index $n$ in the data indices such that $n \notin \bigcup_{k=1}^{K} \blockfa_{k}$, where the union is $\emptyset$ if $K = 0$. It is possible that no features contain $n$. So we further note that there exists some minimum index $m$ such that $m \notin \bigcup_{j=1}^{K} \blockfa_{j}$ but $m$ is contained in some feature of the allocation. By construction, we must have $m \ge N$. Let $K_{m}$ be the number of features containing $m$; $K_{m}$ is finite by definition of a feature allocation. Let $(U_{k})$ denote a sequence of iid uniform random variables, independent of the random feature allocation. Assign $U_{K+1}, \ldots, U_{K + K_{m}}$ to these new features and determine their order of appearance by the order of these random variables. While features remain to be labeled, continue the recursion with $N$ now equal to $m$ and $K$ now equal to $K + K_{m}$.

\[ex:oa\_6\] Consider the feature allocation $$\label{eq:feat_6_oa}
    \detfa_{6} = \{\{2,5,4\},\{3,4\},\{6,4\},\{3\},\{3\}\}.$$ And consider the random variables $$U_{1}, U_{2}, U_{3}, U_{4}, U_{5} \iid \unif[0,1].$$ We see from $\detfa_{6}$ that index $1$ has no features. Index $2$ has exactly one feature, so we assign this feature, $\{2,5,4\}$, to have order-of-appearance label 1. While $U_{1}$ is associated with this feature, we 