  Polymer models are a widely used tool to study the prebiotic formation of
metabolism at the origins of life. Counts of the number of reactions in these
models are often crucial in probabilistic arguments concerning the emergence of
autocatalytic networks. In the first part of this paper, we provide the first
exact description of the number of reactions under widely applied model
assumptions. Conclusions from earlier studies rely on either approximations or
asymptotic counting, and we show that the exact counts lead to similar, though
not always identical, asymptotic results. In the second part of the paper, we
investigate a novel model assumption whereby polymers are invariant under
spatial rotation. We outline the biochemical relevance of this condition and
again give exact enumerative and asymptotic formulae for the number of
reactions.
