Inversions in Split Trees and Conditional Galton-Watson Trees

Authors Xing Shi Cai , Cecilia Holmgren , Svante Janson , Tony Johansson, Fiona Skerman

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Xing Shi Cai
  • Department of Mathematics, Uppsala University, Uppsala, Sweden
Cecilia Holmgren
  • Department of Mathematics, Uppsala University, Uppsala, Sweden
Svante Janson
  • Department of Mathematics, Uppsala University, Uppsala, Sweden
Tony Johansson
  • Department of Mathematics, Uppsala University, Uppsala, Sweden
Fiona Skerman
  • Department of Mathematics, Uppsala University, Uppsala, Sweden

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Xing Shi Cai, Cecilia Holmgren, Svante Janson, Tony Johansson, and Fiona Skerman. Inversions in Split Trees and Conditional Galton-Watson Trees. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 15:1-15:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We study I(T), the number of inversions in a tree T with its vertices labeled uniformly at random. We first show that the cumulants of I(T) have explicit formulas. Then we consider X_n, the normalized version of I(T_n), for a sequence of trees T_n. For fixed T_n's, we prove a sufficient condition for X_n to converge in distribution. For T_n being split trees [Devroye, 1999], we show that X_n converges to the unique solution of a distributional equation. Finally, when T_n's are conditional Galton-Watson trees, we show that X_n converges to a random variable defined in terms of Brownian excursions. Our results generalize and extend previous work by Panholzer and Seitz [Panholzer and Seitz, 2012].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probabilistic algorithms
  • inversions
  • random trees
  • split trees
  • Galton-Watson trees
  • permutation
  • cumulant


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