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.
Area: 6.
Perimeter: 14.
Size: 2x5.
Is rectangular: no.
Is convex: yes.
Holes: 0.
Order: 2.
Square order: 6.
Odd order: 21.
Prime rectangles: ≥ 8.
Smallest rectangle (2x6):
Smallest square (6x6):
Smallest odd rectangle (9x14):
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 (6Lx6):
$N(w; h)$ - 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 $A(w; h)$
$G(N(2); x) = \frac{1}{1 - 2x^6} \tag{1}$
$G(N(4); x) = \frac{1}{1 - 4x^6} \tag{2}$
$G(N(6); x) = \frac{1}{1 - 2x^2 - 8x^6} \tag{3}$
$G(N(7); x) = \frac{1 - 144x^{12} - 3648x^{24} + 292864x^{36} - 13500416x^{48} + 1480589312x^{60} + 30601641984x^{72} - 2456721293312x^{84} + 42880953483264x^{96} - 2269391999729664x^{108} - 36028797018963968x^{132}}{1 - 444x^{12} - 18048x^{24} + 747264x^{36} + 15785984x^{48} + 5163712512x^{60} + 178660573184x^{72} - 6268504768512x^{84} - 202791175847936x^{96} - 12055045486936064x^{108} - 226939199972966400x^{120} - 36028797018963968x^{132} - 3602879701896396800x^{144}} \tag{4}$
