Polyomino

A polyomino is a polyform consisting of a finite nonempty edge-connected union of cells of the square tessellation. Its squares are joined along complete sides, and holes are allowed. Polyominos were originally called "super-dominoes" by Gardner (1957). A polyomino with squares, where is a positive integer, is known as an -polyomino or "-omino."

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

Free polyominoes identify translations, rotations, and reflections. One-sided polyominoes identify translations and rotations, while fixed polyominoes (also called "lattice animals") identify only translations.

When the type of polyomino being dealt with is not specified, it is usually assumed that they are free. There is a single 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

If is the number of free -polyominoes, then the limit

exists and is the same for free, one-sided, and fixed polyominoes (Klarner 1967). Bounds on , rather than bounds on valid for every , include

The lower bound is due to Barequet et al. (2016), and the upper bound to Bui (2026).

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