User:B jonas/Polyminoes
This page records the result of enumerating polyminoes (polyminos, polyominoes, polyominos) and other similar species of animals (polyforms), modulo different symmetries.
The first types of animals live on the square grid. A polymino is a set of points on the square grid that is connected through edges between rook-adjacent points. Similarly a polyking (aka polyplet) is a set of points on the square grid that is connected through edges between king-adjacent points.
We also consider species with the weighted template. A weighted animal aka animal with multiplicity is a multiset of points connected in a similar way to the base species, except any point is also considered to be adjacent to itself. (If a point isn't adjacent to itself then you have to subtract one from all but the first element of the sequence.)
We always consider an object equivalent to its translations, and we may consider it equivalent to some of its rotated or mirrored images too. I give multiple counts depending on which rotations or mirrorings are considered equivalent. On a square grid, the available movement groups are the following:
- none
- none (translations only)
- central
- central symnetry (180° rotation).
- horz
- north-south mirroring.
- diag
- northeast-southwest mirroring.
- rect
- north-south mirroring, west-east mirroring, central symmetry.
- diamond
- northeast-southwest mirroring, northwest-southeast mirroring, central symmetry.
- rotate
- quarter turn (90°) rotations, central symmetry
- square
- all symmetries of the square grid
I always give the number of equivalence classes of the objects as a sequence of the number of points in the objects, starting with 1 point. Some of these sequences are
| species | symmetries | count | OEIS |
|---|---|---|---|
| polymino | none | 1,2,6,19,63,216,760,2725,9910,36446,135268,505861,1903890 | A001168 |
| polymino | central | 1,2,4,13,35,120,392,1405,4998,18378,67792,253509,952534,3604624 | A151522 |
| polymino | horz | 1,2,4,12,35,116,392,1390,4998,18321,67791,253288,952527,3603761 | A151525 |
| polymino | diag | 1,1,4,10,34,110,388,1369,4982,18246,67727,253014,952275,3602743 | A182645 |
| polymino | rect | 1,2,3,9,21,68,208,730,2542,9287,34053,127112,476849,1803636 | A056780 |
| polymino | diamond | 1,1,3,7,20,62,204,709,2526,9212,33989,126838,476597,1802618 | A056783 |
| polymino | rotate | 1,1,2,7,18,60,196,704,2500,9189,33896,126759,476270,1802312 | A000988 |
| polymino | square | 1,1,2,5,12,35,108,369,1285,4655,17073,63600,238591,901971,3426576 | A000105 |
| weighted polymino | none | 1,3,11,44,184,792,3484,15592,70745,324561,1502511 | A113174 |
| weighted polymino | central | 1,3,7,28,98,422,1768,7912,35489,162809,751789,3506415 | |
| weighted polymino | horz | 1,3,8,29,106,433,1821,7998,35818,163398,753770,3510195 | |
| weighted polymino | diag | 1,2,7,24,98,405,1768,7837,35488,162468,751776,3504828 | |
| weighted polymino | rect | 1,3,6,21,63,248,963,4158,18190,82522,378409,1759438 | |
| weighted polymino | diamond | 1,2,5,16,55,220,910,3997,17860,81592,376415,1754071 | |
| weighted polymino | rotate | 1,2,4,15,50,212,885,3959,17747,81407,375897,1753218 | |
| weighted polymino | square | 1,2,4,12,35,129,495,2101,9154,41356,189466,880156,4120515 | A331621 |
| polyking | none | 1,4,20,110,638,3832,23592,147941,940982 | A006770 |
| polyking | central | 1,4,12,66,330,1982,11862,74383,470904,3029236 | |
| polyking | horz | 1,3,12,60,330,1945,11862,74147,470902,3027698 | |
| polyking | diag | 1,3,12,60,330,1944,11861,74138,470890,3027626 | |
| polyking | rect | 1,3,8,38,176,1020,5997,37368,235863,1515726 | |
| polyking | diamond | 1,3,8,38,176,1019,5996,37359,235851,1515654 | |
| polyking | rotate | 1,2,6,34,166,991,5931,37196,235456,1514618 | A030233 |
| polyking | square | 1,2,5,22,94,524,3031,18770,118133,758381,4915652 | A030222 |
| weighted polyking | none | 1,5,29,183,1215,8343,58679,420186,3051091 | |
| weighted polyking | central | 1,5,17,107,623,4277,29445,210844,1526297 | |
| weighted polyking | horz | 1,4,18,102,634,4251,29548,210716,1527215 | |
| weighted polyking | diag | 1,4,18,102,634,4250,29545,210697,1527159 | |
| weighted polyking | rect | 1,4,12,64,338,2218,14931,106045,764818 | |
| weighted polyking | diamond | 1,4,12,64,338,2217,14928,106026,764762 | |
| weighted polyking | rotate | 1,3,9,55,313,2140,14724,105429,763155 | |
| weighted polyking | square | 1,3,8,38,183,1149,7569,53328,383219,2806574 |
TODO: graphical visualization. polyaboloes, polyhexes, polyiamonds, 3-d and 4-d polyminos, animals on the 4-d dense sphere packing grid.