DMTCS Proceedings, Discrete Random Walks, DRW'03

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DMTCS Conference vol AC (2003), pp. 155-170


Discrete Random Walks, DRW'03

Cyril Banderier and Christian Krattenthaler (eds.)

DMTCS Conference Volume AC (2003), pp. 155-170

author: Guy Louchard
title: The number of distinct part sizes of some multiplicity in compositions of an Integer. A probabilistic Analysis
keywords: Mellin transforms, urns models, Poissonization, saddle point method, generating functions
abstract: Random compositions of integers are used as theoretical models for many applications. The degree of distinctness of a composition is a natural and important parameter. A possible measure of distinctness is the number
of distinct parts (or components). This parameter has been analyzed in several papers. In this article we consider a variant of the distinctness: the number
of distinct parts of multiplicity
that we call the
-distinctness. A first motivation is a question asked by Wilf for random compositions: what is the asymptotic value of the probability that a randomly chosen part size in a random composition of an integer
has multiplicity
. This is related to
, which has been analyzed by Hitczenko, Rousseau and Savage. Here, we investigate, from a probabilistic point of view, the first full part, the maximum part size and the distribution of
. We obtain asymptotically, as
ν→ ∞
, the moments and an expression for a continuous distribution
, the (discrete) distribution of
being computable from
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reference: Guy Louchard (2003), The number of distinct part sizes of some multiplicity in compositions of an Integer. A probabilistic Analysis, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 155-170
bibtex: For a corresponding BibTeX entry, please consider our BibTeX-file.
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