Column-convex polygons were first counted by area several decades ago, and the result was found to be a simple, rational, generating function. In this work we generalize that result. Let a p-column polyomino be a polyomino whose columns can have 1, 2, . . . , p connected components. Then columnconvex polygons are equivalent to 1-convex polyominoes. The area generating function of even the simpl...

We introduce a computability model { called shape grammar { based on adjoining polyominoes. A computation in a shape grammar is correct when: (1) it starts by using a speci ed polyomino; (2) it ends when a rectangle is obtained (without holes); (3) at any step the current picture is connected; and (4) a sequencing mapping is given, so that at every step (except the rst one) we use a polyomino d...

We consider the class of L-convex polyominoes, i.e. those polyominoes in which any two cells can be connected with an “L” shaped path in one of its four cyclic orientations. The paper proves bijectively that the number fn of L-convex polyominoes with perimeter 2(n + 2) satisfies the linear recurrence relation fn+2 = 4 fn+1 − 2 fn , by first establishing a recurrence of the same form for the car...

The Randić index R(G) of an organic molecule whose molecular graph is G is the sum of the weights (dudv)− 1 2 of all edges uv of G, where du(or dv) denote the degree of vertex u(or v). Efficient formulas for calculating the Randić index of polyomino chains are provided. Mathematics Subject Classification: 05C50, 05C12, 05C05

The present note is devoted to establish some extremal results for the zerothorder general Randić index of cacti, characterize the extremal polyomino chains with respect to the aforementioned index, and hence to generalize two already reported results.

The homology group of a tiling introduced by M. Reid is studied for certain topological tilings. As in the planar case, finite square grids on surfaces, method groups, namely non-triviality some specific element allows "coloring proof" impossibility tiling. Several results about non-existence polyomino tilings square-tiled surfaces are proved paper.

We consider staged self-assembly, in which squareshaped Wang tiles can be added to bins in several stages. Within these bins the tiles may connect to each other, depending on the glue types of their edges. In general, self-assembly constructs complex (polyomino-shaped) structures from a limited set of square tiles. Previous work by Demaine et al. considered a setting in which assembly proceeds ...