  An m-ballot path of size n is a path on the square grid consisting of north
and east unit 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 T_n^(m), which
generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was
introduced by F. Bergeron in connection with the study of coinvariant spaces.
He conjectured several intriguing formulas dealing with the enumeration of
intervals in this lattice. One of them states that the number of intervals in
T_n^(m) is $$ \frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}. $$ This conjecture
was proved recently, but in a non-bijective way, while its form strongly
suggests a connection with plane trees. Here, we prove another conjecture of
Bergeron, which deals with the number of labelled, intervals. An interval [P,Q]
of T_n^(m) is labelled, if the north steps of Q are labelled from 1 to n in
such a way the labels increase along any sequence of consecutive north steps.
We prove that the number of labelled intervals in T_n^(m) is $$
{(m+1)^n(mn+1)^{n-2}}. $$ The form of these numbers suggests a connection with
parking functions, but our proof is non-bijective. It is based on a recursive
description of intervals, which translates into a functional equation satisfied
by the associated generating function. This equation involves a derivative and
a divided difference, taken with respect to two additional variables. Solving
this equation is the hardest part of the paper. Finding a bijective proof
remains an open problem.
