Problem 37: Counting Polyominoes

Statement

How many polyominoes on n squares are there? A polyomino is a connected interior-disjoint union of axis-aligned unit squares joined edge-to-edge, in other words, an edge-connected union of cells in the planar square lattice. The order of a polyomino is the number of unit squares forming it. The problem asks for the number of polyominoes of order n. The key constraint here is that polyominoes must be edge-connected. There are three variations on the problem, depending on whether two polyominoes are considered equivalent by factoring out just translations (fixed polyominoes), rotations and translations (chiral polyominoes), or reflections, rotations, and translations (free polyominoes).

Origin

To quote Klarner [Kla97]: ``Polyominoes have a long history, going back to the start of the 20th century, but they were popularized in the present era initially by Solomon Golomb, then by Martin Gardner in his Scientific American columns.''

Status/Conjectures

Open.

Partial and Related Results

Asymptotically, results of Klarner et al. [Kla97, Thm. 12.3.1] show that the number of fixed n-ominoes (factoring out just translations), denoted t(n), satisfies

$$\lim_{{n \to \infty}}^{}$$[t(n)]^1/n = $$\theta$$

(roughly, t(n) is around n^$\theta$) for a constant $\theta$ with 3.9 < $\theta$ < 4.65, but the precise value of $\theta$ remains open.

The exact counts have been computed for small n. See [Slo] for the number of fixed n-ominoes for n $\leq$ 28 and for related references. The current record is n = 46 by Jensen [Jen01]. See also [Epp] for related links.

Related Open Problems

There are many related problems involving polyiamonds (edge-to-edge unions of unit equilateral triangles), polyhexes (edge-to-edge unions of unit regular hexagons), polyabolos (edge-to-edge unions of unit right isosceles triangles), polycubes (face-to-face unions of unit cubes), etc. All of these problems are also open.

Appearances

[Kla97]

Categories

combinatorial geometry

Entry Revision History

E. Demaine & J. O'Rourke, 30 Nov. 2001; E. Demaine, 28 Aug. 2002.

Bibliography

Epp

David Eppstein.
Polyominoes and other animals.
http://www1.ics.uci.edu/~eppstein/junkyard/polyomino.html.

Jen01

Iwan Jensen.
Enumerations of lattice animals and trees.
J. Statistical Physics, 102:865-881, 2001.

Kla97

David A. Klarner.
Polyominoes.
In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 12, pages 225-242. CRC Press LLC, Boca Raton, FL, 1997.

Slo

Neil J. A. Sloane.
Sequence A000105.
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000105.