# 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 tetromino¶

Prime rectangles: 16.

## Smallest rectangle tilings¶

Smallest rectangle (1x7):

Smallest square (4x4):

## 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 $A(w; h)$

$G(N(1); x) = \frac{2x^7 + 2x^8 + 2x^9 + x^{10}}{1 + x + x^2 - x^3 - 3x^4 - 3x^5 - 2x^6 + x^7 + 2x^8 + 2x^9 + x^{10}} \tag{1}$

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

$G(N(3); x) = \frac{2x^5 - 3x^6 + 9x^7 - 14x^8 - x^9 + 4x^{10} - 17x^{11} + 10x^{12} + x^{13} - 31x^{14} + 53x^{15} - 2x^{16} - 18x^{17} + 48x^{18} - 49x^{19} + 22x^{20} + x^{21} - 10x^{22} + 15x^{23} - 3x^{24} + x^{25} - x^{26} - x^{27}}{1 - 3x + 3x^2 - 5x^3 + 4x^4 + 8x^5 - 8x^6 + x^7 + 11x^8 - 30x^9 + 36x^{10} - 14x^{11} - 46x^{12} + 67x^{13} - 80x^{14} + 49x^{15} + 23x^{16} - 46x^{17} + 61x^{18} - 46x^{19} + 9x^{20} + 3x^{21} - 11x^{22} + 16x^{23} - 2x^{24} + x^{25} - x^{26} - x^{27}} \tag{3}$