Polyking
A polyking (or polyplet, or hinged polyomino) is a plane geometric figure formed by joining one or more equal squares edge to edge or corner to corner at 90°. It is a polyform with square cells. The polyominoes are a subset of the polykings.
The name "polyking" refers to the king in chess. The n-kings are the n-square shapes which could be occupied by a king on an infinite chessboard in the course of legal moves.
Enumeration of polykings
Free, one-sided, and fixed polykings
There are three common ways of distinguishing polyominoes and polykings for enumeration:[1][2]
- free polykings are distinct when none is a rigid transformation (translation, rotation, reflection or glide reflection) of another (pieces that can be picked up and flipped over).
- one-sided polykings are distinct when none is a translation or rotation of another (pieces that cannot be flipped over).
- fixed polykings are distinct when none is a translation of another (pieces that can be neither flipped nor rotated).
The following table shows the numbers of polykings of various types with n cells.
| n |
free |
one-sided |
fixed |
| 1 |
1 |
1 |
1 |
| 2 |
2 |
2 |
4 |
| 3 |
5 |
6 |
20 |
| 4 |
22 |
34 |
110 |
| 5 |
94 |
166 |
638 |
| 6 |
524 |
991 |
3832 |
| 7 |
3,031 |
5,931 |
23,592 |
| 8 |
18,770 |
37,196 |
147,941 |
| 9 |
118,133 |
235,456 |
940,982 |
| 10 |
758,381 |
1,514,618 |
6,053,180 |
| 11 |
4,915,652 |
9,826,177 |
39,299,408 |
| 12 |
32,149,296 |
64,284,947 |
257,105,146 |
| OEIS |
A030222 |
A030233 |
A006770 |
| Free polykings |
|
|
The 3,031 free heptakings.
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Notes
External links