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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).
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.”
Open.
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} = \lambda (roughly, t(n) is around n^\lambda) for “Klarner’s constant” \lambda, with 4.0025 < \lambda < 4.5685, but the precise value of \lambda remains open. The lower bound of >4 is established in [BRS16], and the upper bound in [BB15].
The exact counts have been computed for small n. See [Sloa] for the number of fixed n-ominoes for n \leq 28 and for related references. The current record is n=56 by Jensen [Jen03]. See also [Epp] for related links.
There are many related problems involving polyominoes with restricted geometric shapes (e.g., [Slob]), 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.
combinatorial geometry
E. Demaine & J. O’Rourke, 30Nov2001; E. Demaine, 28Aug2002; G. Barequet, 4Dec2015.
Gill Barequet and Ronnie Barequet. An improved upper bound on the growth constant of polyominoes. Electronic Notes in Discrete Mathematics, 49:167–172, 2015.
Gill Barequet, Günter Rote, and Mira Shalah. \lambda > 4: an improved lower bound on the growth constant of polyominoes. Comm. ACM, 59(7):88–95, July 2016.
David Eppstein. Polyominoes and other animals. http://www1.ics.uci.edu/~eppstein/junkyard/polyomino.html.
Iwan Jensen. Counting polyominoes: A parallel implementation for cluster computing. In Computational Science—ICCS 2003, pages 203–212. Springer, 2003.
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.
Neil J. A. Sloane. Sequence A000105. https://oeis.org/A000105.
Neil J. A. Sloane. Sequence A204804. https://oeis.org/A204804.