Limiting Distributions for the Number of Inversions in Labelled Tree Families

Panholzer, Alois; Seitz, Georg

doi:10.1007/s00026-012-0164-3

Cited by 6 publications

(11 citation statements)

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“…where the constant κ depends on the particular tree family, i.e., on the degree-weight sequence, and is given in [64]. Consequently, an application of Lemma 2 and taking into account Example 3 yields the following result, which adds to the results of [64] the characterization of the limiting distribution as a mixed Poisson distribution. Corollary 6.…”

Section: The Number Of Inversions In Labelled Tree Familiesmentioning

confidence: 91%

“…We discuss several families of random trees where a mixed Poisson law arises as the limit law of a discrete random variable X n,j . The parameter n ∈ N usually measures the size of the investigated trees, and j denotes an additional parameter measuring or marking a certain aspect of the combinatorial structure, i.e., a node with a certain label j of interest, often satisfying a natural constraint of the type 1 ≤ j ≤ n, see [47,49,51,64]. In the limit n → ∞, with j = j(n), phase transitions were observed according to the relative growth of j with respect to n, e.g., j = 1, 2, .…”

Section: Examples and Applicationsmentioning

confidence: 99%

“…Our main motivation to study random variables with a given sequence of factorial moments (6) stems from the analysis of combinatorial structures. In many cases, amongst others the analysis of inversions in labelled tree families [64], stopping times in urn models [51,53], node degrees in increasing trees [49], block sizes in k-Stirling permutations [51], descendants in increasing trees [47], ancestors and descendants in evolving k-tree models [65], pairs of random variables X and Y arise as limiting distributions for certain parameters of interest associated to the combinatorial structures. The random variable X can usually be determined via its (power) moment sequence (µ s ) s∈N , and the random variable Y in terms of the sequence of factorial moments satisfying relation (6).…”

Section: Introductionmentioning

confidence: 99%

“…An open problem was to understand in more detail the nature of the random variable Y . In [53,64] a few results in this direction were obtained. The goal of this work is twofold: first, to survey the properties of mixed Poisson distributions, and second to discuss their appearances in combinatorics and the analysis of random discrete structures, complementing existing results; in fact, we will add several entirely new results.…”

Section: Introductionmentioning

confidence: 99%

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On moment sequences and mixed Poisson distributions

Kuba¹,

Panholzer²

2016

Probab. Surveys

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Abstract. In this article we survey properties of mixed Poisson distributions and probabilistic aspects of the Stirling transform: given a non-negative random variable X with moment sequence (µs) s∈N we determine a discrete random variable Y , whose moment sequence is given by the Stirling transform of the sequence (µs) s∈N , and identify the distribution as a mixed Poisson distribution. We discuss properties of this family of distributions and present a simple limit theorem based on expansions of factorial moments instead of power moments. Moreover, we present several examples of mixed Poisson distributions in the analysis of random discrete structures, unifying and extending earlier results. We also add several entirely new results: we analyse triangular urn models, where the initial configuration or the dimension of the urn is not fixed, but may depend on the discrete time n. We discuss the branching structure of plane recursive trees and its relation to table sizes in the Chinese restaurant process. Furthermore, we discuss root isolation procedures in Cayley-trees, a parameter in parking functions, zero contacts in lattice paths consisting of bridges, and a parameter related to cyclic points and trees in graphs of random mappings, all leading to mixed Poisson-Rayleigh distributions. Finally, we indicate how mixed Poisson distributions naturally arise in the critical composition scheme of Analytic Combinatorics.

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Section: The Number Of Inversions In Labelled Tree Familiesmentioning

confidence: 91%

Section: Examples and Applicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On moment sequences and mixed Poisson distributions

Kuba¹,

Panholzer²

2016

Probab. Surveys

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“…While analysing linear probing hashing, Flajolet, Poblete and Viola [23] noticed that the numbers of inversions in Cayley trees with uniform random labelling converge to an Airy distribution. Panholzer and Seitz [46] showed that this is true for conditional Galton-Watson trees, which encompass the case of Cayley trees.…”

Section: Inversions In a Fixed Treementioning

confidence: 93%

Inversions in Split Trees and Conditional Galton–Watson Trees

Cai¹,

Holmgren²,

Janson³

et al. 2018

Combinator. Probab. Comp.

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We study I(T ), the number of inversions in a tree T with its vertices labeled 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].

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Embedding Small Digraphs and Permutations in Binary Trees and Split Trees

et al. 2020

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We investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of inversions in randomly labelled trees [3]. We consider complete binary trees as well as random split trees a large class of random trees of logarithmic height introduced by Devroye [5]. 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 probability tending to one as the number of balls increases, 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.

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Limiting Distributions for the Number of Inversions in Labelled Tree Families

Cited by 6 publications

References 18 publications

On moment sequences and mixed Poisson distributions

On moment sequences and mixed Poisson distributions

Inversions in Split Trees and Conditional Galton–Watson Trees

Embedding Small Digraphs and Permutations in Binary Trees and Split Trees

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