Polyomino count by type.
Area | Free | Free without holes | Free with holes | One-sided | Fixed | Without symmetry | Mirror (90°) symmetry | Mirror (45°) symmetry | C₂ symmetry | D₂ (90°) symmetry | D₂ (45°) symmetry | C₄ symmetry | D₄ symmetry |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
2 | 1 | 1 | 0 | 1 | 2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
3 | 2 | 2 | 0 | 2 | 6 | 0 | ? | ? | 0 | ? | ? | 0 | 0 |
4 | 5 | 5 | 0 | 7 | 19 | 1 | ? | ? | 1 | ? | ? | 0 | 1 |
5 | 12 | 12 | 0 | 18 | 63 | 5 | ? | ? | 1 | ? | ? | 0 | 1 |
6 | 35 | 35 | 0 | 60 | 216 | 20 | ? | ? | 5 | ? | ? | 0 | 0 |
7 | 108 | 107 | 1 | 196 | 760 | 84 | ? | ? | 4 | ? | ? | 0 | 0 |
8 | 369 | 363 | 6 | 704 | 2725 | 316 | ? | ? | 18 | ? | ? | 1 | 1 |
9 | 1285 | 1248 | 37 | 2500 | 9910 | 1196 | ? | ? | 19 | ? | ? | 0 | 2 |
10 | 4655 | 4460 | 195 | 9189 | 36446 | 4461 | ? | ? | 73 | ? | ? | 0 | 0 |
11 | 17073 | 16094 | 979 | 33896 | 135268 | 16750 | ? | ? | 73 | ? | ? | 0 | 0 |
12 | 63600 | 58937 | 4663 | 126759 | 505861 | 62878 | ? | ? | 278 | ? | ? | 3 | 3 |
13 | 238591 | 217117 | 21474 | 476270 | 1903890 | 237394 | ? | ? | 283 | ? | ? | 2 | 2 |
14 | 901971 | 805475 | 96496 | 1802312 | 7204874 | 899265 | ? | ? | 1076 | ? | ? | 0 | 0 |
15 | 3426576 | 3001127 | 425449 | 6849777 | 27394666 | 3422111 | ? | ? | 1090 | ? | ? | 0 | 0 |
16 | 13079255 | 11230003 | 1849252 | 26152418 | 104592937 | 13069026 | ? | ? | 4125 | ? | ? | 12 | 5 |
17 | 50107909 | 42161529 | 7946380 | 10020319 | 400795844 | 50091095 | ? | ? | 4183 | ? | ? | 7 | 4 |
18 | 192622052 | 158781106 | 33840946 | 385221143 | 1540820542 | 192583152 | ? | ? | 15939 | ? | ? | 0 | 0 |
19 | 742624232 | 599563893 | 143060339 | 1485200848 | 5940738676 | 742560511 | ? | ? | 16105 | ? | ? | 0 | 0 |
20 | 2870671950 | 2269506062 | 601165888 | 5741256764 | 22964779660 | 2870523142 | ? | ? | 61628 | ? | ? | 44 | 12 |
21 | 11123060678 | 8609442688 | 2513617990 | 22245940545 | 88983512783 | 11122817672 | ? | ? | 62170 | ? | ? | 25 | 7 |
22 | 43191857688 | 32725637373 | 10466220315 | 86383382827 | 345532572678 | 43191285751 | ? | ? | 239388 | ? | ? | 0 | 0 |
23 | 168047007728 | 124621833354 | 43425174374 | 336093325058 | 1344372335524 | 168046076423 | ? | ? | 240907 | ? | ? | 0 | 0 |
24 | 654999700403 | 475368834568 | 179630865835 | 1309998125640 | 5239988770268 | 654997492842 | ? | ? | 932230 | ? | ? | 165 | 20 |
25 | 2557227044764 | 1816103345752 | 741123699012 | 5114451441106 | 20457802016011 | 2557223459805 | ? | ? | 936447 | ? | ? | 90 | 11 |
26 | 9999088822075 | 6948228104703 | 3050860717372 | 19998172734786 | 79992676367108 | 9999080270766 | ? | ? | 3641945 | ? | ? | 0 | 0 |
27 | 39153010938487 | 26618671505989 | 12534339432498 | 78306011677182 | 313224032098244 | 39152997087077 | ? | ? | 3651618 | ? | ? | 0 | 0 |
28 | 153511100594603 | 102102788362303 | 51408312232300 | 307022182222506 | 1228088671826973 | 153511067364760 | ? | ? | 14262540 | ? | ? | 603 | 45 |
29 | 602621953061978 | ? | ? | 1205243866707468 | 4820975409710116 | ? | ? | ? | ? | ? | ? | ? | ? |
30 | 2368347037571252 | ? | ? | 4736694001644862 | 18946775782611174 | ? | ? | ? | ? | ? | ? | ? | ? |
31 | 9317706529987950 | ? | ? | 18635412907198670 | 74541651404935148 | ? | ? | ? | ? | ? | ? | ? | ? |
32 | 36695016991712879 | ? | ? | 73390033697855860 | 293560133910477776 | ? | ? | ? | ? | ? | ? | ? | ? |
33 | 144648268175306702 | ? | ? | 289296535756895985 | 1157186142148293638 | ? | ? | ? | ? | ? | ? | ? | ? |
34 | 570694242129491412 | ? | ? | 1141388483146794007 | 4565553929115769162 | ? | ? | ? | ? | ? | ? | ? | ? |
35 | 2253491528465905342 | ? | ? | 4506983054619138245 | 18027932215016128134 | ? | ? | ? | ? | ? | ? | ? | ? |
36 | 8905339105809603405 | ? | ? | 17810678207278478530 | 71242712815411950635 | ? | ? | ? | ? | ? | ? | ? | ? |
37 | 35218318816847951974 | ? | ? | 70436637624668665265 | 281746550485032531911 | ? | ? | ? | ? | ? | ? | ? | ? |
38 | 139377733711832678648 | ? | ? | 278755467406691820628 | 1115021869572604692100 | ? | ? | ? | ? | ? | ? | ? | ? |
39 | 551961891896743223274 | ? | ? | 1103923783758183428889 | 4415695134978868448596 | ? | ? | ? | ? | ? | ? | ? | ? |
40 | 2187263896664830239467 | ? | ? | 4374527793263174673335 | 17498111172838312982542 | ? | ? | ? | ? | ? | ? | ? | ? |
41 | 8672737591212363420225 | ? | ? | 17345475182286431485513 | 69381900728932743048483 | ? | ? | ? | ? | ? | ? | ? | ? |
42 | 34408176607279501779592 | ? | ? | 68816353214298169362691 | 275265412856343074274146 | ? | ? | ? | ? | ? | ? | ? | ? |
43 | 136585913609703198598627 | ? | ? | 273171827218863802383383 | 1092687308874612006972082 | ? | ? | ? | ? | ? | ? | ? | ? |
44 | 542473001706357882732070 | ? | ? | 1084946003411691009916361 | 4339784013643393384603906 | ? | ? | ? | ? | ? | ? | ? | ? |
45 | 2155600091107324229254415 | ? | ? | 4311200182212516601049225 | 17244800728846724289191074 | ? | ? | ? | ? | ? | ? | ? | ? |
46 | ? | ? | ? | ? | 68557762666345165410168738 | ? | ? | ? | ? | ? | ? | ? | ? |
47 | ? | ? | ? | ? | 272680844424943840614538634 | ? | ? | ? | ? | ? | ? | ? | ? |
48 | ? | ? | ? | ? | 1085035285182087705685323738 | ? | ? | ? | ? | ? | ? | ? | ? |
49 | ? | ? | ? | ? | 4319331509344565487555270660 | ? | ? | ? | ? | ? | ? | ? | ? |
50 | ? | ? | ? | ? | 17201460881287871798942420736 | ? | ? | ? | ? | ? | ? | ? | ? |
51 | ? | ? | ? | ? | 68530413174845561618160604928 | ? | ? | ? | ? | ? | ? | ? | ? |
52 | ? | ? | ? | ? | 273126660016519143293320026256 | ? | ? | ? | ? | ? | ? | ? | ? |
53 | ? | ? | ? | ? | 1088933685559350300820095990030 | ? | ? | ? | ? | ? | ? | ? | ? |
54 | ? | ? | ? | ? | 4342997469623933155942753899000 | ? | ? | ? | ? | ? | ? | ? | ? |
55 | ? | ? | ? | ? | 17326987021737904384935434351490 | ? | ? | ? | ? | ? | ? | ? | ? |
56 | ? | ? | ? | ? | 69150714562532896936574425480218 | ? | ? | ? | ? | ? | ? | ? | ? |
$P_{\text{free}}(n)$ - 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
$P_{\text{holeless}}(n) = P_{\text{free}}(n) - P_{\text{holey}}(n)$
$P_{\text{free}}(n) = P_{\text{none}}(n) + P_{\text{mirror 90°}}(n) + P_{\text{mirror 45°}}(n) + P_{\text{C2}}(n) + P_{\text{D2 90°}}(n) + P_{\text{D2 45°}}(n) + P_{\text{C4}}(n) + P_{\text{D4}}(n)$
$P_{\text{one-sided}}(n) = 2P_{\text{none}}(n) + P_{\text{mirror 90°}}(n) + P_{\text{mirror 45°}}(n) + 2P_{\text{C2}}(n) + P_{\text{D2 90°}}(n) + P_{\text{D2 45°}}(n) + 2P_{\text{C4}}(n) + P_{\text{D4}}(n)$
$P_{\text{fixed}}(n) = 8P_{\text{none}}(n) + 4(P_{\text{mirror 90°}}(n) + P_{\text{mirror 45°}}(n) + P_{\text{C2}}(n)) + 2(P_{\text{D2 90°}}(n) + P_{\text{D2 45°}}(n) + P_{\text{C4}}(n)) + P_{\text{D4}}(n)$