial extra step weight for the non-intersectingSchr¨oder paths. In general, one would want to associate weights α, β, γ respectively to up,down and horizontal steps. However, by simple rescaling of u, v we may restrict withoutloss of generality to only a weight γ per horizontal step. Theorem 8.5. The partition function for domino tilings of the Aztec triangle of order n ,with an extra weight γ per horizontal step in the non-intersecting Schr¨oder path formulationreads: Z DT,γn = det ≤ i,j ≤ n − (cid:16) f DT,γ ( u, v ) (cid:12)(cid:12)(cid:12) u i v j (cid:17) (8.5) f DT,γ ( u, v ) = 1 + u − v − γ ) uv − u v + γ u v (8.6) Proof. We adapt the proof of Theorem 3.3. By the Lindstr¨om-Gessel-Viennot theorem, thepartition function function of the non-intersecting Schr¨oder paths with the new weights isgiven by Z DT,γn = det(( M γ ) n ), where the subscript n indicates the truncation to the first n rows and columns of the infinite matrix M γ whose entries ( M γ ) i,j are the partition functionsfor a single path form (0 , i ) to (2 j + 1 , j + 1), counted with a multiplicative weight γ perhorizontal step. We simply have to show that these entries are generated by: f M γ ( u, v ) = 1 + u − v − γ ) uv − u v + γ u v = f DT,γ ( u, v )To show this, we start with the function σ γ ( u, v ) := − u − v − γuv the partition function forarbitrary finite length Schr¨oder paths from the origin, with weight u per up step, v per downstep, and γuv per horizontal step. Accordingly the partition function (Σ γ ) i,j of paths fromthe origin with fixed endpoint ( i, j ), with i − j even, is given by (Σ γ ) i,j = σ γ ( u, v ) | u i + j v i − j ,or equivalently by a straightforward generalization of (3.2). Noting again that ( M γ ) i,j =(Σ γ ) j +1 , j +1 − i , we get( M γ ) i,j = I dx iπx dy iπy x j +1 − i y i − x − y − γxy = I du iπu dv iπv v j +1 u i − v − uv − γuv after again changing integration variables to u = y/x, v = x . Using the same trick (3.3) asin previous section, we conclude that: f M γ ( u, v ) = − (1+ u ) √ v − γuv − u ) √ v − γuv √ v = 1 + u (1 − γuv ) − v (1 + u ) and the Theorem follows. (cid:3) The result above is easily extended to the refined domino tiling of Section. 6. Thepartition function Z DT,γn ( t ) := P nk =1 Z DT,γn,k t k − now including a multiplicative weight γ per horizontal step in the non-intersecting Schr¨oder path formulation of the refined tilingproblem, can be directly calculated as follows (we leave the proof to the reader as a straight-forward exercise). WENTY VERTEX MODEL AND DOMINO TILINGS OF THE AZTEC TRIANGLE 49 Theorem 8.6. The generating function Z DT ( t, γ ) reads: Z DT ( t, γ ) = det ≤ i,j ≤ n − (cid:16) f ref DT,γ ( u, v ) (cid:12)(cid:12)(cid:12) u i v j (cid:17) f ref DT,γ ( u, v ) = f DT,γ ( u, v ) + v n − α + ( γ ; u ) n p γ ) u + u ( t − uα + ( γ ; u ) − tu where α + ( γ, u ) = (cid:0) u + p γ ) u + u (cid:1) and f DT,γ ( u, v ) as in (8.6) . It is interesting to notice that the upper triangular matrix L involved in the proofs ofTheorems 4.3 and 6.2 can be modified to obtain an alternative expression for the partitionfunction generating function Z DT,γ ( t ) above. Indeed, let L γ be the infinite matrix generatedby: f L γ ( u, v ) := 1 + 2 γu − γu − u − vu (1 + γu )Writing Z DT ( t, γ ) = det (cid:0) ( L γ ˜ M ( n ) γ ( t )) n (cid:1) , where ˜ M ( n ) γ ( t ) is generated by f ref DT,γ ( u, v ) above,we finally get: Theorem 8.7. The generating function Z DT ( t, γ ) reads: Z DT ( t, γ ) = det (cid:16) ¯ f ref DT,γ ( u, v ) (cid:12)(cid:12) u i v j (cid:17) ¯ f ref DT,γ ( u, v ) = (1 + γu )(1 + 2 γu − γu )(1 − γ u v )((1 − u ) − v (1 + γu ) ) + v n − (cid:18) γu − u (cid:19) n ( t − u − tu Proof. We use the infinite matrix ¯ M ( n ) γ ( t ) := L γ ˜ M ( n ) γ ( t ), whose generating function is easilycomputed via the formula f L γ ˜ M ( n ) γ ( t ) ( u, v ) = γu − γu − u f ref DT,γ (cid:0) u γu − u , v (cid:1) = ¯ f ref DT,γ ( u, v ). (cid:3) Note also that (1 + γu )(1 + 2 γu − γu )(1 − γ u v )((1 − u ) − v (1 + γu ) ) = (cid:0) γu − u (cid:1) − v (cid:0) γu − u (cid:1) − γ u − vγ u This γ -deformation of (2.22) is very suggestive, however we have not been able to find acounterpart of the parameter γ in the 20V model.8.4. Conclusion. Despite the great progress on proving the conjectures of Ref. [DFG20],we are still only understanding part of the sequence of correspondences between the 20Vmodel on the domains P n,k and the domino tiling of the domains T n,k . One way to gain abetter global understanding of these would be to include deformation (spectral?) parame-ters in the correspondence of partition functions on Q n and T n respectively. However, thenumber of deformation parameters at hand is limited. 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