Séminaire Lotharingien de Combinatoire, B89b (2025), 18 pp.
Jean-Luc Baril, Sergey Kirgizov and Mehdi Naima
A Lattice on Dyck Paths Close to the Tamari Lattice
Abstract.
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 to 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.
Received: August 29, 2023.
Accepted: April 25, 2025.
Final Version: May 5, 2025.
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