A306205 Number of lonely (gene tree, species tree) pairs with n+1 leaves.

1, 6, 102, 3420, 191700, 16291800, 1966015800, 321188943600, 68482943802000, 18508629682620000, 6191158589749980000, 2512773755138326680000, 1216899394130358698760000, 693258145152338464498800000

Offset: 1

Author

Noah A Rosenberg, Jan 29 2019

Keywords

• nonn

Comments

A (gene tree, species tree) pair consisting of leaf-labeled binary trees whose leaves are labeled by the same label set is said to be lonely if and only if the pair has exactly one coalescent history. The sequence a(n) gives the number of distinct lonely (gene tree, species tree) pairs, considering all possible pairs of binary trees with n+1 leaves, bijectively labeled by the same set of n+1 distinguishable leaf labels.

Examples

			For n+1=2, the only (gene tree, species tree) pair ((A,B), (A,B)) with n+1=2 leaves is lonely and a(1)=1. For n+1=3, there are a(2)=6 lonely pairs with n+1=3 leaves: (((A,C),B), ((A,B),C)), (((B,C),A), ((A,B),C)), (((A,B),C), ((A,C),B)), (((B,C),A), ((A,C),B)), (((A,B),C), ((B,C),A)), and (((A,C),B), ((B,C),A)).
		

Links

• N. A. Rosenberg, Enumeration of lonely pairs of gene trees and species trees by means of antipodal cherries, Adv. Appl. Math., 102 (2019), 1-17.

Crossrefs

Lonely pairs are tabulated among pairs of leaf-labeled binary trees (A001818, or the square of A001147), where both trees in the pair are bijectively labeled by the same label set.

Programs

Mathematica

b[n_] := Sum[Binomial[n, p] T[p] T[n - p]/2^KroneckerDelta[p, n - p] Sum[
   Factorial[p] Factorial[
     n - p] Factorial[
      n - 2]/(2^(k - 1) Factorial[k] Factorial[p - k] Factorial[
        n - p - k] Factorial[k - 1]), {k, 1, p}], {p, 1, Floor[n/2]}]
a[n_] := b[n+1]
Table[a[n], {n, 1, 30}]

Formula

a(n-1) = Sum_{p=1..floor(n/2)} Sum__{k=1..p} (2n-2p-2)! (2p-2)! n! (n-2)! / (2^(n+k-3) (p-k)! (n-p-k)! (n-p-1)! (p-1)! k! (k-1)! 2^(KroneckerDelta(p,n-p)) ).