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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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