An algebraic-combinatorial approach for studying coloured Dyck-Schroder paths

We characterize coloured Dyck and Schroder paths in both an algebraic and combinatorial way. In fact, we give algebraic and combinatorial proofs that, starting from the definition of such paths, we obtain a generating functions and, from this, the corresponding recurrence. Finally, by using a generalization of a beautiful bijection of Sulanke, we give a combinatorial proof of this recurrence, thus closing the ideal cycle connecting all the basic properties of the combinatorial objects considered.