  In earlier work we proposed a generating function that encodes the Jordan
block spectrum of the integrable Hypereclectic spin chain, related to the
one-loop dilatation operator of the dynamical fishnet quantum field theory. We
significantly improve the expressions for these generating functions, rendering
them much more explicit and elegant. In particular, we treat the case of the
full spin chain without imposing any cyclicity constraints on the states, as
well as the case of cyclic states. The latter involves the P\'olya enumeration
theorem in conjunction with q-binomial coefficients.
