On atomic and unsplitable endofunctions

In the study of a cocommutative Hopf algebra PFSym built on parking functions, the author presented two free generating sets of PFSym which are indexed by atomic parking functions and unsplitable parking functions. The notions of atomic parking function and unsplitable parking function are introduced via two binary operations called the slash product and the split product respectively. It follows that the set of atomic parking functions of length n is equinumerous with the set of unsplitable parking functions of length n . In this paper, we will give a bijective proof of this fact in the more general setting of endofunctions. We generalize the slash product and the split product to endofunctions, and then propose the notions of atomic endofunction and unsplitable endofunction. We show that this leads to two free monoid structures on endofunctions, whose free generators are atomic endofunctions and unsplitable endofunctions respectively. Moreover, it is shown that for each n ≥ 1 , the set of atomic endofunctions of length n is in bijection with the set of unsplitable endofunctions of length n .