  Kingman derived the Ewens sampling formula for random partitions describing
the genetic variation in a neutral mutation model defined by a Poisson process
of mutations along lines of descent governed by a simple coalescent process,
and observed that similar methods could be applied to more complex models.
M{\"o}hle described the recursion which determines the generalization of the
Ewens sampling formula in the situation when the lines of descent are governed
by a $\Lambda$-coalescent, which allows multiple mergers. Here we show that the
basic integral representation of transition rates for the $\Lambda$-coalescent
is forced by sampling consistency under more general assumptions on the
coalescent process. Exploiting an analogy with the theory of regenerative
partition structures, we provide various characterizations of the associated
partition structures in terms of discrete-time Markov chains.
