A class of random discrete distributions P is introduced by means of a recursive splitting of unity. Assuming supercritical branching, we show that for partitions induced by sampling from such P a power growth of the number of blocks is typical. Some known and some new partition structures appear when P is induced by a Dirichlet splitting.
Alexander V. Gnedin. Yuri Yakubovich. "Recursive partition structures." Ann. Probab. 34 (6) 2203 - 2218, November 2006. https://doi.org/10.1214/009117906000000584