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

Louchard, Guy

Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03), DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03) (2003).

Résumé

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 $φ$ .

Détail

DOI : 10.46298/dmtcs.3347