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A306266 Number of reciprocally monophyletic coalescence sequences for 2n lineages, n each in 2 species.

%I A306266 #20 Apr 30 2024 18:43:11
%S A306266 1,2,54,6480,2268000,1837080000,2970558360000,8650265944320000,
%T A306266 42040292489395200000,321608237543873280000000,
%U A306266 3696886690566823353600000000,61486619437507406017075200000000,1433990938521547723130227814400000000
%N A306266 Number of reciprocally monophyletic coalescence sequences for 2n lineages, n each in 2 species.
%C A306266 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 reciprocally monophyletic coalescence sequence is a sequence in which all n red leaves have a common ancestor node that is not the ancestor of any blue leaves, and all n blue leaves have a common ancestor node that is not the ancestor of any red leaves.
%H A306266 N. A. Rosenberg, <a href="https://doi.org/10.1111/j.0014-3820.2003.tb00355.x">The shapes of neutral gene genealogies in two species: probabilities of monophyly, paraphyly, and polyphyly in a coalescent model</a>, Evolution 57 (2003), 1465-1477.
%F A306266 a(n) = (n!)^2*(2*n-2)! / 2^(2*n-2).
%F A306266 a(n) ~ 4*exp(-4*n)*n^(4*n-1/2)*Pi^(3/2). - _Stefano Spezia_, Apr 30 2024
%e A306266 For n=2, consider two red leaves R1 and R2 and two blue leaves B1 and B2. In a reciprocally monophyletic coalescence sequence, the pair of red leaves must coalesce, and the pair of blue leaves must coalesce. These events can occur in either of two orders (red first or blue first), so that a(2)=2.
%p A306266 a:= n-> n!^2*(2*n-2)!/2^(2*n-2):
%p A306266 seq(a(n), n=1..15);  - _Alois P. Heinz_, Feb 07 2019
%t A306266 Table[(n!)^2*(2n - 2)!/2^(2n - 2), {n, 1, 30}]
%Y A306266 The total number of coalescence sequences for n leaves, from among which the reciprocally monophyletic coalescence sequences are identified, follows A006472.
%K A306266 nonn
%O A306266 1,2
%A A306266 _Noah A Rosenberg_, Feb 01 2019