{"id":"palomar/PALOMAR-2026-10-29-000020","url":"https://github.com/cenozoic-works/rhone-theory/tree/babec8449a724c270450c7c0e2cd3985ccc2a989","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-29-000020-v1.json","repo":"https://github.com/cenozoic-works/rhone-theory","tags":["q-bio.PE","math.CO","msc:92D15","msc:60J90","msc:05C05","msc:05A15"],"score":0,"title":"Combinatorics and probabilities of Kingman coalescent trees: histories, shapes, topologies, root splits, Colless, and Sackin","trust":"high","domain":"q-bio.PE","source":"palomar","authors":["X. Lee","N. Volkov"],"reviews":0,"subject":"q-bio.PE","summary":"This entry formalizes combinatorics and probabilities of trees under Kingman's coalescent, through its embedded jump chain, in which each pair of the k current lineages is the next to merge with probability 1/binomial(k, 2). A ranked labelled history on n leaves is a sequence of n − 1 merges starting from n singletons; there are n!(n − 1)!/2^(n−1) of them, and each has probability 2^(n−1)/(n!(n − 1)!). Ranked and unranked tree shapes are defined as classes of histories under relabelling of the leaves; their numbers satisfy André's recurrence for alternating permutations, shifted by one, and…","category":"research","comments":0,"disputed":false,"division":"q-bio","mentions":0,"verified":true,"discussed":0,"published":"2026-10-29","reactions":{},"formalized":true,"formalizes":[{"title":"Probability Models for DNA Sequence Evolution","identifier":"doi:10.1007/978-0-387-78168-6","relationship":"adapts"},{"title":"Sur les permutations alternees","identifier":"","relationship":"adapts"},{"title":"The functional equation g(x^2) = 2 alpha x + [g(x)]^2","identifier":"doi:10.2307/1967710","relationship":"formalizes"},{"title":"Exact formulas for the variance of several balance indices under the Yule model","identifier":"doi:10.1007/s00285-012-0615-9","relationship":"adapts"},{"title":"On the genealogy of nested subsamples from a haploid population","identifier":"doi:10.2307/1427285","relationship":"formalizes"}]}