POLYOMINO TILINGS

Polyomino Tilings

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

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

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

Red number (C) - 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 \ h1234567N>0
10
211P22C
3033C0
411C55C1111C3636C
5088C09595C0
611C1313C4141C281281C11831183C67286728C
702121C0781781C03152931529C0
811C3434C153153C22452245C1482414824C167089167089C12926971292697Call
905555C063366336C0817991817991C02k
1011C8989C571571C1806118061C185921185921C42131334213133C5317551753175517Call
110144144C05120551205C02100179921001799C02k
1211C233233C21312131C145601145601C23320972332097C106912793106912793C2.18897811×10¹⁰2188978117Call
130377377C0413351413351C0536948224536948224C02k
1411C610610C79537953C11745001174500C2925316029253160C2.72024663×10¹⁰2720246633C9.01241674×10¹¹90124167441Call
150987987C033356513335651C01.37043005×10¹¹13704300553C02k
1611C15971597C2968129681C94759019475901C366944287366944287C6.92892889×10¹¹69289288909C3.71070820×10¹³3710708201969Call
N>02kall2kall2kall2k

Smallest prime reptiles

Smallest prime reptile (2Ix2):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
I domino15P41P2245C185921P106912793C90124167441P

Smallest tori tilings

Smallest torus and smallest odd torus (1x2):

Smallest square torus (2x2):

Tori tilings' solutions count (including translations)

w \ h12345678
100
22288
300141400
42236365050272272
50082820072272200
62220020022422431083108992299229017690176
7004784780010082100820040199840199800
82211561156105810583995239952155682155682311386031138601968153819681538≥230000000≥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 $A(w; h)$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

See Also

L2 15-ominoL1 20-omino