uu.seUppsala University Publications

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Inversions in Split Trees and Conditional Galton-Watson TreestPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true}); PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt905",{id:"formSmash:j_idt905",widgetVar:"widget_formSmash_j_idt905",onLabel:"Hide others and affiliations",offLabel:"Show others and affiliations"});
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2019 (English)In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 28, no 3, p. 335-364Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2019. Vol. 28, no 3, p. 335-364
##### National Category

Probability Theory and Statistics Computer Sciences
##### Identifiers

URN: urn:nbn:se:uu:diva-382243DOI: 10.1017/S0963548318000512ISI: 000462848800002OAI: oai:DiVA.org:uu-382243DiVA, id: diva2:1316041
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##### Funder

Knut and Alice Wallenberg FoundationSwedish Research CouncilAvailable from: 2019-05-15 Created: 2019-05-15 Last updated: 2019-05-15Bibliographically approved

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 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. As an application, we identify the limit of X-n for complete b-ary trees. For T(n )being split trees [16], 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. 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].

doi
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