Determine the generating function for Class 1509

Determine the generating function for inversion sequences avoiding the relation triple $(-,\geq,>)$, equivalently the patterns $(100,110,120,210)$, and establish whether the generating function is algebraic.

Background

For Class 1509, the paper constructs a succession rule using the lengths of two consecutive runs of zeros in an auxiliary class of inversion sequences. This yields a functional equation for the counting generating function.

The authors provide asymptotic estimates for the sequence but explicitly state that they cannot solve the corresponding generating function and do not believe it is algebraic. Thus both an exact generating-function description and a rigorous classification remain open.

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 3.9, subsection “Class 1509: $(-,\geq,>) \equiv (100,110,120,210)$”