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Table 5 Formulae for the expected numbers of triplets after three events

From: Distinguishing successive ancient polyploidy levels based on genome-internal syntenic alignment

Model \(M\in \mathscr {M}\)   
Triplet \(\Delta \in \mathscr {T}\)(2,2,2)(3,2,2)(2,3,2)
{t1,t1,t1}-u(1+v)3(1+w)3-
{t1,t1,t2}2uv(1+v)(1+w)32(3u+u)v(1+v)(1+w)32u(3v+v)
   ×(1+2v+v)(1+w)3
{t1,t2,t2}---
{t2,t2,t2}--(1+u)v(1+w)3
{t2;t2;t3}2(1+u)vw(1+w)2(1+2u+u)vw(1+w)2(1+u)(3v+v)w(1+w)
{t2,t3,t3}---
{t3,t3,t3}---
{t1,t1,t3}2u(1+v)2w(1+w)2(3u+u)(1+v)2w(1+w)2u(1+2v+v)2w(1+w)
{t1,t3,t3}---
{t1,t2,t3}---
model \(M\in \mathscr {M}\)   
triplet \(\Delta \in \mathscr {T}\)(2,2,3)(2,3,3)(3,2,3)
{t1,t1,t1}--u(1+v)3
   ×(1+2w+w)3
{t1,t1,t2}2uv(1+v)2u(3v+v)(1+2v+v)2(3u+u)v(1+v)
 ×(1+2w+w)3×(1+2w+w)3×(1+2w+w)3
{t1,t2,t2}---
{t2,t2,t2}-(1+u)v-
  ×(1+2w+w)3 
{t2;t2;t3}2(1+u)v2(1+u)(3v+v)2(1+2u+u)v
 ×(3w+w)(1+2w+w)×(3w+w)(1+2w+w)×(3w+w)(1+2w+w)
{t2,t3,t3}---
{t3,t3,t3}(1+u)(1+v)w(1+u)(1+2v+v)w(1+2u+u)(1+v)w
{t1,t1,t3}2u(1+v)22u(1+2v+v)22(3u+u)(1+v)2
 ×(3w+w)(1+2w+w)×(3w+w)(1+2w+w)×(3w+w)(1+2w+w)
{t1,t3,t3}---
{t1,t2,t3}---
model \(M\in \mathscr {M}\)   
triplet \(\Delta \in \mathscr {T}\)(3,3,2)(3,3,3) 
{t1,t1,t1}u(1+2v+v)3(1+w)3u(1+2v+v)3(1+2w+w)3 
{t1,t1,t2}2(3u+u)(3v+v)(1+2v+v)(1+w)32(3u+u)(3v+v)(1+2v+v)(1+2w+w)3 
{t1,t2,t2}-- 
{t2,t2,t2}(1+2u+u)v(1+w)3(1+2u+u)v(1+2w+w)3 
{t2;t2;t3}2(1+2u+u)(3v+v)w(1+w)2(1+2u+u)(3v+v)(3w+w)(1+2w+w) 
{t2,t3,t3}-- 
{t3,t3,t3}-(1+2u+u)(1+2v+v)w 
{t1,t1,t3}2(3u+u)(1+2v+v)2w(1+w)2(3u+u)(1+2v+v)2(3w+w)(1+2w+w) 
{t1,t3,t3}-- 
{t1,t2,t3}-- 
  1. u= probability that two progeny survive after the first polyploidization event. u= probability that three survive. Similarly v and v are the probabilities that two or three progeny survive, respectively, after the second event. w and w are the probabilities that two or three progeny survive after the third event