Site-Frequency Spectra under Multiple Ancestral Mergers

Definitions and mathematical statement

Research problem. Determine how multiple mergers of ancestral lineages affect the full site-frequency spectrum, including the number of mutations affecting exactly $k$ sampled lineages for each $k$. For a sample of $n$ labeled genomes, let $\xi_k^{(n)}$ count mutations carried by exactly $k$ sampled descendants. In the usual unfolded segregating-site convention the spectrum is $(\xi_1^{(n)},\ldots,\xi_{n-1}^{(n)})$; mutations carried by all $n$ descendants require a separate fixed-site or ancestral-branch convention. The count $k$ refers to sampled descendants of a mutated branch, not to the number of ancestral blocks participating in one merger event.

A conditional coalescent formulation uses a partition process $\Pi_t$ of $\{1,\ldots,n\}$, whose blocks record common ancestors. A multiple merger combines more than two blocks, while simultaneous distinct mergers require additional specification. If $T$ is the most recent common-ancestor time, define\[L_k^{(n)}=\int_0^T\#\{B\in\Pi_t:|B|=k\}\,dt.\] Under a specified infinite-sites model with independent Poisson mutations at rate $\rho$ per unit branch length, the conditional spectrum consists of independent Poisson counts with means $\rho L_k^{(n)}$. In particular, when these expectations are finite, $\mathbb E\xi_k^{(n)}=\rho\,\mathbb E L_k^{(n)}$. These conventions explain one precise connection between merger mechanisms and frequency spectra; they are not supplied assumptions of every model in the question. A full joint distribution requires the joint branch-length law, not merely its expectations.

The source does not specify the merger rates or their $\Lambda$- or $\Xi$-coalescent class, the mutation mechanism, population-size time scaling, sample size regime, unfolded versus folded data, or whether full means all expected entries or the joint law. Nor is a comparison with binary mergers normalized. Consequently no universal direction of change or single numerical formula is asserted for every merger mechanism.

The supplied triage checked 2026-08-21 labels the problem partially solved, reporting computed site-frequency spectra in several multiple-merger models while describing a general treatment for arbitrary mechanisms as active. No specific models or theorem hypotheses are supplied, so this dated assessment requires independent verification.

Short English statement

Describe the mutation-frequency counts across all descendant sample sizes under multiple-merger genealogies. The mechanism and mutation model must be fixed, and expectations should be distinguished from the full joint spectrum. A mutation's frequency counts sampled descendants, not the number of merging ancestors.

Sources

• Local UnsolvedMath registry, permanent ID 30000305, code catalogue entry; Site-Frequency Spectra under Multiple Ancestral Mergers, Oberwolfach Reports Open Problems.
• Mathematical Population Genetics (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/40, pp. 2241--2304. https://doi.org/10.4171/owr/2005/40. Proposed year and year tag: 2005; no proposer named.
• Supplied literature label original Oberwolfach report: https://doi.org/10.4171/owr/2005/40. No separate references identifying the computed special models are supplied.
• Literature triage checked 2026-08-21, status partially_solved; record updated 2026-09-16. The corrected_verified field concerns notation for the lineage-count variable. Verify dated claims independently.