Abstract: 
We introduce a graded Hopf algebra based on the set of parking
functions (hence of dimension $(n+1)^{n-1}$ in degree $n$). 
This algebra can be embedded into a noncommutative polynomial algebra
in infinitely many variables.
We determine its structure, and show that it admits natural quotients
and subalgebras whose graded components have dimensions respectively given
by the Schr\"oder numbers (plane trees), the Catalan numbers, and powers of 3.
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These smaller algebras are always bialgebras and belong to some family of di-
or trialgebras occurring in the works of Loday and Ronco.

Moreover, the fundamental notion of parkization allows one to endow
the set of parking functions of fixed length with an associative
multiplication (different from the one coming from the Shi arrangement),
leading to a generalization of the internal product of symmetric functions.
Several of the intermediate algebras are stable under this operation.
Among them, one finds the Solomon descent algebra but also a new algebra
based on a Catalan set, admitting the Solomon algebra as a left ideal.

1. Institut Gaspard Monge
   Université de Marne-la-Vallée
   5 Boulevard Descartes
   Champs-sur-Marne
   77454 Marne-la-Vallée Cedex 2, France
   
2. Institut Gaspard Monge
   Université de Marne-la-Vallée
   5 Boulevard Descartes
   Champs-sur-Marne
   77454 Marne-la-Vallée Cedex 2, France