POLYOMINO TILINGS

Polyomino Count

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
?
?
?
?
?
?
?
?
66
?
?
?
?
?
?
?
?
?
?
?
?
?
70
?
?
?
?
?
?
?
?
?
?
?
?
?

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).