involved: (1) two singletons of the same sampling group coalesce (up to $m$ possible moves for the chain), (2) two singletons of different sampling groups coalesce (up to $m (m-1)/2$ possible moves), (3) one singleton lineage and one vintaged lineage coalesce (up to $m$ possible moves), or (4) two vintaged lineages coalesce (only one possibility because for vintages, the sampling information is irrelevant). Each pair coalesces with the same probability and the transition probabilities at coalescent times are thus given by $$\begin{aligned}
\label{transition}
P\Big[({\textbf{a}}{(t_i)}, &b(t_i))\Big| ({\textbf{a}}{(t_i-)},b(t_i-))\Big]\\
& =\left\{
\begin{array}{ll}
\frac{\mathlarger\prod_{j=1}^m \dbinom{a_j(t_i-)}{a_j(t_i-)-a_j(t_i)}}{\dbinom{\sum_{j=1}^m a_j(t_i-)+|b(t_i-)|}{2}} \,\ \,\ \,\ \,\ & \text{if} \ \,\  ({\textbf{a}}{(t_i)},b(t_i)) \prec  ({\textbf{a}}{(t_i-)},b(t_i-))\nonumber\\
\,\ \\
0 \,\ \,\ \,\ \,\  \,\ \,\ \,  \,\ \,\  & \text{otherwise}
\end{array}
\right.\end{aligned}$$ where $({\textbf{a}}{(t_i)},b(t_i)) \prec  ({\textbf{a}}{(t_i-)},b(t_i-))$ means that $({\textbf{a}}{(t_i)},b(t_i))$ can be obtained by merging two lineages of $({\textbf{a}}{(t_i-)},b(t_i-))$ and $|b|$ denotes the cardinality of the set $b$.

Observe that the quantity $\sum_{j=1}^m a_j(t_i-)+|b(t_i-)|$ appearing in corresponds to the total number of extant lineages just before the event at $t_i$. Furthermore, since only two lineages coalesce at time $t_i$, at most two terms in the product appearing in the numerator of are not equal to one. Finally, if $m=1$, degenerates into the transition probabilities 