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.
Area: 6.
Size: 3x3.
Holes: 0.
Order: 2.
Square order: 24.
Prime rectangles: ≥ 1.
Smallest rectangle (3x4):
Smallest square (12x12):
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.
$N(w; h)$ - 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 $A(w; h)$
$N(1; n) = T(1; n) = 0, \qquad n \geq 1 \tag{1}$
$N(2; n) = T(2; n) = 0, \qquad n \geq 1 \tag{2}$
$N(3; n) = 2 \times N(3; n - 4), \qquad n \geq 5 \tag{3}$
$N(4; n) = 2 \times N(4; n - 3), \qquad n \geq 4 \tag{4}$
$N(5; n) = T(5; n) = 0, \qquad n \geq 1 \tag{5}$
$N(6; n) = 4 \times N(6; n - 4), \qquad n \geq 5 \tag{6}$
$N(7; n) = 256 \times N(7; n - 12), \qquad n \geq 13 \tag{7}$
$N(8; n) = 4 \times N(8; n - 3), \qquad n \geq 4 \tag{8}$
$N(9; n) = 8 \times N(9; n - 4), \qquad n \geq 5 \tag{9}$
$N(10; n) = 3072 \times N(10; n - 12), \qquad n \geq 13 \tag{10}$
$N(11; n) = 6144 \times N(11; n - 12), \qquad n \geq 13 \tag{11}$
$N(12; n) = 8 \times N(12; n - 3) + 16 \times N(12; n - 4), \qquad n \geq 5 \tag{12}$
$N(13; n) = 32768 \times N(13; n - 12), \qquad n \geq 13 \tag{13}$
$N(14; n) = 98304 \times N(14; n - 12), \qquad n \geq 13 \tag{14}$
$N(15; n) = 128 \times N(15; n - 4) - 6144 \times N(15; n - 8) + 262144 \times N(15; n - 12) - 5242880 \times N(15; n - 16) + 83886080 \times N(15; n - 20) - 536870912 \times N(15; n - 24), \qquad n \geq 25 \tag{15}$
$N(16; n) = 48 \times N(16; n - 3) - 768 \times N(16; n - 6) + 4096 \times N(16; n - 9) + 327680 \times N(16; n - 12) - 4194304 \times N(16; n - 15) + 25165824 \times N(16; n - 18), \qquad n \geq 19 \tag{16}$
$G(N(3); x) = \frac{1}{1 - 2x^4} \tag{17}$
$G(N(4); x) = \frac{1}{1 - 2x^3} \tag{18}$
$G(N(6); x) = \frac{1}{1 - 4x^4} \tag{19}$
$G(N(7); x) = \frac{1}{1 - 256x^{12}} \tag{20}$
$G(N(8); x) = \frac{1}{1 - 4x^3} \tag{21}$
$G(N(9); x) = \frac{1}{1 - 8x^4} \tag{22}$
$G(N(10); x) = \frac{1}{1 - 3072x^{12}} \tag{23}$
$G(N(11); x) = \frac{1}{1 - 6144x^{12}} \tag{24}$
$G(N(12); x) = \frac{1}{1 - 8x^3 - 16x^4} \tag{25}$
$G(N(13); x) = \frac{1}{1 - 32768x^{12}} \tag{26}$
$G(N(14); x) = \frac{1}{1 - 98304x^{12}} \tag{27}$
$G(N(15); x) = \frac{1 - 96x^4 + 3072x^8 - 32768x^{12}}{1 - 128x^4 + 6144x^8 - 262144x^{12} + 5242880x^{16} - 83886080x^{20} + 536870912x^{24}} \tag{28}$
$G(N(16); x) = \frac{1 - 32x^3 + 256x^6}{1 - 48x^3 + 768x^6 - 4096x^9 - 327680x^{12} + 4194304x^{15} - 25165824x^{18}} \tag{29}$
$G(N(17); x) = \frac{1 - 65536x^{12}}{1 - 1376256x^{12} + 68719476736x^{24}} \tag{30}$
$G(N(18); x) = \frac{1 - 192x^4 + 12288x^8 - 262144x^{12}}{1 - 256x^4 + 24576x^8 - 3670016x^{12} + 260046848x^{16} - 10267656192x^{20} + 141733920768x^{24}} \tag{31}$
$G(N(19); x) = \frac{1 - 14680064x^{12} + 13606456393728x^{24} + 7266557998762295296x^{36} - 36397492092765200808148992x^{48} + 48372700463207372582351232040960x^{60} - 25541004588578434450176025192877260800x^{72} + 5911894040165929663836097612456999671300096x^{84} + 558798723226961312422803433363423879486385422336x^{96} - 45437589222420710052913953165631870656795481565298688x^{108} - 1082161848769781680155464365438617400285212588475452227584x^{120}}{1 - 20447232x^{12} - 31954556682240x^{24} + 17043872789783642112x^{36} + 103129548010101336675713024x^{48} - 140569290715267987926936864161792x^{60} + 131406197612185736373182868273561075712x^{72} - 56806242864046038263184132070577665722023936x^{84} + 8567148247845973072418431577152182913246712299520x^{96} - 466677208780852039795671841064694715794885161592553472x^{108} + 162965336363250360902692504375440886350206777825016414208x^{120} + 283682235683905648762674050613540919780366768793308948747780096x^{132}} \tag{1}$
$G(N(20); x) = \frac{1 - 128x^3 + 6144x^6 - 147456x^9 + 2097152x^{12} - 37748736x^{15} + 1744830464x^{18} - 47244640256x^{21} + 326417514496x^{24} + 5497558138880x^{27} - 92358976733184x^{30} + 281474976710656x^{33} - 4503599627370496x^{36}}{1 - 160x^3 + 10240x^6 - 344064x^9 - 8912896x^{12} + 1488977920x^{15} - 58317602816x^{18} + 1206885810176x^{21} - 19396072308736x^{24} + 475538779013120x^{27} - 15256823347019776x^{30} + 322148110845345792x^{33} - 1787929052066086912x^{36} - 41072828601618923520x^{39} + 637565592047586377728x^{42} - 1992248359960631574528x^{45} + 28629346802397224108032x^{48}} \tag{1}$
