Strictness of the multiple-rank-sort hierarchy

Determine whether the hierarchy obtained by allowing a fixed number of successive rank sorts in weighted-rank polyregular presentations is strict, and in particular establish whether two successive rank-sort layers suffice to realize the inverse Haglund zeta map ζ^{-1}.

Background

Weighted-rank polyregular maps allow one global sort of selected atoms by unbounded additive integer ranks. The paper establishes that this one-layer model contains the zeta map but cannot realize its inverse.

The authors propose extending the model by permitting a fixed number of successive rank sorts and ask whether each additional layer increases expressive power. The specific first test is whether two layers can realize ζ^{-1}.