Tree/Endofunction Bijections and Concentration Inequalities
Abstract
We demonstrate a method for proving precise concentration inequalities in uniformly random trees on $n$ vertices, where $n\geq1$ is a fixed positive integer. The method uses a bijection between mappings $f\colon\{1,\ldots,n\}\to\{1,\ldots,n\}$ and doubly rooted trees on $n$ vertices. The main application is a concentration inequality for the number of vertices connected to an independent set in a uniformly random tree, which is then used to prove partial unimodality of its independent set sequence. So, we give probabilistic arguments for inequalities that often use combinatorial arguments.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 DOI:
 10.48550/arXiv.2006.06724
 arXiv:
 arXiv:2006.06724
 Bibcode:
 2020arXiv200606724H
 Keywords:

 Mathematics  Probability;
 Mathematics  Combinatorics
 EPrint:
 15 pages, 3 figures