A pseudo-polyomino, also called a polyking, 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.
Golomb uses the term pseudo-polyomino referring to kingwise-connected sets of squares.
Enumeration of polykings
Free, one-sided, and fixed polykings There are three common ways of distinguishing polyominoes and polykings for enumeration: 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
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