  Consider the Markov process taking values in the partitions of N such that
each pair of blocks merges at rate one, and each integer is eroded, i.e.,
becomes a singleton block, at rate d. This is a special case of exchangeable
fragmentation-coalescence process called Kingman's coalescent with erosion. We
provide a new construction of the stationary distribution of this process as a
sample from a standard flow of bridges. This allows us to give a representation
of the asymptotic frequencies of this stationary distribution in terms of a
sequence of hierarchically independent diffusions. Moreover, we introduce a new
process called Kingman's coalescent with immigration, where pairs of blocks
coalesce at rate one, and new blocks of size one immigrate at rate d. By
coupling Kingman's coalescents with erosion and with immigration, we are able
to show that the size of a block chosen uniformly at random from the stationary
distribution of the restriction of Kingman's coalescent with erosion to
{1,...,n} converges to the total progeny of a critical binary branching
process.
