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A306266 Number of reciprocally monophyletic coalescence sequences for 2n lineages, n each in 2 species. 2
1, 2, 54, 6480, 2268000, 1837080000, 2970558360000, 8650265944320000, 42040292489395200000, 321608237543873280000000, 3696886690566823353600000000, 61486619437507406017075200000000, 1433990938521547723130227814400000000 (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 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.
a(n) = n! n! (2n-2)! / 2^(2n-2).
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.
a:= n-> n!^2*(2*n-2)!/2^(2*n-2):
seq(a(n), n=1..15); - Alois P. Heinz, Feb 07 2019
Table[n! n! (2 n - 2)!/2^(2 n - 2), {n, 1, 30}]
The total number of coalescence sequences for n leaves, from among which the reciprocally monophyletic coalescence sequences are identified, follows A006472.
Sequence in context: A122418 A069788 A283678 * A117681 A221603 A359183
Noah A Rosenberg, Feb 01 2019

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Last modified May 28 16:21 EDT 2023. Contains 363019 sequences. (Running on oeis4.)