Enumeration of symmetry classes of convex polyominoes in the square lattice

Abstract

This paper concerns the enumeration of rotation-type and congruence-type convex polyominoes on the square lattice. These can be defined as orbits of the groups C4, of rotations, and D4, of symmetries of the square acting on (translation- type) polyominoes. In virtue of Burnside's Lemma, it is sufficient to enumerate the various symmetry classes (fixed points) of polyominoes defined by the elements of C4 and D4. Using the Temperley--Bousquet-Melou methodology, we solve this problem and provide explicit or recursive formulas for their generating functions according to width, height and area. We also enumerate the class of asymmetric convex polyominoes, using Moebius inversion, and prove that their number is asymptotically equivalent to the number of convex polyominoes, a fact which is empirically evident.