A307698 Number of evolutionary duplication-loss-histories with n leaves of the caterpillar species tree with 4 leaves.

4, 39, 495, 7235, 115303, 1948791, 34379505, 626684162, 11722058693, 223870302588, 4349161774626, 85701267415112, 1709101664822416, 34432888701965454, 699810795294490974, 14331183304458656628, 295434131968070459359, 6125911207605272841753, 127680054133385458855845

Offset

1,1

Comments

An evolutionary history of size n is an ordered rooted (incomplete) binary tree with n leaves describing the evolution of a gene family of a species in phylogenomics. The caterpillar species tree S of size k is a binary tree with k leaves, where any left child is a leaf. Any node of the history is associated to a unique node of S, where specifically every leaf is associated to a leaf of S. A history is created by the following process (note that intermediate trees in this process may not be valid histories): Start with a root node associated to the root of S. For a given tree in the growth process, choose a leaf and perform a duplication, speciation, or (speciation-)loss event. A duplication event creates two children both associated to the same node as its parent. A speciation or (speciation-)loss event can only occur if the node is associated to an internal node in S. In that case, a speciation event creates two children associated to the children of the node in S. A (speciation-)loss event creates only a left or right child, associated to the left or right child in S, respectively.

Links

Table of n, a(n) for n=1..19.

C. Chauve, Y. Ponty, M. Wallner, Counting and sampling gene family evolutionary histories in the duplication-loss and duplication-loss-transfer models, arXiv preprint arXiv:1905.04971 [math-CO], 2019.

Formula

G.f.: 1/2 - (1/2)*sqrt(-4 - t*v + 3*t + 3*v) where t = sqrt(1 - 4*z), u = sqrt(-5 + 6*t + 4*z) and v = sqrt(-t*u + 3*t + 3*u - 4).

Example

The caterpillar species tree with 4 leaves is equal to
        a
       / \
      b   4
     / \
    c   3
   / \
  1   2
For convenience the internal nodes are labeled by a,b,c, and the leaves by 1,2,3,4. The associated nodes in the histories will be denoted by the same labels.
The a(1)=4 histories with n=1 leaf are created by the following growth process:
        a     a     a    a
       /     /     /      \
      b     b     b        4
     /     /       \
    c     c         3
   /       \
  1         2
after three loss events each.

Prog

(PARI) my(z = 'z + O('z^25), t = sqrt(1-4*z), u = sqrt(-5+6*t+4*z), v = sqrt(-t*u+3*t+3*u-4)); Vec(1/2-(1/2)*sqrt(-4-t*v+3*t+3*v)) \\ Michel Marcus, May 07 2019

Crossrefs

Caterpillar species tree sequences: A000108 (1 leaf), A307696 (2 leaves), A307697 (3 leaves), A307700 (5 leaves).

Sequence in context: A275517 A300187 A068187 * A319177 A323323 A300188

Adjacent sequences: A307695 A307696 A307697 * A307699 A307700 A307701

Keyword

nonn

Author

Michael Wallner, Apr 22 2019

Status

approved