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Polyominoes determined by involutions

Abstract : A permutomino of size n is a polyomino determined by particular pairs $(\pi_1, \pi_2)$ of permutations of length $n$, such that $\pi_1(i) \neq \pi_2(i)$, for $1 \leq i \leq n$. In this paper we consider the class of convex permutominoes which are symmetric with respect to the diagonal $x = y$. We determine the number of these permutominoes according to the dimension and we characterize the class of permutations associated to these objects as particular involutions of length $n$.
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Filippo Disanto, Simone Rinaldi. Polyominoes determined by involutions. 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.189-202. ⟨hal-01185173⟩

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