Solve the functional equation for Class 214

Solve the functional equation for inversion sequences avoiding the relation triple $(-,\geq,\geq)$, equivalently the patterns $(000,010,100,110,120,210)$, to determine the associated generating function and its algebraic status.

Background

Class 214 is represented by inversion sequences avoiding six length-three patterns. The paper introduces an auxiliary class and a succession rule based on the lengths of two possible consecutive runs of zeros, then derives a functional equation for the corresponding bivariate generating function.

Although the equation has a relatively simple form, the authors do not solve it and conjecturally expect the generating function not to be algebraic. The exact enumeration remains unresolved despite numerical asymptotic estimates.

References

However unfortunately we have been unable to solve this equation.

— Completing the enumeration of inversion sequences avoiding triples of relations  (2512.21943 - Britt et al., 26 Dec 2025) in Section 3.8, subsection “Class 214: $(-,\geq,\geq) \equiv (000,010,100,110,120,210)$”