Minimum maximal matchings in permutahedra

We prove that the minimal size $M(\pi_n)$ of a maximal matching in the permutahedron $\pi_n$ is asymptotically $n!/3$. On the one hand, we obtain a lower bound $M(\pi_n) \ge n! (n-1) / (3n-2)$ by considering $4$-cycles in the permutahedron. On the other hand, we obtain an asymptotical upper bound $M(\pi_n) \le n!(1/3+o(1))$ by multiple applications of Hall's theorem (similar to the approach of Forcade (1973) for the hypercube) and an exact upper bound $M(\pi_n) \le n!/3$ by an explicit construction. We also derive bounds on minimum maximal matchings in products of permutahedra.