About Half Permutations

In 2011, Beaton et al. analytically proved that the number of directed columnconvex permutominoes of size n is given by (n+ 1)!/2. In this paper, we provide a different proof of this statement using a bijective method. More precisely, we present a bijective correspondence between the class Dn of directed column-convex permutominoes of size n and a set of permutations (called dcc-permutations) of length n + 1, which we prove to be counted by (n + 1)!/2. The class of dcc-permutations is a new class of permutations counted by half factorial numbers, and here we show some combinatorial characterizations of this class, using the concept of logical formulas determined by a permutation and the notion of mesh pattern.