# 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 triomino and I heptomino¶

Prime rectangles: 29.

## Smallest rectangle tilings¶

Smallest rectangle (1x10):

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 $A(w; h)$

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

$G(N(2); x) = \frac{4x^{10} + 4x^{11} + 4x^{12} + x^{13} + x^{14} - 3x^{15} - 2x^{16} - 13x^{17} - 14x^{18} - 14x^{19} - 14x^{20} - 14x^{21} - 7x^{22} - 7x^{23} + 4x^{24} + 6x^{25} + 6x^{26} + 9x^{27} + 10x^{28} + 6x^{29} + 7x^{30} + 2x^{31} - 2x^{34} - 2x^{35} - x^{36} - x^{37}}{1 + x + x^2 - 2x^3 - 2x^4 - 3x^5 - 5x^7 - 3x^8 - 4x^9 + 5x^{10} + 4x^{11} + 8x^{12} + 3x^{13} + 13x^{14} + 9x^{15} + 10x^{16} + x^{18} - 5x^{19} - 4x^{20} - 15x^{21} - 12x^{22} - 13x^{23} - 7x^{24} - 5x^{25} - x^{26} + 2x^{27} + 8x^{28} + 6x^{29} + 7x^{30} + 4x^{31} + 2x^{32} + x^{33} - x^{34} - 2x^{35} - x^{36} - x^{37}} \tag{2}$