In problems of species counts, the interest is more on the number of different species and their relative abundance rather than on counts of representatives of specific species. These kinds of problems originated in Genetics, where species are alleles of a gene, but are also common in other applied sciences. Ewens sampling formula is the first and most famous devoted probability distribution, obtained by a geneticist, in this area of research. From a statistical point of view, Ewens sampling formula is an example of distribution of a random integer partition, i.e. a random list of multiplicities m_1 , ., m_n such that m 1 + 2m 2 + . . . nm_n = n. In this paper, Ewens and other distributions of symmetric random integer partitions are obtained without reference to biological evolutionary models. Then, their mixtures and limits are studied, obtaining some interesting relationships and some new examples