Combinatorial characterization of algebraic realizability for linear recurrent divisibility sequences

Determine a combinatorial characterization of algebraic realizability for integer sequences realized by group automorphisms, and, specifically, identify which linear recurrent divisibility sequences classified by Bézivin, Pethő, and van der Poorten are algebraically realizable.

Background

A sequence of non-negative integers (a_n) is realizable if there exists a map T: X → X such that a_n equals the number of points fixed by T^n. Beyond purely combinatorial realizability, one can ask when a sequence is realized by a group automorphism (“algebraic realizability”), a stronger property implying the sequence is a divisibility sequence.

Pat Moss showed equivalences for nilpotent realizability via local conditions, but a general combinatorial characterization for algebraic realizability remains unresolved. Even within the restricted class of linear recurrent divisibility sequences—already classified in earlier work—the question of which are algebraically realizable is not settled.