A377461 Number of ranked labeled trees compatible with the 2-leaf perfect phylogeny of sample size n that possesses the largest number of compatible ranked labeled trees.

1, 1, 2, 9, 54, 540, 6480, 113400, 2268000, 61236000, 1837080000, 70727580000, 2970558360000, 154469034720000, 8650265944320000, 583892951241600000, 42040292489395200000, 3573424861598592000000, 321608237543873280000000, 33608060823334757760000000

Offset: 2

Author

Noah A Rosenberg, Jan 03 2025

Keywords

• nonn

Comments

The 2-leaf perfect phylogeny of sample size n that possesses the largest number of compatible ranked labeled trees is (floor(n/2), ceiling(n/2)); a(n) is the number of ranked labeled trees for this perfect phylogeny.

Links

• J. A. Palacios, A. Bhaskar, F. Disanto and N. A. Rosenberg, Enumeration of binary trees compatible with a perfect phylogeny, J. Math. Biol. 84 (2022), 54.

Crossrefs

Cf. A001405, A006472, A306266.

Programs

Mathematica

a[n_] := ((n - 2)!/((Floor[n/2] - 1)! (n - 1 - Floor[n/2])!)) Product[Binomial[i, 2], {i, 2, Floor[n/2]}] Product[Binomial[i, 2], {i, 2, Ceiling[n/2]}]
a[n_] := ((n - 2)!/((Floor[n/2] - 1)! (n - 1 - Floor[n/2])!)) Floor[n/2]! (Floor[n/2] - 1)! Ceiling[n/2]! (Ceiling[n/2] - 1)! /(2^(Floor[n/2] - 1) 2^(Ceiling[n/2] - 1))

Formula

a(n) = ((n-2)! / ((floor(n/2)-1)! (n-1-floor(n/2))!)) * (floor(n/2))! (floor(n/2)-1)! (ceiling(n/2))! (ceiling(n/2)-1)! / (2^(floor(n/2)-1) 2^(ceiling(n/2)-1)).

a(n) = A001405(n-2)*A006472(floor(n/2))*A006472(ceiling(n/2)).

a(2n) = A306266(n).