Posetted Trees and Baker-Campbell-Hausdorff Product

In this paper, we use some typical tools of algebraic operads and homotopy transfer theory, in order to give a simple and elementary combinatorial description of the Baker-Campbell-Hausdorff formula. More precisely, we exploit the usual operadic notion of planar rooted trees, enriched with the notion of subroots, and we define a posetted tree as a planar rooted tree endowed with a monotone labelling of leaves, with elements in a partially ordered set. The main result of this paper is an explicit expression of the Baker-Campbell-Hausdorff product as a sum of iterated brackets over an indexing set of posetted binary trees.