# 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.

## X pentomino and F hexomino¶

Prime rectangles: ≥ 120.

## Smallest rectangle tilings¶

Smallest rectangle and smallest square (7x7):

## 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.

## 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 $A(w; h)$

$G(N(7); x) = \frac{2x^7 + 64x^{11} - 40x^{24} - 180x^{25} + 5760x^{29} + 25920x^{30}}{1 - 2x^7 - 64x^{11} - 576x^{12} - 20x^{17} - 90x^{18} + 576x^{19} + 18432x^{23} + 82984x^{24} + 180x^{25} + 5760x^{29} + 25920x^{30} - 11520x^{36} - 51840x^{37}} \tag{1}$

$G(N(8); x) = \frac{1440x^{47} + 12672x^{50} - 5760x^{63} - 36000x^{71} - 316800x^{74} - 115200x^{77} - 5760x^{79} - 1013760x^{80} - 50688x^{82} + 144000x^{87} + 92160x^{101} + 144000x^{103} + 811008x^{104} + 1267200x^{106}}{1 - 12x^3 + 48x^6 - 64x^9 - 20x^{13} + 108x^{16} + 96x^{19} - 832x^{22} - 5x^{23} - 25x^{24} - 4x^{26} + 300x^{27} + 32x^{29} - 1280x^{30} + 1212x^{32} + 2560x^{33} - 3792x^{35} - 3840x^{36} + 500x^{37} - 192x^{38} + 5140x^{39} - 2700x^{40} + 256x^{41} - 160x^{42} - 900x^{43} + 400x^{45} + 10480x^{46} - 1315x^{47} - 448x^{48} + 10560x^{49} - 6812x^{50} - 256x^{51} + 26880x^{52} + 50288x^{53} + 3136x^{54} + 20x^{55} - 29980x^{56} - 768x^{57} + 16x^{58} + 105440x^{59} + 3072x^{60} + 192x^{61} - 198080x^{62} + 1164x^{63} - 7424x^{64} + 512000x^{65} - 19040x^{66} + 19280x^{67} - 10000x^{69} + 8256x^{70} + 36000x^{71} + 11200x^{72} - 8448x^{73} + 172800x^{74} + 6800x^{75} - 21504x^{76} - 1152320x^{77} - 76480x^{78} + 5260x^{79} + 552704x^{80} - 21760x^{81} + 27248x^{82} - 4063552x^{83} + 56320x^{84} - 207552x^{85} + 162304x^{86} - 144000x^{87} + 185600x^{88} - 414720x^{89} + 576000x^{90} - 512000x^{91} - 320x^{99} - 92160x^{101} - 256x^{102} - 144000x^{103} - 442368x^{104} + 17408x^{105} - 691200x^{106} + 3244032x^{107} - 45056x^{108} + 5068800x^{109}} \tag{2}$