\begin{algorithm}\label{alg:sbo1}
To place the blocks of a partition in size-biased order, sample uniformly without replacement from the underlying set, then list the blocks in the order they are discovered. In other words, given $A$ and $P$, and $B(u), u\in A$ as above, let $n=|A|$ and $u:\{1,\dots,n\}\to A$ be a uniform random bijection, then let $B_1=B(u(1))$ and for $i>1$, $I_i:=\min\{i\colon u(i) \notin \bigcup_{j<i}B_j\}$ and $B_i = B(u(I_i))$. The property quoted in Definition \ref{def:sbo} follows from the fact that, conditioned on $u(1),\dots,u(I_{i-1})$, $u(I_i)$ is uniform on $A \setminus \bigcup_{j<i}B_j$.
\end{algorithm}

\begin{algorithm}
To place the blocks of a partition in size-biased order, sample uniformly without replacement from the underlying set, then list the blocks in the order they are discovered. In other words, given $A$ and $P$, and $B(u), u\in A$ as above, let $n=|A|$ and $u:\{1,\dots,n\}\to A$ be a uniform random bijection, then let $B_1=B(u(1))$ and for $i>1$, $I_i:=\min\{i\colon u(i) \notin \bigcup_{j<i}B_j\}$ and $B_i = B(u(I_i))$. The property quoted in Definition \ref{def:sbo} follows from the fact that, conditioned on $u(1),\dots,u(I_{i-1})$, $u(I_i)$ is uniform on $A \setminus \bigcup_{j<i}B_j$.
\end{algorithm}
{
    "title": "takeover, fixation and identifiability in finite neutral genealogy   models",
    "id": "2104.00193",
    "abstract": "for neutral genealogy models in a finite, possibly non-constant population, there is a convenient ordered rearrangement of the particles, known as the lookdown representation, that greatly simplifies the analysis of the family trees. by introducing the dual notions of forward and backward neutrality, we give a more intuitive implementation of this rearrangement. we also show that the lookdown arranges subtrees in size-biased order of the number of their descendants, a property that is familiar in other settings but appears not to have been previously established in this context. in addition, we use the lookdown to study three properties of finite neutral models, as a function of the sequence of unlabelled litter sizes of the model: uniqueness of the infinite path (fixation), existence of a single lineage to which almost all individuals can trace their ancestry (takeover) and whether or not we can infer the lookdown rearrangement by examining the unlabelled genealogy model (identifiability). identifiability of the spine path in size-biased galton-watson trees was previously studied, so we also discuss connections to those results, by relating the spinal decomposition to the lookdown.",
    "categories": "math.pr",
    "doi": "",
    "created": "2021-03-31",
    "updated": "2024-01-12",
    "authors": [
        "eric foxall",
        "jen labossiere"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2104.00193"
}
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