Size-biased permutation is motivated by applications in species sampling. In the 1960s, biologists in population genetics were interested in inferring the distribution of alleles in a population through sampling. Size-biased
permutation models the outcome of successive sampling, where one samples without replacement from the population and records the abundance of newly discovered species in the order that they appear. To account for the occurrence of new types of alleles through mutation and migration, biologists considered random abundance sequences and did not assume an upper limit to the number of possible types. This leads to the study of size-biased permutation of an
infinite, summable sequence of i.i.d. random variables, in other words, jumps of a subordinator. We will head towards major results of this theory, starting with the case of finitely many terms. These are lecture notes for the Vietnam 2016 Spring School on Combinatorial Stochastic Processes. The lecture notes are largely based on Jim’s textbook ‘Combinatorial Stochastic Processes’, Bertoin’s textbook ‘Random fragmentation and coagulation processes’, and on my paper with Jim. This is part of an ongoing effort to update Jim’s textbook and the status of the open problems in that text