# 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.

## L1 20-omino¶

Area: 20.

Perimeter: 18.

Size: 3x10.

Is rectangular: no.

Is convex: yes.

Holes: 0.

Order: 12.

Square order: 20.

Prime rectangles: ≥ 3.

## Smallest rectangle tilings¶

Smallest rectangle (12x20):

Smallest square (20x20):

## Rectangle tilings' solutions count (including symmetric)¶

Blue number (P) - strongly prime rectangle (which cannot be divided into two or more number of rectangles tileable by this set).

Green number (W) - weakly prime rectangle (which cannot be divided into two rectangles tileable by this set, but which can be divided into three or more rectangles).

Red number (C) - 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.

w \ h1-111213-151617-1920N>0
1-110
1200
13-15000
160000
17-1900000
20022P022P022P
210000000?
220000000?
230000000?
240000066C?
250000000?
260000000?
270000000?
28000001010C?
290000000?
300000000?
310000000?
32000001414C?
330000000?
340000000?
350000000?
36000002626C?
370000000?
380000000?
390000000?
40044C044C04646C?
410000000?
420000000?
430000000?
44000007474C?
450000000?
460000000?
470000000?
4800000126126C?
490000000?
500000000?
510000000?
5200000218218C?
530000000?
540000000?
550000000?
5600000366366C?
570000000?
580000000?
590000000?
60088C088C0618618C?
610000000?
620000000?
630000000?
640000010541054C?
650000000?
660000000?
670000000?
680000017861786C?
690000000?
700000000?
710000000?
720000030223022C?
730000000?
740000000?
750000000?
760000051305130C?
770000000?
780000000?
790000000?
8001616C01616C087028702C?
810000000?
820000000?
830000000?
84000001474614746C?
850000000?
860000000?
870000000?
88000002500625006C?
890000000?
900000000?
910000000?
92000004241042410C?
930000000?
940000000?
950000000?
96000007190271902C?
970000000?
980000000?
990000000?
10003232C03232C0121914121914C?
N>0x20kx20kx?

## 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 $A(w; h)$

$N(12; n) = 2 \times N(12; n - 20), \qquad n \geq 21 \tag{1}$

$N(16; n) = 2 \times N(16; n - 20), \qquad n \geq 21 \tag{2}$