3-dimensional polygons determined by permutations

In this paper we introduce the notion of $d$-dimensional permupolygons on $\mathbb{Z}^d$, with $d \geq 2$ 2-dimensional permupolygons, also called permutominides, where introduced by Incitti et al. By using an encoding of permupolygons inspired by the encoding given for convex polyominoes by Bousquet-Mélou and Guttmann, 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.

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Received 15 February 2014

Published 5 January 2018