# Geometry of random permutation factorizations

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Time: 10:30 -- Location: Salle Philippe Flajolet du LIX

We study random minimal factorizations of the $$n$$-cycle into transpositions, that is, factorizations of the cycle $$(1 2...n)$$ into a product of $$n-1$$ transpositions. It is known that these factorizations are in bijection with Cayley trees of size $$n$$, and therefore that there are $$n^{n-2}$$ of them. Moreover, they are naturally coded by sequences of sets of non-crossing chords in the unit disk. We establish the existence of a limit for such sets of chords, when they code a uniform random minimal factorization of the cycle. We show in particular how this random limit is naturally encoded in a Brownian excursion. The key tool of this study is an explicit bijection between minimal factorizations of the $$n$$-cycle and a family of trees with $$n$$ labelled vertices. Finally, we explain how this framework can be extended to factorizations of $$(1 2 ... n)$$ into cycles that are not necessarily transpositions, giving birth to new random structures.

Category: seminars
Tags: Combi seminar combinatorics