Enumeration of Tilings and Related Problems

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1 Enumeration of Tilings and Related Problems Tri Lai Institute for Mathematics and its Applications Minneapolis, MN Discrete Mathematics Seminar University of British Columbia Vancouver February 2016

2 Outline 1 Introduction to the Enumeration of Tilings. 2 Electrical networks 3 Tiling expression of minors 4 Future work.

3 Definition of Tilings A lattice divides the plane into disjoint pieces, called fundamental regions.

4 Definition of Tilings A lattice divides the plane into disjoint pieces, called fundamental regions. A tile is a union of any two fundamental regions sharing an edge.

5 Definition of Tilings A lattice divides the plane into disjoint pieces, called fundamental regions. A tile is a union of any two fundamental regions sharing an edge. A tiling of a region is a covering of the region by tiles so that there are no gaps or overlaps.

6 Definition of Tilings A lattice divides the plane into disjoint pieces, called fundamental regions. A tile is a union of any two fundamental regions sharing an edge. A tiling of a region is a covering of the region by tiles so that there are no gaps or overlaps. The number of tilings of a 8 8 chessboard is 12,988,816.

7 Tilings We would like to find the number of tilings of certain regions.

8 Tilings We would like to find the number of tilings of certain regions. Tilings can carry weights, we also care about the weighted sum of tilings: T wt(t ), called the tiling generating function.

9 Kasteleyn Temperley Fisher Theorem Theorem (Kasteleyn, Temperley and Fisher 1961) The number of tilings of a 2m 2n chessboard equals m n j=1 k=1 ( ( 4 cos 2 jπ ) ( + 4 cos 2 kπ ) ). 2m + 1 2n + 1

10 Tilings and dimer coverings

11 Pfaffian Let A = (a i,j ) be a 2N 2N skew-symmetric matrix. The Pfaffian of A is defined by the equation Pf(A) = 1 2 N N! σ S 2N sign(σ) N i=1 a σ(2i 1),σ(2i) where S 2N is the symmetric group and sign(σ) is the signature of σ.

12 Pfaffian Let G m,n is an orientation of the grid graph of size 2m 2n.

13 Pfaffian Let G m,n is an orientation of the grid graph of size 2m 2n. The adjacent matrix of G m,n is A = (a i,j ) 2N 2N (N = mn) defined as: 1 if i j, a i,j = 1 if j i, 0 if {i, j} / E.

14 Pfaffian Let G m,n is an orientation of the grid graph of size 2m 2n. The adjacent matrix of G m,n is A = (a i,j ) 2N 2N (N = mn) defined as: 1 if i j, a i,j = 1 if j i, 0 if {i, j} / E. There exists an orientation so that Pf(A) = M(G m,n ).

15 Pfaffian and Determinant Theorem (Muir) det A = (Pf(A)) 2. (det(a)) 2 = m n j=1 k=1 ( ( 4 cos 2 jπ ) ( + 4 cos 2 kπ ) ) 4 2m + 1 2n + 1

16 Connections and Applications to Other Fields Statistical Mechanics: Dimer model, double-dimer model, square ice model, 6-vertex model, fully packed loops configuration... Probability: Arctic Curves. Graph Theory: Bijection between tilings and perfect matchings, Temperley s correspondence. Cluster Algebras: Combinatorial interpretation of cluster variables. Electrical networks:... Other topics in combinatorics: Alternating sign matrices, monotone triangles, plane partitions, lattice paths, symmetric functions...

17 Aztec Diamond Theorem (Elkies, Kuperberg, Larsen and Propp 1991) The Aztec diamond of order n has 2 n(n+1)/2 domino tilings. Figure: The Aztec diamond of order 5 and one of its tilings.

18 An Aztec Temple

19 Urban renewal Definition M(G) = wt(π) π M(G) Lemma (Spider Lemma) M(G) = (xz + yt) M(G ) Lemma (Vertex-spitting Lemma) M(G) = M(G )

20 Proof using Urban renewal

21 Quasi-hexagon a=5 b=3 c=3 c=3 a=5 b=3 In 1999, James Propp collected 32 open problems in enumeration of tilings.

22 Quasi-hexagon a=5 b=3 c=3 c=3 a=5 b=3 In 1999, James Propp collected 32 open problems in enumeration of tilings. Problem 16 on the list asks for the number of tilings of a quasi-hexagon.

23 Quasi-hexagon a=5 b=3 c=3 c=3 a=5 b=3 Theorem (L. 2014) The number of tilings of a quasi-hexagon is a power of 2 times the number of tilings of a semi-regular hexagon.

24 MacMahon s Formula Theorem (MacMahon 1900) The number of (lozenge) tilings of a semi-regular hexagon of sides a, b, c, a, b, c on the triangular lattice is a b c i=1 j=1 t=1 i + j + t 1 H(a) H(b) H(c) H(a + b + c) = i + j + t 2 H(a + b) H(b + c) H(c + a), where the hyperfactorial H(n) = 0! 1! 2!... (n 1)!. a=4 b=5 c=6 c=6 b=5 a=4

25 Tilings and non-intersecting lattice paths Bijection between lozenge tilings and non-intersecting lattice paths

26 Lindström-Gessel-Viennot Lemma Lemma G = (E, V ) is an acyclic directed graph, A = {a 1, a 2,..., a k } and B = {b 1, b 2,..., b k } are disjoint subsets of V. Assume that any pair of paths connecting a i and b j, and a j and b i are intersected. The number of families of non-intersecting lattice paths connecting A and B is given by P(A B) = det(p i,j ) k k where p i,j is the number of paths from a i to b j.

27 Connection to Plane Partitions a=10 O j k i b=10 c=10 c=10 b=10 a=10

28 MacMahon s q-formula Theorem (MacMahon) π q vol(π) = H q(a) H q (b) H q (c) H q (a + b + c) H q (a + b) H q (b + c) H q (c + a), where the sum is taken over all stacks π fitting in an a b c box. q-integer [n] q := 1 + q + q q n 1 q-factorial [n] q! := [1] q [2] q [3] q... [n] q q-hyperfactorial H q (n) := [0] q! [1] q! [2] q!... [n 1] q!.

29 Generalizing MacMahon s Formula a z+b+c x m y j O k i a z+b+c x m y t m x c c m a y+b b t+c t m a+b x c a z m q volume of the stack =? stacks c m y+b 3 b 1 m 2 z t+c a+b

30 Generalizing MacMahon s Formula Theorem (L ) For non-negative integers x, y, z, t, m, a, b, c q vol(π) H q( + x + y + z + t) = H q( + x + y + t) H q( + x + y + z) π Hq( + x + t) Hq( + x + y) Hq( + y + z) Hq( ) H q( + z + t) H q( + x) H q( + y) Hq(m + b + c + z + t) Hq(m + a + c + x) Hq(m + a + b + y) H q(m + b + y + z) H q(m + c + x + t) H q(c + x + t) H q(b + y + z) H q(a + c + x) H q(a + b + y) H q(b + c + z + t) Hq(m)3 H q(a) 2 H q(b) H q(c) H q(x) H q(y) H q(z) H q(t) H q(m + a) 2 H q(m + b) H q(m + c) H q(x + t) H q(y + z), where = m + a + b + c.

31 Generalizing MacMahon s Formula c z+b x+y+d+e+f z+c b k O j i c z+b 7 x+y+d+e+f 6 z+c b x+a f e y+a x+a f e y+a 5 3 y+z+d+e+f d x+z+d+e+f y+z+d+e+f 8 4 d 2 x+z+d+e+f x+c a (a) y+b x+c a (b) y+b

32 Blum s Conjecture and Hexagonal Dungeon 2a=4 b=6 a=2 a=2 b=6 2a=4 Theorem (Blum s (ex-)conjecture) The number of tilings of the hexagonal dungeon of side-lengths a, 2a, b, a, 2a, b (b 2a) is 13 2a2 14 a2 2. The conjecture was proven by Ciucu and L. (2014).

33 Kuo condensation Theorem (Kuo)

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