How many ways are there to tile a rectangle with polyominoes?

polyominoes?
Aleksandrov N.M., Askerova A.A., Dzhuraev A.A., Kaniber V.V., Kruzhkov D.O.,

board
Tiling of a plane
Graphs and polyominoes on a lattice

squares with adjacent sides, a “spring” appears between them.
Percentage of

Number of springs

d=3
lattice with holes
surface of a “g” kind containing the lattice

General

Aim:
finding the statistical weight for macrostates n1, n2, n3, n4
n1: a spring between an outer node of primitive 1 and an inner node of primitive 2
n2: a spring between an inner node of primitive 1 and an outer node of primitive 2
n3: a spring between an inner node of primitive 1 and an inner node of primitive 2
n4: a spring between an outer node of primitive 1 and an outer node of primitive 2

square lattice
1
2
…
N
1
2
…
M
M x N nodes
M x N squares
N\M
2
1
…
1
2
…
(n1,n2,n3)=(6,2,4)

of possible arrangements within the lattice,
k is the Boltzmann constant.
DS

polymer solute

low molecular weight solute

Two-dimensional lattice model of solubility

formed by joining equal squares.
A graph covering of a

Suppose a (kxh)*(nxh) field, r monominoes and b L-minoes are given.
Enclose every polyomino into a hxh field and consider it as a whole.
The size of the initial field changes to kxn.

(P)
c is a number of pairs
a number of

set of polyominoes are given.
Construct all the possible partitions

Tiling of a 2x14 stripe with L-minoes and dominoes

Introduced methods

+ Triomino
L-mino
3xn stripe
Domino
Triomino
n
n
n

.
n
2
2
1,
1
[ n=0 ]
[n=0]
1
Derivation of the generating function

planar graph G
Pfaffian method
Pfaffian orientation
Let A denote the signed

The matrix A is called skew-symmetric if Aij = −Aji for all i, j. A is skew-symmetric.
Let A be a 2n × 2n skew-symmetric matrix. The Pfaffian of A is defined as the sum taken over all permutations of the set {1, ..., 2n}.



Given Muir’s Identity, Kastelyn’s Theorem can be reformulated as follows.
Let G be a graph equipped with any Pfaffian orientation. Let A be the corresponding signed adjacency matrix. Then

Muir’s Identity

1

1 Kastelyn’s Formula for Perfect Matchings. Rich Schwartz

Ways to tile a MxN field with dominoes

forming a spring
– with forming a

– mirroring
– rotation
– translation

Coloured digraphs method

χ

γ↑

γ

R

T

M

of a MxN field into arbitrary tiles is

2MN-M-N

to the whole class of similar problems
Cluster characterization (coloured digraphs

Conclusion

1918

adjacent/non-adjacent squares on a lattice
Paul Klee. Personal diaries, 1921-1931.
Investigating colour

three different musical representations: Notes on the musical staff, letter

The 12 musical notes repeat in a cyclic fashion, and therefore can be represented as 12 positions on a circle.

Michael Keith. From Polychords to Polya. Adventures in Musical Combinatorics. 1991. Vinculum Press, Princenton.

Piano Sonata Op. 79 (1808)
Claude Debussy, L’Apres-Midi d’un Faune (1892)
George

Chords

Michael Keith. From Polychords to Polya. Adventures in Musical Combinatorics. 1991. Vinculum Press, Princenton.

is the number of n-note chords in a scale of L notes having exactly a adjacencies

Graphs of a(t) for excerpts from several compositions

beginning represents a fragment from Debussy’s “Mazurka”.
Here, the t-axis shows

board
Tiling of a plane
Graphs and polyominoes on a lattice