Solve the generating-function equations for Class 830

Solve the functional equations for the generating functions associated with inversion sequences avoiding the relation triple $(\neq,>,\geq)$, equivalently the patterns $(010,120,210)$, in order to determine their counting generating function and establish whether it is algebraic.

Background

Class 830 concerns inversion sequences avoiding the triple of binary relations $(\ne,>,\geq)$, which is equivalent to avoiding the patterns $(010,120,210)$. The paper constructs a generating tree whose labels track the sequence length, maximum, and premaximum, and translates this construction into coupled functional equations for the generating functions $S(z,x,y)$ and $T(z,x,y)$.

The authors obtain asymptotic estimates for the associated counting sequence but do not solve the functional equations. They also state that they do not expect the generating function to be algebraic, making the exact solution and algebraic-status determination unresolved.