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Rauzy-type dynamics are group actions on a collection of combinatorialobjects. The first and best known example (the Rauzy dynamics) concerns anaction on permutations, associated to interval exchange transformations (IET)for the Poincar\'e map on compact orientable translation surfaces. Theequivalence classes on the objects induced by the group action have beenclassified by Kontsevich and Zorich in [KZ03] and correspond bijectively to theconnected components of the strata of the moduli space of abeliandifferentials.  In a paper [Boi14] Boissy proposed a Rauzy-type dynamics that acts on asubset of the permutations (the standard permutations) and conjectured that theRauzy classes of this dynamics are exactly the Rauzy classes of the Rauzydynamics restricted to standard permutations. In this paper, we apply thelabelling method introduced in [D18] to classify this dynamics thus provingBoissy's conjecture.  Finally, this paper conclude our serie of three papers on the study of theRauzy dynamics by presenting two new results on the Rauzy classes: An quadraticalgorithm for outputting a path between two connected permutations and a tight$\Theta(n)$ bound on the diameter of the Rauzy classes for the alternatingdistance.

A Combinatorial Approach to Rauzy-type Dynamics III: The Sliding Dynamics, Diameter and Algorithm