This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set S with asymptotic density σ and, on the other hand, permutations selected according to the Ewens distribution with parameter σ. In particular we show that the asymptotic expected number of cycles of random permutations of [n] with all cycles...

Theory of Kingman’s partition structures has two culminating points • the general paintbox representation, relating finite partitions to hypothetical infinite populations via a natural sampling procedure, • a central example of the theory: the Ewens-Pitman two-parameter partitions. In these notes we further develop the theory by • passing to structures enriched by the order on the collection of...

Kingman derived the Ewens sampling formula for random partitions from the genealogy model defined by a Poisson process of mutations along lines of descent governed by a simple coalescent process. Möhle described the recursion which determines the generalization of the Ewens sampling formula when the lines of descent are governed by a coalescent with multiple collisions. In [7] authors exploit a...

This paper discusses asymptotic behavior of oneand two-parameter PoissonDirichlet models, that is, Ewens models and its two parameter extensions by Pitman, and show that their asymptotic behavior are very different. The paper shows asymptotic properties of a class of oneand twoparameter Poisson-Dirichlet distribution models are drastically different. Convergence behavior is expressed in terms o...

Ewens’s sampling formula exemplifies the harmony of mathematical theory, statistical application, and scientific discovery. The formula not only contributes to the foundations of evolutionary molecular genetics, the neutral theory of biodiversity, Bayesian nonparametrics, combinatorial stochastic processes, and inductive inference but also emerges from fundamental concepts in probability theory...

We examine the variance of a linear statistic defined on symmetric group endowed with Ewens probability. Despite dependence summands, it can be bounded from above by constant multiple sum variances summands. find exact value this constant. The analysis appearing quadratic forms and eigenvalue search is built upon exponential matrices discrete Hahn’s polynomials.