Solve the generating function for Class 1509

Solve the generating function associated with inversion sequences avoiding the relation triple $(-,\geq,>)$, equivalently the patterns $(100,110,120,210)$, and determine whether the resulting generating function is algebraic.

Background

The paper develops a succession rule for Class 1509 by tracking the lengths of two consecutive runs of zeros. This construction leads to a functional equation for the generating function of an auxiliary class containing the desired avoidance class.

The authors explicitly report that they cannot solve the resulting functional equation and conjecture that the generating function is not algebraic. Thus both an exact solution and a definitive algebraicity classification remain unresolved.

References

As with class 214, we can construct a fairly simple succession rule for ${I}(-,\geq,>)$, but we have been unable to solve the corresponding generating function, and we do not believe it is algebraic.

— Completing the enumeration of inversion sequences avoiding triples of relations  (2512.21943 - Britt et al., 26 Dec 2025) in Section 3, Subsection 3.5, Class 1509: $(-,\geq,>) \equiv (100,110,120,210)$