  Let $R$ be the polynomial ring $K[x_{i,j}]$ where $1 \le i \le r$ and $j \in
\mathbb{N}$, and let $I$ be an ideal of $R$ stable under the natural action of
the infinite symmetric group $S_{\infty}$. Nagel--R\"omer recently defined a
Hilbert series $H_I(s,t)$ 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.
