A note on Tamari intervals

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.