# 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.

## Dominoes and I triomino¶

Prime rectangles: ≥ 9.

## Smallest rectangle tilings¶

Smallest rectangle (1x5):

Smallest square (3x3):

## 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 (2Ix2, 3Ix2):

polyomino \ n²
I domino
?
?
?
?
I triomino
?
?
?
?

## Smallest common multiples¶

Smallest common multiple (area 6):

area
6
12
solutions
4
391

## 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 $A(w; h)$

$N(1; n) = -N(1; n - 1) + N(1; n - 2) + 3\times N(1; n - 3) + 2\times N(1; n - 4) - N(1; n - 5) - 2\times N(1; n - 6) - N(1; n - 7) \tag{1}$

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

$N(2; n) = 2\times N(2; n - 1) + N(2; n - 2) + N(2; n - 3) - 5\times N(2; n - 4) - 2\times N(2; n - 5) - N(2; n - 6) + 2\times N(2; n - 7) + N(2; n - 8) + N(2; n - 9) + N(2; n - 10) \tag{3}$