Bart Jacobs

The probabilistic approach to mutations was initiated in the 1970s by Warren Ewens and others in population biology. It led to the famous Ewens' sampling formula, giving parameterised families of discrete probability distributions on (multiset) partitions, for which the length of a partition turned out to be a sufficient statistic. The current paper takes a fresh (mathematical) look at this area and generalises it from partitions (as certain multisets) to six basic datatypes that are used in mathematics and computer science, such as lists, multisets, subsets, set partitions (covers), multiset partitions, and numbers. These datatypes are organised in a triangular prism diagram, with basic transformations between them. This generalisation builds on (combinatorial) relationships between these datatypes and develops probabilistic mutation operation for all of them. They lead to mutation distributions on each of the different datatypes, via iterated mutations. Moreover, each of the transformations in (a restricted version of) the prism turns out to be a sufficient statistic. The paper thus provides a `multi-datatype' generalisation of the groundbraking work of Ewens on mutations.