Abstract
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.
Funding Statement
During the preparation of this work MM was supported by EPSRC Fellowship EP/N004833/1.
Acknowledgements
We thank Nicolas Broutin for useful discussions, Adrien Segovia for pointing out the reference [23], and the anonymous referees for useful comments.
Citation
Louigi Addario-Berry. Arthur Blanc-Renaudie. Serte Donderwinkel. Mickaël Maazoun. James B. Martin. "The Foata–Fuchs proof of Cayley’s formula, and its probabilistic uses." Electron. Commun. Probab. 28 1 - 13, 2023. https://doi.org/10.1214/23-ECP523
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