A Hopf algebra of parking functions

2003/12/05 by Jean-Christophe Novelli, Novelli, Jean-Christophe, Jean‐Yves Thibon +2 · 5 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #math.CO

paper · pdf · doi:10.48550/arxiv.math/0312126

AmsLatex, 14 pages

arxiv created 2003/12/05 · arxiv updated 2009/12/01

Abstract

If the moments of a probability measure on \R are interpreted as a specialization of complete homogeneous symmetric functions, its free cumulants are, up to sign, the corresponding specializations of a sequence of Schur positive symmetric functions (f_n). We prove that (f_n) is the Frobenius characteristic of the natural permutation representation of \SG_n on the set of prime parking functions. This observation leads us to the construction of a Hopf algebra of parking functions, which we study in some detail.

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• A Hopf algebra on permutations with a coupling product
• Symmetry in Tree Parking Distributions
• Polynomial realizations of some trialgebras
• Parking functions and triangulation of the associahedron
• An instance of umbral methods in representation theory: the parking function module

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