  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.
