# 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.

## Monominoes and Dominoes¶

Prime rectangles: ≥ 4.

## Smallest rectangle tilings¶

Smallest rectangle (1x3):

Smallest square (2x2):

## 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.

## Smallest prime reptiles¶

Smallest prime reptiles (1Ox2, 2Ix2):

polyomino \ n²
O monomino
?
?
?
?
I domino
?
?
?
?

## Smallest common multiples¶

Smallest common multiple (area 2):

area
2
solutions
1

## 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 $A(w; h)$

$G(N(1); x) = \frac{2x^3 + x^4}{1 - x - 2x^2 + x^3 + x^4} \tag{1}$

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

$G(N(3); x) = \frac{2x + 8x^2 + 12x^3 - 23x^4 - 66x^5 + 12x^6 + 53x^7 - x^8 - 14x^9 + x^{11}}{1 - 5x - 14x^2 + 34x^3 + 51x^4 - 71x^5 - 51x^6 + 55x^7 + 14x^8 - 14x^9 - x^{10} + x^{11}} \tag{3}$

$G(N(4); x) = \frac{3x + 32x^2 + 15x^3 - 212x^4 - 181x^5 + 417x^6 + 465x^7 - 391x^8 - 315x^9 + 167x^{10} + 58x^{11} - 27x^{12} - 2x^{13} + x^{14}}{1 - 11x - 27x^2 + 163x^3 + 159x^4 - 660x^5 - 200x^6 + 1012x^7 - 112x^8 - 504x^9 + 123x^{10} + 83x^{11} - 25x^{12} - 3x^{13} + x^{14}} \tag{4}$