The number of intervals in the m-Tamari lattices

Abstract

An m-ballot path of size n is a path on the square grid consisting of north and east steps, starting at (0,0), ending at (mn,n), and never going below the line {x = my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by Tn(m), which generalizes the usual Tamari lattice Tn obtained when m = 1. We prove that the number of intervals in this lattice is. This formula was recently conjectured by Bergeron in connection with the study of diagonal coinvariant spaces. The case m = 1 was proved a few years ago by Chapoton. Our proof is based on a recursive description of intervals, which translates into a functional equation satisfied by the associated generating function. The solution of this equation is an algebraic series, obtained by a guess-and-check approach. Finding a bijective proof remains an open problem.