In this paper, we give a decomposition of triply rooted trees into three doubly rooted trees. This leads to a combinatorial interpretation of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory, and proved by Younsi by using the Hurwitz identity on multivariate Abel polynomials. We also give a bijection between the set of functions from [n+1] to [n] and the set of triply rooted trees on [n], which leads to the refined enumeration of functions from [n+1] to [n] with respect to the number of elements in the orbit of n+1 and the number of periodic points.
@article{10_37236_3016,
author = {William Y.C. Chen and Janet F.F. Peng and Harold R.L. Yang},
title = {Decomposition of triply rooted trees},
journal = {The electronic journal of combinatorics},
year = {2013},
volume = {20},
number = {2},
doi = {10.37236/3016},
zbl = {1267.05022},
url = {http://geodesic.mathdoc.fr/articles/10.37236/3016/}
}
TY - JOUR
AU - William Y.C. Chen
AU - Janet F.F. Peng
AU - Harold R.L. Yang
TI - Decomposition of triply rooted trees
JO - The electronic journal of combinatorics
PY - 2013
VL - 20
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/3016/
DO - 10.37236/3016
ID - 10_37236_3016
ER -
%0 Journal Article
%A William Y.C. Chen
%A Janet F.F. Peng
%A Harold R.L. Yang
%T Decomposition of triply rooted trees
%J The electronic journal of combinatorics
%D 2013
%V 20
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/3016/
%R 10.37236/3016
%F 10_37236_3016
William Y.C. Chen; Janet F.F. Peng; Harold R.L. Yang. Decomposition of triply rooted trees. The electronic journal of combinatorics, Tome 20 (2013) no. 2. doi: 10.37236/3016
