Higher Trivariate Diagonal Harmonics via generalized Tamari Posets

TL;DR

This paper explores the structure of higher trivariate diagonal harmonic polynomials, revealing connections with Tamari posets and parking functions, and provides new formulas for their Hilbert series and Frobenius characteristics.

Contribution

It introduces novel combinatorial formulas and links between higher trivariate harmonic modules, Tamari posets, and parking functions, expanding understanding of their algebraic and combinatorial properties.

Findings

Derived formulas for Hilbert series of trivariate harmonic modules

Established connections between harmonic modules and Tamari posets

Presented new combinatorial formulas related to parking functions

Abstract

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.