A lattice on Dyck paths close to the Tamari Lattice
Résumé
We introduce a new poset structure on Dyck paths where the covering relation
is a particular case of the relation inducing the Tamari lattice. We prove that the transitive
closure of this relation endows Dyck paths with a lattice structure. We provide a trivariate
generating function counting the number of Dyck paths with respect to the semilength, the
numbers of outgoing and incoming edges in the Hasse diagram. We deduce the numbers of
coverings, meet and join irreducible elements. As a byproduct, we present a new involution
on Dyck paths that transports the bistatistic of the numbers of outgoing and incoming edges
into its reverse. Finally, we give a generating function for the number of intervals, and we
compare this number with the number of intervals in the Tamari lattice.