Foata-Fuchs对Cayley公式的证明，以及它的概率应用

Pub Date : 2021-07-20 DOI:10.1214/23-ecp523

L. Addario-Berry, Serte Donderwinkel, M. Maazoun, James Martin

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引用次数: 3

摘要

由Foata和Fuchs（1970）给出了Cayley公式的一个非常简单的双射证明。当通过概率透镜观察时，这种双射是非常有用的；我们解释了它可以用来推导具有给定度的随机树（包括随机d元树）的概率恒等式、边界和生长过程的一些方法。我们还引入了有根树度序列上的一个偏序，并推测它在给定度的随机有根树的高度上诱导了一个随机偏序。

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The Foata–Fuchs proof of Cayley’s formula, and its probabilistic uses

We present a very simple bijective proof of Cayley's formula due to Foata and Fuchs (1970). This bijection turns out to be very useful when seen through a probabilistic lens; we explain some of the ways in which it can be used to derive probabilistic identities, bounds, and growth procedures for random trees with given degrees, including random d-ary trees. We also introduce a partial order on the degree sequences of rooted trees, and conjecture that it induces a stochastic partial order on heights of random rooted trees with given degrees.

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