Classification of polyregular additive level sorts

Classify which additive level sorts are polyregular in terms of the step weights and tie-order, separating degenerate cases such as ν(U)=ν(D)=1, which yield the identity map, from mixed-sign sweep maps such as the height-sweep bijection H.

Background

An additive level sort assigns each input step an integer level obtained by a running weighted sum and outputs the steps in increasing level order, using a specified tie-order. The paper shows that important mixed-sign examples, including zeta-like and height-sweep constructions, lie outside PolyReg.

However, unbounded levels alone do not imply non-polyregularity: equal positive step weights produce levels equal to position indices and hence the identity transformation. A complete classification by weights and tie-order is left unresolved.

References

Which additive level sorts are polyregular? Unbounded levels alone do not force a lower bound: if $\nu(U)=\nu(D)=1$, then the level of position $i$ is $i-1$, and $\Phi_\nu$ is the identity map. A classification by the step weights and tie-order would separate such degenerate cases from examples such as $H$ and other mixed-sign sweep maps.

— A Computational Obstruction to Swapping Area and Dinv: An Automata-Theoretic View of the $q,t$-Catalan Symmetry  (2609.05005 - Baek et al., 4 Sep 2026) in Section “Related work and open problems”, item 4