Hilbert series of symmetric ideals in infinite polynomial rings via formal languages

Let R be the polynomial ring K[xi,j] where 1≤i≤r and j∈N, and let I be an ideal of R stable under the natural action of the infinite symmetric group S∞. Nagel-Römer recently defined a Hilbert series HI(s,t) of I and proved that it is rational. We give a much shorter proof of this theorem using tools from the theory of formal languages and a simple algorithm that computes the series.