License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.AofA.2018.9
URN: urn:nbn:de:0030-drops-89025
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8902/
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Albert, Michael ; Holmgren, Cecilia ; Johansson, Tony ; Skerman, Fiona

Permutations in Binary Trees and Split Trees

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Abstract

We investigate the number of permutations that occur in random node labellings of trees. This is a generalisation of the number of subpermutations occuring in a random permutation. It also generalises some recent results on the number of inversions in randomly labelled trees [Cai et al., 2017]. We consider complete binary trees as well as random split trees a large class of random trees of logarithmic height introduced by Devroye [Devroye, 1998]. Split trees consist of nodes (bags) which can contain balls and are generated by a random trickle down process of balls through the nodes.
For complete binary trees we show that asymptotically the cumulants of the number of occurrences of a fixed permutation in the random node labelling have explicit formulas. Our other main theorem is to show that for a random split tree with high probability the cumulants of the number of occurrences are asymptotically an explicit parameter of the split tree. For the proof of the second theorem we show some results on the number of embeddings of digraphs into split trees which may be of independent interest.

BibTeX - Entry

@InProceedings{albert_et_al:LIPIcs:2018:8902,
  author =	{Michael Albert and Cecilia Holmgren and Tony Johansson and Fiona Skerman},
  title =	{{Permutations in Binary Trees and Split Trees}},
  booktitle =	{29th International Conference on Probabilistic,  Combinatorial and Asymptotic Methods for the Analysis of Algorithms  (AofA 2018)},
  pages =	{9:1--9:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-078-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{110},
  editor =	{James Allen Fill and Mark Daniel Ward},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8902},
  URN =		{urn:nbn:de:0030-drops-89025},
  doi =		{10.4230/LIPIcs.AofA.2018.9},
  annote =	{Keywords: random trees, split trees, permutations, inversions, cumulant}
}

Keywords: random trees, split trees, permutations, inversions, cumulant
Collection: 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)
Issue Date: 2018
Date of publication: 18.06.2018

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