cp's OEIS Frontend

A307700 Number of evolutionary duplication-loss-histories with n leaves of the caterpillar species tree with 5 leaves.

%I A307700 #25 Jul 31 2019 11:13:47
%S A307700 5,69,1230,24843,541315,12426996,296546600,7292489761,183702242491,
%T A307700 4719659859582,123261298705663,3263950145153931,87452457544863592,
%U A307700 2366980142343757033,64628573978046899555,1778185743733577832862,49254755849062502247446,1372455474283175885070422
%N A307700 Number of evolutionary duplication-loss-histories with n leaves of the caterpillar species tree with 5 leaves.
%C A307700 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.
%H A307700 C. Chauve, Y. Ponty, M. Wallner, <a href="https://arxiv.org/abs/1905.04971">Counting and sampling gene family evolutionary histories in the duplication-loss and duplication-loss-transfer models</a>, arXiv preprint arXiv:1905.04971 [math-CO], 2019.
%F A307700 G.f.: 1/2 - (1/2)*sqrt(-4 - t*w + 3*t + 3*w) where t = sqrt(1 - 4*z), u = sqrt(-5 + 6*t + 4*z), v = sqrt(-t*u + 3*t + 3*u - 4) and w = sqrt(-t*v + 3*t + 3*v - 4).
%e A307700 The caterpillar species tree with 5 leaves is equal to
%e A307700           a
%e A307700          / \
%e A307700         b   5
%e A307700        / \
%e A307700       c   4
%e A307700      / \
%e A307700     d   3
%e A307700    / \
%e A307700   1   2
%e A307700 For convenience the internal nodes are labeled by a,b,c,d, and the leaves by 1,2,3,4,5. The associated nodes in the histories will be denoted by the same labels.
%e A307700 The a(1)=5 histories with n=1 leaf are created by the following growth process:
%e A307700           a     a     a     a    a
%e A307700          /     /     /     /      \
%e A307700         b     b     b     b        5
%e A307700        /     /     /       \
%e A307700       c     c     c         4
%e A307700      /     /       \
%e A307700     d     d         3
%e A307700    /       \
%e A307700   1         2
%e A307700 after four loss events each.
%o A307700 (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), w = sqrt(-t*v+3*t+3*v-4)); Vec(1/2-(1/2)*sqrt(-4-t*w+3*t+3*w)) \\ _Michel Marcus_, May 07 2019
%Y A307700 Caterpillar species tree sequences: A000108 (1 leaf), A307696 (2 leaves), A307697 (3 leaves), A307698 (4 leaves).
%K A307700 nonn
%O A307700 1,1
%A A307700 _Michael Wallner_, Apr 22 2019