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Inversions in Split Trees and Conditional Galton–Watson Trees

Part of: Combinatorial probability

Published online by Cambridge University Press:  31 October 2018

XING SHI CAI ,
CECILIA HOLMGREN ,
SVANTE JANSON ,
TONY JOHANSSON  and
FIONA SKERMAN
XING SHI CAI
Affiliation:
Department of Mathematics, Uppsala University, Lägerhyddsvägen 1, 752 37 Uppsala, Sweden (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected])
CECILIA HOLMGREN
Affiliation:
Department of Mathematics, Uppsala University, Lägerhyddsvägen 1, 752 37 Uppsala, Sweden (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected])
SVANTE JANSON
Affiliation:
Department of Mathematics, Uppsala University, Lägerhyddsvägen 1, 752 37 Uppsala, Sweden (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected])
TONY JOHANSSON
Affiliation:
Department of Mathematics, Uppsala University, Lägerhyddsvägen 1, 752 37 Uppsala, Sweden (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected])
FIONA SKERMAN
Affiliation:
Department of Mathematics, Uppsala University, Lägerhyddsvägen 1, 752 37 Uppsala, Sweden (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected])
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Abstract

We study I(T), the number of inversions in a tree T with its vertices labelled uniformly at random, which is a generalization of inversions in permutations. We first show that the cumulants of I(T) have explicit formulas involving the k-total common ancestors of T (an extension of the total path length). Then we consider Xn, the normalized version of I(Tn), for a sequence of trees Tn. For fixed Tn's, we prove a sufficient condition for Xn to converge in distribution. As an application, we identify the limit of Xn for complete b-ary trees. For Tn being split trees [16], we show that Xn converges to the unique solution of a distributional equation. Finally, when Tn's are conditional Galton–Watson trees, we show that Xn converges 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].

MSC classification

Primary: 60C05: Combinatorial probability
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 3 , May 2019 , pp. 335 - 364
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

This work was partially supported by two grants from the Knut and Alice Wallenberg Foundation, a grant from the Swedish Research Council, and the Swedish Foundations' starting grant from the Ragnar Söderberg Foundation.

§

Work partially conducted while affiliated with the Heilbronn Institute for mathematical research at the University of Bristol.

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