Geodesic
On an identity for the cycle indices of rooted tree automorphism groups
The electronic journal of combinatorics, Tome 13 (2006)
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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 $k$.
DOI : 10.37236/1152
Classification : 05C30, 05A15, 05A19
Mots-clés : enumeration 
@article{10_37236_1152,
     author = {Stephan G. Wagner},
     title = {On an identity for the cycle indices of rooted tree automorphism groups},
     journal = {The electronic journal of combinatorics},
     year = {2006},
     volume = {13},
     doi = {10.37236/1152},
     zbl = {1113.05050},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1152/}
}
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Stephan G. Wagner. On an identity for the cycle indices of rooted tree automorphism groups. The electronic journal of combinatorics, Tome 13 (2006). doi: 10.37236/1152

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