Discrete Random Walks, DRW'03
Cyril Banderier and Christian Krattenthaler (eds.)
DMTCS Conference Volume AC (2003), pp. 155170
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
X
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
X(m)
of distinct parts of multiplicity
m
that we call the
m
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
ν
has multiplicity
m
. This is related to
E(X(m))
, 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
X(m)
. We obtain asymptotically, as
ν→ ∞
, the moments and an expression for a continuous
distribution
φ
, the (discrete) distribution of
X(m,ν)
being computable from
φ
.

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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. 155170 
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