Ewens' sampling formula and related formulae: combinatorial proofs, extensions to variable population size and applications to ages of alleles

A connection is explored between the distribution of age-ordered frequencies and ladder indices and heights in a sequence of random variables. In a sample of n genes the connection is with ladder heights and indices in a sequence of draws from an urn containing balls labelled 1,2,…,n; and in the population the connection is with ladder heights and indices in a sequence of independent uniform random variables.