  To every partial order P, one associates a polynomial $\mathbb{D}_P$ in 4
variables that enumerates the intervals of P according to 4 parameters. Some
symmetry properties of this polynomial are obtained for a specific family of
posets, the Tamari lattices. A ternary symmetry is proved for the polynomial in
3 variables obtained by setting one variable to 1. Another global symmetry is
conjectured. The set of synchronized intervals is described using a facet of
the Newton polytope. A relation to the statistics of the canopy of binary
planar trees is described.
