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On square permutations

Abstract : Severini and Mansour introduced $\textit{square polygons}$, as graphical representations of $\textit{square permutations}$, that is, permutations such that all entries are records (left or right, minimum or maximum), and they obtained a nice formula for their number. In this paper we give a recursive construction for this class of permutations, that allows to simplify the derivation of their formula and to enumerate the subclass of square permutations with a simple record polygon. We also show that the generating function of these permutations with respect to the number of records of each type is algebraic, answering a question of Wilf in a particular case.
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Enrica Duchi, Dominique Poulalhon. On square permutations. Fifth Colloquium on Mathematics and Computer Science, 2008, Kiel, Germany. pp.207-222, ⟨10.46298/dmtcs.3565⟩. ⟨hal-01194689⟩

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