We introduce a graded Hopf algebra based on the set of parking functions (hence of dimension 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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm193-3-1,
author = {Jean-Christophe Novelli and Jean-Yves Thibon},
title = {Hopf algebras and dendriform structures arising from parking functions},
journal = {Fundamenta Mathematicae},
volume = {193},
year = {2007},
pages = {189-241},
zbl = {1127.16033},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm193-3-1}
}
Jean-Christophe Novelli; Jean-Yves Thibon. Hopf algebras and dendriform structures arising from parking functions. Fundamenta Mathematicae, Tome 193 (2007) pp. 189-241. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm193-3-1/