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Weighted Aztec diamond graphs revisited

TRI LAI

Indiana UniversityDepartment of Mathematics

Bloomington, IN 47405, USA

February 28, 2018

Abstract

Kamioka (Journal of Combinatorial Theory, Series A, 2014 ) considereda certain weighted Aztec diamond graph when presenting a new proof forAztec diamond theorem by Elkies, Kuperberg, Larsen and Propp. In thispaper, we generalize Kamioka’s result by investigating several new weightedAztec diamonds. We also prove a generalization for a result due to Mills,Robins, Rumsey on holey Aztec rectangle. The result implies a q-analog ofa MacMahon’s theorem on rhombus tilings. In addition, we prove a simpleproduct formula for the number of perfect matching for a certain family ofweighted graphs obtained by connecting two Aztec rectangles.

Keywords: perfect matchings, tilings, dual graphs, Aztec diamonds, Aztecrectangles.

1 Introduction

Given a lattice in the plane, a (lattice) region is a finite connected union of funda-mental regions of that lattice. A tile is the union of two fundamental regions sharingan edge. A tiling of the region R is a covering of R by tiles so that there are nogaps or overlaps.

A perfect matching of a graph G is a collection of edges such that each vertex ofG is adjacent to precisely one edge in the collection.

The Aztec diamond region of order n is defined to be the union of all the unitsquares with inside the contour |x|+ |y| = n+1 in the Cartesian coordinate system(see Figure 1.1 for an example of Aztec diamond region of order 4). The tiles of anAztec diamond region are dominoes, and a (domino) tiling of the region here is acovering by dominoes. For each tilings T of ADn, we denote v(T ) by haft numberof vertical dominoes, and r(T ) the rank of T which will be defined as follows. Thetiling T0 consisting of all horizontal dominoes has rank 0; and the rank r(T ) of T is

1

2

Figure 1.1: The Aztec diamond region of order 4 (left) and its dual graph, the Aztecdiamond graph of order 4 (right).

Figure 1.2: The elementary moves: rotation of a two-by-two block of two vertical orhorizontal dominoes.

the minimal number of elementary moves required to reach T from T0 (see Figure1.2 for two types of the elementary moves).

Define ADn(t, q) =∑

T tv(T )qr(T ), where the sum is taken over all tilings of theAztec diamond region of order n. Elkies, Kuperberg, Larsen and Propp [6] provethat

Theorem 1.1 (Aztec diamond theorem [6]).

ADn(t, q) =n−1∏

k=0

(1 + tq2k+1)n−k. (1.1)

Especially, when t = q = 1, we imply the number tilings of ADn is 2n(n+1)/2.Further proof of the Aztec diamond and its special cases are given by several authors([10],[7],[19],[2],[11]).

Tiles (resp., edges) of a region (resp., a graph) can carry weights, and we definethe weight of a tiling (resp., a perfect matching) to be the product of weights ofits constituent tiles (resp., edges). The tiling generating function (resp., matchinggenerating function) of a region (resp., a graph) is defined to be the sum of weightsof all its tilings (resp., perfect matching).

Denote by M(G) the matching generating function of weighted graph G. Thetilings of a region R can be naturally identified with the perfect matchings of itsdual graph (i.e., the graph whose vertices are the fundamental regions of R, andwhose edges connect two fundamental regions precisely when they share an edge;each edge of the dual graph of the region is assigned the weight of its correspondingtile). In the view of this, we denote by M(R) the tiling generating function of aweighted region R. We call the dual graph of an Aztec diamond region an Aztecdiamond graph, denoted by ADn.

Color the square in an Aztec diamond regions by black and white, so that theretwo neighbor square have opposite colors, and the square along the northwest side arewhite. We have four types of dominoes to distinguish: odd vertical, even vertical,

3

odd vertical even vertical odd horizontal even horizontal

Figure 1.3: Four types of dominoes.

odd horizontal, and even horizontal dominoes (see figure 1.3). Assign each evenhorizontal and odd vertical domino a weight 1, each odd horizontal domino on levelk (from the bottom of the Aztec diamond region; the bottom is at level 0) a weighttq2k, and each even vertical domino on k a weight q2k+1. Kamioka [11] showed that,the tiling generating function of the weighted Aztec diamond region of order n isgiven by the following simple product

q2(n−1)n(n+1)

3

n∏

k=1

(t+ q2k−1)n−k+1. (1.2)

Weighted Aztec diamond graphs has been investigated by several authors (e.g.,see [1], [3], [14], [21]). However, we only considered the weighted Aztec diamondgraphs whose weight assignment is periodic in the following sense: there exists asimple the weight pattern A so that the weight assignment of the Aztec diamondgraphs is obtained by translating the pattern A southwest, northeast, southeast,or northwest. One readily sees that the weighted Aztec diamond graph yielded byKamioka’s weight assignment is not of the above type.

The goals of this paper are to generalize the above weight assignment on Aztec di-amond regions/graphs and investigate several new families of weighted regions/graphs.

This paper is organized as follows. In Section 2, we investigate several gener-alizations of Kamioka’s weight assignment. In particular, we prove a q-analogue ofa Stanley’s result on weighted Aztec diamond graph (see[21] and [3]) by using acertain reduction theorem due to Propp [19].

Section 3 concerns a certain family of weighted holey Aztec rectangle graphs. Inparticular, we prove a generalization of a related result due to Mills, Robins andRumsey (Theorem 2 in [18]). The result yields a q-analog of MacMahon’s theoremon rhombus tilings. In Section 4, we investigate a counterpart of Aztec diamondtheorem on Aztec rectangle regions with defects. Finally, we use a result in Section3 to prove a simple product formula for the matching generating function of a newfamily of weighted graphs.

2 Weighted Aztec diamond graphs

The goal of this section is to consider several generalizations of Kamioka’s weightassignment mentioned in Section 1.

Assume that a, b, c, d, q are five positive real numbers. We consider a weightassignment to the dominoes of an Aztec diamond region inspired by Kamioka’sassignment as follows. Assign each even horizontal domino a weight b, each oddvertical domino a weight a, each odd horizontal domino on level k a weight cqk,and each even vertical domino on k a weight dqk. This yields a weighted Aztec

4

diamond graph, and denote by wta,bc,d(q) the weight assignment to the graph. Wedenote ADn(wt) by the Aztec diamond graph of order n with edges weighted by aweight assignment wt.

Theorem 2.1. Assume that a, b, c, d, q are positive real numbers. For any positiveintegers n

M(ADn

(wta,bcd (q)

))= q

(n−1)n(n+1)3

n−1∏

k=0

(adqk + bc

)n−k+1. (2.1)

We notice that by setting a = b = 1, c = t, d = q, and replacing q by q2, we getKamioka’s result from Theorem 2.1.

Besides Aztec diamond graphs, we are also interesting on following three familiesof graphs. Consider a (2m+1)× (2n+1) rectangular chessboard B and suppose thecorners are black. The Aztec rectangle graph ARm,n is the graph whose vertices arethe white squares and whose edges connect precisely those pairs of white squaresthat are diagonally adjacent (see graph on the left of Figure 2.6). Repeat the aboveprocess with the chess board B for black squares, we get a new graph called oddAztec rectangle graph, denoted by ORm,n (see graph on the right of Figure 2.3).If one removes all bottommost vertices of ARm,n, the resulting graph is denotedby ARm− 1

2,n, and called a baseless Aztec rectangle graph (see graph on the right of

Figure 2.5). One can view the Aztec diamond graph of order n as the special Aztecrectangle graph ARm,n when m = n.

If the centers of edges of a Aztec rectangle graph of order m×n form a 2m×2n-array. The entries of the array are the weights of these edges. We call the arraythe weight matrix of the Aztec rectangle. We denote by wtA the weight assignmentdetermined by the weight matrix A.

Consider a family of 2m× 2n matrices

Am,n =

a b a b . . . a b

cqm−1 dqm−1 cqm dqm . . . cqm+n−2 dqm+n−2

a b a b . . . a b

cqm−2 dqm−2 cqm−1 dqm−1 . . . cqm+n−3 dqm+n−3

......

......

. . ....

...a b a b . . . a b

c d cq dq . . . cqn−1 dqn−1

. (2.2)

One readily see that we weighted Aztec diamond graph ADn

(wta,bc,d(q)

)has weight

matrix An,n (i.e. wta,bc,d(q) ≡ wtAn,n). We denote ARm,n

(wta,bc,d(q)

)by the weighted

Aztec rectangle ARm,n with the weight matrix Am,n (see the graph on the left of

Figure 2.5 for AR3,4

(wta,bc,d(q)

)).

We present several preliminary results about subgraph replacements in the nextthree lemmas, that we will employ in the proof of Theorem 2.1.

5

v v′

x

v′′

H K H K

Figure 2.1: Vertex splitting.

A

B

C

D

A

B

C

D

x y

zty/∆ x/∆

t/∆z/∆

∆ = xz + yt

Figure 2.2: Urban renewal.

Lemma 2.2 (Vertex-Splitting Lemma). Let G be a graph, v be a vertex of it, anddenote the set of neighbors of v by N(v). For any disjoint union N(v) = H ∪ K,let G′ be the graph obtained from G \ v by including three new vertices v′, v′′ and xso that N(v′) = H ∪ {x}, N(v′′) = K ∪ {x}, and N(x) = {v′, v′′} (see Figure 2.1).Then M(G) = M(G′).

Lemma 2.3 (Star Lemma). Let G be a weighted graph, and let v be a vertex of G.Let G′ be the graph obtained from G by multiplying the weights of all edges incidentto v by t > 0. Then M(G′) = tM(G).

The following result is a generalization (due to Propp) of the “urban renewal”trick first observed by Kuperberg.

Lemma 2.4 (Spider Lemma). Let G be a weighted graph containing the subgraphK shown on the left in Figure 2.2 (the labels indicate weights, unlabeled edges haveweight 1). Suppose in addition that the four inner black vertices in the subgraph K,different from A,B,C,D, have no neighbors outside K. Let G′ be the graph obtainedfrom G by replacing K by the graph K shown on right in Figure 2.2, where the dashedlines indicate new edges, weighted as shown. Then M(G) = (xz + yt)M(G′).

A forced edge of a graph is an edge contained in every perfect matching of G.Assume that G is a weighted graph with weight function wt on its edges, and G′

is obtained from G by removing forced edges e1, . . . , ek, and removing the verticesincident to those edges. Then one clearly has

M(G) = M(G′)

k∏

i=1

wt(ei).

6

c d

a b a b a b a b

cq dq cq2 dq2 cq3 dq3

a b a b a b a b

a b a b a b a b

cq dq cq2 dq2 cq3 dq3

cq2 dq2 cq3 dq3 cq4 dq4 cq5 dq5

cq4 dq4

aq b aq b aq b

aq b aq b aq b

cq dq cq2 dq2 cq3 dq3

aq b aq b aq b

cq2 dq2 cq3 dq3 cq4 dq4

cq3 dq3 cq4 dq4 cq5 dq5

Figure 2.3: Illustrating the transformation in Lemma 2.5. The white circles indicatethe vertices v1, v2, . . . , vn, v

′1, v

′2, . . . , v

′n.

Hereafter, whenever we remove some forced edges, we remove also the vertices inci-dent to them.

Denote by ARm− 12,n

(wta,bc,d(q)

)the weighted baseless Aztec rectangle graph ob-

tained from ARm,n

(wta,bc,d(q)

)by removing all its bottommost vertices (see the graph

on the right of Figure 2.5 for AR3− 12,3

(wtaq,bc,d (q)

)); and denote by ORm,n

(wta,bc,d(q)

)

the weighted odd Aztec rectangle graph obtained from ARm,n−1

(wta,bc,d(q)

)by re-

moving all its bottommost and topmost vertices (see the graph on the right of Figure

2.6 for OR4,4

(wtaq,bc,d (q)

)).

The connected sum G#G′ of two disjoint graphs G and G′ along the orderedsets of vertices {v1, . . . , vn} ⊂ V (G) and {v′1, . . . , v

′n} ⊂ V (G′) is the graph obtained

from G and G′ by identifying vertices vi and v′i, for i = 1, . . . , n.

Lemma 2.5. Let G be a graph and let {v1, . . . , vn, vn+1, . . . , v2n} be an ordered subsetof its vertices. Then

M(||ARm,n

(wta,bc,d(q)

)#G

)= (ad+ bc)mq

m(m−1)2 M

(ORm,n

(wtaq,bc,d (q)

)#G

),

(2.3)

where||ARm,n

(wta,bc,d(q)

)is obtained from ARm,n

(wta,bc,d(q)

)by appending vertical

edges to it bottommost and topmost vertices; and where the connected sum acts on

G along {v1, v2 . . . , v2n}, and on||ARm,n

(wta,bc,d(q)

)and ORm,n

(wtaq,bc,d (q)

)along

their n topmost vertices, and then their n bottommost vertices (ordered from left toright).

Proof. The proof is based on Figure 2.4, for m = 3 and n = 4. First, we apply

the Vertex-Splitting Lemma to vertices of||ARm,n

(wta,bc,d(q)

)in the graph on the

7

(a) (b)

(c)(d)

d∆

c∆

d∆

c∆

d∆

aq2∆

bq3∆

aq3∆

bq4∆

aq4∆

bq5∆

c∆

d∆

c∆

d∆

c∆

d∆

aq∆

bq2∆

aq2∆

bq3∆

aq3∆

bq4∆

c∆

d∆

c∆

d∆

c∆

d∆

a∆

bq∆

aq∆

bq2∆

aq2∆

bq3∆

aq b aq b aq b

aq b aq b aq b

cq dq cq2 dq2 cq3 dq3

aq b aq b aq b

cq2 dq2 cq3 dq3 cq4 dq4

cq3 dq3 cq4 dq4 cq5 dq5c∆

Figure 2.4: Illustrating the proof of Lemma 2.5.

8

c d

a b a b a b a b

cq dq cq2 dq2 cq3 dq3

a b a b a b a b

a b a b a b a b

cq dq cq2 dq2 cq3 dq3

cq2 dq2 cq3 dq3 cq4 dq4 cq5 dq5

cq4 dq4

aq b aq b aq b

aq b aq b aq b

cq dq cq2 dq2 cq3 dq3

aq b aq b aq b

cq2 dq2 cq3 dq3 cq4 dq4

Figure 2.5: Illustrating the transformation in (2.7) of Lemma 2.6. The white circlesindicate the vertices v1, v2, . . . , vn.

left hand side of (2.3) as in Figures 2.4(a) and (b); the sides of shaded diamond areweighted as in Figure 2.3(a). Denote by G1 the resulting graph.

Next, we apply Spider Lemma to all shaded diamond with leg in G1, and removedallm leftmost andm rightmost horizontal edges which are forced (see Figure 2.4(b)).We get the graph G2 = ORm,n(wt

′)#G, where ORm,n(wt′) is a weighted version of

ORm,n with edges weighted as in Figure 2.4(c), and where ∆ = ad+ bc.Finally, we use Star lemma to change the weights on the graph G2. Divide the

graph ORm,n(wt′) into m(n − 1) X-graphs (restricted by dotted squares in Figure

2.4(c)). Apply Star Lemma with factor qi+j−1∆ to the X-graph in row i and columnj. We obtain the graph on the right hand side of (2.3).

By Vertex-splitting, Spider, and Star Lemmas, we get

M(||ARm,n

(wta,bc,d(q)

)#G

)= M(G1) (2.4)

= M(G2)∏

1≤i,j≤n

(qi+j−2∆) (2.5)

= M(ORm,n(wt

ap,bc,d (q))

) ∏

1≤i≤m,1≤j≤n−1

(qi+j−1∆

)−1∏

1≤i,j≤n

(qi+j−2∆

), (2.6)

which implies (2.3).

Similar to Lemma 2.5, we have the following two transformations.

Lemma 2.6. Let G be a graph and let {v1, . . . , vn} be an ordered subset of itsvertices. Then

M(|ARm,n

(wta,bc,d(q)

)#G

)= (ad+ bc)mq

m(m−1)2 M

(ARm− 1

2,n−1

(wtaq,bc,d (q)

)#G

),

(2.7)and

M(ARm,n

(wta,bc,d(q)

)#G

)= (ad+ bc)mq

m(m−1)2 M

(|ARm− 1

2,n−1

(wtaq,bc,d (q)

)#G

),

(2.8)

9

c d

a b a b a b a b

cq dq cq2 dq2 cq3 dq3

a b a b a b a b

a b a b a b a b

cq dq cq2 dq2 cq3 dq3

cq2 dq2 cq3 dq3 cq4 dq4 cq5 dq5

cq4 dq4

aq b aq b aq b

aq b aq b aq b

cq dq cq2 dq2 cq3 dq3

aq b aq b aq b

cq2 dq2 cq3 dq3 cq4 dq4

Figure 2.6: Illustrating the transformation in (2.8) of Lemma 2.6. The white circlesindicate the vertices v1, v2, . . . , vn.

where |ARm,n

(wta,bc,d(q)

)and |ARm− 1

2,n−1

(wta,bc,d(q)

)are obtained from ARm,n

(wta,bc,d(q)

)

and ARm− 12,n−1

(wta,bc,d(q)

)by appending vertical edges from their bottommost ver-

tices, respectively; and where the connected sum acts on G along {v1, . . . , vn}, andon other summands along their bottommost vertices (ordered from left to right).

The transformations in (2.7) and (2.8) are illustrated by Figures 2.5 and 2.6, form = 3 and n = 4, respectively. The proofs for those transformation are essentiallythe same as that of Lemma 2.5, and are omitted.

Proof of Theorem 2.1. Apply the transformation in (2.8) for the graph G consisting

of n isolated vertices. We get the graph ARm− 12,n−1

(wtaq,bc,d (q)

), by removing all

vertical forced edges at the bottom, we get graph ADn−1

(wtaq,bcq,dq(q)

), and

M(ADn

(wta, b

c, d(q)))

=

n−1∏

k=0

(∆0q

k)M(ADn−1

(wtaq, bcq, dq(q)

)), (2.9)

where ∆i = adqi + bc, for i = 0, 1, . . . , n− 1.Similarly, we get

M(ADn−i

(wtaq

i, bcqi, dqi(q)

))=

n−1∏

k=i

(∆iqk)M

(ADn−i−1

(wtaq

i+1, bcqi+1, dqi+1(q)

)), (2.10)

for any i = 0, 1, . . . , n− 1.From the above recurrence, we obtain

M(ADn

(wta, b

c, d(q)))

=

n−1∏

i=0

n−1∏

k=i

(∆iq

k), (2.11)

which implies (2.1).

10

Next, we consider another generalization of Kamioka’s weight assignment forAztec diamond graph as follows. Assume that h1, h2, . . . , hk are k positive inte-gers with the sum n. The Aztec diamond graph (rotated 900) can be partitionedinto k parts, the i-th part (in order from bottom to top) consisting of hi rows ofshaded diamonds. Assume in addition that the i-th part has weight assignmentwtai,bicqαi−1 ,dqαi−1 (q), where α0 = 0 and αi =

∑ij=1 hj. In particular, the weight matrix

of the Aztec diamond graph is the block matrix A =

A1

A2...Ak

, where Ai is the following

2hi × 2n matrix

ai bi . . . ai bicqαi−1+m−1 dqαi−1+m−1 . . . cqαi−1+m+n−2 dqαi−1+m+n−2

ai bi . . . ai bicqαi−1+m−2 dqαi−1+m−2 . . . cqαi−1+m+n−3 dqαi−1+m+n−3

......

. . ....

...ai bi . . . ai bi

cqαi−1 dqαi−1 . . . cqαi−1+n−1 dqαi−1+n−1

.

Denote by ADn

(c, d, q,

[(a1, b1) (a2, b2) . . . (ak, bk)h1 h2 . . . hk

])the resulting weighted

Aztec diamond graph.

Theorem 2.7. Assume that c, d, q, a1, a2, . . . , ak, b1, b2, . . . , bk are positive real num-bers. Assume in addition that n, h1, h2, . . . , hk are positive integers so that

∑ki hi =

n. Then

M

(ADn

(c, d, q,

[(a1, b1) (a2, b2) . . . (ak, bk)h1 h2 . . . hk

]))=

q(n−1)n(n+1)

3

k∏

i=1

(αi−1−1∏

j=0

∆hi

i,j ·

hi−1∏

j=0

∆hi−ji,αi−1+j

), (2.12)

where ∆i,j = aidqj + bic.

First, we notice that Theorem 2.1 is obtained from Theorem 2.7 by letting(a1, b1) = (a2, b2) = . . . = (ak, bk) = (a, b).

Proof of Theorem 2.7. Apply Vertex-splitting Lemma to the vertices along the bor-der of two consecutive parts (see the dotted lines in Figure 2.7(a)). Apply thesuitable transformation in (2.7) in Lemma 2.6 to the top and bottom parts, andapply the transformation in (2.3) of Lemma 2.5 to all k − 2 remaining parts withlegs appended at the top and bottom (see Figures 2.7(b) and (c); the parts above,between and below the dotted lines in graph (b) are replayed by the parts above,between below those lines in graph (c), respectively). After removing vertical forced

11

(a)(b)

(c)

h3 = 2

h2 = 2

h1 = 1

Figure 2.7: Illustrating the proof of Theorem 2.7.

edges from the top and bottom of the resulting graph, we get the weighted Aztecdiamond graph

ADn−1

(cq, dq, q,

[(a1q, b1) (a2q, b2) . . . (akq, bk)h1 − 1 h2 . . . hk

]),

where

ADn−1

(cq, dq, q,

[(a1q, b1) (a2q, b2) . . . (akq, bk)h1 − 1 h2 . . . hk

])≡

ADn−1

(xcq, dq, q,

[(a2q, b2) . . . (akq, bk)

h2 . . . hk

]),

and we obtain

M

(Dn

(c, d, q,

[(a1, b1) (a2, b2) . . . (ak, bk)h1 h2 . . . hk

]))= q

n(n−1)2

k∏

i=1

∆hi

i,0

×M

(ADn−1

(cq, dq, q,

[(a1q, b1) (a2q, b2) . . . (akq, bk)h1 − 1 h2 . . . hk

])). (2.13)

Then the theorem follows from induction on n.

Before going to the next generalization of Kamioka’s result, we quote the fol-lowing Stanley’s theorem on weighted Aztec diamond graphs (see [3] or Section 2.3in [21]). Consider the Aztec diamond graph of order n, and its edges are weightedas follows. The odd rows of its weight matrix are all

[x1 y1 x2 y2 . . . xn yn

];

12

the even rows are all[t1 z1 t2 z2 . . . tn zn

], i.e. the weight matrix is

S =

x1 y1 x2 y2 . . . xn ynt1 z1 t2 z2 . . . tn znx1 y1 x2 y2 . . . xn ynt1 z1 t2 z2 . . . tn zn...

......

.... . .

......

x1 y1 x2 y2 . . . xn ynt1 z1 t2 z2 . . . tn zn

.

Then we have

Theorem 2.8 (Stanley).

M(ADn (wtS)) =∏

1≤i≤j≤n

(xizj + yjti). (2.14)

Consider a 2n × 2n matrix V (q) defined as follows. The odd rows of V (q) arestill

[x1 y1 x2 y2 . . . xn yn

]; however the 2k-th row is

[t1q

(n−k) z1q(n−k) t2q

(n−k) z2q(n−k) . . . tnq

(n−k) znq(n−k)

],

i.e. V (q) is

x1 y1 x2 y2 . . . xn ynt1q

(n−1) z1q(n−1) t2q

(n−1) z2q(n−1) . . . tnq

(n−1) znq(n−1)

x1 y1 x2 y2 . . . xn ynt1q

(n−2) z1q(n−2) t2q

(n−2) z2q(n−2) . . . tnq

(n−2) znq(n−2)

......

......

. . ....

...x1 y1 x2 y2 . . . xn ynt1 z1 t2 z2 . . . tn zn

.

Theorem 2.9 (q-analog of Stanley’s Theorem). For any positive integer n andpositive real number q

M(ADn

(wtV (q)

))= q

(n−1)n(n+1)3

∏

1≤i≤j≤n

(xizj + yjti). (2.15)

Proof. Denote by V0 := V (q). We apply the same process as in the proof of Lemma2.5 based on Figures 2.8 and 2.4. First, we apply Vertex-splitting Lemma to alltopmost and bottommost vertices of the weighted graph ADn(wtV0) (see Figure2.8(a)). Apply the same process as in Figure 2.4 to the weighted subgraph betweentwo dotted lines in Figure 2.8(a) (the edges are weighted by the assignment wtV0).We replace the weighted subgraph by a weighted odd Aztec rectangle graph asin Figure 2.8(b). Finally, we remove all the the vertical forced edges, we get the

13

(a)(b)

(c)

Figure 2.8: Illustrating the proof of Theorem 2.9.

weighted Aztec rectangle graph ADn−1(wtN) (see Figure 2.8(c)). By calculatingexplicitly the edge-weights, based on Spider Lemma, we get N is

x1q−(n−1)

♦1,1

y2q−(n−1)

♦2,2

x2q−(n−1)

♦2,2

y3q−(n−1)

♦3,3. . .

xn−1q−(n−1)

♦n−1,n−1

ynq−(n−1)

♦n,n

t1♦1,1

z2♦2,2

t2♦2,2

z3♦3,3

. . .tn−1

♦n−1,n−1

zn♦n,n

x1q−(n−2)

♦1,1

y2q−(n−2)

♦2,2

x2q−(n−2)

♦2,2

y3q−(n−2)

♦3,3. . .

xn−1q−(n−2)

♦n−1,n−1

ynq−(n−2)

♦n,n

t1♦1,1

z2♦2,2

t2♦2,2

z3♦3,3

. . .tn−1

♦n−1,n−1

zn♦n,n

......

......

. . ....

...x1q−1

♦1,1

y2q−1

♦2,2

x2q−1

♦2,2

y3q−1

♦3,3. . .

xn−1q−1

♦n−1,n−1

ynq−1

♦n,n

t1♦1,1

z2♦2,2

t2♦2,2

z3♦3,3

. . .tn−1

♦n−1,n−1

zn♦n,n

,

where ♦i,j = (xizj + yjti), and

M (ADn (wtV0)) =

n∏

i=1

(xizi + yiti)nqn(i−1) M(ADn−1 (wtN)) . (2.16)

Partition the resulting weighted Aztec diamond graph into n−1 〈-graphs on left,n− 1 〉-graphs on right, and (n− 2)(n− 1) X-graphs as in Figure 2.8 (c) (restrictedby dotted rectangles). Each of these graphs gives us a chance to apply Star Lemma.In particular, we apply the Star Lemma with factor ♦1,1 to all 〈-graphs; apply theStar Lemma with factor ♦n,n to all 〉-graphs; and apply the Star Lemma with factor

14

♦k,k to all X-graphs in column k, for 1 < k < n. We get the matrix

V1 =

x1 y2 x2 y3 . . . xn−1 ynt1q

(n−2) z2q(n−2) t2q

(n−2) z3q(n−2) . . . tn−1q

(n−2) znq(n−2)

x1 y2 x2 y3 . . . xn−1 ynt1q

(n−3) z2q(n−3) t2q

(n−3) z3q(n−3) . . . tn−1q

(n−3) znq(n−3)

......

......

. . ....

...x1 y2 x2 y3 . . . xn−1 ynt1 z2 t2 z3 . . . tn−1 zn

.

By Lemma 2.5, we obtain

M (ADn−1 (wtN)) =n∏

i=1

♦−(n−1)i,i

n−1∏

i=1

q−(n−1)i M(ADn−1 (wtV1)) . (2.17)

Evaluate two equalities (2.16) and (2.17), we have

M (ADn (wtV0)) =

n∏

i=1

♦i,iqi−1M(ADn−1 (wtV1)) . (2.18)

Similarly, we get

M (ADn−k (wtVk)) =

n−k∏

i=1

♦i,i+kqi−1M

(ADn−k−1

(wtVk+1

)), (2.19)

where the 2(n− k)× 2(n− k) matrix Vk is

x1 yk+1 . . . xn−k ynt1q

(n−k−1) zk+1q(n−k−1) . . . tn−kq

(n−k−1) znq(n−k−1)

x1 yk+1 . . . xn−k ynt1q

(n−k−2) zk+1q(n−k−2) . . . tn−kq

(n−k−2) znq(n−k−2)

......

. . ....

...x1 yk+1 . . . xn−k ynt1 zk+1 . . . tn−k zn

.

Repeated application of the recurrence (2.19) implies the theorem.

3 Weighted Holey Aztec Rectangle

An Aztec rectangle graph ARm,n does not have perfect matchings itself if m 6= n.However, when some of its bottommost vertices have been removed, its perfectmatchings are enumerated by a simple product formula.

First, we quote a result by Mills, Robins and Rumsey (Theorem 2 in [18]).

15

Lemma 3.1. Denote by ARm,n(s1, s2, . . . , sm) the graph obtained from the Aztecrectangle graph ARm,n by removing all its bottommost vertices, except for the s1-st,the s2-nd, . . . , and the sm-th ones. Then

M(ARm,n(s1, s2, . . . , sm)) = 2(m+1

2 )∏

1≤i<j≤m

sj − si

j − i. (3.1)

Next, we have a variant of Lemma 3.1 (see Lemma 3 in [9]).

Lemma 3.2. Label the bottom vertices of the baseless Aztec rectangle ARm− 12,n−1

from left to right by 1, . . . , n, and denote by ARm− 12,n−1(s1, s2, . . . , sm) the graph

obtained from it by deleting the vertices with labels in the set {s1, . . . , sm}, where1 ≤ s1 < · · · < sm ≤ n are given integers. Then

M(ARm− 1

2,n−1(s1, s2, . . . , sm)

)= 2(

m

2 )∏

1≤i<j≤m

sj − si

j − i. (3.2)

The next result is due to Cohn, Larsen and Propp (see [5], Proposition 2.1), seealso Lemma 2 in [9].

Lemma 3.3. Denote by SHa,b(s1, s2, . . . , sa) the the region obtained from the tophalf of a lozenge hexagon of side-lengths b, a, a, b, a, a (clockwise from top) onthe triangular lattice by removing the si-th up-pointing triangles from its base, for1 ≤ s1 < s2 < . . . < sa ≤ a+ b. Then

M(SHa,b(s1, s2, . . . , sa)) =∏

1≤i<j≤a

sj − si

j − i. (3.3)

Denote [a]q :=1− qa

1− q=∑

0≤i<a qi and [a]q! = [a]q[a−1]q . . . [1]q. The q-binomial

coefficient is defined by [a

k

]

q

=[a]q!

[a− k]q![k]q!.

We have the following identity for the q-binomial coefficient

n−1∏

i=0

(1 + qix) =

n−1∑

k=0

qk(k+1)/2

[n

k

]

q

xk. (3.4)

Next, we quote an identity due to Krattenthaler (see identity (3.12) of Theorem26 in [12]).

Lemma 3.4. Let n ba a nonnegative integer, and let L1, L2, . . . , Ln, and A be inde-terminants. Then

det1≤i,j≤n

(qiLj

[A

Lj + i

]

q

)= q

∑ni=1 iLi

∏1≤i<j≤n[Li − Lj]q∏n

i=1[Li + n]q!

∏ni=1[A + i− 1]q!∏ni=1[A− Li − 1]q!

. (3.5)

16

Vertical Left Right

Figure 3.1: Three types of rhombi.

We have a generalization of Lemma 3.1 as follows.

Theorem 3.5. The matching generating function of the weighted Aztec rectangle

ARm,n

(wta,bc,d(q)

), where the r1-st, the r2-nd, . . . , and the rn−m-th bottommost ver-

tices have been removed, equals

q(m−1)m(m+1)

3+∑n−m

i=1(i+m−ri)(i+m−ri+3)

2 a∑n−m

i=1 (m+i−ri)b∑n−m

i=1 (i+ri)

m−1∏

k=0

∆m−kk (3.6)

×

∏1≤i<j≤n−m[rj − ri]q∏n−m

i=1 [n− ri]q!

∏n−mi=1 [m+ i− 1]q!∏n−m

i=1 [ri − 1]q!, (3.7)

where ∆k = adqk + bc.

Proof. Denote by G the dual graph of the region. We consider the graph G′ obtained

from ARm,n

(wta,bc,d(t, q)

)by adding a vertical leg at the ri-th bottommost vertex,

for i = 1, 2, . . . , n −m (see Figure 3.2(a), for m = 3, n = 5, r1 = 1, r2 = 4). Thenby the fact about forced edges, M(G) = M(G′).

Next, we apply a m-step transforming process based on Figure 3.2 as follows.First, apply the transformation in (2.8) in Lemma 2.6 as in Figures 3.2(a) and (b);the part above the dotted line in graph (a) is replaced by the part above that linein graph (b). Second, we apply the same same transformation to replace the partabove the upper dotted line in graph (b) by the part above the lower dotted line ingraph (c). Keep doing this process until we eliminate all rows of diamonds on thetop of the resulting graph. Denote by G′′ the final graph (see Figure 3.2(d)).

By removing vertical forced edges at the bottom of G′′, we get a the dual graph

G of a weighted semi-hexagon SHm,n−m, where the ri-th up-pointing triangles havebeen removed from its base, for i = 1, 2, . . . , n − m (see Figure 3.2(e); the whitecircles indicate the vertices removed from the base). In particular, the left rhombion the level k (from bottom; the bottom is considered to be the level 0) are weightedby aqk+1, all right rhombi are weighted by b, and all vertical rhombi have weight 1(see Figure 3.1 for three types of rhombi). By Lemma 2.6, we obtain

M(ARm,n(wta,bc,d(q))) = q

(m−1)m(m+1)3

(m−1∏

k=0

∆m−kk

)M(G). (3.8)

We have a bijection between tilings of the above weighted semi-hexagon andfamilies of n − m disjoint rhombi paths as follows (see Figures 3.3(a) and (b)).

17

(a) (b)

(c) (d)

(e)

c d

a b a b a b a b

cq dq cq2 dq2 cq3 dq3

a b a b a b a b

a b a b a b a b

cq dq cq2 dq2 cq3 dq3

cq2 dq2 cq3 dq3 cq4 dq4 cq5 dq5

cq4 dq4

aq b aq b aq b

aq b aq b aq b

cq dq cq2 dq2 cq3 dq3

aq b aq b aq b

cq2 dq2 cq3 dq3 cq4 dq4

a b

a b

a b

cq6 dq6

cq5 dq5

cq4 dq4

aq b

aq b

aq b

cq4 dq4

cq5 dq5

aq b aq b aq b aq b

aq2 b aq2 b baq2

cq2 dq2 cq3 dq3 cq4 dq4

aq2 b aq2 b baq2

aq b aq b aq b aq b

aq2 b aq2 b baq2

aq3 b aq3 b

aq b aq

aq2 b aq2 b baq2

aq3 b aq3 b

Figure 3.2: Transform a holey Aztec rectangle into the dual graph of a semi-hexagonwith defects.

18

Label the centers of the horizontal lattice segments on the top of the semi-hexagonby u1, u2, . . . , un−m from left to right; and label the centers of horizontal latticesegments on the bottom of the region by v1, v2, . . . , vn−m. Given any tiling T ofthe region. For any given ui, there is only one rhombus that has horizontal sidecontaining ui, this is the first rhombus of the i-th rhombi path. The second rhombusof the path is the rhombus that shares a horizontal side with the first one, and staysbelow the first one; the third rhombus shares a horizontal side with the second one,and stays below the second one, and so on. The path of rhombi stops when reachingone of vj ’s. Doing the same process for all other ui’s, we have n−m disjoint rhombipaths. We notice that all rhombi of T that do not belong to one of the above rhombipaths are vertical. This yields a bijection between the tilings of the region and thefamilies of n−m disjoint rhombi paths.

Next, each rhombi path can be identified with the linear path passing the cen-ters of horizontal sides of its rhombi (see the dashed lines in Figure 3.3(b)). Con-sider the 600 coordinate system with the x-axis passing the center of the leftmostlattice segment on top of the region, and the origin is at the position of u1 (seeFigure 3.3(b)). Deform the coordinate system into a orthogonal coordinate sys-tem, each family of linear paths becomes a family of non-intersecting lattice pathsP = (P1, P2, . . . , Pn−m) using (1, 0) and (0, 1) steps, and connecting two sets of ver-tices {u1, u2, . . . , un − m} and {v1, v2, . . . , vn−m} in the plane Z

2. It is easy to seethat ui has coordinate (−i+1, i− 1), and vj has coordinate (m− rj +1, rj − 1), for1 ≤ i, j ≤ n − m (see Figure 3.3(c)). To make sure the bijection above is weight-preserved, we assign weights to each level step in Z

2 as follows: the level steps havingcenters on y = −x− 2l−1

2are weighted by q(m−l+1), for 1 ≤ l ≤ m (see Figure 3.3(d)

for the weighted P2).By Lindstrom-Gessel-Viennot theorem ([16], Lemma 1; [20] Theorem 1.2), we

haveM(G) =

∑

P

w(P) = det1≤i,j≤n−m

(si,j) , (3.9)

where the sum is taken over all families of the non-intersecting lattice paths in Z2 us-

ing (0,1) and (1,0) steps, and starting at u1, u2, . . . , un−m and ending at v1, v2, . . . , vn−m; and where si,j =

∑Pi,j

w(Pi,j), the sum is taken over all lattice paths Pi,j in Z2

connecting ui = (−i+ 1, i− 1) and vj = (m− rj + 1, rj − 1).One ready see that each path Pi,j has exactly i + m − rj level steps, say the

l1-st,the l2-nd, . . . , the li+m−rj -th steps from right to left in Pi,j are level. Then

w(Pi,j) = brj−i∏i+m−rj

t=1 aq(y+lt). Thus, si,j is the coefficient of xi+m−rj in the poly-nomial

∏mi=1 (b+ aqix).

19

����

������

������

��������

����

���

���

����

���

���

���

���

��������

������

������

����

���

���

������

������

���

���

���

���

(a) (b)

(c)(d)

O

x y

O

y

x

u1 u2 u3 u4

v1 v2 v3 v4

u1

u2u3

u4v1

v2

v3

v4

y

x

b

aq2aq3b

aq5aq6u2

v2

O

Figure 3.3: Bijection between tilings of a semihexagon and families of non-intersecting lattice paths in Z

2.

20

By identity (3.4), we get

m∏

i=1

(b+ aqix

)= bm

m−1∏

i=0

(1 + qi

(aqxb

))(3.10)

= bmm−1∑

k=0

qk(k+1)

2

[m

k

]

q

(aqxb

)k(3.11)

=

m−1∑

k=0

qk(k+3)

2 akbm−k

[m

k

]

q

xk. (3.12)

It means that si,j = q(i+m−rj)(i+m−rj+3)

2 ai+m−rjbrj−i

[m

i+m− rj

]

q

. Therefore, we get

M(G) = det1≤i,j≤n−m

(q

(i+m−rj )(i+m−rj+3)

2 ai+m−rjbrj−i

[m

i+m− rj

]

q

). (3.13)

Rewrite

q(i+m−rj)(i+m−rj+3)

2 ai+m−rjbrj−i = qi(m−rj)(q

i2+3i2 aib−i

)(q

(m−rj )2+3(m−rj )

2 am−rjbrj).

Thus, we can factor out qi2+3i

2 aib−i from the i-th row, and q(m−rj )

2+3(m−rj )

2 am−rjbrj

from the j-th column of the matrix on the right hand side of (3.13), for 1 ≤ i, j ≤n−m. Then the determinant in (3.13) equals

q∑n−m

i=1i2+3i

2 (ab)(n−m)(n−m+1)

2 q∑n−m

j=1

(m−rj )2+3(m−rj )

2 a∑n−m

i=1 (m−ri)b∑n−m

i=1 ri

× det1≤i,j≤n−m

(qi(m−rj)

[m

i+m− rj

]

q

). (3.14)

By Lemma 3.4, we get

det1≤i,j≤n−m

(qi(m−rj)

[m

i+m− rj

]

q

)=

q∑n−m

i=1 i(m−ri)

∏1≤i<j≤n−m[rj − ri]q∏n−m

i=1 [n− ri]q!

∏n−mi=1 [m+ i− 1]q!∏n−m

i=1 [ri − 1]q!. (3.15)

Thus, the theorem follows from (3.14) and (3.15).

Remark 1. From Lemmas 3.1, 3.2, and 3.3, we have an interesting relation betweenthree families of graphs and regions:

M(ARm,n(s1, s2, . . . , sm)) = 2mM(ARm− 12,n−1(s1, s2, . . . , sm))

= 2m(m+1)

2 M(SHm,n−m(s1, s2, . . . , sm)). (3.16)

By letting a = b = q = 1, the transforming process in Figure 3.2 explains the hiddenrelation in (3.16).

21

Figure 3.4: The hexagon H3,3,2 is obtained from a semi-hexagon by removing forcedvertical rhombi.

Let n = u + v + w, m = v + w, a = q−1, b = 1, si = v + i, then the hexagonHu,v,w of sides u, v, w, u, v, w (in cyclic order, starting from the northern side) onthe triangular lattice is obtained from the semi-hexagon SHm,n−m, where the si-thup-pointing triangles have been removed from its base, by removing several verticalforced rhombi (see Figure 3.4). Thus, the argument in proof of Theorem 3.5 impliesthe following q-analog of MacMahon’s Theorem about rhombus.

Corollary 3.6 (q-analog of MacMahon’s Theorem). Assume H(q)u,v,w is the weighted

version of the hexagon Hu,v,w, where all the left rhombi on level k are weighted byqk (from the bottom; the bottom is at level 0), and all other rhombi are weighted by1. Then

M(H(q)

u,v,w

)= q

uw(w+1)2

[H(u)]q[H(v)]q[H(w)]q[H(u+ v + w)]q[H(u+ v)]q[H(v + w)]q[H(w + u)]q

, (3.17)

where [H(n)]q is the q-hyperfactorial function defined by

[H(n)]q =

{[0]q![1]q![2]q! . . . [n− 1]q! if n ≥ 1;

1 if n = 0.(3.18)

4 A counterpart of Aztec diamond theorem

In Section 4, we consider a counterpart of Aztec diamond theorem 1.1 for certainholey Aztec rectangle regions defined as follows. Assume thatm ≤ n are two positiveinteger, and the Aztec rectangle region of order (m,n) is defined to be a portion ofthe Aztec diamond region of order n as Figure 4.1. Denote by ARm,n the resultingregion. One readily sees that the dual graph of ARm,n is the Aztec rectangle graphARm,n. The Aztec rectangle region ARm,n does not have any tilings if m < n. Weare interested in the number of tilings of the Aztec rectangle region, where the r1-th,the r2-nd, . . . , and the rn−m-th white squares, from the bottom to top, have beenremove from the southeastern boundary (see the region on the left of Figure 4.2 foran example with m = 4, n = 7, r1 = 3, r2 = 4, and r3 = 6).

22

m=2n=4

Figure 4.1: The Aztec rectangle region as a portion of an Aztec diamond region

T P = (P1, P2, P3, P4, P5)

P1

P2

P3

P4

P5

Figure 4.2: Bijection between tilings of a holey Aztec rectangle region and familiesof non-intersection Schroder paths.

23

odd vertical even vertical odd horizontal even horizontal

Figure 4.3: Drawing the steps of the Schroder paths.

For any tiling T of the region, we define two statistics v(T ) and r(T ) similarlyto their counterparts in the Aztec diamond region as follows. We define v(T ) isthe number of even vertical dominoes in T 1. Similar to the case of Aztec diamondregions, any given two tilings T and T ′ of the region can be obtained from eachother by a finite sequence of elementary moves. We have also a bijection betweentilings of the region and families of non-intersecting (partial) Schroder paths P =(P1, P2, . . . , Pm), where Pi connects the i-th vertical steps on the southwestern andsouthwestern boundaries of the region (see Figure 4.2). The way to draw the familyof Schroder paths P from the tilings T is shown in Figure 1.3.

Denote by Sm the set of above families of non-intersecting Schroder paths P =(P1, P2, . . . , Pm). Assume that A1 is on the x-axis of a standard coordinate system.Denote by area(Pi) is the area underneath Pi, and define area(P) =

∑mi=1 area(Pi).

Assume that T0 is the tiling so that its corresponding family of non-intersectingSchroder paths P has minimal underneath area area(P). We call T0 minimal tiling.Assign r(T0) := 0, and for any other tiling T , r(T ) is defined to be the minimalnumber of elementary moves required to reach T from tiling T0.

In the Aztec diamond regions, the tiling T0 consists of all horizontal dominoes.However, in this case T0 contains some vertical dominoes in general, and the defini-tion of T0 is not trivial. We investigate the minimal T0 the next paragraph.

In order to define T0, we define its corresponding family of Schroder paths P∗ =(P ∗

1 , P∗2 , . . . , P

∗m). First, we define P ∗

1 as follows. If B1 is on the same level with A1,then P ∗

1 consists of an odd horizontal domino. If B1 is on a higher level than A1, wego up until reaching the B1’s level, and then we finish P ∗

1 by an odd vertical domino.Next, for any 1 ≤ k < m, P ∗

k+1 consists of two parts: the first part obtained from

P ∗k by shifting it along vector

−−−−−→AkAk+1. Assume that Ck+1 is the end of the first path

of P ∗k+1. Draw the zigzag path passing B1 parallel to the southwestern boundary of

the region. We start over the process for P ∗1 above to connect Ck+1 and Bk+1 (see

the Figure 4.4 for an example). It is easy to see that P ∗ has the smallest underneatharea among all elements of Sm. Thus, T0 is indeed the minimal tiling.

We assign the dominoes in the holey Aztec rectangle region based on Kamioka’sweights assignment. We recall that, in the Kamioka’s weight assignment, each evenhorizontal and odd vertical domino a weight 1, each odd horizontal domino on levelk (from the bottom of the Aztec diamond region) a weight tq2k, and each evenvertical domino on k a weight q2k+1. The weight of each tiling of the region can bewritten as a product txqy, called the standard expression of the weight of the tiling.Denote by β(P) the exponent of q in the standard expression of w(P) = w(T ). Wedenote level(Pi), down(Pi) and up(Pi) by the numbers of level, down and up stepsin the path Pi, respectively. Define level(P) =

∑mi=1 level(Pi). Similar to Lemma

1In the case of Aztec diamond regions, v(T ) is equal to half of the number of vertical dominoesbecause the numbers of odd and even vertical dominoes are the same.

24

T∗

P∗ = (P ∗

1 , P∗2 , P

∗3 , P

∗4 , P

∗5 )

A1

A2

A3

A4

A5

B1

B2

B3

B4

B5

Figure 4.4: Minimal tiling

Figure 4.5: The elementary moves rise the rank of the tiling T by one (left-to-right,respectively) if only if the exponent of q in w(T ) increases by one.

7.1 in [11], we have following facts.

Lemma 4.1. Denote by ARm,n({s1, s2, . . . , sm}) the region obtained from ARm,n byremoving all white squares, except for the s1-st, the s2-nd, . . . , and the sm-th ones.Assume that T is a tiling of the region and an n-tuple P = (P1, P2, . . . , Pm) ∈ Sm ofnon-intersecting Schroder paths are in the on-to-one correspondence by the bijection.Then

v(T ) + level(P) =m(m+ 1)

2, (4.1)

and

β(P)− r(T ) = β(P∗) =m∑

k=1

k∑

i=1

2(si − 1 + k − i). (4.2)

Proof. The difference up(Pi)−down(Pi) is a constant independent of Pi that is equalto (si− i). Thus, by adding (si− i) down steps to the end of Pi, we have a Schroderpath P ′

i connecting (−i, i) and (i + 2(si − i), i). Moreover, we get down(P ′i ) =

up(P ′i ) = up(Pi), level(P

′i ) = level(Pi), down(P

′i ) = down(Pi)+(si−i), and up(P ′

i )+down(P ′

i ) + 2 level(P ′i ) = 2si. This implies that

down(Pi) + level(Pi) = i, (4.3)

for any i = 1, 2, . . . , m. By adding m equalities in (4.3), for i = 1, 2, . . . , m, weobtain (4.1).

25

Divide the set of elementary moves into to types as in Figure 4.5. We notice thatthe elementary moves (from left to right) increase simultaneously the rank r(T ) andβ(P) by one. This implies the first equality in (4.2).

One can see that w(P ∗1 ) = tq2(s1−1), and by the definition of P ∗

k+1, we get

w(P ∗k+1) = q2kw(P ∗

k )tq2(sk+1−1).

Thus, by induction, we get w(P ∗k ) =

∏ki=1 tq

2(si−1+k−i), for k = 1, 2, . . . , m. There-fore,

w(T0) = w(P∗) = tn(n+1)/2q∑m

k=1

∑ki=1 2(si−1+k−i),

which impliesthe second equality in (4.2).

We have the following counterpart of Aztec diamond 1.1 for holey Aztec rectangleregion.

Theorem 4.2. Denote by ARm,n({s1, s2, . . . , sm})(t, q) :=∑

T tv(T )qr(T ), where thesum is taken over all tilings of the Aztec rectangle region ARm,n where all whitesquares, except for the s1-th, the s2-nd, . . . , and the sm-th ones (ordered from thebottom to top), have been remove from the southeastern boundary. Then

ARm,n({s1, s2, . . . , sm})(t, q) = tm(m+1)/2 (4.4)

× q2(m−1)m(m+1)

3+∑n−m

i=1 (i+m−ri)(i+m−ri+3)−∑m

k=1

∑ki=1 2(si−1+k−i) (4.5)

×m−1∏

k=0

(t−1 + q2k+1)m−k

∏1≤i<j≤n−m[rj − ri]q2∏n−m

i=1 [n− ri]q2!

∏n−mi=1 [m+ i− 1]q2 !∏n−m

i=1 [ri − 1]q2!, (4.6)

where the order set {r1, r2, . . . , rn−m} = {1, 2, . . . , n} − {s1, s2, . . . , sm}.

Proof. Denote by AR(t, q) the Aztec rectangle graph with the Kamioka’s assign-ment, and with the bottommost vertices at the positions of ri’s removed. First, bythe above bijection between tilings of holey Aztec rectangle and families of non-intersecting Schoder paths, we obtain

M(AR(t, q)) =∑

P∈Sm

w(P) =∑

P∈Sm

tlevel(P)qβ(P).

By Lemma 4.1, we have

ARm,n({s1, s2, . . . , sm})(t, q) =∑

T

tv(T )qr(T ) (4.7)

= q−∑m

k=1

∑ki=1 2(si−1+k−i)tm(m+1)/2

∑

P∈Sm

t−level(P)qβ(P)

(4.8)

= q−∑m

k=1

∑ki=1 2(si−1+k−i)tm(m+1)/2 M(AR(t−1, q)),

(4.9)

and the theorem follows from Theorem 3.5 (by letting a = b = 1, c = t, d = q, andreplacing q by q2).

26

5 Weighted Double Aztec Rectangle graphs

We consider new family of graphs as follows. We start with two Aztec rectanglegraph ARm,n and ARm′,n′, so that n−m = n′ −m′ > 0. Assume that c is a positiveinteger, so that n − m ≤ c ≤ min(n, n′). Consider a graph obtained from thetwo Aztec rectangle graphs by connecting the c rightmost vertices at the bottom ofARm,n to the c leftmost vertices at the top of ARm′,n′ by c vertical edges. Denote by

DR(m,n)(m′,n′),c the resulting graph, called a double Aztec rectangle (see Figure 5.1(a) for

DR(5,8)(4,7),4). Assign the weight assignment wta,bc,d(q) to ARm,n, and assign the weight

assignment wt∗ a,bc,d(q) = wtV ∗ to ARm′,n′, where

V ∗ =

c d cq−1 dq−1 . . . cq1−n′

dq1−n′

b aq b aq . . . b aq

cq−1 dq−1 cq−2 dq−2 . . . cq−n′

dq−n′

b aq b aq . . . b aq...

......

.... . .

......

cq1−m′

dq1−m′

cq−m′

dq−m′

. . . cq2−m′−n′

dq2−m′−n′

b aq b aq . . . b aq

. (5.1)

We notice that ARm′,n′(wt∗ a,bc,d(q)) is obtained from ARm′,n′(wtaq,bc,d (q−1)) by rotating

1800.Assume in addition that all the vertical edges connecting the two Aztec rectangle

graphs are weighted by 1. Denote by DA(m,n)(m′,n′),c(q) the resulting weighted double

Aztec rectangle.

Theorem 5.1.

M(DRm′,n′

m,n (q)) = q(n−m)(c−n+m)(c−n+m−2m′+3)

2+m(m−1)(m+1)−m′(m′

−1)(m′+1)3

× a(n−m)(c−n+m)b(n−m)(n+n′+1−c)

m−1∏

i=0

(adqi + bc)m−i

m′−1∏

i=0

(adq−i+1 + bc)m′−i

×[H(n−m)]q[H(c− n+m)]q[H(m+m′)]q[H(m+m′ − c)]q

[H(c)]q[H(n+m′)]q[H(n′ +m)]q. (5.2)

Proof. Denote by SHa,b,qx,y the weighted region obtained from SHa,b by weighted

each left rhombus on level k a weight aqk+1, each right rhombus a weight b, andeach vertical rhombus a weight 1 (the bottom is at level 0).

Similar to the proof of Theorem 3.5, we apply m times the transformation in

(2.8) of Lemma 2.6, we transform the ARm,n

(wta,bc,d(q)

)to the weighted graph G1

obtained from the dual graph of semi-hexagon SHa,b,qm,n−m by adding n vertical edges to

its bottommost vertices. The total factor of this process is qm(m−1)(m+1)

3

∏m−1i=0 (adqi+

bc)m−i.

27

(a) (b) (c)

Figure 5.1:

View ARm′,n′(wt∗ a,bc,d(q)) as ARm′,n′(wtaq,bc,d (q−1)) after rotated 1800. We apply

the same process to transform ARm′,n′(wt∗ a,bc,d(q)) to the graph G2 obtained from the

dual graph of the weighted semi-hexagon SHm′,n′−m′(aq, b, q−1) by adding n′ verticaledges to its bottommost vertices, and rotating the resulting graph 1800. The total

factor of this process is q−m′(m′

−1)(m′+1)3

∏m′−1i=0 (adq−i+1+bc)m

′−i. The process is shownin Figures 5.1(a) and (b); the part above (resp., below) the dotted line in graph (a)is replaced by the part above (resp., below) those lines in graph (b).

Finally, by applying Vertex-splitting Lemma (in reverse), we get the dual graphof the hexagonal region Hn−m,m+n′−c,c−n+m where all the left rhombi on the level iare weighted q−m′+1+i, all right rhombi are weighted by b, and all vertical rhombiare weighted by 1 (see Figures 5.1(b) and (c); the bottom is at level 0).

Let ri =: m + n′ − c + i, for i = 1, 2, . . . , n − m. By considering forced verti-cal rhombus, the above region have the same tiling generating function as that of

SHaq−m′

,b,qm+m′,n−m({ri}

n−m1 ), the region obtained from SH

aq−m′

,b,qm+m′,n−m by removing all up-

pointing triangles, except for the (m+ n′ − c+ i)-th ones, for 1 ≤ i ≤ n−m, fromits base. Thus,

M(DR

(m,n)(m′,n′),c(q)

)= q

m(m−1)(m+1)3

−m′(m′

−1)(m′+1)3

m−1∏

i=0

(acqi + bd)m−i

×

m′−1∏

i=0

(acq−i+1 + bd)m′−i M

(SH

aq−m′

,b,qm+m′,n−m({ri}

n−m1 )

). (5.3)

By the same argument in the proof of Theorem 3.5, we obtain

M(SH

aq−m′

,b,qm+m′,n−m({ri}

n−m1 )

)= det

1≤i,j≤n−m(si,j) , (5.4)

where si,j is the coefficient of xi+(m+m′)−ri in the polynomial∏m+m′

i=1

(b+ aq−m′

qix),

and where ri = m+ n′ − c+ i. Apply the same process as in the proof of Theorem

28

3.5, we get

det1≤i,j≤n−m

(si,j) =q∑n−m

i=1(i+m+m′

−ri)(i+m+m′−ri+3)

2 (aq−m′

)∑n−m

i=1 (m+m′+i−ri)b∑n−m

i=1 (i+ri)

(5.5)

×

∏1≤i<j≤n−m[rj − ri]q∏n−mi=1 [n+m′ − ri]q!

∏n−mi=1 [m+m′ + i− 1]q!∏n−m

i=1 [ri − 1]q!. (5.6)

By evaluating (5.3), (5.4), and (5.5), we get the theorem.

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