A Hopf algebra of parking functions
Abstract
If the moments of a probability measure on 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 (fn). We prove that (fn) is the Frobenius characteristic of the natural permutation representation of 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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