Enumeration of L-convex polyominoes

In this paper, we consider the class of L-convex polyominoes, i.e. the convex polyominoes in which any two cells can be connected by a path of cells in the polyomino that switches direction between the vertical and the horizontal at most once. Using the ECO method, we prove that the number fn of L-convex polyominoes with perimeter 2(n+2) satisfies the rational recurrence relation fn =4fn−1 −2fn−2, with f0 =1, f1 =2, f2 =7. Moreover, we give a combinatorial interpretation of this statement. In the last section, we present some open problems.