Identification with shifted Dyck-path polynomials

Determine whether the identity $N_{n,1}(q,t)=qt\,B_{n-1,3}(q,t)$ holds for every $n$, and explain the geometric meaning of the path base in the corresponding shifted Dyck-path model on $Y_n$.

Background

For n=2,3,4, the opposite-parity polynomial at m=1 agrees, up to the factor qt, with a shifted Dyck-path polynomial Bn−1,3(q,t)B_{n-1,3}(q,t) introduced in Lingxi Lu’s thesis. The agreement also holds for the scalar and principal specializations for every n.

The paper does not establish the full polynomial identity for arbitrary n, nor does it identify what the base parameter of the shifted paths represents geometrically on the type-B Nakajima quiver variety YnY_n.

References

It would be interesting to know whether the equality holds for all $n$ and what the base of the paths means on $Y_n$.

— 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 $N_{n,m}$”