Otterʹs Method and the Homology of Homeomorphically Irreduciblek-Trees

Let T(k)ndenote the collection of trees withnk+2 labelled leaves which have the property that every internal node has degreemk+2 for somem⩾1. Partially order the elements of T(k)nby definingT⩽SifTcan be obtained fromSby contracting some collection of internal edges. Then T(k)nis a Cohen–Macaulay poset with an action ofSnk+2. We show that the top homology of T(k)nas anSnk+2-module is[formula]whereLie(k)Nis the action ofSNon the 1N-homogeneous piece of the free Liek-algebra. This generalizes the result obtained by Sarah Whitehouse [W] in the casek=1.