# 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.

## Dominoes¶

Area: 2.

Perimeter: 6.

Size: 1x2.

Is rectangular: yes.

Is convex: yes.

Holes: 0.

Order: 1.

Square order: 2.

Odd order: 1.

Prime rectangles: ≥ 1.

## Smallest rectangle tilings¶

Smallest rectangle and smallest odd rectangle (1x2):

Smallest square (2x2):

## 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
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
N>0
1
0
2
1
2
3
0
3
0
4
1
5
11
36
5
0
8
0
95
0
6
1
13
41
281
1183
6728
7
0
21
0
781
0
31529
0
8
1
34
153
2245
14824
167089
1292697
≥1
9
0
55
0
6336
0
817991
0
≥1
0
10
1
89
571
18061
185921
4213133
53175517
≥1
≥1
≥1
11
0
144
0
51205
0
21001799
0
≥1
0
≥1
0
12
1
233
2131
145601
2332097
106912793
2.18897811×10¹⁰
≥1
≥1
≥1
≥1
≥1
13
0
377
0
413351
0
536948224
0
≥1
0
≥1
0
≥1
0
14
1
610
7953
1174500
29253160
2.72024663×10¹⁰
9.01241674×10¹¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
15
0
987
0
3335651
0
1.37043005×10¹¹
0
≥1
0
≥1
0
≥1
0
≥1
0
16
1
1597
29681
9475901
366944287
6.92892889×10¹¹
3.71070820×10¹³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
17
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
18
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
19
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
20
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
21
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
N>0
2k
all
2k
all
2k
all
2k
all
2k
all
2k
all
2k
all
2k
all
?
?
?
?

## Smallest prime reptiles¶

Smallest prime reptile (2Ix2):

polyomino \ n²
I domino
1
5
41
2245
185921
106912793
90124167441

## Smallest tori tilings¶

Smallest torus and smallest odd torus (1x2):

Smallest square torus (2x2):

w \ h
1
2
3
4
5
6
7
8
1
0
2
2
8
3
0
14
0
4
2
36
50
272
5
0
82
0
722
0
6
2
200
224
3108
9922
90176
7
0
478
0
10082
0
401998
0
8
2
1156
1058
39952
155682
3113860
19681538
≥230000000

## Smallest Baiocchi figures¶

Smallest Baiocchi figure (area 4):

## 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 $A(w; h)$

$N(1; n) = N(1; n - 2) \tag{1}$

$N(2; n) = N(2; n - 1) + N(2; n - 2) \tag{2}$

$N(3; n) = 4\times N(3; n - 2) - N(3; n - 4) \tag{3}$

$N(4; n) = N(4; n - 1) + 5\times N(4; n - 2) + N(4; n - 3) - N(4; n - 4) \tag{4}$

$N(5; n) = 15\times N(5; n - 2) - 32\times N(5; n - 4) + 15\times N(5; n - 6) - N(5; n - 8) \tag{5}$

$N(6; n) = N(6; n - 1) + 20\times N(6; n - 2) + 10\times N(6; n - 3) - 38\times N(6; n - 4) - 10\times N(6; n - 5) + 20\times N(6; n - 6) - N(6; n - 7) - N(6; n - 8) \tag{6}$

$N(7; n) = 56 \times N(7; n - 2) - 672 \times N(7; n - 4) + 2632 \times N(7; n - 6) - 4094 \times N(7; n - 8) + 2632 \times N(7; n - 10) - 672 \times N(7; n - 12) + 56 \times N(7; n - 14) - N(7; n - 16) \tag{7}$

$G(N(1); x) = \frac{1}{1 - x^2} \tag{8}$

$G(N(2); x) = \frac{1}{1 - x - x^2} \tag{9}$

$G(N(3); x) = \frac{1 - x^2}{1 - 4x^2 + x^4} \tag{10}$

$G(N(4); x) = \frac{1 - x^2}{1 - x - 5x^2 - x^3 + x^4} \tag{11}$

$G(N(5); x) = \frac{1 - 7x^2 + 7x^4 - x^6}{1 - 15x^2 + 32x^4 - 15x^6 + x^8} \tag{12}$

$G(N(6); x) = \frac{1 - 8x^2 - 2x^3 + 8x^4 - x^6}{1 - x - 20x^2 - 10x^3 + 38x^4 + 10x^5 - 20x^6 + x^7 + x^8} \tag{13}$

$G(N(7); x) = \frac{1 - 35x^2 + 277x^4 - 727x^6 + 727x^8 - 277x^{10} + 35x^{12} - x^{14}}{1 - 56x^2 + 672x^4 - 2632x^6 + 4094x^8 - 2632x^{10} + 672x^{12} - 56x^{14} + x^{16}} \tag{14}$

$G(T; x; y) = \frac{xy(x y+x+y)}{\left(1-x^2\right)\left(1-y^2\right)} \tag{15}$