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Hopf algebras and dendriform structures arising from parking functions

Jean-Christophe Novelli; Jean-Yves Thibon — 2007

Fundamenta Mathematicae

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 Schröder numbers (plane trees), the Catalan numbers, and powers of 3. These smaller algebras are always bialgebras and belong to some family...

Natural endomorphisms of quasi-shuffle Hopf algebras

Jean-Christophe Novelli; Frédéric Patras; Jean-Yves Thibon — 2013

Bulletin de la Société Mathématique de France

The Hopf algebra of word-quasi-symmetric functions ( $𝐖𝐐𝐒𝐲𝐦$ ), a noncommutative generalization of the Hopf algebra of quasi-symmetric functions, can be endowed with an internal product that has several compatibility properties with the other operations on $𝐖𝐐𝐒𝐲𝐦$ . This extends constructions familiar and central in the theory of free Lie algebras, noncommutative symmetric functions and their various applications fields, and allows to interpret $𝐖𝐐𝐒𝐲𝐦$ as a convolution algebra of linear endomorphisms of quasi-shuffle...