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Commutative Hopf algebras of permutations and trees

F. Hivert·J. -C. Novelli·J. -Y. Thibon
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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 non-commutative dual is realized in three different ways, in particular as the Grossman-Larson algebra of heap ordered trees. Extensions to endofunctions, parking functions, 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.

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F. Hivert, J. -C. Novelli, J. -Y. Thibon. 2005-02-22. Commutative Hopf algebras of permutations and trees. https://arxiv.org/abs/math/0502456

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