Medical Service Quality in the Situation of Healthcare System

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