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The twoparameter PoissonDirichlet distribution derived from a stable subordinator.
, 1995
"... The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov ..."
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Cited by 221 (37 self)
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The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to VershikShmidtIgnatov, are generalized to the twoparameter case. The sizebiased random permutation of pd(ff; `) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 ! ff ! 1, pd(ff; 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...
Random Mapping Statistics
 IN ADVANCES IN CRYPTOLOGY
, 1990
"... Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of ..."
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Cited by 78 (6 self)
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Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of about twenty characteristic parameters of random mappings is carried out: These parameters are studied systematically through the use of generating functions and singularity analysis. In particular, an open problem of Knuth is solved, namely that of finding the expected diameter of a random mapping. The same approach is applicable to a larger class of discrete combinatorial models and possibilities of automated analysis using symbolic manipulation systems ("computer algebra") are also briefly discussed.
Probabilistic bounds on the coefficients of polynomials with only real zeros
 J. Combin. Theory Ser. A
, 1997
"... The work of Harper and subsequent authors has shown that nite sequences (a 0;;an) arising from combinatorial problems are often such that the polynomial A(z): = P n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coe cients, Stirling numbers of the rst and sec ..."
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Cited by 20 (0 self)
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The work of Harper and subsequent authors has shown that nite sequences (a 0;;an) arising from combinatorial problems are often such that the polynomial A(z): = P n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coe cients, Stirling numbers of the rst and second kinds, and Eulerian numbers. Assuming the ak are nonnegative, A(1)> 0 and that A(z) is not constant, it is known that A(z) has only real zeros i the normalized sequence (a 0=A(1);;an=A(1)) is the probability distribution of the Research supported in part by N.S.F. Grant MCS9404345 1 number of successes in n independent trials for some sequence of success probabilities. Such sequences (a 0;;an) are also known to be characterized by total positivity of the in nite matrix (ai,j) indexed by nonnegative integers i and j. This papers reviews inequalities and approximations for such sequences, called Polya frequency sequences which follow from their probabilistic representation. In combinatorial examples these inequalities yield a number of improvements of known estimates.
Order statistics for decomposable combinatorial structures
 Random Structures and Algorithms
, 1994
"... Summary. In this paper we consider the component structure of decomposable combinatorial objects, both labeled and unlabeled, from a probabilistic point of view. In both cases we show that when the generating function for the components of a structure is a logarithmic function, then the joint distr ..."
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Cited by 13 (3 self)
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Summary. In this paper we consider the component structure of decomposable combinatorial objects, both labeled and unlabeled, from a probabilistic point of view. In both cases we show that when the generating function for the components of a structure is a logarithmic function, then the joint distribution of the normalized order statistics of the component sizes of a random object of size n converges to the PoissonDirichlet distribution on the simplex ∇ = {{xi} : � xi =1,x1 ≥ x2 ≥... ≥ 0}. This result complements recent results obtained by Flajolet and Soria [9] on the total number of components in a random combinatorial structure.
GENERALIZED STIRLING PERMUTATIONS, FAMILIES OF INCREASING TREES AND URN MODELS
"... ABSTRACT. Bona [6] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley [14]. Recently, Janson [18] showed the connection between Stirling permutations and plane recursive trees and proved a joint normal law for the parame ..."
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Cited by 7 (5 self)
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ABSTRACT. Bona [6] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley [14]. Recently, Janson [18] showed the connection between Stirling permutations and plane recursive trees and proved a joint normal law for the parameters considered by Bona. Here we will consider generalized Stirling permutations extending the earlier results of [6], [18], and relate them with certain families of generalized plane recursive trees, and also (k + 1)ary increasing trees. We also give two different bijections between certain families of increasing trees, which both give as a special case a bijection between ternary increasing trees and plane recursive trees. In order to describe the (asymptotic) behaviour of the parameters of interests, we study three (generalized) Pólya urn models using various methods. 1.
On The Number Of Predecessors In Constrained Random Mappings
 Stat. Probab. Letters
, 1997
"... We consider random mappings from an nelement set into itself with constraints on coalescence as introduced by Arney and Bender. A local limit theorem for the distribution of the number of predecessors of a random point in such a mapping is presented by using a generating function approach and sing ..."
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Cited by 3 (3 self)
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We consider random mappings from an nelement set into itself with constraints on coalescence as introduced by Arney and Bender. A local limit theorem for the distribution of the number of predecessors of a random point in such a mapping is presented by using a generating function approach and singularity analysis. 1.
Mixed Moments of Random Mappings and Chaotic Dynamical Systems
, 2000
"... Some statistical characteristics of completely random mappings and of random mappings with an absorbing or an attracting centre are calculated. Results are applied to validation of some phenomenological models of computer simulations of dynamical systems. Permanent address: Institute for Informati ..."
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Cited by 1 (1 self)
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Some statistical characteristics of completely random mappings and of random mappings with an absorbing or an attracting centre are calculated. Results are applied to validation of some phenomenological models of computer simulations of dynamical systems. Permanent address: Institute for Information Transmission Problems, Russian Academy of Science, 19 Bolshoi Karetny lane, Moscow 101447, Russia. Contents 1 Introduction 3 2 Definitions 6 2.1 Mixed moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Random mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Mixed moments of scaled components size 10 3.1 Completely random mappings . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Random mappings with a heavy centre . . . . . . . . . . . . . . . . . . . 12 4 Mixed moments of scaled cycle lengths 15 4.1 Completely random mappings . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Random mappings with a heavy centre . . . . ....
Properties of Nonequiprobable Random Transformations
"... With a given transformation on a finite domain, we associate a threedimensional distribution function describing the component size, cycle length, and trajectory length of each point in the domain. We then consider a random transformation on the domain, in which images of points are independent and ..."
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With a given transformation on a finite domain, we associate a threedimensional distribution function describing the component size, cycle length, and trajectory length of each point in the domain. We then consider a random transformation on the domain, in which images of points are independent and identically distributed. The threedimensional distribution function associated with this random transformation is itself random. We show that, under a simple homogeneity condition on the distribution of images, and with a suitable scaling, this random distribution function has a limit law as the number of points in the domain tends to infinity. The proof is based on a Poisson approximation technique for matches in an urn model. The result helps to explain the behavior of computer implementations of chaotic dynamical systems. Permanent address: Institute for Information Transmission Problems, Russian Academy of Science, 19 Bolshoi Karetny Lane, Moscow 101447, Russia. Contents 1 Motivat...
Australasian Journal of Combinatorics 10(1994). 00.211224 ON THE WIENER INDEX OF TREES FROM CERTAIN FAMILIES
"... ABSTRACT. The Wiener index W ( G) of a connected graph G is the sum of the distances between all pairs of vertices of G. We determine the expected value of W(Tn) for trees Tn from certain families. 1. ..."
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ABSTRACT. The Wiener index W ( G) of a connected graph G is the sum of the distances between all pairs of vertices of G. We determine the expected value of W(Tn) for trees Tn from certain families. 1.