A Closed Formula for the Number of Convex Permutominoes

In this paper we determine a closed formula for the number of convex permutominoes of size n. We reach this goal by providing a recursive generation of all convex permutominoes of size n+1 from the objects of size n, according to the ECO method, and then translating this construction into a system of functional equations satisfied by the generating function of convex permutominoes. As a consequence we easily obtain also the enumeration of some classes of convex polyominoes, including stack and directed convex permutominoes. 1 Basic definitions and contents of the paper A polyomino is a finite union of elementary cells of the lattice Z×Z, whose interior is connected (see Figure 1 (a)). Polyominoes are defined up to a translation. A polyomino is said to be column convex (resp. row convex) if all its columns (resp. rows) are connected (see Figure 1 (b)). A polyomino is said to be convex, if it is both row and column convex (see Figure 1 (c)). Delest and Viennot [13] determined the number cn of convex polyominoes with semiperimeter n + 2, cn+2 = (2n + 11)4 n − 4(2n + 1) ( 2n n ) , n ≥ 0; c0 = 1, c1 = 2, (1) sequence A005436 in [18], the first few terms being: 1, 2, 7, 28, 120, 528, 2344, 10416, . . . . Università di Siena, Dipartimento di Scienze Matematiche e Informatiche, Pian dei Mantellini 44, 53100 Siena, Italy ([email protected]). Università di Firenze, Dipartimento di Sistemi e Informatica, viale Morgagni 65, 50134 Firenze, Italy ([frosini, pinzani]@dsi.unifi.it). the electronic journal of combinatorics 14 (2007), #R57 1