A Motzkin filter in the Tamari lattice

The Tamari lattice of order n can be defined on the set T-n, 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 M-n, 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 M-n endowed with an order relation based on a restricted rotation is then isomorphic to a ranked join-semilattice recently defined in Baril and Pallo (2014). As a consequence, we deduce an upper bound for the rotation distance between two Motzkin trees in T-n which gives the exact value for some specific pairs of Motzkin trees. (C) 2015 Elsevier B.V. All rights reserved.