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The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles

Ashkan Nikeghbali, Dirk Zeindler (2013)

Annales de l'I.H.P. Probabilités et statistiques

The goal of this paper is to analyse the asymptotic behaviour of the cycle process and the total number of cycles of weighted and generalized weighted random permutations which are relevant models in physics and which extend the Ewens measure. We combine tools from combinatorics and complex analysis (e.g. singularity analysis of generating functions) to prove that under some analytic conditions (on relevant generating functions) the cycle process converges to a vector of independent Poisson variables...

Universality of separoids

Jaroslav Nešetřil, Ricardo Strausz (2006)

Archivum Mathematicum

A separoid is a symmetric relation † ⊂ 2 S 2 defined on disjoint pairs of subsets of a given set S such that it is closed as a filter in the canonical partial order induced by the inclusion (i.e., A † B ⪯ A ' † B ' ⇔ A ⊆ A ' and B ⊆ B ' ). We introduce the notion of homomorphism as a map which preserve the so-called “minimal Radon partitions” and show that separoids, endowed with these maps, admits an embedding from the category of all finite graphs. This proves that separoids constitute a countable universal partial order. Furthermore,...

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