# M\"obius functions of directed restriction species and free operads, via the generalised Rota formula

## TL;DR

This paper develops tools to apply the generalized Rota formula, enabling the computation of Möbius functions for incidence algebras of directed restriction species, free operads, and free monads on finitary polynomial monads.

## Contribution

It introduces methods to utilize the generalized Rota formula for calculating Möbius functions in new algebraic structures.

## Key findings

- Computed Möbius functions for incidence algebras of directed restriction species
- Extended the application of the generalized Rota formula to free operads
- Provided a framework for free monads on finitary polynomial monads

## Abstract

We present some tools for providing situations where the generalised Rota formula of arXiv:1801.07504 applies. As an example of this, we compute the M\"obius function of the incidence algebra of any directed restriction species, free operad, or more generally free monad on a finitary polynomial monad.