Motzkin subposets and Motzkin geodesics in Tamari lattices

The Tamari lattice of order n can be defined by the set Dn of Dyck words endowed with the partial order relation induced by the well-known rotation transformation. In this paper, we study this rotation on the restricted set of Motzkin words. An upper semimodular join semilattice is obtained and a shortest path metric can be defined. We compute the corresponding distance between two Motzkin words in this structure. This distance can also be interpreted as the length of a geodesic between these Motzkin words in a Tamari lattice. So, a new upper bound is obtained for the classical rotation distance between two Motzkin words in a Tamari lattice. For some specific pairs of Motzkin words, this bound is exactly the value of the rotation distance in a Tamari lattice. Finally, enumerating results are given for join and meet irreducible elements, minimal elements and coverings.