  The method of 'coupling from the past' permits exact sampling from the
invariant distribution of a Markov chain on a finite state space. The coupling
is successful whenever the stochastic dynamics are such that there is
coalescence of all trajectories. The issue of the coalescence or
non-coalescence of trajectories of a finite state space Markov chain is
investigated in this note. The notion of the 'coalescence number' $k(\mu)$ of a
Markovian coupling $\mu$ is introduced, and results are presented concerning
the set $K(P)$ of coalescence numbers of couplings corresponding to a given
transition matrix $P$.
