# POLYOMINO TILINGS

## Polyomino Tilings

Select polyominoes for a set (currently 1 or 2), for which tilings should be shown.

Then click "Show" button.

You may also see list of all polyomino sets for which data is available here.

## J1 octomino¶

Area: 8.

Size: 3x4.

Holes: 0.

Order: 2.

Square order: 2.

Prime rectangles: ≥ 1.

## Smallest rectangle tilings¶

Smallest rectangle and smallest square (4x4):

## Rectangle tilings' solutions count (including symmetric)¶

Blue number - strongly prime rectangle (which cannot be divided into two or more number of rectangles tileable by this set).

Green number - weakly prime rectangle (which cannot be divided into two rectangles tileable by this set, but which can be divided into three or more rectangles).

Purple number - prime rectangle (unknown if weakly or strongly prime).

Red number - composite rectangle (which can be divided into two rectangles tileable by this set).

Gray number - it is unknown whether rectangle is prime or composite.

Question mark (?) - solution count is unknown.

Click on underlined numbers to view picture with one solution.

w \ h
1-3
4
5-7
8
9-11
12
13
14
15
16
17
18
19
20
N>0
1-3
0
4
0
4
5-7
0
0
0
8
0
16
0
256
9-11
0
0
0
0
0
12
0
64
0
4096
0
262144
13
0
0
0
0
0
0
?
14
0
0
0
0
0
0
?
?
15
0
0
0
0
0
0
?
?
?
16
0
256
0
65536
0
16777216
?
?
?
≥1
17
0
0
0
0
0
0
?
?
?
?
?
18
0
0
0
0
0
0
?
?
?
?
?
?
19
0
0
0
0
0
0
?
?
?
?
?
?
?
20
0
1024
0
1048576
0
1.07374182×10¹⁰
?
?
?
≥1
?
?
?
≥1
21
0
0
0
0
0
0
?
?
?
?
?
?
?
?
?
22
0
0
0
0
0
0
?
?
?
?
?
?
?
?
?
23
0
0
0
0
0
0
?
?
?
?
?
?
?
?
?
24
0
4096
0
16777216
0
6.87194767×10¹¹
?
?
?
≥1
?
?
?
≥1
?
25
0
0
0
0
0
0
?
?
?
?
?
?
?
?
?
26
0
0
0
0
0
0
?
?
?
?
?
?
?
?
?
27
0
0
0
0
0
0
?
?
?
?
?
?
?
?
?
28
0
16384
0
268435456
0
4.39804651×10¹³
?
?
?
≥1
?
?
?
≥1
?
29
0
0
0
0
0
0
?
?
?
?
?
?
?
?
?
30
0
0
0
0
0
0
?
?
?
?
?
?
?
?
?
31
0
0
0
0
0
0
?
?
?
?
?
?
?
?
?
32
0
65536
0
4.29496729×10¹⁰
0
2.81474976×10¹⁵
?
?
?
≥1
?
?
?
≥1
?
N>0
x
4k
x
4k
x
4k
?
?
?
?
?
?
?
?

## Formulas¶

$N(w; h)$ - number of ways to tile $w\times h$ rectangle (including symmetric solutions)

$T(w; h) = \begin{cases} 1, & N(w; h) \geq 1 \\ 0, & \text{else} \end{cases}$ - tileability function, $1$ if tiles rectangle, $0$ otherwise

$A(w; h) = \left(N(w; h)\right)^{\frac{1}{wh}}$ - average number of ways to tile cell in $w\times h$ rectangle (including symmetric solutions)

$G(T; x; y) = \sum_{w=1}^{\infty}\sum _{h=1}^{\infty}T(w; h)x^wy^h$ - bivariate generating function of $T(w; h)$

$G(A; x; y) = \sum_{w=1}^{\infty}\sum _{h=1}^{\infty}A(w; h)x^wy^h$ - bivariate generating function of $A(w; h)$

$N(1; n) = T(1; n) = 0, \qquad n \geq 1 \tag{1}$

$N(2; n) = T(2; n) = 0, \qquad n \geq 1 \tag{2}$

$N(3; n) = T(3; n) = 0, \qquad n \geq 1 \tag{3}$

$N(4; n) = 4 \times N(4; n - 4), \qquad n \geq 5 \tag{4}$

$N(8; n) = 16 \times N(8; n - 4), \qquad n \geq 5 \tag{5}$

$N(12; n) = 64 \times N(12; n - 4), \qquad n \geq 5 \tag{6}$

## See Also¶

I2 octominoL1 octomino