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We present some tools for providing situations where the generalised Rota formula of Carlier (Algebr Geometr Topol 20:169–213, 2020) applies. As an example of this, we compute the Möbius function of the incidence algebra of any directed restriction species, free operad, or more generally free monad on a finitary polynomial monad.

Möbius Functions of Directed Restriction Species and Free Operads, via the Generalised Rota Formula