Given a connected (quiver) nonselfinjective Nakayama algebra with a circle as a quiver and at least two points. Such an algebra is determined by the (Kupisch) sequence $[c_0,c_1,...,c_{n-1}]$, when the algebra has $n$ simples and $c_i$ is the dimension of the projective module at point $i$. We can furthermore always rotate and assume $c_{n-1}=c_0+1$. Call two Kupisch series (and the corresponding Nakayama algebras) $[c_0,c_1,...,c_{n-1}]$ and $[e_0,e_1,...,e_{m-1}]$ in the same difference class in case $n=m$ and $e_i=c_i$ mod $n$ for all $i$. I can prove that the dominant dimension is always the same for two algebras in the same difference class. Now the same seems to hold for the finitistic dimension, but to prove that I have to use the result about the dominant dimension. Is there a more direct proof? Note also that the global or Gorenstein dimension is not independent of the difference class.