The Foata-Fuchs proof of Cayley's formula, and its probabilistic uses
Combinatorics
2022-11-21 v2 Probability
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.
Keywords
Cite
@article{arxiv.2107.09726,
title = {The Foata-Fuchs proof of Cayley's formula, and its probabilistic uses},
author = {Louigi Addario-Berry and Serte Donderwinkel and Mickaël Maazoun and James Martin},
journal= {arXiv preprint arXiv:2107.09726},
year = {2022}
}
Comments
11 pages, 2 figures