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A306350 Number of paraphyletic coalescence sequences for 2n lineages, n each in 2 species. 1
0, 4, 162, 23328, 9072000, 7873200000, 13367512620000, 40367907740160000, 201793403949096960000, 1578804075215377920000000, 18484433452834116768000000000, 312162837144268369009766400000000, 7374810540967959718955457331200000000 (list; graph; refs; listen; history; text; internal format)



Consider a binary tree evolving in time from a single node until the tree has 2n labeled leaves. Color the 2n leaves in 2 colors, red and blue, assigning n leaves to each color. Suppose coalescences of pairs of leaves happen at distinct times (i.e., no simultaneous mergers). A coalescence sequence is a sequence of coalescence events backward in time, tracing the reduction of the 2n leaves to the single ancestral node. A paraphyletic coalescence sequence is a sequence in which (1) all n red leaves have a common ancestor node that is not the ancestor of any blue leaves; or (2) all n blue leaves have a common ancestor node that is not the ancestor of any red leaves; but not both (1) and (2).


Table of n, a(n) for n=1..13.

N. A. Rosenberg, The shapes of neutral gene genealogies in two species: probabilities of monophyly, paraphyly, and polyphyly in a coalescent model, Evolution 57 (2003), 1465-1477.


a(n) = 3 n! n! (2n-2)! (n-1) / ( (n+1) 2^(2n-3) ).


For n=2, consider two red leaves R1 and R2 and two blue leaves B1 and B2. The a(2)=4 paraphyletic coalescence sequences, separated by semicolons, are (R1,R2), ((R1,R2),B1), (((R1,R2),B1),B2); (R1,R2), ((R1,R2),B2), (((R1,R2),B2),B1); (B1,B2), ((B1,B2),R1), (((B1,B2),R1),R2); and (B1,B2), ((B1,B2),R2), (((B1,B2),R2),R1).


Table[3 n! n! (2 n - 2)! (n - 1)/((n + 1) (2^(2 n - 3))), {n, 1, 30}]


The total number of coalescence sequences for n leaves, from among which the paraphyletic coalescence sequences are identified, follows A006472. Reciprocally monophyletic coalescence sequences, in which conditions (1) and (2) above both hold, follow A306266.

Sequence in context: A157463 A041779 A222394 * A125532 A229405 A211157

Adjacent sequences:  A306347 A306348 A306349 * A306351 A306352 A306353




Noah A Rosenberg, Feb 09 2019



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Last modified October 16 19:46 EDT 2021. Contains 348044 sequences. (Running on oeis4.)