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A catalanization map on the symmetric group

: In this article we investigate a self-map of the symmetric group. We show that the set of plus-indecomposable permutations is stable under our map. Furthermore, the ﬁxed points are precisely the 231-avoiding permutations. Finally, by using our map we obtain a nested sequence of polytopes that starts with the permutahedron and ends with the associahedron.