UC Berkeley

This question is related to recent work on fragmentation of partitions of the set [n] = {1,2,...,n}. After reviewing the known results about partitions, we show how to use generating functions to pass from partitions to forests and determine the offspring distributions for which this fragmentation model works. We will also discuss certain interesting combinatorial interpretations of some of the offspring distributions. Finally, we describe how to go in the reverse direction, and determine how trees in a conditioned Galton-Watson forest should coalesce in order to preserve the conditioned structure.

This is joint work with N. Berestycki and J. Pitman.