Toric ideals to hierarchical models are invariant under the action of
a product of symmetric groups. Taking the number of factors, say,
, 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
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 coefficients.
A key is to construct finite automata that recognize languages corresponding to invariant filtrations.
