Measures of distinctness for partitions and compositions

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, population genetics, nonparametric statistics, etc. We shall present a new notion of ``measures of distinctness'' for partitions and compositions and give some quantitative results. Special cases include the number and the sum of distinct parts. While it is generally true that the theory of partitions is more difficult than the theory of compositions, the problems that we encounter here all reveal the contrary: both generating functions and asymptotic analysis are technically more involved in compositions than in partitions. Multiplicative counterparts will also be briefly indicated.