Higher Trivariate Diagonal Harmonics via generalized Tamari Posets

By François Bergeron, Louis-François Préville-Ratelle. 2011

127 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/84d0b3c5d4dedaf827714bd5c9617306a6ef50c7

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

The representation of the symmetric group on m-Tamari intervals (2012, unknown): “In the case of three sets of variables, Bergeron and Préville-Ratelle [5] conjecture that, as representations of S n , DR m where Tam m ( n ) is the m -Tamari representation of S n .”
Associahedra for finite‐type cluster algebras and minimal relations between g‐vectors (2019, background): “The associahedron appears as a fundamental structure in several mathematical theories, such as moduli spaces and topology [54, 97], operads and rewriting theory [71, 99], combinatorial Hopf algebras [27, 62, 77], diagonal harmonics [21, 84], mathematical physics [4], and so on.”