Abstract
We studyI(T), the number of inversions in a treeTwith its vertices labelled uniformly at random, which is a generalization of inversions in permutations. We first show that the cumulants ofI(T) have explicit formulas involving thek-total common ancestors ofT(an extension of the total path length). Then we considerXn, the normalized version ofI(Tn), for a sequence of treesTn. For fixedTn's, we prove a sufficient condition forXnto converge in distribution. As an application, we identify the limit ofXnfor completeb-ary trees. ForTnbeing split trees [16], we show thatXnconverges to the unique solution of a distributional equation. Finally, whenTn's are conditional Galton–Watson trees, we show thatXnconverges to a random variable defined in terms of Brownian excursions. By exploiting the connection between inversions and the total path length, we are able to give results that significantly strengthen and broaden previous work by Panholzer and Seitz [46].
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.