A Motzkin filter in the Tamari lattice

The Tamari lattice of order n can be defined on the set Tn of binary trees endowed with the partial order relation induced by the well-known rotation transformation. In this paper, we restrict our attention to the subset Mn of Motzkin trees. This set appears as a filter of the Tamari lattice. We prove that its diameter is 2n − 5 and that its radius is n − 2. Enumeration results are given for join and meet irreducible elements, minimal elements and coverings. The set Mn endowed with an order relation based on a restricted rotation is then isomorphic to a ranked join-semilattice recently defined in [2]. As a consequence, we deduce an upper bound for the rotation distance between two Motzkin trees in Tn which gives the exact value for some specific pairs of Motzkin trees.