DOI: 10.1017/S0963548302005163 Printed in the United Kingdom Poisson{Dirichlet and GEM Invariant Distributions for Split-and-Merge Transformations of an Interval Partition

This paper introduces a split-and-merge transformation of interval partitions which com-bines some features of one model studied by Gnedin and Kerov [12, 11] and another studied by Tsilevich [30, 31] and Mayer-Wolf, Zeitouni and Zerner [21]. The invariance under this split-and-merge transformation of the interval partition generated by a suitable Poisson process yields a simple proof of the recent result of [21] that a Poisson{Dirichlet distri-bution is invariant for a closely related fragmentation{coagulation process. Uniqueness and convergence to the invariant measure are established for the split-and-merge transformation of interval partitions, but the corresponding problems for the fragmentation{coagulation process remain open. 1