  We consider the number of blocks involved in the last merger of a
$\Lambda$-coalescent started with $n$ blocks. We give conditions under which,
as $n \to \infty$, the sequence of these random variables a) is tight, b)
converges in distribution to a finite random variable or c) converges to
infinity in probability. Our conditions are optimal for $\Lambda$-coalescents
that have a dust component. For general $\Lambda$, we relate the three cases to
the existence, uniqueness and non-existence of quasi-invariant measures for the
dynamics of the block-counting process, and in case b) investigate the
time-reversal of the block-counting process back from the time of the last
merger.
