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A Hopf algebra of parking functions
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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.
Forward citations
Cited by 3 Pith papers
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A Hopf algebra on permutations with a coupling product
A new coupling product and draw coproduct make the vector space of permutations into a graded, connected, cocommutative, free Hopf algebra with a monomial-basis-like presentation.
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Symmetry in Tree Parking Distributions
Tree parking distributions on m-regular caterpillars yield a q,t-Fuss-Catalan generating function in which the lucky-car and first-spot statistics have a symmetric joint distribution.
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The free and parking quasi-symmetrizing actions
New symmetric group actions make FQSym* and PQSym* into invariant spaces, and their r-parameter deformations give a nested chain of Hopf subalgebras with explicit bases and Hilbert series.
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