OFFSET
1,2
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.
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.
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
EXAMPLE
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.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Noah A Rosenberg, Feb 01 2019
STATUS
approved