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Bounding Boxes for Polyominoes

I've computed the number of polyominoes (unique with respect to 3-d rotation) of various sizes which fit inside rectangles of various sizes. Here are my results. All results were computed on a PowerMac G3/400 in January 2000.

Size 1 Polyominoes
Dimensions Number of polyominoes
1 x 1
1

Size 2 Polyominoes
Dimensions Number of polyominoes
2 x 1
1

Size 3 Polyominoes
Dimensions Number of polyominoes
2 x 2
1
3 x 1
1

Size 4 Polyominoes
Dimensions Number of polyominoes
2 x 2
1
3 x 2
3
4 x 1
1

Size 5 Polyominoes
Dimensions Number of polyominoes
3 x 2
2
3 x 3
6
4 x 2
3
5 x 1
1

Size 6 Polyominoes
Dimensions Number of polyominoes
3 x 2
1
3 x 3
7
4 x 2
6
4 x 3
15
5 x 2
5
6 x 1
1

Size 7 Polyominoes
Dimensions Number of polyominoes
3 x 3
7
4 x 2
2
4 x 3
39
4 x 4
18
5 x 2
11
5 x 3
25
6 x 2
5
7 x 1
1

Size 8 Polyominoes
Dimensions Number of polyominoes
3 x 3
3
4 x 2
1
4 x 3
59
4 x 4
77
5 x 2
10
5 x 3
96
5 x 4
61
6 x 2
19
6 x 3
35
7 x 2
7
8 x 1
1

Size 9 Polyominoes
Dimensions Number of polyominoes
3 x 3
1
4 x 3
42
4 x 4
181
5 x 2
3
5 x 3
210
5 x 4
383
5 x 5
73
6 x 2
22
6 x 3
188
6 x 4
97
7 x 2
28
7 x 3
49
8 x 2
7
9 x 1
1

Size 10 Polyominoes
Dimensions Number of polyominoes
4 x 3
21
4 x 4
266
5 x 2
1
5 x 3
255
5 x 4
1304
5 x 5
529
6 x 2
15
6 x 3
550
6 x 4
822
6 x 5
240
7 x 2
52
7 x 3
332
7 x 4
155
8 x 2
40
8 x 3
63
9 x 2
9
10 x 1
1

Size 11 Polyominoes
Dimensions Number of polyominoes
4 x 3
4
4 x 4
251
5 x 3
212
5 x 4
2847
5 x 5
2413
6 x 2
3
6 x 3
954
6 x 4
3548
6 x 5
2366
6 x 6
255
7 x 2
45
7 x 3
1231
7 x 4
1551
7 x 5
410
8 x 2
90
8 x 3
529
8 x 4
220
9 x 2
53
9 x 3
81
10 x 2
9
11 x 1
1

Size 12 Polyominoes
Dimensions Number of polyominoes
4 x 3
1
4 x 4
168
5 x 3
103
5 x 4
4441
5 x 5
7375
6 x 2
1
6 x 3
1184
6 x 4
10323
6 x 5
13161
6 x 6
2835
7 x 2
21
7 x 3
2800
7 x 4
8239
7 x 5
4738
7 x 6
908
8 x 2
119
8 x 3
2406
8 x 4
2680
8 x 5
646
9 x 2
158
9 x 3
800
9 x 4
313
10 x 2
69
10 x 3
99
11 x 2
11
12 x 1
1

Size 13 Polyominoes
Dimensions Number of polyominoes
4 x 4
66
5 x 3
33
5 x 4
5008
5 x 5
17041
6 x 3
964
6 x 4
21995
6 x 5
51133
6 x 6
18533
7 x 2
4
7 x 3
4634
7 x 4
29442
7 x 5
30998
7 x 6
11952
7 x 7
950
8 x 2
73
8 x 3
6818
8 x 4
16821
8 x 5
8683
8 x 6
1553
9 x 2
257
9 x 3
4313
9 x 4
4327
9 x 5
979
10 x 2
238
10 x 3
1142
10 x 4
415
11 x 2
86
11 x 3
121
12 x 2
11
13 x 1
1

Size 14 Polyominoes
Dimensions Number of polyominoes
4 x 4
20
5 x 3
6
5 x 4
4168
5 x 5
30320
6 x 3
546
6 x 4
36035
6 x 5
153122
6 x 6
86974
7 x 2
1
7 x 3
5497
7 x 4
79155
7 x 5
143230
7 x 6
89212
7 x 7
13402
8 x 2
28
8 x 3
14182
8 x 4
71742
8 x 5
65236
8 x 6
23215
8 x 7
3417
9 x 2
237
9 x 3
14722
9 x 4
31576
9 x 5
14894
9 x 6
2555
10 x 2
505
10 x 3
7171
10 x 4
6634
10 x 5
1415
11 x 2
360
11 x 3
1580
11 x 4
551
12 x 2
106
12 x 3
143
13 x 2
13
14 x 1
1

Size 15 Polyominoes
Dimensions Number of polyominoes
4 x 4
3
5 x 3
1
5 x 4
2439
5 x 5
42670
6 x 3
187
6 x 4
45748
6 x 5
366832
6 x 6
323331
7 x 3
4745
7 x 4
167354
7 x 5
517302
7 x 6
483329
7 x 7
111296
8 x 2
4
8 x 3
22011
8 x 4
230998
8 x 5
348090
8 x 6
193911
8 x 7
54983
8 x 8
3473
9 x 2
119
9 x 3
36920
9 x 4
156007
9 x 5
126496
9 x 6
42154
9 x 7
6003
10 x 2
591
10 x 3
28762
10 x 4
55084
10 x 5
24214
10 x 6
3965
11 x 2
895
11 x 3
11304
11 x 4
9751
11 x 5
1991
12 x 2
498
12 x 3
2107
12 x 4
698
13 x 2
127
13 x 3
169
14 x 2
13
15 x 1
1

Size 16 Polyominoes
Dimensions Number of polyominoes
4 x 4
1
5 x 4
1008
5 x 5
47344
6 x 3
47
6 x 4
45289
6 x 5
720244
6 x 6
991660
7 x 3
2833
7 x 4
285375
7 x 5
1524442
7 x 6
2097534
7 x 7
675981
8 x 2
1
8 x 3
26097
8 x 4
594488
8 x 5
1459163
8 x 6
1182179
8 x 7
500827
8 x 8
59728
9 x 2
36
9 x 3
70760
9 x 4
585305
9 x 5
763678
9 x 6
390229
9 x 7
105600
9 x 8
12859
10 x 2
429
10 x 3
84925
10 x 4
310890
10 x 5
229573
10 x 6
72565
10 x 7
10001
11 x 2
1353
11 x 3
52245
11 x 4
91127
11 x 5
37708
11 x 6
5985
12 x 2
1493
12 x 3
16997
12 x 4
13852
12 x 5
2715
13 x 2
690
13 x 3
2752
13 x 4
885
14 x 2
151
14 x 3
195
15 x 2
15
16 x 1
1

Size 17 Polyominoes
Dimensions Number of polyominoes
5 x 4
271
5 x 5
41330
6 x 3
6
6 x 4
34112
6 x 5
1168734
6 x 6
2567828
7 x 3
1173
7 x 4
395338
7 x 5
3766042
7 x 6
7630113
7 x 7
3334120
8 x 3
22883
8 x 4
1256623
8 x 5
5039358
8 x 6
5799584
8 x 7
3356103
8 x 8
587349
9 x 2
5
9 x 3
106490
9 x 4
1764700
9 x 5
3630653
9 x 6
2633896
9 x 7
1047667
9 x 8
241977
9 x 9
13006
10 x 2
172
10 x 3
193669
10 x 4
1331013
10 x 5
1547398
10 x 6
737198
10 x 7
191974
10 x 8
22827
11 x 2
1248
11 x 3
177886
11 x 4
577936
11 x 5
395198
11 x 6
119428
11 x 7
16025
12 x 2
2709
12 x 3
89146
12 x 4
143749
12 x 5
56623
12 x 6
8689
13 x 2
2343
13 x 3
24660
13 x 4
19119
13 x 5
3630
14 x 2
902
14 x 3
3504
14 x 4
1085
15 x 2
176
15 x 3
225
16 x 2
15
17 x 1
1


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Copyright © 2004-2006 Kevin L. Gong