POLYOMINO TILINGS

Polyomino Tilings

Select polyominoes for a set (currently 1 or 2), for which tilings should be shown.

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You may also see list of all polyomino sets for which data is available here.

L1 9-omino¶

Area: 9.

Size: 2x8.

Is rectangular: no.

Is convex: yes.

Holes: 0.

Order: 2.

Square order: 36.

Odd order: 33.

Prime rectangles: ≥ 7.

Smallest rectangle tilings¶

Smallest rectangle (2x9):

Smallest square (18x18):

Smallest odd rectangle (11x27):

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 reptile (9L1x8):

Reptile tilings' solutions count (including symmetric)¶

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 $A(w; h)$

$G(N(2); x) = \frac{1}{1 - 2x^9} \tag{1}$

$G(N(4); x) = \frac{1}{1 - 4x^9} \tag{2}$

$G(N(6); x) = \frac{1}{1 - 8x^9} \tag{3}$

$G(N(8); x) = \frac{1}{1 - 16x^9} \tag{4}$

$G(N(9); x) = \frac{1}{1 - 2x^2} \tag{5}$

$G(N(10); x) = \frac{1 - 96x^9 + 2048x^{18}}{1 - 128x^9 + 5112x^{18} - 64768x^{27} - 16528x^{36}} \tag{6}$

$G(N(11); x) = \frac{1 - 392x^{18} - 1045540x^{36} + 8355872x^{54} - 4194304x^{63} - 100663296x^{72} + 805306368x^{81} + 14596308992x^{90} - 12884901888x^{108}}{1 - 6536x^{18} - 1024x^{27} + 1330076x^{36} + 393216x^{45} + 6444425248x^{54} + 1061154816x^{63} - 17767071488x^{72} + 8254390272x^{81} + 169756721152x^{90} - 824667275264x^{99} - 6712228577280x^{108} + 1649267441664x^{117} - 103079215104x^{126}} \tag{7}$

See Also¶