OFFSET
9,1
COMMENTS
a(n) is the number of nonequivalent root ancestral configurations associated with a sequence of pairs (G(n), S(n)), where G(n) is a gene tree and S(n) is a bijectively labeled species tree, G(n) and S(n) both have n leaves, and G(n) = S(n). The sequence of pairs possesses certain (binary, rooted) trees for n = 9, n = 10, and n = 11, as shown in Figure 5 of Disanto and Rosenberg (2019); for n >= 12, the tree G(n) = S(n) is formed by appending the tree G(n-3) and a 3-leaf binary rooted tree to a shared root.
For 9 <= n <= 20, a(n) tabulates the number of nonequivalent root ancestral configurations for the pair (G, S) with the largest number of nonequivalent root ancestral configurations (among pairs with n leaves and G = S). Disanto and Rosenberg (2019) conjecture that a(n) provides this maximum for all n >= 9.
LINKS
F. Disanto and N. A. Rosenberg, On the number of non-equivalent ancestral configurations for matching gene trees and species trees, Bull. Math. Biol. 81 (2019), 384-407.
FORMULA
a(n) = floor(3^(floor(n/3)) * (2*(n-3*floor(n/3))^2 + 8*(n-3*floor(n/3)) + 23)/27).
CROSSREFS
KEYWORD
nonn
AUTHOR
Noah A Rosenberg, Jun 19 2022
STATUS
approved