Some applications arising from the interactions between the theory of Catalan-like numbers and the ECO method

In previous work of Ferrari and Pinzani, the ECO method and Aigner’s theory of Catalan-like numbers are compared, showing that it is often possible to translate a combinatorial situation from one theory into the other by means of a standard change of basis in a suitable vector space. In the present work we emphasize the soundness of such an approach by finding some applications suggested by the above mentioned translation. More precisely, we describe a presumably new bijection between two classes of lattice paths and we give a combinatorial interpretation to an integer sequence not appearing in the "Encyclopedia of Integer Sequences" (oeis.org).