Constructive Hall proofs and symmetric chain decompositions for permutahedra

Develop a simple or constructive proof of conditions (ii) and (iii) in Proposition regarding the neighborhood expansion of inversion-level sets in the permutahedron, and determine whether the permutahedron admits a symmetric chain decomposition.

Background

The paper proves an asymptotically tight upper bound for minimum maximal matchings in the permutahedron by applying Hall’s theorem to the partition of permutations by inversion number. The resulting proof establishes the required neighborhood inequalities, but it is nonconstructive because Hall’s theorem guarantees the relevant matchings without explicitly producing them.

For the hypercube, analogous matchings can be constructed using a symmetric chain decomposition. The authors note that it is unknown whether a comparable decomposition exists for the permutahedron and whether the Hall conditions can be proved constructively. A constructive solution would provide an explicit algorithmic realization of the Hall-based upper-bound argument.