  We study the block counting process and the fixation line of exchangeable
coalescents. Formulas for the infinitesimal rates of both processes are
provided. It is shown that the block counting process is Siegmund dual to the
fixation line. For exchangeable coalescents restricted to a sample of size n
and with dust we provide a convergence result for the block counting process as
n tends to infinity. The associated limiting process is related to the
frequencies of singletons of the coalescent. Via duality we obtain an analog
convergence result for the fixation line of exchangeable coalescents with dust.
The Dirichlet coalescent and the Poisson-Dirichlet coalescent are studied in
detail.
