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%I #27 Jul 31 2019 11:13:33
%S 4,39,495,7235,115303,1948791,34379505,626684162,11722058693,
%T 223870302588,4349161774626,85701267415112,1709101664822416,
%U 34432888701965454,699810795294490974,14331183304458656628,295434131968070459359,6125911207605272841753,127680054133385458855845
%N Number of evolutionary duplication-loss-histories with n leaves of the caterpillar species tree with 4 leaves.
%C 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 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 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).
%e The caterpillar species tree with 4 leaves is equal to
%e a
%e / \
%e b 4
%e / \
%e c 3
%e / \
%e 1 2
%e 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.
%e The a(1)=4 histories with n=1 leaf are created by the following growth process:
%e a a a a
%e / / / \
%e b b b 4
%e / / \
%e c c 3
%e / \
%e 1 2
%e after three loss events each.
%o (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
%Y Caterpillar species tree sequences: A000108 (1 leaf), A307696 (2 leaves), A307697 (3 leaves), A307700 (5 leaves).
%K nonn
%O 1,1
%A _Michael Wallner_, Apr 22 2019
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