Markov degrees of hierarchical models determined by Betti numbers of Stanley-Reisner ideals

There are two seemingly unrelated objects associated to a simplicial complex: a hierarchical model and a Stanley-Reisner ring. A hierarchical model gives rise to a toric ideal, its generators providing a Markov basis for the model - a relationship that is a staple of algebraic statistics. In this note, we explore the connection between minimal generators of this toric ideal and syzygies of the Stanley-Reisner ideal. We propose a precise conjecture, supported by extensive computations, and we prove it in several cases, most notably for vertex-decomposable and decomposable complexes. As a first result establishing a relationship between the Stanley-Reisner ring and the ideal of the model, we hope this inspires a further study of the effect of the algebraic and geometric invariants of the classical combinatorial object on the underlying algebraic statistical model.