Commutative combinatorial Hopf algebras

Authors:
Florent Hivert;Jean-Christophe Novelli;Jean-Yves Thibon
Affiliations:
Laboratoire d'Informatique de l'Institut Gaspard Monge, Université Paris-Est, Champs-sur-Marne, France 77454;Laboratoire d'Informatique de l'Institut Gaspard Monge, Université Paris-Est, Champs-sur-Marne, France 77454;Laboratoire d'Informatique de l'Institut Gaspard Monge, Université Paris-Est, Champs-sur-Marne, France 77454
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2008

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Abstract

We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its noncommutative dual is realized in three different ways, in particular, as the Grossman---Larson algebra of heap-ordered trees. Extensions to endofunctions, parking functions, set compositions, set partitions, planar binary trees, and rooted forests are discussed. Finally, we introduce one-parameter families interpolating between different structures constructed on the same combinatorial objects.