the internal node labels as *vintages* (*i.e.*, rankings).

The Tajima heterochronous $n$-coalescent is the process $({\textbf{a}}{}(t), b(t))_{t \geq 0}$ that keeps track of ${\textbf{a}}{} (t)$, a vector of length $m$ whose $j$-th position indicates the number of singletons (*i.e.*, lineages that have not been involved in a coalescence event) from sampling group $s_j$ at time $t$, and $b (t)$ is the set of vintaged lineages at time $t$. The process starts at $t=0$ in state $({\textbf{a}}{}(0)=(n_1,0,\dots,0), b(0)=\emptyset)$, jumps deterministically at every sampling time and jumps stochastically at every random coalescent time until it reaches the unique absorbing state $({\textbf{a}}{(t_2)}=(0,\ldots, 0), b(t_2)=\{n-1\})$ at time $t_2$, when all $n$ samples have a single most recent common ancestor at the root (Figure \[fig:hetTaj\]). At each sampling time $s_i$, the state of the Tajima coalescent jumps deterministically as follows: $$({\textbf{a}}{}(s_i), b(s_i))=({\textbf{a}}{}(s_i-)+ n_i\textbf{e}_i, b(s_i-)),$$ where $f(s_i-)$ denotes the left-limit of the function $f$ at $s_i$ and $\textbf{e}_i$ is the $i$-th unit vector.

Let us now turn to the embedded jump chain at coalescent times. At time $t_i$, two extant lineages coalesce to create a new lineage with vintage $n+1-i$. Four types of coalescence transitions are possible depending on which and how many sampling groups are 