ENUMERATION OF SYMMETRY CLASSES OF CONVEX POLYOMINOES IN THE SQUARE LATTICE

This paper concerns the enumeration of rotation-type and congruence-ty pe convex polyominoes on the square lattice. These can be defined as o rbits of the groups C-4, of rotations, and D-4, of symmetries, of the square, acting on (translation-type) polyominoes. By virtue of Burnsid e's lemma, it is sufficient to enumerate the various symmetry classes (fixed points) of polyominoes defined by the elements of C-4 and D-4. Using the Temperley-Bousquet-Melou methodology, we solve this problem and provide explicit or recursive formulas for their generating functi ons according to width, height, and area. We also enumerate the class of asymmetric convex polyominoes, using Mobius inversion, and prove th at their number is asymptotically equivalent to the number of convex p olyominoes, a fact which is empirically evident.