On an Identity for the Cycle Indices of Rooted Tree Automorphism Groups

This note deals with a formula due to G. Labelle for the summed cycle indices of all rooted trees, which resembles the well-known formula for the cycle index of the symmetric group in some way. An elementary proof is provided as well as some immediate corollaries and applications, in particular a new application to the enumeration of k-decomposable trees. A tree is called k-decomposable in this context if it has a spanning forest whose components are all of size k.

• Type: Journal
• Year: 2006
• Where: COMBINATORICS
• Authors: Stephan G. Wagner
• Keywords: COMBINATORICS 2006, Cycle Index, Summed Cycle Indices, Well-known Formula