  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, which generalizes the usual Tamari lattice
obtained when m=1. We prove that the number of intervals in this lattice is $$
\frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}. $$ This formula was recently
conjectured by Bergeron in connection with the study of 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.
