Coalescent random forests,

"... We study moments and asymptotic distributions of the construction cost, measured as the total displacement, for hash tables using linear probing. Four different methods are employed for different ranges of the parameters; together they yield a complete description. This extends earlier results by F ..."

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We study moments and asymptotic distributions of the construction cost, measured as the total displacement, for hash tables using linear probing. Four different methods are employed for different ranges of the parameters; together they yield a complete description. This extends earlier results by Flajolet, Poblete and Viola. The average cost of unsuccessful searches is considered too.

A family of random trees with random edge lengths

by David Aldous, Jim Pitman , 1999

"... We introduce a family of probability distributions on the space of trees with I labeled vertices and possibly extra unlabeled vertices of degree 3, whose edges have positive real lengths. Formulas for distributions of quantities such asdegree sequence, shape, and total length are derived. An interpr ..."

Abstract - Cited by 12 (8 self) - Add to MetaCart

We introduce a family of probability distributions on the space of trees with I labeled vertices and possibly extra unlabeled vertices of degree 3, whose edges have positive real lengths. Formulas for distributions of quantities such asdegree sequence, shape, and total length are derived. An interpretation is given in terms of sampling from the inhomogeneous continuum random tree of Aldous and Pitman (1998). Key words and phrases. Continuum tree, enumeration, random tree, spanning tree, weighted tree, Cayley's multinomial expansion.

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The cut-tree of large Galton-Watson trees and the Brownian CRT. The Annals of Applied Probability

by Jean Bertoin, Universite ́ Paris-sud

"... ar ..."

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Abel-Cayley-Hurwitz multinomial expansions associated with random mappings, forests, and subsets

by Jim Pitman , 1998

"... Extensions of binomial and multinomial formulae due to Abel, Cayley and Hurwitz are related to the probability distributions of various random subsets, trees, forests, and mappings. For instance, an extension of Hurwitz's binomial formula is associated with the probability distribution of the r ..."

Abstract - Cited by 10 (9 self) - Add to MetaCart

Extensions of binomial and multinomial formulae due to Abel, Cayley and Hurwitz are related to the probability distributions of various random subsets, trees, forests, and mappings. For instance, an extension of Hurwitz's binomial formula is associated with the probability distribution of the random set of vertices of a fringe subtree in a random forest whose distribution is defined by terms of a multinomial expansion over rooted labeled forests which generalizes Cayley's expansion over unrooted labeled trees. Contents 1 Introduction 2 Research supported in part by N.S.F. Grant DMS97-03961 2 Probabilistic Interpretations 5 3 Cayley's multinomial expansion 11 4 Random Mappings 14 4.1 Mappings from S to S : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 4.2 The random set of cyclic points : : : : : : : : : : : : : : : : : : : : : : : 18 5 Random Forests 19 5.1 Distribution of the roots of a p-forest : : : : : : : : : : : : : : : : : : : : 19 5.2 Conditioning on the set...

Pointed and multi-pointed partitions of type A and B

by F. Chapoton, B. Vallette , 2005

"... The aim of this paper is to define and study pointed and multi-pointed partition posets of type A and B (in the classification of Coxeter groups). We compute their characteristic polynomials, incidence Hopf algebras and homology groups. As a corollary, we show that some operads are Koszul over Z. ..."

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The aim of this paper is to define and study pointed and multi-pointed partition posets of type A and B (in the classification of Coxeter groups). We compute their characteristic polynomials, incidence Hopf algebras and homology groups. As a corollary, we show that some operads are Koszul over Z.

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The Multinomial Distribution on Rooted Labeled Forests

by Jim Pitman , 1997

"... For a probability distribution (p s ; s 2 S) on a finite set S, call a random forest F of rooted trees labeled by S (with edges directed away from the roots) a p-forest if given F has m edges the vector of out-degrees of vertices of F has a multinomial distribution with parameters m and (p s ; s 2 ..."

Abstract - Cited by 7 (7 self) - Add to MetaCart

For a probability distribution (p s ; s 2 S) on a finite set S, call a random forest F of rooted trees labeled by S (with edges directed away from the roots) a p-forest if given F has m edges the vector of out-degrees of vertices of F has a multinomial distribution with parameters m and (p s ; s 2 S), and given also these out-degrees the distribution of F is uniform on all forests with the given out-degrees. The family of distributions of p-forests is studied, and shown to be closed under various operations involving deletion of edges. Some related enumerations of rooted labeled forests are obtained as corollaries. 1 Introduction Let F(S) denote the set of all forests of rooted trees labeled by a finite set S of size jSj. Each f 2 F(S) is a directed graph labeled by S, that is a subset of S \Theta S, such that each Research supported in part by N.S.F. Grant DMS97-03961 connected component of the graph is a tree with edges directed away from some root vertex. The notation v f ...

Stationary Markov Processes Related To Stable Ornstein-Uhlenbeck Processes And The Additive Coalescent

by Steven N. Evans, Jim Pitman , 1998

"... We consider some classes of stationary, counting-measure-valued Markov processes and their companions under time-reversal. Examples arise in the Levy-Ito decomposition of stable Ornstein-Uhlenbeck processes, the large-time asymptotics of the standard additive coalescent, and extreme value theory. T ..."

Abstract - Cited by 4 (4 self) - Add to MetaCart

We consider some classes of stationary, counting-measure-valued Markov processes and their companions under time-reversal. Examples arise in the Levy-Ito decomposition of stable Ornstein-Uhlenbeck processes, the large-time asymptotics of the standard additive coalescent, and extreme value theory. These processes share the common feature that points in the support of the evolving counting-measure are born or die randomly, but each point follows a deterministic flow during its life-time.

The asymptotic behavior of the Hurwitz binomial distribution

by Jim Pitman - Combinatorics, Probability and Computing , 1998

"... Hurwitz's extension of Abel's binomial theorem defines a probability distribution on the set of integers from 0 to n. This is the distribution of the number of non-root vertices of a fringe subtree of a suitably defined random tree with n + 2 vertices. The asymptotic behaviour of this dist ..."

Abstract - Cited by 4 (4 self) - Add to MetaCart

Hurwitz's extension of Abel's binomial theorem defines a probability distribution on the set of integers from 0 to n. This is the distribution of the number of non-root vertices of a fringe subtree of a suitably defined random tree with n + 2 vertices. The asymptotic behaviour of this distribution is described in a limiting regime where the distribution of the delabeled fringe subtree approaches that of a Galton-Watson tree with a mixed Poisson offspring distribution. 1 Introduction and statement of results Hurwitz [10] discovered the following identity of polynomials in n + 2 variables x; y and z s ; s 2 [n] := f1; : : : ; ng, which reduces to the binomial expansion of (x + y) n when Research supported in part by N.S.F. Grant DMS97-03961 z s j 0: X A`[n] x(x + z A ) jAj\Gamma1 (y + z ¯ A ) j ¯ Aj = (x + y + z [n] ) n (1) where the sum is over all 2 n subsets A of [n], with the notations z A := P s2A z s , and jAj for the number of elements of A, and ¯ A := [n] \...

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Coalescent random forests, (1999)

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