Boundary between P-recursive and non-P-recursive run constraints

Characterize the boundary between constraints on tableau run lengths that preserve P-recursiveness and constraints that destroy P-recursiveness for standard Young tableaux of three-row rectangular shape.

Background

The paper contrasts its statistic, which constrains only the first run and yields a P-recursive sequence, with work of Kauers and Zeilberger conjecturing that constraining the lengths of all runs in a three-row rectangular tableau produces non-P-recursive sequences.

The unresolved question is to identify where the transition occurs between these two behaviors: which classes or combinations of run-length constraints lead to P-recursive counting sequences and which do not.