This paper derives basic probabilistic properties of random polynomials over finite fields that are of interest in the study of polynomial factorization algorithms. We show that the main characteristics of random polynomial can be treated systematically by methods of "analytic combinatorics" based on the combined use of generating functions and of singularity analysis. Our object of study is the classical factorization chain which is described in Fig. 1 and which, despite its simplicity, does not appear to have been totally analysed so far. In this paper, we provide a complete average-case analysis.

A random integer N, drawn uniformly from the set {1,2,..., n), has a prime factorization of the form N = a1a2...aM where ax ^ a2>... ^ aM. We establish the asymptotic distribution, as «-» • oo, of the vector A(«) = (loga,/logiV: i:> 1) in a transparent manner. By randomly re-ordering the components of A(«), in a size-biased manner, we obtain a new vector B(n) whose asymptotic distribution is the GEM distribution with parameter 1; this is a distribution on the infinite-dimensional simplex of vectors (xv x2,...) having non-negative components with unit sum. Using a standard continuity argument, this entails the weak convergence of A(/i) to the corresponding Poisson-Dirichlet distribution on this simplex; this result was obtained by Billingsley [3]. 1.

Summary. In this paper we consider the component structure of decomposable combi-natorial 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 Poisson-Dirichlet distribu-tion 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.

A class of models for riffle shuffles ("f \Gammashuffles") related to certain expansive mappings of the unit interval is studied. The main results concern the cycle structure of the resulting random permutations in Sn when n is large. The first result describes the asymptotic distribution of the number of cycles of a given length, relating this distribution to dynamical properties of the associated mapping; this result generalizes a recent result of Diaconis, McGrath, and Pitman. The second result describes the "local " structure of the large cycles, and suggests that these are similar to the large cycles of completely random permutations. 1 Introduction The cycle structure of a random permutation chosen from the uniform distribution on the permutation group S n is reasonably well understood. When n !1, the joint distribution of the "large cycles" is governed by "Poisson-Dirichlet" asymptotics (see [8], [9]), and the number of "short cycles" of a given length j is approximately Poisso...

Markov chains are used to give a purely probabilistic way of understanding the conjugacy classes of the finite symplectic and orthogonal groups in odd characteristic. As a corollary of these methods one obtains a probabilistic proof of Steinberg’s count of unipotent matrices and generalizations of formulas of Rudvalis and Shinoda. 1

This extended abstract proposes a surprisingly simple framework for the random generation of combinatorial configurations based on Boltzmann models. Random generation of possibly complex structured objects is performed by placing an appropriate measure spread over the whole of a combinatorial class. The resulting algorithms can be implemented easily within a computer algebra system, be analysed mathematically with great precision, and, when suitably tuned, tend to be efficient in practice, as they often operate in linear time.

This booklet develops in nearly 200 pages the basics of combinatorial enumeration through an approach that revolves around generating functions. The major objects of interest here are words, trees, graphs, and permutations, which surface recurrently in all areas of discrete mathematics. The text presents the core of the theory with chapters on unlabelled enumeration and ordinary generating functions, labelled enumeration and exponential generating functions, and finally multivariate enumeration and generating functions. It is largely oriented towards applications of combinatorial enumeration to random discrete structures and discrete mathematics models, as they appear in various branches of science, like statistical physics, computational biology, probability theory, and, last not least, computer science and the analysis of algorithms.

Using representation theoretic work on the Whitehouse module, a formula is obtained for the cycle structure of a riffle shuffle followed by a cut. It is proved that the use of cuts does not speed up the convergence rate of riffle shuffles to randomness. Type A affine shuffles are compared with riffle shuffles followed by a cut. Although these probability measures on the symmetric group Sn are different, they both satisfy a convolution property. Strong evidence is given that when the underlying parameter q satisfies gcd(n, q −1) = 1, the induced measures on conjugacy classes of the symmetric group coincide. This gives rise to interesting combinatorics concerning the modular equidistribution by major index of permutations in a given conjugacy class and with a given number of cyclic descents. Generating functions for the first pile size in patience sorting from decks with repeated values are derived. This relates to random matrices.

The Poisson-Dirichlet distribution and its marginals are studied, in particular the largest component, that is Dickman's distribution. Size-biased sampling and the GEM distribution are considered. Ewens sampling formula and random permutations, generated by the Chinese restaurant process, are also investigated. The used methods are elementary and based on properties of the finite-dimensional Dirichlet distribution. Keywords: Chinese restaurant process; Dickman's function; Ewens sampling formula; GEM distribution; Hoppe's urn; random permutations; residual allocation models; size-biased sampling ams 1991 subject classification: primary 60g57 secondary 60c05, 60k99 Running title: The Poisson--Dirichlet distribution revisited 1

We give a precise average-case analysis of a complete polynomial factorization chain over finite fields by methods based on generating functions and singularity analysis. Polynomes al'eatoires et factorisation de polynomes R'esum'e Nous donnons une analyse en moyenne pr'ecise d'une chaine compl`ete de factorisation de polynomes sur les corps finis par des m'ethodes fond'ees sur les fonctions g'en'eratrices et l'analyse de singularit'es. To appear in Automata, Languages and Programming, Proceedings of the 23rd ICALP colloquium, Paderborn, July 1996, F. Meyer auf der Heide, Ed., in Lecture Notes in Computer Science. Random Polynomials and Polynomial Factorization Philippe Flajolet, 1 Xavier Gourdon, 1 and Daniel Panario 2 1 Algorithms Project, INRIA Rocquencourt, F-78153 Le Chesnay, France. 2 Department of Computer Science, University of Toronto, Toronto, Canada M5S-1A4. E-mails: Philippe.Flajolet@inria.fr, Xavier.Gourdon@inria.fr, daniel@cs.toronto.edu. Abstract. We give a pr...