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Table 2 Conditions required for a 4-state Markovian process to be lumpable (in terms of s and π), and transformations to obtain π ˜ and s ˜ such that the lumpability holds.

From: Statistical tests to identify appropriate types of nucleotide sequence recoding in molecular phylogenetics

S Lumpability conditions ( π ˜ , s ˜ )
{{A, G}, C, T} s12 = s23
s14 = s34
s ˜ 13 = s ˜ 23 = ( ŝ 13 + ŝ 23 ) / 2
s ˜ 14 = s ˜ 34 = ( ŝ 14 + ŝ 34 ) / 2
{A, G, {C, T}} s12 = s14
s23 = s34
s ˜ 12 = s ˜ 14 = ( ŝ 12 + ŝ 14 ) / 2
s ˜ 23 = s ˜ 34 = ( ŝ 23 + ŝ 34 ) / 2
{A, {C, G}, T} s12 = s13
s24 = s34
s ˜ 12 = s ˜ 13 = ( ŝ 12 + ŝ 13 ) / 2
s ˜ 24 = s ˜ 34 = ( ŝ 24 + ŝ 34 ) / 2
{C, G, {A, T}} s12 = s24
s13 = s34
s ˜ 12 = s ˜ 24 = ( ŝ 12 + ŝ 24 ) / 2
s ˜ 13 = s ˜ 34 = ( ŝ 13 + ŝ 34 ) / 2
{{A, C}, G, T} s13 = s23
s14 = s24
s ˜ 13 = s ˜ 23 = ( ŝ 13 + ŝ 23 ) / 2
s ˜ 14 = s ˜ 24 = ( ŝ 14 + ŝ 24 ) / 2
{A, C, {G, T}} s13 = s14
s12 = s13
s ˜ 13 = s ˜ 14 = ( ŝ 13 + ŝ 14 ) / 2
s ˜ 23 = s ˜ 24 = ( ŝ 23 + ŝ 24 ) / 2
{A, {C, G, T}} s12 = s13
s13 = s14
s ˜ 12 = s ˜ 13 = s ˜ 14 = ( ŝ 12 + ŝ 13 + ŝ 14 ) / 3
{C, {A, G, T}} s12 = s23
s23 = s24
s ˜ 12 = s ˜ 23 = s ˜ 24 = ( ŝ 12 + ŝ 23 + ŝ 24 ) / 3
{G, {A, C, T}} s13 = s23
s23 = s34
s ˜ 13 = s ˜ 23 = s ˜ 34 = ( ŝ 13 + ŝ 23 + ŝ 34 ) / 3
{T, {A, C, G}} s14 = s24
s24 = s34
s ˜ 14 = s ˜ 24 = s ˜ 34 = ( ŝ 14 + ŝ 24 + ŝ 34 ) / 3
{{A, G}, {C, T}} s12π2 + s14π4 = s23π2+ s34π4
s12π1 + s23π3 = s14π1+ s34π3
s ˜ 23 = s ˜ 12 ( π ^ 2 π ^ 3 - π ^ 1 π ^ 4 ) + s ˜ 14 π ^ 4 ( π ^ 1 + π ^ 3 ) π ^ 3 ( π ^ 2 + π ^ 4 )
s ˜ 34 = s ˜ 12 π ^ 1 + s ˜ 23 π ^ 3 - s ˜ 14 π ^ 1 π ^ 3
{{A, T}, {C, G}} s12π2 + s13π3 = s24π2+ s34π3
s12π1 + s24π4 = s13π1+ s34π4
s ˜ 13 = s ˜ 12 ( π ^ 1 π ^ 3 - π ^ 2 π ^ 4 ) + s ˜ 24 π ^ 4 ( π ^ 2 + π ^ 3 ) π ^ 3 ( π ^ 1 + π ^ 4 )
s ˜ 34 = s ˜ 12 π ^ 2 + s ˜ 13 π ^ 3 - s ˜ 24 π ^ 2 π ^ 3
{{A, C}, {G, T}} s13π3 + s14π4 = s23π3 + s24π4
s13π1 + s23π2 = s14π1 + s24π4
s ˜ 23 = s ˜ 13 ( π ^ 2 π ^ 3 - π ^ 1 π ^ 4 ) + s ˜ 14 π ^ 4 ( π ^ 1 + π ^ 2 ) π ^ 2 ( π ^ 3 + π ^ 4 )
s ˜ 24 = s ˜ 13 π ^ 1 + s ˜ 23 π ^ 2 - s ˜ 14 π ^ 1 π ^ 2