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

1, 2, 54, 6480, 2268000, 1837080000, 2970558360000, 8650265944320000, 42040292489395200000, 321608237543873280000000, 3696886690566823353600000000, 61486619437507406017075200000000, 1433990938521547723130227814400000000

Offset: 1

Author

Noah A Rosenberg, Feb 01 2019

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• 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.

Crossrefs

The total number of coalescence sequences for n leaves, from among which the reciprocally monophyletic coalescence sequences are identified, follows A006472.

Programs

Maple

a:= n-> n!^2*(2*n-2)!/2^(2*n-2):
seq(a(n), n=1..15);  - Alois P. Heinz, Feb 07 2019

Mathematica

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

Formula

a(n) = (n!)^2*(2*n-2)! / 2^(2*n-2).

a(n) ~ 4*exp(-4*n)*n^(4*n-1/2)*Pi^(3/2). - Stefano Spezia, Apr 30 2024