3-dimensional polygons determined by permutations

In this paper we introduce the notion of d-dimensional permupolygons on Z(d), with d >= 2. 2-dimensional permupolygons, also called permutominides, where introduced by Incitti et al. [12]. By using an encoding of permupolygons inspired by the encoding given for convex polyominoes by Bousquet-Melou and Guttmann [6], we easily recover enumerative results about 2-dimensional parallelogram, unimodal, column convex and convex permupolygons. Moreover, we extend these results for dimension d = 3. Finally, we study combinatorial characterizations of permutations defining 3-dimensional permupolygons. We show some necessary and sufficient conditions for a triple of 2-dimensional permutations (pi(1), pi(2), pi(3)) to define a 3-dimensional permupolygon.