Polyomino

A polyomino is a generalization of the domino to a collection of squares of equal size arranged with coincident sides. Polyominos were originally called "super-dominoes" by Gardner (1957). A polyomino with squares is known as an -polyomino or "-omino."

Polyominoes may be conveniently represented and visualized in the Wolfram Language using ArrayMesh.

Free polyominoes can be picked up and flipped, so mirror image pieces are considered identical. One-sided polyominoes may not be flipped, but may be rotated, so different rotational orientations are the same, but pieces having different chiralities are considered distinct. Fixed polyominoes (also called "lattice animals") are considered distinct if they have different chirality or orientation.

When the type of polyomino being dealt with is not specified, it is usually assumed that they are free. There is a single unique 2-omino (the domino), and two distinct 3-ominoes (the straight- and -triominoes). The 4-ominoes (tetrominoes) are known as the straight, L, T, square, and skew tetrominoes. The 5-ominoes (pentominoes) are called , , , , , , , , , , , and (Golomb 1995). Another common naming scheme replaces , , , and with , , , and so that all letters from O to Z are used (Berlekamp et al. 1982).

The first few polyominoes with holes are illustrated above (Myers).

Redelmeier (1981) computed the number of free and fixed polyominoes for , and Mertens (1990) gives a simple computer program. The following table gives the number of free (Lunnon 1971, 1972; Read 1978; Redelmeier 1981; Ball and Coxeter 1987; Conway and Guttmann 1995; Goodman and O'Rourke 1997, p. 229), fixed (Redelmeier 1981), and one-sided polyominoes (Redelmeier 1981; Golomb 1995; Goodman and O'Rourke 1997, p. 229), as well as the number containing holes (Parkin et al. 1967, Madachy 1969, Golomb 1994) for the first few .

	name	free	one-sided	fixed	with holes
Sloane		A000105	A000988	A001168	A001419
1	monomino	1	1	1	0
2	domino	1	1	2	0
3	triomino	2	2	6	0
4	tetromino	5	7	19	0
5	pentomino	12	18	63	0
6	hexomino	35	60	216	0
7	heptomino	108	196	760	1
8	octomino	369	704	2725	6
9		1285	2500	9910	37
10		4655	9189	36446	195
11		17073	33896	135268	979
12		63600	126759	505861	4663
13		238591	476270	1903890	21474
14		901971	1802312	7204874	96496
15		3426576	6849777	27394666	425449

The best currently known bounds on the number of -polyominoes are

(Eden 1961, Klarner 1967, Klarner and Rivest 1973, Ball and Coxeter 1987).

Generalizations of polyominoes to other base shapes other that squares are known as polyforms, with the best-known examples being the polyiamonds and polyhexes.