POLYOMINO TILINGS

Polyomino Tilings

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

U pentomino and X pentomino¶

Prime rectangles: ≥ 8.

Smallest rectangle tilings¶

Smallest rectangle (3x5):

Smallest square (15x15):

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 $A(w; h)$

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

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

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

$G(N(8); x) = \frac{2x^{15}}{1 - 2x^{15}} \tag{4}$

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

$G(N(10); x) = \frac{x^3}{1 - x^3} \tag{6}$

$G(N(11); x) = \frac{3x^{15}}{1 - 3x^{15}} \tag{7}$

$G(N(12); x) = \frac{x^5}{1 - x^5} \tag{8}$

$G(N(13); x) = \frac{3x^{15}}{1 - 3x^{15}} \tag{9}$

$G(N(14); x) = \frac{4x^{15}}{1 - 4x^{15}} \tag{10}$

$G(N(15); x) = \frac{x^3 + x^5}{1 - x^3 - x^5} \tag{11}$

$G(N(16); x) = \frac{6x^{15}}{1 - 6x^{15}} \tag{12}$

$G(N(18); x) = \frac{x^5 - 4x^{10} + 10x^{15} - 12x^{20} + 13x^{25} - 8x^{30} + 2x^{35}}{1 - 5x^5 + 10x^{10} - 14x^{15} + 13x^{20} - 13x^{25} + 8x^{30} - 2x^{35}} \tag{13}$

Attributions¶

1. Prime rectangle list taken from http://www.recmath.com/PolyPages/PolyPages/index.htm?5pairs.htm

See Also¶