We introduce an anticyclic operad V given by a ternary generator and a
quadratic relation. We show that it admits a natural basis indexed by planar
binary trees. We then relate this construction to the familly of Tamari
lattices (Y_n) for n>=0 by defining an isomorphism between V(2n+1) and the
Grothendieck group of the category mod Y_n. This isomorphism maps the basis of
V(2n+1) to the classes of projective modules and sends the anticyclic map of
the operad V to the Coxeter transformation of the derived category of mod Y_n.