Higher Trivariate Diagonal Harmonics via generalized Tamari Posets

We consider the graded $\S_n$-modules of higher diagonally harmonic polynomials in three sets of variables (the trivariate case), and show that they have interesting ties with generalizations of the Tamari poset and parking functions. In particular we get several nice formulas for the associated Hilbert series and graded Frobenius characteristic. This also leads to entirely new combinatorial formulas.