Partial automorphisms and injective partial endomorphisms of a finite undirected path

In this paper, we study partial automorphisms and, more generally, injective endomorphisms of a finite undirected path from Semigroup Theory perspective. Our main objective is to give formulas for the ranks monoids $$\mathrm {IEnd}(P_n)$$ and {PAut}(P_n)$$ all $$P_n$$ with n vertices. We also describe Green’s relations calculate their cardinals.