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The number of distinct part sizes of some multiplicity in compositions of an Integer. A probabilistic Analysis

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 firstmotivation 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 $\mathbb{E}(X(m))$, 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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Guy Louchard. The number of distinct part sizes of some multiplicity in compositions of an Integer. A probabilistic Analysis. Discrete Random Walks, DRW'03, 2003, Paris, France. pp.155-170. ⟨hal-01183943⟩

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