Equivariant Hilbert series for hierarchical Models

Toric ideals to hierarchical models are invariant under the action of a product symmetric groups. Taking number factors, say m, into account, we introduce and study filtrations their equivariant Hilbert series. We present condition that guarantees series is rational function in m+1 variables with coefficients. Furthermore give explicit formulas for functions coefficients field an algorithm determining A key construct finite automata recognize languages corresponding filtrations.