Tiling Shuffling Phenomenon Tri Lai University of Nebraska Lincoln - PowerPoint PPT Presentation

Tiling Shuffling Phenomenon Tri Lai University of Nebraska – Lincoln Lincoln, NE 68588 Dimers 2020 University of Michigan August 11, 2020 Tri Lai Tiling Shuffling Phenomenon

MacMahon’s Theorem Theorem (MacMahon) a b c 1 − q i + j + t − 1 q vol ( π ) = � � � � PP q ( a , b , c ) := 1 − q i + j + t − 2 , π i =1 j =1 t =1 where the sum is taken over all plane partitions π fitting in an a × b × c box. k O b b i j 6 a a c c 4 3 2 2 1 2 1 c c a a 1 0 0 0 b b Tri Lai Tiling Shuffling Phenomenon

MacMahon’s Theorem Theorem ( MacMahon 1900) The number of (lozenge) tilings of a centrally symmetric hexagon Hex ( a , b , c ) of sides a , b , c , a , b , c on the triangular lattice is a b c i + j + t − 1 � � � PP( a , b , c ) := i + j + t − 2 i =1 j =1 t =1 k O b b i j 6 a a c c 4 3 2 2 1 2 1 c c a a 1 0 0 0 b b Tri Lai Tiling Shuffling Phenomenon

Punctured Hexagon: James Propp’s Problem n n n n + + 1 1 + 1 1 + n n n n n n n+1 n+1 Open Problem (Propp 1997) Find an explicit formula for the number of tilings of a hexagon of sides n , n + 1 , n , n + 1 , n , n + 1 with the central unit triangle removed. This is Problem 2 on his list of 20 open problems in the field of enumeration of tilings. Tri Lai Tiling Shuffling Phenomenon

Ciucu–Eisenk¨ olbl–Krattenthaler–Zare’s cored hexagon x z+m y+m m y z x+m Ciucu–Eisenk¨ olbl–Krattenthaler–Zare (2001) generalized the above results by extending the size of the hole. Unit triangle is replaced by a triangle of an abritrary side. The triangular hole is at the ‘center’ of the hexagon of sides a , b + m , c , a + m , c + m , b . Tri Lai Tiling Shuffling Phenomenon

Example 2 6 · 3 2 · 5 3 · 7 · 13 3 · 17 3 · 19 Tri Lai Tiling Shuffling Phenomenon

Example 2 5 · 7 · 11 2 · 13 3 · 17 3 · 19 · 71 Tri Lai Tiling Shuffling Phenomenon

Example 2 6 · 11 · 13 3 · 17 3 · 19 · 281 Tri Lai Tiling Shuffling Phenomenon

Example The left tiling number: 2 5 · 3 · 7 3 · 11 3 · 13 4 · 17 Tri Lai Tiling Shuffling Phenomenon

Example The left tiling number: 2 5 · 3 · 7 3 · 11 3 · 13 4 · 17 The right tiling number: 2 6 · 7 3 · 11 · 13 4 · 17 · 2683 Tri Lai Tiling Shuffling Phenomenon

Shuffling Phenomenon The tiling number of punctured regions are not given by simple product formulas. Tri Lai Tiling Shuffling Phenomenon

Shuffling Phenomenon The tiling number of punctured regions are not given by simple product formulas. A small modification (in the position, orientation, side-length, etc.) of the region would lead to unpredictable change in the tiling number. Tri Lai Tiling Shuffling Phenomenon

Shuffling Phenomenon The tiling number of punctured regions are not given by simple product formulas. A small modification (in the position, orientation, side-length, etc.) of the region would lead to unpredictable change in the tiling number. However, in certain cases, our modifications change the tiling number by only a simple multiplicative factor. Tri Lai Tiling Shuffling Phenomenon

First Example: Doubly-dented hexagon x+d x+d u y + + u y y u + + y u l l y + y d d + d + + y d y x+u x+u (a) (b) Position set of upper holes U = { s 1 , s 2 , . . . , s u } ⊂ [ x + y + u + d ] Position set of lower holes D = { t 1 , t 2 , . . . , t d } ⊂ [ x + y + u + d ] Assume U ∩ D = ∅ . Doubly-dented hexagon: H x , y ( U , D ) Tri Lai Tiling Shuffling Phenomenon

Shuffling the holes x+d x+d y+u y+u y+u y+u l l y+d y+d y+d y+d x+u x+u (a) (b) U = { s 1 , s 2 , . . . , s u } → U ′ = { s ′ 1 , s ′ 2 , . . . , s ′ u } D = { t 1 , t 2 , . . . , t d } → D ′ = { t ′ 1 , t ′ 2 , . . . , t ′ d } H x , y ( U , D ) → H x , y ( U ′ , D ′ ) Tri Lai Tiling Shuffling Phenomenon

Shuffling the holes x+d x+d y+u y+u y+u y+u l l y+d y+d y+d y+d x+u x+u (a) (b) Tiling number of H x , y ( U , D ) : 2 9 · 3 5 · 5 3 · 7 4 · 20107 Tri Lai Tiling Shuffling Phenomenon

Shuffling the holes x+d x+d y+u y+u y+u y+u l l y+d y+d y+d y+d x+u x+u (a) (b) Tiling number of H x , y ( U , D ) : 2 9 · 3 5 · 5 3 · 7 4 · 20107 Tiling number of H x , y ( U ′ , D ′ ) : 2 11 · 3 3 · 5 3 · 7 5 · 20107 Tri Lai Tiling Shuffling Phenomenon

Shuffling the holes x+d x+d y+u y+u y+u y+u l l y+d y+d y+d y+d x+u x+u (a) (b) Tiling number of H x , y ( U , D ) : 2 9 · 3 5 · 5 3 · 7 4 · 20107 Tiling number of H x , y ( U ′ , D ′ ) : 2 11 · 3 3 · 5 3 · 7 5 · 20107 The ratio of tilings: 2 − 2 · 3 2 · 7 − 1 Tri Lai Tiling Shuffling Phenomenon

Shuffling Theorem x+d x+d y+u y+u y+u y+u l l y+d y+d y+d y+d x+u x+u (a) (b) Theorem (Shuffling Theorem) For U = { s 1 , s 2 , . . . , s u } , D = { t 1 , t 2 , . . . , t d } , U ′ = { s ′ 1 , s ′ 2 , . . . , s ′ u } , D ′ = { t ′ d } of [ x + y + n ] , such that U ∪ D = U ′ ∪ D ′ and 1 , t ′ 2 , . . . , t ′ U ∩ D = U ′ ∩ D ′ = ∅ M( H x , y ( U , D )) s j − s i t j − t i � � · M( H x , y ( U ′ , D ′ )) = (1) s ′ j − s ′ t ′ j − t ′ i i 1 ≤ i < j ≤ u 1 ≤ i < j ≤ d Tri Lai Tiling Shuffling Phenomenon

q -Shuffling Theorem A right lozenge is weighted by q t , where t is the distance to the base of the hexagon. x+d 15 15 15 14 14 14 13 13 12 u y + + u y 11 11 11 10 10 10 9 l 8 8 8 6 5 5 4 4 4 y 4 d + + d y 3 3 2 2 2 2 1 1 1 x+u Tri Lai Tiling Shuffling Phenomenon

q -Shuffling Theorem A right lozenge is weighted by q t , where t is the distance to the base of the hexagon. A tiling is weighted by the product of weights of all lozenges x+d 15 15 15 14 14 14 13 13 12 u y + + u y 11 11 11 10 10 10 9 l 8 8 8 6 5 5 4 4 4 y 4 d + + d y 3 3 2 2 2 2 1 1 1 x+u Tri Lai Tiling Shuffling Phenomenon

q -Shuffling Theorem A right lozenge is weighted by q t , where t is the distance to the base of the hexagon. A tiling is weighted by the product of weights of all lozenges M q ( R ) is the sum of weights of all tilings of R x+d 15 15 15 14 14 14 13 13 12 u y + + u y 11 11 11 10 10 10 9 l 8 8 8 6 5 5 4 4 4 y 4 d + + d y 3 3 2 2 2 2 1 1 1 x+u Tri Lai Tiling Shuffling Phenomenon

q -Shuffling Theorem Theorem (L. –Rohatgi 2019) q s j − q s i q t j − q t i M q ( H x , y ( U , D )) M q ( H x , y ( U ′ , D ′ )) = q C · � � i · q s ′ j − q s ′ q t ′ j − q t ′ i 1 ≤ i < j ≤ u 1 ≤ i < j ≤ d Tri Lai Tiling Shuffling Phenomenon

Schur functions Theorem (Shuffling Theorem) For U = { s 1 , s 2 , . . . , s u } , D = { t 1 , t 2 , . . . , t d } , U ′ = { s ′ 1 , s ′ 2 , . . . , s ′ u } , d } of [ x + y + n ] , such that U ∪ D = U ′ ∪ D ′ and D ′ = { t ′ 1 , t ′ 2 , . . . , t ′ U ∩ D = U ′ ∩ D ′ = ∅ s j − s i t j − t i � � · j − i j − i M( H x , y ( U , D )) 1 ≤ i < j ≤ u 1 ≤ i < j ≤ d M( H x , y ( U ′ , D ′ )) = s ′ j − s ′ t ′ j − t ′ i i � � · j − i j − i 1 ≤ i < j ≤ u 1 ≤ i < j ≤ d All products can be expressed in terms of special Schur functions. Tri Lai Tiling Shuffling Phenomenon

Schur functions Theorem (Shuffling Theorem) s j − s i t j − t i � � · j − i j − i M( H x , y ( U , D )) 1 ≤ i < j ≤ u 1 ≤ i < j ≤ d M( H x , y ( U ′ , D ′ )) = s ′ j − s ′ t ′ j − t ′ i i � � · j − i j − i 1 ≤ i < j ≤ u 1 ≤ i < j ≤ d s j − s i � j − i = s λ ( s 1 ,..., s u ) (1 , 1 , . . . ), where 1 ≤ i < j ≤ u λ ( s 1 , . . . , s u ) = ( s u − u + 1 , s u − 1 − u + 2 , . . . , s 3 − 2 , s 2 − 1 , s 1 ) Tri Lai Tiling Shuffling Phenomenon

Schur functions Theorem (Shuffling Theorem) s j − s i t j − t i � � · j − i j − i M( H x , y ( U , D )) 1 ≤ i < j ≤ u 1 ≤ i < j ≤ d M( H x , y ( U ′ , D ′ )) = s ′ j − s ′ t ′ j − t ′ i i � � · j − i j − i 1 ≤ i < j ≤ u 1 ≤ i < j ≤ d s j − s i � j − i = s λ ( s 1 ,..., s u ) (1 , 1 , . . . ), where 1 ≤ i < j ≤ u λ ( s 1 , . . . , s u ) = ( s u − u + 1 , s u − 1 − u + 2 , . . . , s 3 − 2 , s 2 − 1 , s 1 ) s λ ( U ) (1 , 1 ,... ) s λ ( D ) (1 , 1 ,... ) RHS = s λ ( U ′ ) (1 , 1 ,... ) s λ ( D ′ ) (1 , 1 ,... ) Tri Lai Tiling Shuffling Phenomenon

Schur functions x+d x+d y+u y+u y+u y+u l y+d y+d y+d y+d x+u (a) x+u (b) M( H x , y ( U , D )) = � S ⊆ ( U ∪ D ) c M( S x + d , y + u ( U ∪ S ))M( S x + u , y + d ( D ∪ S )) | S | = y Tri Lai Tiling Shuffling Phenomenon

Schur functions x+d x+d y+u y+u y+u y+u l y+d y+d y+d y+d x+u (a) x+u (b) M( H x , y ( U , D )) = � S ⊆ ( U ∪ D ) c M( S x + d , y + u ( U ∪ S ))M( S x + u , y + d ( D ∪ S )) | S | = y (Cohn–Larsen –Propp) M( S x + d , y + u ( U ∪ S )) = s λ ( U ∪ S ) (1 , 1 , . . . ) Tri Lai Tiling Shuffling Phenomenon