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Bounding Boxes for Translation Polyominoes I've computed the number of translation polyominoes 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 2 Size 3 Polyominoes Dimensions Number of polyominoes 2 x 2 4 3 x 1 2 Size 4 Polyominoes Dimensions Number of polyominoes 2 x 2 1 3 x 2 16 4 x 1 2 Size 5 Polyominoes Dimensions Number of polyominoes 3 x 2 12 3 x 3 25 4 x 2 24 5 x 1 2 Size 6 Polyominoes Dimensions Number of polyominoes 3 x 2 2 3 x 3 44 4 x 2 36 4 x 3 100 5 x 2 32 6 x 1 2 Size 7 Polyominoes Dimensions Number of polyominoes 3 x 3 32 4 x 2 16 4 x 3 308 4 x 4 120 5 x 2 76 5 x 3 166 6 x 2 40 7 x 1 2 Size 8 Polyominoes Dimensions Number of polyominoes 3 x 3 9 4 x 2 2 4 x 3 424 4 x 4 584 5 x 2 64 5 x 3 752 5 x 4 460 6 x 2 132 6 x 3 248 7 x 2 48 8 x 1 2 Size 9 Polyominoes Dimensions Number of polyominoes 3 x 3 1 4 x 3 316 4 x 4 1396 5 x 2 20 5 x 3 1568 5 x 4 3052 5 x 5 497 6 x 2 176 6 x 3 1500 6 x 4 776 7 x 2 204 7 x 3 346 8 x 2 56 9 x 1 2 Size 10 Polyominoes Dimensions Number of polyominoes 4 x 3 124 4 x 4 2038 5 x 2 2 5 x 3 1974 5 x 4 10308 5 x 5 4180 6 x 2 100 6 x 3 4266 6 x 4 6552 6 x 5 1864 7 x 2 384 7 x 3 2632 7 x 4 1204 8 x 2 292 8 x 3 460 9 x 2 64 10 x 1 2 Size 11 Polyominoes Dimensions Number of polyominoes 4 x 3 24 4 x 4 1952 5 x 3 1556 5 x 4 22656 5 x 5 18944 6 x 2 24 6 x 3 7604 6 x 4 28384 6 x 5 18916 6 x 6 1924 7 x 2 340 7 x 3 9606 7 x 4 12388 7 x 5 3182 8 x 2 720 8 x 3 4228 8 x 4 1760 9 x 2 396 9 x 3 590 10 x 2 72 11 x 1 2 Size 12 Polyominoes Dimensions Number of polyominoes 4 x 3 2 4 x 4 1232 5 x 3 740 5 x 4 35196 5 x 5 58665 6 x 2 2 6 x 3 9244 6 x 4 82392 6 x 5 104976 6 x 6 22576 7 x 2 144 7 x 3 22254 7 x 4 65648 7 x 5 37872 7 x 6 7176 8 x 2 900 8 x 3 18980 8 x 4 21408 8 x 5 5076 9 x 2 1216 9 x 3 6368 9 x 4 2460 10 x 2 516 10 x 3 736 11 x 2 80 12 x 1 2 Size 13 Polyominoes Dimensions Number of polyominoes 4 x 4 488 5 x 3 202 5 x 4 39824 5 x 5 135325 6 x 3 7628 6 x 4 175960 6 x 5 408932 6 x 6 147656 7 x 2 28 7 x 3 36556 7 x 4 235232 7 x 5 247304 7 x 6 95596 7 x 7 7265 8 x 2 584 8 x 3 54504 8 x 4 134568 8 x 5 69452 8 x 6 12424 9 x 2 2004 9 x 3 34084 9 x 4 34588 9 x 5 7690 10 x 2 1904 10 x 3 9132 10 x 4 3320 11 x 2 652 11 x 3 898 12 x 2 88 13 x 1 2 Size 14 Polyominoes Dimensions Number of polyominoes 4 x 4 116 5 x 3 30 5 x 4 32880 5 x 5 241550 6 x 3 4136 6 x 4 287732 6 x 5 1223718 6 x 6 695088 7 x 2 2 7 x 3 43650 7 x 4 632252 7 x 5 1145410 7 x 6 713028 7 x 7 107052 8 x 2 196 8 x 3 112764 8 x 4 573496 8 x 5 521232 8 x 6 185640 8 x 7 27164 9 x 2 1824 9 x 3 117510 9 x 4 252156 9 x 5 119112 9 x 6 20312 10 x 2 3940 10 x 3 56918 10 x 4 53032 10 x 5 11184 11 x 2 2816 11 x 3 12600 11 x 4 4356 12 x 2 804 12 x 3 1076 13 x 2 96 14 x 1 2 Size 15 Polyominoes Dimensions Number of polyominoes 4 x 4 16 5 x 3 2 5 x 4 19316 5 x 5 339488 6 x 3 1416 6 x 4 365984 6 x 5 2933876 6 x 6 2584696 7 x 3 37342 7 x 4 1337680 7 x 5 4135338 7 x 6 3866360 7 x 7 887992 8 x 2 32 8 x 3 175928 8 x 4 1847984 8 x 5 2784544 8 x 6 1551288 8 x 7 439844 8 x 8 27288 9 x 2 924 9 x 3 294042 9 x 4 1247456 9 x 5 1010724 9 x 6 337204 9 x 7 47718 10 x 2 4728 10 x 3 230044 10 x 4 440672 10 x 5 193700 10 x 6 31720 11 x 2 7060 11 x 3 89786 11 x 4 77972 11 x 5 15734 12 x 2 3984 12 x 3 16852 12 x 4 5584 13 x 2 972 13 x 3 1270 14 x 2 104 15 x 1 2 Size 16 Polyominoes Dimensions Number of polyominoes 4 x 4 1 5 x 4 7784 5 x 5 376953 6 x 3 298 6 x 4 361472 6 x 5 5758252 6 x 6 7930294 7 x 3 22250 7 x 4 2280600 7 x 5 12192846 7 x 6 16777072 7 x 7 5406161 8 x 2 2 8 x 3 207712 8 x 4 4754120 8 x 5 11670048 8 x 6 9456828 8 x 7 4005140 8 x 8 477464 9 x 2 256 9 x 3 565242 9 x 4 4680140 9 x 5 6108738 9 x 6 3120596 9 x 7 844752 9 x 8 102580 10 x 2 3332 10 x 3 677792 10 x 4 2486312 10 x 5 1835380 10 x 6 580400 10 x 7 79712 11 x 2 10672 11 x 3 417534 11 x 4 728328 11 x 5 301616 11 x 6 47704 12 x 2 11780 12 x 3 135296 12 x 4 110768 12 x 5 21532 13 x 2 5440 13 x 3 21968 13 x 4 7020 14 x 2 1156 14 x 3 1480 15 x 2 112 16 x 1 2 Size 17 Polyominoes Dimensions Number of polyominoes 5 x 4 2100 5 x 5 328380 6 x 3 36 6 x 4 272896 6 x 5 9347396 6 x 6 20537800 7 x 3 8972 7 x 4 3160524 7 x 5 30117946 7 x 6 61038620 7 x 7 26662289 8 x 3 182684 8 x 4 10052984 8 x 5 40313676 8 x 6 46396672 8 x 7 26848492 8 x 8 4694728 9 x 2 36 9 x 3 849530 9 x 4 14114392 9 x 5 29038218 9 x 6 21070684 9 x 7 8378416 9 x 8 1935788 9 x 9 102745 10 x 2 1376 10 x 3 1549076 10 x 4 10648104 10 x 5 12378964 10 x 6 5897584 10 x 7 1535772 10 x 8 182616 11 x 2 9884 11 x 3 1420372 11 x 4 4622464 11 x 5 3159524 11 x 6 955388 11 x 7 127722 12 x 2 21672 12 x 3 713104 12 x 4 1149992 12 x 5 452972 12 x 6 69512 13 x 2 18580 13 x 3 196360 13 x 4 152908 13 x 5 28786 14 x 2 7216 14 x 3 28028 14 x 4 8680 15 x 2 1356 15 x 3 1706 16 x 2 120 17 x 1 2

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