Centralizers of maps as automorphism groups of Yang--Baxter solutions

For a partition X = Y Z of a set X and an arbitrary map g Y → HomSet (Z, \, Z), we construct a solution Rg X × X → X × X of the braid equation. We characterize intrinsically the solutions which are isomorphic to some Rg and we give a necessary and sufficient condition for two solutions Rg and Rg' to be isomorphic. Consequently, the number of isomorphism classes of the solutions Rg on a set with n elements is 1 + ∑k = 1ⁿ⁻¹ o (n - k, \, k), where o (m, \, k) is the number of orbits of the symmetric group Sₘ, acting by conjugation, on the set of multisets with k elements of maps \1, …, m\ → \1, …, m\. This construction is then used to obtain a solution of the braid equation whose automorphism group is exactly the centralizer Aut (f₁, …, fₖ) of a given family f₁, …, fₖ X → X of maps, k ≥ 1. For $k = 2$, combined with a result of Agore, Chirvasitu and the author, this gives another proof of the fact that any group is isomorphic to the automorphism group of a solution of the braid equation (equivalently, of the set-theoretic Yang--Baxter equation).