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.
Prime rectangles: ≥ 13.
Smallest rectangle (10x12):
Smallest square (12x12):
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.
$N(w; h)$ - 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 $A(w; h)$
$G(N(10); x) = \frac{2x^{12} - 46x^{18} + 450x^{24} - 2310x^{30} + 5022x^{36} + 13406x^{42} - 147762x^{48} + 568230x^{54} - 1128668x^{60} + 79620x^{66} + 7862212x^{72} - 31891404x^{78} + 80603384x^{84} - 152394036x^{90} + 227687984x^{96} - 273438500x^{102} + 262264654x^{108} - 193268142x^{114} + 95720870x^{120} - 10357734x^{126} - 36343486x^{132} + 43967390x^{138} - 29934334x^{144} + 12875190x^{150} - 2242914x^{156} - 1523738x^{162} + 1660768x^{168} - 892188x^{174} + 329722x^{180} - 88858x^{186} + 17384x^{192} - 2360x^{198} + 200x^{204} - 8x^{210}}{1 - 60x^6 + 1630x^{12} - 26847x^{18} + 303403x^{24} - 2527992x^{30} + 16307640x^{36} - 84446149x^{42} + 361024584x^{48} - 1303066286x^{54} + 4042715944x^{60} - 10938268282x^{66} + 26113184245x^{72} - 55523659222x^{78} + 105940134904x^{84} - 182472216969x^{90} + 285052784353x^{96} - 405345974688x^{102} + 526128321757x^{108} - 624575848664x^{114} + 679016257585x^{120} - 676538333124x^{126} + 617874949283x^{132} - 517081356960x^{138} + 396188686741x^{144} - 277551734780x^{150} + 177447292275x^{156} - 103280946006x^{162} + 54560026056x^{168} - 26062629610x^{174} + 11207405848x^{180} - 4315207904x^{186} + 1478113060x^{192} - 446929358x^{198} + 118155647x^{204} - 26989413x^{210} + 5246424x^{216} - 850686x^{222} + 111930x^{228} - 11480x^{234} + 861x^{240} - 42x^{246} + x^{252}} \tag{1}$
