Measures of Distinctness for Random Partitions and Compositions of an Integer

Abstract

Partitions and compositions of integers are, besides their intrinsic interests, usually used as theoretical models for evolutionary processes in different contexts: statistical mechanics, theory of quantum strings, populaw x tion biology, nonparametric statistics, etc.; cf. 1, 4, 8, 10, 12, 30, 49, 54 . Also parameters in partitions often have natural interpretations in terms w x Ž of characters in symmetric groups; cf. 15, 47 . Thus properties statistical, . algebraic, analytic, . . . of these objects received constant attention in the literature. In many situations, the notion of ‘‘degree of distinctness’’ naturally arises. The classical birthday paradox states that one needs on the average ) 23 people to discover two that have the same birthday with probability w x ) 1r2, assuming all birth dates to be equally likely; cf. 16 . The coupon collector problem is similar: what is the expected number of coupons one needs to gather before a full collection, under suitable probability assumptions on the issuing of the coupons? In applications in which only the first Ž . product element, particle, . . . is ‘‘expensive’’ and the ‘‘cost’’ of the remaining reproductions is negligible, the study of the measures of distinctness becomes meaningful and important. The number of distinct outcomes Ž . in a sequence of multinomial trials the classical occupancy problem has w x w x wide applications; see, for example, Knuth 34 , Johnson and Kotz 28 , w x w x w x Kolchin et al. 35 , Arato and Benczur 5 , and Vitter and Chen 50 . The number of distinct sites visited by a random walk plays an important role