POLYOMINO TILINGS

Polyomino Count

Polyomino count by type.

AreaFreeFree without holesFree with holesOne-sidedFixed Without symmetryMirror (90°) symmetryMirror (45°) symmetryC₂ symmetry D₂ (90°) symmetryD₂ (45°) symmetryC₄ symmetryD₄ symmetry
11101100000001
21101200001000
3220260??0??00
45507191??1??01
51212018635??1??01
6353506021620??5??00
7108107119676084??4??00
836936367042725316??18??11
91285124837250099101196??19??02
10465544601959189364464461??73??00
1117073160949793389613526816750??73??00
126360058937466312675950586162878??278??33
13238591217117214744762701903890237394??283??22
149019718054759649618023127204874899265??1076??00
15342657630011274254496849777273946663422111??1090??00
16130792551123000318492522615241810459293713069026??4125??125
17501079094216152979463801002031940079584450091095??4183??74
18192622052158781106338409463852211431540820542192583152??15939??00
1974262423259956389314306033914852008485940738676742560511??16105??00
20287067195022695060626011658885741256764229647796602870523142??61628??4412
211112306067886094426882513617990222459405458898351278311122817672??62170??257
224319185768832725637373104662203158638338282734553257267843191285751??239388??00
23168047007728124621833354434251743743360933250581344372335524168046076423??240907??00
2465499970040347536883456817963086583513099981256405239988770268654997492842??932230??16520
25255722704476418161033457527411236990125114451441106204578020160112557223459805??936447??9011
2699990888220756948228104703305086071737219998172734786799926763671089999080270766??3641945??00
273915301093848726618671505989125343394324987830601167718231322403209824439152997087077??3651618??00
28153511100594603102102788362303514083122323003070221822225061228088671826973153511067364760??14262540??60345
29602621953061978??12052438667074684820975409710116????????
302368347037571252??473669400164486218946775782611174????????
319317706529987950??1863541290719867074541651404935148????????
3236695016991712879??73390033697855860293560133910477776????????
33144648268175306702??2892965357568959851157186142148293638????????
34570694242129491412??11413884831467940074565553929115769162????????
352253491528465905342??450698305461913824518027932215016128134????????
368905339105809603405??1781067820727847853071242712815411950635????????
3735218318816847951974??70436637624668665265281746550485032531911????????
38139377733711832678648??2787554674066918206281115021869572604692100????????
39551961891896743223274??11039237837581834288894415695134978868448596????????
402187263896664830239467??437452779326317467333517498111172838312982542????????
418672737591212363420225??1734547518228643148551369381900728932743048483????????
4234408176607279501779592??68816353214298169362691275265412856343074274146????????
43136585913609703198598627??2731718272188638023833831092687308874612006972082????????
44542473001706357882732070??10849460034116910099163614339784013643393384603906????????
452155600091107324229254415??431120018221251660104922517244800728846724289191074????????
46????68557762666345165410168738????????
47????272680844424943840614538634????????
48????1085035285182087705685323738????????
49????4319331509344565487555270660????????
50????17201460881287871798942420736????????
51????68530413174845561618160604928????????
52????273126660016519143293320026256????????
53????1088933685559350300820095990030????????
54????4342997469623933155942753899000????????
55????17326987021737904384935434351490????????
56????69150714562532896936574425480218????????

Formulas

$P_{\text{free}}(n)$ - number of free $n$-ominoes

$P_{\text{holeless}}(n)$ - number of free $n$-ominoes without holes

$P_{\text{holey}}(n)$ - number of free $n$-ominoes with holes

$P_{\text{one-sided}}(n)$ - number of one-sided $n$-ominoes

$P_{\text{fixed}}(n)$ - number of fixed $n$-ominoes

$P_{\text{none}}(n)$ - number of $n$-ominoes without symmetry

$P_{\text{mirror 90°}}(n)$ - number of $n$-ominoes with one axis of reflection symmetry at 90° to the gridlines

$P_{\text{mirror 45°}}(n)$ - number of $n$-ominoes with one axis of reflection symmetry at 45° to the gridlines

$P_{\text{C2}}(n)$ - number of $n$-ominoes with rotational symmetry of order 2

$P_{\text{D2 90°}}(n)$ - number of $n$-ominoes with with two axis of reflection symmetry at 90° to the gridlines

$P_{\text{D2 45°}}(n)$ - number of $n$-ominoes with with two axis of reflection symmetry at 45° to the gridlines

$P_{\text{C4}}(n)$ - number of $n$-ominoes with rotational symmetry of order 4

$P_{\text{D4}}(n)$ - number of $n$-ominoes with full symmetry of square

$P_{\text{holeless}}(n) = P_{\text{free}}(n) - P_{\text{holey}}(n)$

$P_{\text{free}}(n) = P_{\text{none}}(n) + P_{\text{mirror 90°}}(n) + P_{\text{mirror 45°}}(n) + P_{\text{C2}}(n) + P_{\text{D2 90°}}(n) + P_{\text{D2 45°}}(n) + P_{\text{C4}}(n) + P_{\text{D4}}(n)$

$P_{\text{one-sided}}(n) = 2P_{\text{none}}(n) + P_{\text{mirror 90°}}(n) + P_{\text{mirror 45°}}(n) + 2P_{\text{C2}}(n) + P_{\text{D2 90°}}(n) + P_{\text{D2 45°}}(n) + 2P_{\text{C4}}(n) + P_{\text{D4}}(n)$

$P_{\text{fixed}}(n) = 8P_{\text{none}}(n) + 4(P_{\text{mirror 90°}}(n) + P_{\text{mirror 45°}}(n) + P_{\text{C2}}(n)) + 2(P_{\text{D2 90°}}(n) + P_{\text{D2 45°}}(n) + P_{\text{C4}}(n)) + P_{\text{D4}}(n)$

Attributions

  1. Free polyominoes have been counted by Toshihiro Shirakawa (http://oeis.org/A000105).
  2. One-sided polyominoes have been counted by Toshihiro Shirakawa (http://oeis.org/A000988).
  3. Free polyominoes with holes and polyominoes of each symmetry have been counted by Tomas Oliveira e Silva (http://sweet.ua.pt/tos/animals/a44.html).
  4. Fixed polyominoes have been counted by Iwan Jensen (http://www.ms.unimelb.edu.au/~iwan/animals/Animals_ser.html).