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You may also see list of all polyomino sets for which data is available here.
Area: 2.
Perimeter: 6.
Size: 1x2.
Is rectangular: yes.
Is convex: yes.
Holes: 0.
Order: 1.
Square order: 2.
Odd order: 1.
Prime rectangles: ≥ 1.
Smallest rectangle and smallest odd rectangle (1x2):
Smallest square (2x2):
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 reptile (2Ix2):
Smallest torus and smallest odd torus (1x2):
Smallest square torus (2x2):
Smallest Baiocchi figure (area 4):
$N(w; h)$ - 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 $A(w; h)$
$N(1; n) = N(1; n - 2) \tag{1}$
$N(2; n) = N(2; n - 1) + N(2; n - 2) \tag{2}$
$N(3; n) = 4\times N(3; n - 2) - N(3; n - 4) \tag{3}$
$N(4; n) = N(4; n - 1) + 5\times N(4; n - 2) + N(4; n - 3) - N(4; n - 4) \tag{4}$
$N(5; n) = 15\times N(5; n - 2) - 32\times N(5; n - 4) + 15\times N(5; n - 6) - N(5; n - 8) \tag{5}$
$N(6; n) = N(6; n - 1) + 20\times N(6; n - 2) + 10\times N(6; n - 3) - 38\times N(6; n - 4) - 10\times N(6; n - 5) + 20\times N(6; n - 6) - N(6; n - 7) - N(6; n - 8) \tag{6}$
$N(7; n) = 56 \times N(7; n - 2) - 672 \times N(7; n - 4) + 2632 \times N(7; n - 6) - 4094 \times N(7; n - 8) + 2632 \times N(7; n - 10) - 672 \times N(7; n - 12) + 56 \times N(7; n - 14) - N(7; n - 16) \tag{7}$
$G(N(1); x) = \frac{1}{1 - x^2} \tag{8}$
$G(N(2); x) = \frac{1}{1 - x - x^2} \tag{9}$
$G(N(3); x) = \frac{1 - x^2}{1 - 4x^2 + x^4} \tag{10}$
$G(N(4); x) = \frac{1 - x^2}{1 - x - 5x^2 - x^3 + x^4} \tag{11}$
$G(N(5); x) = \frac{1 - 7x^2 + 7x^4 - x^6}{1 - 15x^2 + 32x^4 - 15x^6 + x^8} \tag{12}$
$G(N(6); x) = \frac{1 - 8x^2 - 2x^3 + 8x^4 - x^6}{1 - x - 20x^2 - 10x^3 + 38x^4 + 10x^5 - 20x^6 + x^7 + x^8} \tag{13}$
$G(N(7); x) = \frac{1 - 35x^2 + 277x^4 - 727x^6 + 727x^8 - 277x^{10} + 35x^{12} - x^{14}}{1 - 56x^2 + 672x^4 - 2632x^6 + 4094x^8 - 2632x^{10} + 672x^{12} - 56x^{14} + x^{16}} \tag{14}$
$G(T; x; y) = \frac{xy(x y+x+y)}{\left(1-x^2\right)\left(1-y^2\right)} \tag{15}$
