# 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.

## I pentomino and I heptomino¶

Prime rectangles: ≥ 36.

## Smallest rectangle tilings¶

Smallest rectangle (1x12):

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 $A(w; h)$

$G(N(1); x) = \frac{2x^{12} + 2x^{13} + 2x^{14} + 2x^{15} + 2x^{16} + x^{17} + x^{18}}{1 + x + x^2 + x^3 + x^4 - x^5 - x^6 - 3x^7 - 3x^8 - 3x^9 - 2x^{10} - 2x^{11} + x^{12} + x^{13} + 2x^{14} + 2x^{15} + 2x^{16} + x^{17} + x^{18}} \tag{1}$

$G(N(2); x) = \frac{4x^{12} + 8x^{13} + 12x^{14} + 16x^{15} + 20x^{16} + 17x^{17} + 10x^{18} - 8x^{19} - 26x^{20} - 44x^{21} - 58x^{22} - 69x^{23} - 73x^{24} - 70x^{25} - 56x^{26} - 43x^{27} - 31x^{28} - 20x^{29} - 10x^{30} - x^{31} + 4x^{32} + 4x^{33} + 4x^{34} + 4x^{35} + 4x^{36} + 3x^{37} + x^{38}}{1 + 2x + 3x^2 + 4x^3 + 5x^4 + 2x^5 - 2x^6 - 11x^7 - 20x^8 - 29x^9 - 32x^{10} - 32x^{11} - 19x^{12} - 2x^{13} + 25x^{14} + 48x^{15} + 68x^{16} + 76x^{17} + 77x^{18} + 62x^{19} + 42x^{20} + 13x^{21} - 11x^{22} - 31x^{23} - 44x^{24} - 51x^{25} - 48x^{26} - 42x^{27} - 32x^{28} - 23x^{29} - 14x^{30} - 5x^{31} + x^{32} + 3x^{33} + 4x^{34} + 4x^{35} + 4x^{36} + 3x^{37} + x^{38}} \tag{2}$