Decomposition of Triply Rooted Trees
Decomposition of Triply Rooted Trees
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DOI：
10.37236/3016
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发表时间：
2012-12
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影响因子：
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通讯作者：
William Y. C. Chen;Janet F. F. Peng-Janet-F.-F.-Peng-46408159;Harold R. L. Yang
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文献类型：
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作者：
William Y. C. Chen;Janet F. F. Peng-Janet-F.-F.-Peng-46408159;Harold R. L. Yang
本文给出了三重根树分解为三个二重根树的方法。这导致了一个由Lacasse在PAC-Bayesian机器学习理论的研究中提出的恒等式的组合解释，并由Younsi通过使用多变量Abel多项式的Hurwitz恒等式证明。我们还给出了[n+1]到[n]的函数集与[n]上的三重根树集之间的一个双射，从而得到了[n+1]到[n]的函数关于n+1的轨道上的元素个数和周期点个数的精细计数.
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.