Hopf Algebras of Heap Ordered Trees and Permutations

A standard heap ordered tree with n + 1 nodes is a finite rooted tree in which all the nodes except the root are labeled with the natural numbers between 1 and n, and that satisfies the property that the labels of the children of a node are all larger than the label of the node. Denote the set of standard heap ordered trees with n+1 nodes by Tn. Let kT = ⊕ n≥0 kTn. It is known that there are Hopf algebra structures on kT . Let Sn denote the symmetric group on n symbols. Let