Non-Regenerative Sampling Distributions of Ξ-Coalescents: Möhle's Conjecture with a Kingman Component, for Finitely Many Atoms or Boundedly Many Dust-Free Boxes, and Off One Mutation Rate

M. Möhle (Oberwolfach Report 40/2005) conjectured that the sampling distributions of the infinitely-many-alleles model along a coalescent with simultaneous multiple collisions (a \(\Xi\)-coalescent), with mutation rate \(\rho>0\), are never regenerative in the sense of Gnedin and Pitman, except for Kingman's coalescent and the star-shaped coalescent. He had proved this for coalescents with multiple but not simultaneous collisions (\(\Lambda\)-coalescents; Combin. Probab. Comput. 16 (2007)). We prove the conjecture for three classes of \(\Xi\) and for all but at most one mutation rate in general; the conjecture is not settled. Write \(\Xi=a\delta_0+\Xi_0\). The main tool is an identity: the regeneration equation at the partition \((2,2,1^{n-4})\) holds if and only if \(\Phi(n-1)X(n)=\mathrm{Bin}(n)W(n-1)\), where \(X(n)\), \(\mathrm{Bin}(n)\) and \(W(n)\) are the total rates of the mergers of two pairs, of one pair, and of all other kinds, among \(n\) lineages, and \(\Phi(n)\) is \(n\rho\) plus the total merger rate. Consequences: (A) if \(\Xi\) is not a \(\Lambda\)-coalescent, the partition structure is non-regenerative for every \(\rho>0\) except possibly one explicit value \(\rho^*(\Xi)\), and Möhle's theorem is recovered at sample size four; (B) every \(\Xi\)-coalescent with a Kingman component (\(a>0\)), other than Kingman's coalescent, is non-regenerative for every \(\rho>0\); (C, D) the same holds for \(a=0\), the star-shaped coalescent excepted, when \(\Xi_0\) is carried by paintboxes without dust with boundedly many boxes, or is a finite sum of point masses at paintboxes with finitely many boxes. The measure \(\Xi=\frac{17}{1024}\delta_0+\delta_{(1/4,1/8,0,\dots)}\) with \(\rho=\frac{13833}{5120}\) is regenerative at all sample sizes \(n\le5\) but not at \(n=6\), so finitely many sample sizes do not decide the question. Open remains the case \(a=0\), \(\Xi_0\) charging \(\{x_2>0\}\) and outside the classes of (C) and (D), at the single value \(\rho=\rho^*(\Xi)\). The conjecture and the \(\Lambda\) case are Möhle's; his recursion for the sampling probabilities is re-derived. No printed answer to the question was found in the literature searched, which is not a proof of novelty. This is an unrefereed note.