Exceptional Regenerative Composition Structures

Definitions and mathematical statement

A composition of $n$ is an ordered tuple of positive integers summing to $n$. A composition structure is a sampling-consistent family $(C_n)_{n\geq1}$ of random compositions: deleting a uniformly sampled individual, reducing its part by one and removing an empty part, gives the law of $C_{n-1}$. Put $C_0=()$. A usual regenerative convention is that, conditional on the first part of $C_n$ being $m$, the remaining ordered tuple has the law of $C_{n-m}$, for every $n$ and every $m$ of positive conditioning probability. This convention is left-to-right; the source's exact ordering and regeneration conventions must be checked.

The source considers sampling distributions from coalescents allowing simultaneous multiple collisions, parameterized by measures $(\Lambda_j)$ on source-defined simplices $\Delta_j$. Let $\operatorname{Corr}(\Lambda)$ denote all composition structures corresponding to those distributions in the intended sense; the correspondence, collision rates, measure normalization, and simplex definitions are not reproduced.

Define the exceptional parameter condition \[ \mathcal E(\Lambda):\quad \Lambda_j=0\quad(j\geq2),\qquad \left(\Lambda_1\text{ concentrated at }0\right) \ \text{or}\ \left(\Lambda_1\text{ concentrated at }1\right). \] Concentration at one endpoint means support at that singleton, not an arbitrary mixture of mass at both endpoints. The source conjecture is that outside these cases all corresponding structures are nonregenerative: \[ \forall\Lambda\quad\neg\mathcal E(\Lambda) \ \Longrightarrow\ \forall C\in\operatorname{Corr}(\Lambda),\quad C\text{ is not regenerative}. \] Equivalently, the existence of a regenerative corresponding composition structure forces $\mathcal E(\Lambda)$. The wording "except when" specifies this exclusion; by itself it does not say that every possible ordering or every corresponding structure in an exceptional case is regenerative. That converse, and any nonzero-mass convention excluding the zero measure, require source confirmation.

The supplied assessment checked 2026-08-21 identifies this universal nonregenerativity assertion as conjectural and reports no general proof located. The source-verification field says the coalescent family and measure parameters were restored from the report, but does not reproduce the missing construction.

Short English statement

Conjecturally a regenerative composition structure can correspond to these simultaneous-collision sampling laws only when all higher collision measures vanish and the first measure is supported at one endpoint alone. The ordering correspondence and parameter conventions are missing.

Sources

• Local UnsolvedMath registry, permanent ID 30000304, catalogue entry, Exceptional Regenerative Composition Structures.
• Mathematical Population Genetics (2006), Oberwolfach Reports, pp. 2241--2304, DOI https://doi.org/10.4171/owr/2005/40.
• Supplied literature triage checked 2026-08-21 cites only the original report; no separate proof or full definitions of the measure-parametrized sampling construction are supplied.