A Closed Formula for the Number of Convex Permutominoes

Abstract

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.

The number of convex permutominoes

“…In this paper, we provide an explicit formula for the number of convex permutominoes of a given perimeter. Incidentally, we notice that an equivalent formula has been independently obtained in [4], using a totally different technique based on the ECO method.…”

About Half Permutations

“…We point out that (1) was proved independently in [5] and in [9], by using analytical techniques. Recently, a bijective proof of (1) was given in [8] by encoding convex permutominoes in terms of lattice paths.…”

“…These permutations are called the first and the second component of P , respectively. We refer to [3,9,11] for more detailed definitions on polyominoes and on permutominoes, and to [6] for basic definitions on permutations.…”

“…where k = 0.3419111 and h = 1.385933. Exact results have been obtained for some classes of convex permutominoes [3,9,11], and in particular the number of convex permutominoes of size n was proved to be equal to:…”

Tiling the Plane with Permutations

“…During the last years, a particular class of permutominoes, namely the class of convex permutominoes have been widely studied: in [3] the authors provide their enumeration according to the size, while in [2] the authors give a characterization of permutation defining convex permutominoes.…”