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We introduce a class of birth-and-death Polya urns, which allow for bothsampling and removal of observations governed by an auxiliary inhomogeneousBernoulli process, and investigate the asymptotic behaviour of the inducedallelic partitions. By exploiting some embedded models, we show that theasymptotic regimes exhibit a phase transition from partitions with almostsurely infinitely many blocks and independent counts, to stationary partitionswith a random number of blocks. The first regime corresponds to limits ofEwens-type partitions and includes a result of Arratia, Barbour and Tavar\'e(1992) as a special case. We identify the invariant and reversible measure inthe second regime, which preserves asymptotically the dependence betweencounts, and is shown to be a mixture of Ewens sampling formulas, with a tiltedNegative Binomial mixing distribution on the sample size.

Birth-and-death Polya urns and stationary random partitions