of the Tajima isochronous $n$-coalescent; on the other hand if $m=n$, the process degenerates into the Kingman heterochronous $n$-coalescent since all singletons are uniquely labeled by their sampling times. Figure \[fig:hetTaj\] shows a possible realization from the Tajima heterochronous $n$-coalescent.

To define the distribution of the holding times, we introduce the following notation. We denote the intervals that end with a coalescent event at $t_{k}$ by $I_{0,k}$ and the intervals that end with a sampling time within the interval $(t_{k+1},t_{k})$ as $I_{i,k}$ where $i\geq 1$ is an index tracking the sampling events in $(t_{k+1},t_{k})$. More specifically, for every $k\in \{2,\ldots,n\}$, we define $$I_{0,k}=[\max\{t_{k+1},s_{j}\},t_{k}),\quad \text{ where the maximum is taken over all } s_{j}<t_{k},$$ and for every $i\geq 1$ we set $$I_{i,k}=[\max\{t_{k+1},s_{j-i}\},s_{j-i+1}) \text{ with the max taken over all } s_{j-i+1}>t_{k+1} \text{ and } s_{j}<t_{k}.$$ We also let $n_{i,k}$ denote the number of extant lineages during the time interval $I_{i,k}$. For example, in Figure \[fig:hetTaj\], in $(t_9, t_8)$ we have $I_{0,8}=[s_2,t_8)$, $I_{1,8}=[t_9,s_2)$ and no $I_{i,8}$ for $i\geq 2$. The vector of coalescent times [**t**]{} is a random vector whose density with respect to Lebesgue measure on $\mathbb{R}_+^{n-1}$ can be factorized as the product of the conditional densities of $t_{k-1}$ knowing $t_k$, 