Combinatorial realization and interpretation of the parity components

Construct a pair of statistics on lattice paths that realizes the polynomial $P_{n,m}(q,t)$, and determine a geometric or representation-theoretic meaning for its opposite-parity component $N_{n,m}=P_{n,m}-F_{n,m}$.

Background

The localization polynomial Pn,m(q,t)P_{n,m}(q,t) gives the asymptotic type-BnB_n q,t-Fuss–Catalan character, while Nn,mN_{n,m} measures the opposite-parity contribution relative to the unprojected localization polynomial Fn,mF_{n,m}. Low-rank examples exhibit positive combinations of standard q,t strings.

The paper compares these polynomials with shifted Dyck-path enumerators and notes agreement of several specializations, but no general lattice-path model or conceptual interpretation of the opposite-parity part is established.

References

Is there a pair of statistics on lattice paths which realizes $P_{n,m}(q,t)$, and is there a geometric or representation-theoretic meaning of the opposite-parity part $N_{n,m}$?

— Asymptotic $q,t$-Fuss--Catalan numbers for type $B$  (2609.11691 - Oblomkov, 10 Sep 2026) in Section 16, subsection “Open problems,” Question “Combinatorics of $P_{n,m}$ and $N_{n,m}$”