Commutative combinatorial Hopf algebras

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.