OFFSET
1,2
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.
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.
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)) ).
EXAMPLE
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)).
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Noah A Rosenberg, Jan 29 2019
STATUS
approved