We compute the asymptotic probability that two randomly selected compositions of n into parts equal to a or b have the same number of parts. In addition, we provide bijections in the case of parts of sizes 1 and 2 with weighted lattice paths and central Whitney numbers of fence posets. Explicit algebraic generating functions and asymptotic probabilities are also computed in the case of pairs of compositions of n into parts at least d, for any fixed natural number d.
This paper deals with Carlitz compositions of natural numbers (adjacent parts have to be different). The following parameters are analysed: number of parts, number of equal adjacent parts in ordinary compositions, largest part, Carlitz compositions with zeros allowed (correcting an erroneous formula from Carlitz). It is also briefly demonstrated that so-called 1-compositions of a natural number can be treated in a similar style.
We study the asymptotic probability that a random composition of an integer n is gap-free, that is, that the sizes of parts in the composition form an interval. We show that this problem is closely related to the study of the probability that a sample of independent, identically distributed random variables with a geometric distribution is likewise gap-free.
Résumé. Pour chaque entier naturel n, nous déterminons l'ordre moyen α(n) deséléments du groupe cyclique d'ordre n. Nous montrons que plus de la moitié de la contributionà α(n) provient des ϕ(n)éléments primitifs d'ordre n. Il est par conséquent intéressant d'étudierégalement la fonction β(n) = α(n)/ϕ(n). Nous détermi-nons le comportement moyen de α, β, 1/β et considérons aussi ces fonctions dans le cas du groupe multiplicatif d'un corps fini.Abstract. For each natural number n we determine the average order α(n) of the elements in a cyclic group of order n. We show that more than half of the contribution to α(n) comes from the ϕ(n) primitive elements of order n. It is therefore of interest to study also the function β(n) = α(n)/ϕ(n). We determine the mean behavior of α, β, 1/β, and also consider these functions in the multiplicative groups of finite fields.
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