- FindStat

edit this statistic or download as text // json

Identifier

St001695: Standard tableaux ⟶ ℤ

Values

=>
                            Cc0007;cc-rep
                            
                            
                            

                        [[1]]=>0
[[1,2]]=>0
[[1],[2]]=>0
[[1,2,3]]=>0
[[1,3],[2]]=>1
[[1,2],[3]]=>0
[[1],[2],[3]]=>0
[[1,2,3,4]]=>0
[[1,3,4],[2]]=>2
[[1,2,4],[3]]=>1
[[1,2,3],[4]]=>0
[[1,3],[2,4]]=>2
[[1,2],[3,4]]=>0
[[1,4],[2],[3]]=>1
[[1,3],[2],[4]]=>2
[[1,2],[3],[4]]=>0
[[1],[2],[3],[4]]=>0
[[1,2,3,4,5]]=>0
[[1,3,4,5],[2]]=>3
[[1,2,4,5],[3]]=>2
[[1,2,3,5],[4]]=>1
[[1,2,3,4],[5]]=>0
[[1,3,5],[2,4]]=>4
[[1,2,5],[3,4]]=>1
[[1,3,4],[2,5]]=>3
[[1,2,4],[3,5]]=>2
[[1,2,3],[4,5]]=>0
[[1,4,5],[2],[3]]=>2
[[1,3,5],[2],[4]]=>4
[[1,2,5],[3],[4]]=>1
[[1,3,4],[2],[5]]=>3
[[1,2,4],[3],[5]]=>2
[[1,2,3],[4],[5]]=>0
[[1,4],[2,5],[3]]=>2
[[1,3],[2,5],[4]]=>4
[[1,2],[3,5],[4]]=>1
[[1,3],[2,4],[5]]=>3
[[1,2],[3,4],[5]]=>0
[[1,5],[2],[3],[4]]=>1
[[1,4],[2],[3],[5]]=>2
[[1,3],[2],[4],[5]]=>3
[[1,2],[3],[4],[5]]=>0
[[1],[2],[3],[4],[5]]=>0
[[1,2,3,4,5,6]]=>0
[[1,3,4,5,6],[2]]=>4
[[1,2,4,5,6],[3]]=>3
[[1,2,3,5,6],[4]]=>2
[[1,2,3,4,6],[5]]=>1
[[1,2,3,4,5],[6]]=>0
[[1,3,5,6],[2,4]]=>6
[[1,2,5,6],[3,4]]=>2
[[1,3,4,6],[2,5]]=>5
[[1,2,4,6],[3,5]]=>4
[[1,2,3,6],[4,5]]=>1
[[1,3,4,5],[2,6]]=>4
[[1,2,4,5],[3,6]]=>3
[[1,2,3,5],[4,6]]=>2
[[1,2,3,4],[5,6]]=>0
[[1,4,5,6],[2],[3]]=>3
[[1,3,5,6],[2],[4]]=>6
[[1,2,5,6],[3],[4]]=>2
[[1,3,4,6],[2],[5]]=>5
[[1,2,4,6],[3],[5]]=>4
[[1,2,3,6],[4],[5]]=>1
[[1,3,4,5],[2],[6]]=>4
[[1,2,4,5],[3],[6]]=>3
[[1,2,3,5],[4],[6]]=>2
[[1,2,3,4],[5],[6]]=>0
[[1,3,5],[2,4,6]]=>6
[[1,2,5],[3,4,6]]=>2
[[1,3,4],[2,5,6]]=>4
[[1,2,4],[3,5,6]]=>3
[[1,2,3],[4,5,6]]=>0
[[1,4,6],[2,5],[3]]=>4
[[1,3,6],[2,5],[4]]=>7
[[1,2,6],[3,5],[4]]=>3
[[1,3,6],[2,4],[5]]=>5
[[1,2,6],[3,4],[5]]=>1
[[1,4,5],[2,6],[3]]=>3
[[1,3,5],[2,6],[4]]=>6
[[1,2,5],[3,6],[4]]=>2
[[1,3,4],[2,6],[5]]=>5
[[1,2,4],[3,6],[5]]=>4
[[1,2,3],[4,6],[5]]=>1
[[1,3,5],[2,4],[6]]=>6
[[1,2,5],[3,4],[6]]=>2
[[1,3,4],[2,5],[6]]=>4
[[1,2,4],[3,5],[6]]=>3
[[1,2,3],[4,5],[6]]=>0
[[1,5,6],[2],[3],[4]]=>2
[[1,4,6],[2],[3],[5]]=>4
[[1,3,6],[2],[4],[5]]=>5
[[1,2,6],[3],[4],[5]]=>1
[[1,4,5],[2],[3],[6]]=>3
[[1,3,5],[2],[4],[6]]=>6
[[1,2,5],[3],[4],[6]]=>2
[[1,3,4],[2],[5],[6]]=>4
[[1,2,4],[3],[5],[6]]=>3
[[1,2,3],[4],[5],[6]]=>0
[[1,4],[2,5],[3,6]]=>3
[[1,3],[2,5],[4,6]]=>6
[[1,2],[3,5],[4,6]]=>2
[[1,3],[2,4],[5,6]]=>4
[[1,2],[3,4],[5,6]]=>0
[[1,5],[2,6],[3],[4]]=>2
[[1,4],[2,6],[3],[5]]=>4
[[1,3],[2,6],[4],[5]]=>5
[[1,2],[3,6],[4],[5]]=>1
[[1,4],[2,5],[3],[6]]=>3
[[1,3],[2,5],[4],[6]]=>6
[[1,2],[3,5],[4],[6]]=>2
[[1,3],[2,4],[5],[6]]=>4
[[1,2],[3,4],[5],[6]]=>0
[[1,6],[2],[3],[4],[5]]=>1
[[1,5],[2],[3],[4],[6]]=>2
[[1,4],[2],[3],[5],[6]]=>3
[[1,3],[2],[4],[5],[6]]=>4
[[1,2],[3],[4],[5],[6]]=>0
[[1],[2],[3],[4],[5],[6]]=>0
[[1,2,3,4,5,6,7]]=>0
[[1,3,4,5,6,7],[2]]=>5
[[1,2,4,5,6,7],[3]]=>4
[[1,2,3,5,6,7],[4]]=>3
[[1,2,3,4,6,7],[5]]=>2
[[1,2,3,4,5,7],[6]]=>1
[[1,2,3,4,5,6],[7]]=>0
[[1,3,5,6,7],[2,4]]=>8
[[1,2,5,6,7],[3,4]]=>3
[[1,3,4,6,7],[2,5]]=>7
[[1,2,4,6,7],[3,5]]=>6
[[1,2,3,6,7],[4,5]]=>2
[[1,3,4,5,7],[2,6]]=>6
[[1,2,4,5,7],[3,6]]=>5
[[1,2,3,5,7],[4,6]]=>4
[[1,2,3,4,7],[5,6]]=>1
[[1,3,4,5,6],[2,7]]=>5
[[1,2,4,5,6],[3,7]]=>4
[[1,2,3,5,6],[4,7]]=>3
[[1,2,3,4,6],[5,7]]=>2
[[1,2,3,4,5],[6,7]]=>0
[[1,4,5,6,7],[2],[3]]=>4
[[1,3,5,6,7],[2],[4]]=>8
[[1,2,5,6,7],[3],[4]]=>3
[[1,3,4,6,7],[2],[5]]=>7
[[1,2,4,6,7],[3],[5]]=>6
[[1,2,3,6,7],[4],[5]]=>2
[[1,3,4,5,7],[2],[6]]=>6
[[1,2,4,5,7],[3],[6]]=>5
[[1,2,3,5,7],[4],[6]]=>4
[[1,2,3,4,7],[5],[6]]=>1
[[1,3,4,5,6],[2],[7]]=>5
[[1,2,4,5,6],[3],[7]]=>4
[[1,2,3,5,6],[4],[7]]=>3
[[1,2,3,4,6],[5],[7]]=>2
[[1,2,3,4,5],[6],[7]]=>0
[[1,3,5,7],[2,4,6]]=>9
[[1,2,5,7],[3,4,6]]=>4
[[1,3,4,7],[2,5,6]]=>6
[[1,2,4,7],[3,5,6]]=>5
[[1,2,3,7],[4,5,6]]=>1
[[1,3,5,6],[2,4,7]]=>8
[[1,2,5,6],[3,4,7]]=>3
[[1,3,4,6],[2,5,7]]=>7
[[1,2,4,6],[3,5,7]]=>6
[[1,2,3,6],[4,5,7]]=>2
[[1,3,4,5],[2,6,7]]=>5
[[1,2,4,5],[3,6,7]]=>4
[[1,2,3,5],[4,6,7]]=>3
[[1,2,3,4],[5,6,7]]=>0
[[1,4,6,7],[2,5],[3]]=>6
[[1,3,6,7],[2,5],[4]]=>10
[[1,2,6,7],[3,5],[4]]=>5
[[1,3,6,7],[2,4],[5]]=>7
[[1,2,6,7],[3,4],[5]]=>2
[[1,4,5,7],[2,6],[3]]=>5
[[1,3,5,7],[2,6],[4]]=>9
[[1,2,5,7],[3,6],[4]]=>4
[[1,3,4,7],[2,6],[5]]=>8
[[1,2,4,7],[3,6],[5]]=>7
[[1,2,3,7],[4,6],[5]]=>3
[[1,3,5,7],[2,4],[6]]=>9
[[1,2,5,7],[3,4],[6]]=>4
[[1,3,4,7],[2,5],[6]]=>6
[[1,2,4,7],[3,5],[6]]=>5
[[1,2,3,7],[4,5],[6]]=>1
[[1,4,5,6],[2,7],[3]]=>4
[[1,3,5,6],[2,7],[4]]=>8
[[1,2,5,6],[3,7],[4]]=>3
[[1,3,4,6],[2,7],[5]]=>7
[[1,2,4,6],[3,7],[5]]=>6
[[1,2,3,6],[4,7],[5]]=>2
[[1,3,4,5],[2,7],[6]]=>6
[[1,2,4,5],[3,7],[6]]=>5
[[1,2,3,5],[4,7],[6]]=>4
[[1,2,3,4],[5,7],[6]]=>1
[[1,3,5,6],[2,4],[7]]=>8
[[1,2,5,6],[3,4],[7]]=>3
[[1,3,4,6],[2,5],[7]]=>7
[[1,2,4,6],[3,5],[7]]=>6
[[1,2,3,6],[4,5],[7]]=>2
[[1,3,4,5],[2,6],[7]]=>5
[[1,2,4,5],[3,6],[7]]=>4
[[1,2,3,5],[4,6],[7]]=>3
[[1,2,3,4],[5,6],[7]]=>0
[[1,5,6,7],[2],[3],[4]]=>3
[[1,4,6,7],[2],[3],[5]]=>6
[[1,3,6,7],[2],[4],[5]]=>7
[[1,2,6,7],[3],[4],[5]]=>2
[[1,4,5,7],[2],[3],[6]]=>5
[[1,3,5,7],[2],[4],[6]]=>9
[[1,2,5,7],[3],[4],[6]]=>4
[[1,3,4,7],[2],[5],[6]]=>6
[[1,2,4,7],[3],[5],[6]]=>5
[[1,2,3,7],[4],[5],[6]]=>1
[[1,4,5,6],[2],[3],[7]]=>4
[[1,3,5,6],[2],[4],[7]]=>8
[[1,2,5,6],[3],[4],[7]]=>3
[[1,3,4,6],[2],[5],[7]]=>7
[[1,2,4,6],[3],[5],[7]]=>6
[[1,2,3,6],[4],[5],[7]]=>2
[[1,3,4,5],[2],[6],[7]]=>5
[[1,2,4,5],[3],[6],[7]]=>4
[[1,2,3,5],[4],[6],[7]]=>3
[[1,2,3,4],[5],[6],[7]]=>0
[[1,4,6],[2,5,7],[3]]=>6
[[1,3,6],[2,5,7],[4]]=>10
[[1,2,6],[3,5,7],[4]]=>5
[[1,3,6],[2,4,7],[5]]=>7
[[1,2,6],[3,4,7],[5]]=>2
[[1,4,5],[2,6,7],[3]]=>4
[[1,3,5],[2,6,7],[4]]=>8
[[1,2,5],[3,6,7],[4]]=>3
[[1,3,4],[2,6,7],[5]]=>7
[[1,2,4],[3,6,7],[5]]=>6
[[1,2,3],[4,6,7],[5]]=>2
[[1,3,5],[2,4,7],[6]]=>9
[[1,2,5],[3,4,7],[6]]=>4
[[1,3,4],[2,5,7],[6]]=>6
[[1,2,4],[3,5,7],[6]]=>5
[[1,2,3],[4,5,7],[6]]=>1
[[1,3,5],[2,4,6],[7]]=>8
[[1,2,5],[3,4,6],[7]]=>3
[[1,3,4],[2,5,6],[7]]=>5
[[1,2,4],[3,5,6],[7]]=>4
[[1,2,3],[4,5,6],[7]]=>0
[[1,4,7],[2,5],[3,6]]=>5
[[1,3,7],[2,5],[4,6]]=>9
[[1,2,7],[3,5],[4,6]]=>4
[[1,3,7],[2,4],[5,6]]=>6
[[1,2,7],[3,4],[5,6]]=>1
[[1,4,6],[2,5],[3,7]]=>6
[[1,3,6],[2,5],[4,7]]=>10
[[1,2,6],[3,5],[4,7]]=>5
[[1,3,6],[2,4],[5,7]]=>7
[[1,2,6],[3,4],[5,7]]=>2
[[1,4,5],[2,6],[3,7]]=>4
[[1,3,5],[2,6],[4,7]]=>8
[[1,2,5],[3,6],[4,7]]=>3
[[1,3,4],[2,6],[5,7]]=>7
[[1,2,4],[3,6],[5,7]]=>6
[[1,2,3],[4,6],[5,7]]=>2
[[1,3,5],[2,4],[6,7]]=>8
[[1,2,5],[3,4],[6,7]]=>3
[[1,3,4],[2,5],[6,7]]=>5
[[1,2,4],[3,5],[6,7]]=>4
[[1,2,3],[4,5],[6,7]]=>0
[[1,5,7],[2,6],[3],[4]]=>4
[[1,4,7],[2,6],[3],[5]]=>7
[[1,3,7],[2,6],[4],[5]]=>8
[[1,2,7],[3,6],[4],[5]]=>3
[[1,4,7],[2,5],[3],[6]]=>5
[[1,3,7],[2,5],[4],[6]]=>9
[[1,2,7],[3,5],[4],[6]]=>4
[[1,3,7],[2,4],[5],[6]]=>6
[[1,2,7],[3,4],[5],[6]]=>1
[[1,5,6],[2,7],[3],[4]]=>3
[[1,4,6],[2,7],[3],[5]]=>6
[[1,3,6],[2,7],[4],[5]]=>7
[[1,2,6],[3,7],[4],[5]]=>2
[[1,4,5],[2,7],[3],[6]]=>5
[[1,3,5],[2,7],[4],[6]]=>9
[[1,2,5],[3,7],[4],[6]]=>4
[[1,3,4],[2,7],[5],[6]]=>6
[[1,2,4],[3,7],[5],[6]]=>5
[[1,2,3],[4,7],[5],[6]]=>1
[[1,4,6],[2,5],[3],[7]]=>6
[[1,3,6],[2,5],[4],[7]]=>10
[[1,2,6],[3,5],[4],[7]]=>5
[[1,3,6],[2,4],[5],[7]]=>7
[[1,2,6],[3,4],[5],[7]]=>2
[[1,4,5],[2,6],[3],[7]]=>4
[[1,3,5],[2,6],[4],[7]]=>8
[[1,2,5],[3,6],[4],[7]]=>3
[[1,3,4],[2,6],[5],[7]]=>7
[[1,2,4],[3,6],[5],[7]]=>6
[[1,2,3],[4,6],[5],[7]]=>2
[[1,3,5],[2,4],[6],[7]]=>8
[[1,2,5],[3,4],[6],[7]]=>3
[[1,3,4],[2,5],[6],[7]]=>5
[[1,2,4],[3,5],[6],[7]]=>4
[[1,2,3],[4,5],[6],[7]]=>0
[[1,6,7],[2],[3],[4],[5]]=>2
[[1,5,7],[2],[3],[4],[6]]=>4
[[1,4,7],[2],[3],[5],[6]]=>5
[[1,3,7],[2],[4],[5],[6]]=>6
[[1,2,7],[3],[4],[5],[6]]=>1
[[1,5,6],[2],[3],[4],[7]]=>3
[[1,4,6],[2],[3],[5],[7]]=>6
[[1,3,6],[2],[4],[5],[7]]=>7
[[1,2,6],[3],[4],[5],[7]]=>2
[[1,4,5],[2],[3],[6],[7]]=>4
[[1,3,5],[2],[4],[6],[7]]=>8
[[1,2,5],[3],[4],[6],[7]]=>3
[[1,3,4],[2],[5],[6],[7]]=>5
[[1,2,4],[3],[5],[6],[7]]=>4
[[1,2,3],[4],[5],[6],[7]]=>0
[[1,5],[2,6],[3,7],[4]]=>3
[[1,4],[2,6],[3,7],[5]]=>6
[[1,3],[2,6],[4,7],[5]]=>7
[[1,2],[3,6],[4,7],[5]]=>2
[[1,4],[2,5],[3,7],[6]]=>5
[[1,3],[2,5],[4,7],[6]]=>9
[[1,2],[3,5],[4,7],[6]]=>4
[[1,3],[2,4],[5,7],[6]]=>6
[[1,2],[3,4],[5,7],[6]]=>1
[[1,4],[2,5],[3,6],[7]]=>4
[[1,3],[2,5],[4,6],[7]]=>8
[[1,2],[3,5],[4,6],[7]]=>3
[[1,3],[2,4],[5,6],[7]]=>5
[[1,2],[3,4],[5,6],[7]]=>0
[[1,6],[2,7],[3],[4],[5]]=>2
[[1,5],[2,7],[3],[4],[6]]=>4
[[1,4],[2,7],[3],[5],[6]]=>5
[[1,3],[2,7],[4],[5],[6]]=>6
[[1,2],[3,7],[4],[5],[6]]=>1
[[1,5],[2,6],[3],[4],[7]]=>3
[[1,4],[2,6],[3],[5],[7]]=>6
[[1,3],[2,6],[4],[5],[7]]=>7
[[1,2],[3,6],[4],[5],[7]]=>2
[[1,4],[2,5],[3],[6],[7]]=>4
[[1,3],[2,5],[4],[6],[7]]=>8
[[1,2],[3,5],[4],[6],[7]]=>3
[[1,3],[2,4],[5],[6],[7]]=>5
[[1,2],[3,4],[5],[6],[7]]=>0
[[1,7],[2],[3],[4],[5],[6]]=>1
[[1,6],[2],[3],[4],[5],[7]]=>2
[[1,5],[2],[3],[4],[6],[7]]=>3
[[1,4],[2],[3],[5],[6],[7]]=>4
[[1,3],[2],[4],[5],[6],[7]]=>5
[[1,2],[3],[4],[5],[6],[7]]=>0
[[1],[2],[3],[4],[5],[6],[7]]=>0
[[1,2,3,4,5,6,7,8]]=>0
[[1,3,4,5,6,7,8],[2]]=>6
[[1,2,4,5,6,7,8],[3]]=>5
[[1,2,3,5,6,7,8],[4]]=>4
[[1,2,3,4,6,7,8],[5]]=>3
[[1,2,3,4,5,7,8],[6]]=>2
[[1,2,3,4,5,6,8],[7]]=>1
[[1,2,3,4,5,6,7],[8]]=>0
[[1,3,5,6,7,8],[2,4]]=>10
[[1,2,5,6,7,8],[3,4]]=>4
[[1,3,4,6,7,8],[2,5]]=>9
[[1,2,4,6,7,8],[3,5]]=>8
[[1,2,3,6,7,8],[4,5]]=>3
[[1,3,4,5,7,8],[2,6]]=>8
[[1,2,4,5,7,8],[3,6]]=>7
[[1,2,3,5,7,8],[4,6]]=>6
[[1,2,3,4,7,8],[5,6]]=>2
[[1,3,4,5,6,8],[2,7]]=>7
[[1,2,4,5,6,8],[3,7]]=>6
[[1,2,3,5,6,8],[4,7]]=>5
[[1,2,3,4,6,8],[5,7]]=>4
[[1,2,3,4,5,8],[6,7]]=>1
[[1,3,4,5,6,7],[2,8]]=>6
[[1,2,4,5,6,7],[3,8]]=>5
[[1,2,3,5,6,7],[4,8]]=>4
[[1,2,3,4,6,7],[5,8]]=>3
[[1,2,3,4,5,7],[6,8]]=>2
[[1,2,3,4,5,6],[7,8]]=>0
[[1,4,5,6,7,8],[2],[3]]=>5
[[1,3,5,6,7,8],[2],[4]]=>10
[[1,2,5,6,7,8],[3],[4]]=>4
[[1,3,4,6,7,8],[2],[5]]=>9
[[1,2,4,6,7,8],[3],[5]]=>8
[[1,2,3,6,7,8],[4],[5]]=>3
[[1,3,4,5,7,8],[2],[6]]=>8
[[1,2,4,5,7,8],[3],[6]]=>7
[[1,2,3,5,7,8],[4],[6]]=>6
[[1,2,3,4,7,8],[5],[6]]=>2
[[1,3,4,5,6,8],[2],[7]]=>7
[[1,2,4,5,6,8],[3],[7]]=>6
[[1,2,3,5,6,8],[4],[7]]=>5
[[1,2,3,4,6,8],[5],[7]]=>4
[[1,2,3,4,5,8],[6],[7]]=>1
[[1,3,4,5,6,7],[2],[8]]=>6
[[1,2,4,5,6,7],[3],[8]]=>5
[[1,2,3,5,6,7],[4],[8]]=>4
[[1,2,3,4,6,7],[5],[8]]=>3
[[1,2,3,4,5,7],[6],[8]]=>2
[[1,2,3,4,5,6],[7],[8]]=>0
[[1,3,5,7,8],[2,4,6]]=>12
[[1,2,5,7,8],[3,4,6]]=>6
[[1,3,4,7,8],[2,5,6]]=>8
[[1,2,4,7,8],[3,5,6]]=>7
[[1,2,3,7,8],[4,5,6]]=>2
[[1,3,5,6,8],[2,4,7]]=>11
[[1,2,5,6,8],[3,4,7]]=>5
[[1,3,4,6,8],[2,5,7]]=>10
[[1,2,4,6,8],[3,5,7]]=>9
[[1,2,3,6,8],[4,5,7]]=>4
[[1,3,4,5,8],[2,6,7]]=>7
[[1,2,4,5,8],[3,6,7]]=>6
[[1,2,3,5,8],[4,6,7]]=>5
[[1,2,3,4,8],[5,6,7]]=>1
[[1,3,5,6,7],[2,4,8]]=>10
[[1,2,5,6,7],[3,4,8]]=>4
[[1,3,4,6,7],[2,5,8]]=>9
[[1,2,4,6,7],[3,5,8]]=>8
[[1,2,3,6,7],[4,5,8]]=>3
[[1,3,4,5,7],[2,6,8]]=>8
[[1,2,4,5,7],[3,6,8]]=>7
[[1,2,3,5,7],[4,6,8]]=>6
[[1,2,3,4,7],[5,6,8]]=>2
[[1,3,4,5,6],[2,7,8]]=>6
[[1,2,4,5,6],[3,7,8]]=>5
[[1,2,3,5,6],[4,7,8]]=>4
[[1,2,3,4,6],[5,7,8]]=>3
[[1,2,3,4,5],[6,7,8]]=>0
[[1,4,6,7,8],[2,5],[3]]=>8
[[1,3,6,7,8],[2,5],[4]]=>13
[[1,2,6,7,8],[3,5],[4]]=>7
[[1,3,6,7,8],[2,4],[5]]=>9
[[1,2,6,7,8],[3,4],[5]]=>3
[[1,4,5,7,8],[2,6],[3]]=>7
[[1,3,5,7,8],[2,6],[4]]=>12
[[1,2,5,7,8],[3,6],[4]]=>6
[[1,3,4,7,8],[2,6],[5]]=>11
[[1,2,4,7,8],[3,6],[5]]=>10
[[1,2,3,7,8],[4,6],[5]]=>5
[[1,3,5,7,8],[2,4],[6]]=>12
[[1,2,5,7,8],[3,4],[6]]=>6
[[1,3,4,7,8],[2,5],[6]]=>8
[[1,2,4,7,8],[3,5],[6]]=>7
[[1,2,3,7,8],[4,5],[6]]=>2
[[1,4,5,6,8],[2,7],[3]]=>6
[[1,3,5,6,8],[2,7],[4]]=>11
[[1,2,5,6,8],[3,7],[4]]=>5
[[1,3,4,6,8],[2,7],[5]]=>10
[[1,2,4,6,8],[3,7],[5]]=>9
[[1,2,3,6,8],[4,7],[5]]=>4
[[1,3,4,5,8],[2,7],[6]]=>9
[[1,2,4,5,8],[3,7],[6]]=>8
[[1,2,3,5,8],[4,7],[6]]=>7
[[1,2,3,4,8],[5,7],[6]]=>3
[[1,3,5,6,8],[2,4],[7]]=>11
[[1,2,5,6,8],[3,4],[7]]=>5
[[1,3,4,6,8],[2,5],[7]]=>10
[[1,2,4,6,8],[3,5],[7]]=>9
[[1,2,3,6,8],[4,5],[7]]=>4
[[1,3,4,5,8],[2,6],[7]]=>7
[[1,2,4,5,8],[3,6],[7]]=>6
[[1,2,3,5,8],[4,6],[7]]=>5
[[1,2,3,4,8],[5,6],[7]]=>1
[[1,4,5,6,7],[2,8],[3]]=>5
[[1,3,5,6,7],[2,8],[4]]=>10
[[1,2,5,6,7],[3,8],[4]]=>4
[[1,3,4,6,7],[2,8],[5]]=>9
[[1,2,4,6,7],[3,8],[5]]=>8
[[1,2,3,6,7],[4,8],[5]]=>3
[[1,3,4,5,7],[2,8],[6]]=>8
[[1,2,4,5,7],[3,8],[6]]=>7
[[1,2,3,5,7],[4,8],[6]]=>6
[[1,2,3,4,7],[5,8],[6]]=>2
[[1,3,4,5,6],[2,8],[7]]=>7
[[1,2,4,5,6],[3,8],[7]]=>6
[[1,2,3,5,6],[4,8],[7]]=>5
[[1,2,3,4,6],[5,8],[7]]=>4
[[1,2,3,4,5],[6,8],[7]]=>1
[[1,3,5,6,7],[2,4],[8]]=>10
[[1,2,5,6,7],[3,4],[8]]=>4
[[1,3,4,6,7],[2,5],[8]]=>9
[[1,2,4,6,7],[3,5],[8]]=>8
[[1,2,3,6,7],[4,5],[8]]=>3
[[1,3,4,5,7],[2,6],[8]]=>8
[[1,2,4,5,7],[3,6],[8]]=>7
[[1,2,3,5,7],[4,6],[8]]=>6
[[1,2,3,4,7],[5,6],[8]]=>2
[[1,3,4,5,6],[2,7],[8]]=>6
[[1,2,4,5,6],[3,7],[8]]=>5
[[1,2,3,5,6],[4,7],[8]]=>4
[[1,2,3,4,6],[5,7],[8]]=>3
[[1,2,3,4,5],[6,7],[8]]=>0
[[1,5,6,7,8],[2],[3],[4]]=>4
[[1,4,6,7,8],[2],[3],[5]]=>8
[[1,3,6,7,8],[2],[4],[5]]=>9
[[1,2,6,7,8],[3],[4],[5]]=>3
[[1,4,5,7,8],[2],[3],[6]]=>7
[[1,3,5,7,8],[2],[4],[6]]=>12
[[1,2,5,7,8],[3],[4],[6]]=>6
[[1,3,4,7,8],[2],[5],[6]]=>8
[[1,2,4,7,8],[3],[5],[6]]=>7
[[1,2,3,7,8],[4],[5],[6]]=>2
[[1,4,5,6,8],[2],[3],[7]]=>6
[[1,3,5,6,8],[2],[4],[7]]=>11
[[1,2,5,6,8],[3],[4],[7]]=>5
[[1,3,4,6,8],[2],[5],[7]]=>10
[[1,2,4,6,8],[3],[5],[7]]=>9
[[1,2,3,6,8],[4],[5],[7]]=>4
[[1,3,4,5,8],[2],[6],[7]]=>7
[[1,2,4,5,8],[3],[6],[7]]=>6
[[1,2,3,5,8],[4],[6],[7]]=>5
[[1,2,3,4,8],[5],[6],[7]]=>1
[[1,4,5,6,7],[2],[3],[8]]=>5
[[1,3,5,6,7],[2],[4],[8]]=>10
[[1,2,5,6,7],[3],[4],[8]]=>4
[[1,3,4,6,7],[2],[5],[8]]=>9
[[1,2,4,6,7],[3],[5],[8]]=>8
[[1,2,3,6,7],[4],[5],[8]]=>3
[[1,3,4,5,7],[2],[6],[8]]=>8
[[1,2,4,5,7],[3],[6],[8]]=>7
[[1,2,3,5,7],[4],[6],[8]]=>6
[[1,2,3,4,7],[5],[6],[8]]=>2
[[1,3,4,5,6],[2],[7],[8]]=>6
[[1,2,4,5,6],[3],[7],[8]]=>5
[[1,2,3,5,6],[4],[7],[8]]=>4
[[1,2,3,4,6],[5],[7],[8]]=>3
[[1,2,3,4,5],[6],[7],[8]]=>0
[[1,3,5,7],[2,4,6,8]]=>12
[[1,2,5,7],[3,4,6,8]]=>6
[[1,3,4,7],[2,5,6,8]]=>8
[[1,2,4,7],[3,5,6,8]]=>7
[[1,2,3,7],[4,5,6,8]]=>2
[[1,3,5,6],[2,4,7,8]]=>10
[[1,2,5,6],[3,4,7,8]]=>4
[[1,3,4,6],[2,5,7,8]]=>9
[[1,2,4,6],[3,5,7,8]]=>8
[[1,2,3,6],[4,5,7,8]]=>3
[[1,3,4,5],[2,6,7,8]]=>6
[[1,2,4,5],[3,6,7,8]]=>5
[[1,2,3,5],[4,6,7,8]]=>4
[[1,2,3,4],[5,6,7,8]]=>0
[[1,4,6,8],[2,5,7],[3]]=>9
[[1,3,6,8],[2,5,7],[4]]=>14
[[1,2,6,8],[3,5,7],[4]]=>8
[[1,3,6,8],[2,4,7],[5]]=>10
[[1,2,6,8],[3,4,7],[5]]=>4
[[1,4,5,8],[2,6,7],[3]]=>6
[[1,3,5,8],[2,6,7],[4]]=>11
[[1,2,5,8],[3,6,7],[4]]=>5
[[1,3,4,8],[2,6,7],[5]]=>10
[[1,2,4,8],[3,6,7],[5]]=>9
[[1,2,3,8],[4,6,7],[5]]=>4
[[1,3,5,8],[2,4,7],[6]]=>13
[[1,2,5,8],[3,4,7],[6]]=>7
[[1,3,4,8],[2,5,7],[6]]=>9
[[1,2,4,8],[3,5,7],[6]]=>8
[[1,2,3,8],[4,5,7],[6]]=>3
[[1,3,5,8],[2,4,6],[7]]=>11
[[1,2,5,8],[3,4,6],[7]]=>5
[[1,3,4,8],[2,5,6],[7]]=>7
[[1,2,4,8],[3,5,6],[7]]=>6
[[1,2,3,8],[4,5,6],[7]]=>1
[[1,4,6,7],[2,5,8],[3]]=>8
[[1,3,6,7],[2,5,8],[4]]=>13
[[1,2,6,7],[3,5,8],[4]]=>7
[[1,3,6,7],[2,4,8],[5]]=>9
[[1,2,6,7],[3,4,8],[5]]=>3
[[1,4,5,7],[2,6,8],[3]]=>7
[[1,3,5,7],[2,6,8],[4]]=>12
[[1,2,5,7],[3,6,8],[4]]=>6
[[1,3,4,7],[2,6,8],[5]]=>11
[[1,2,4,7],[3,6,8],[5]]=>10
[[1,2,3,7],[4,6,8],[5]]=>5
[[1,3,5,7],[2,4,8],[6]]=>12
[[1,2,5,7],[3,4,8],[6]]=>6
[[1,3,4,7],[2,5,8],[6]]=>8
[[1,2,4,7],[3,5,8],[6]]=>7
[[1,2,3,7],[4,5,8],[6]]=>2
[[1,4,5,6],[2,7,8],[3]]=>5
[[1,3,5,6],[2,7,8],[4]]=>10
[[1,2,5,6],[3,7,8],[4]]=>4
[[1,3,4,6],[2,7,8],[5]]=>9
[[1,2,4,6],[3,7,8],[5]]=>8
[[1,2,3,6],[4,7,8],[5]]=>3
[[1,3,4,5],[2,7,8],[6]]=>8
[[1,2,4,5],[3,7,8],[6]]=>7
[[1,2,3,5],[4,7,8],[6]]=>6
[[1,2,3,4],[5,7,8],[6]]=>2
[[1,3,5,6],[2,4,8],[7]]=>11
[[1,2,5,6],[3,4,8],[7]]=>5
[[1,3,4,6],[2,5,8],[7]]=>10
[[1,2,4,6],[3,5,8],[7]]=>9
[[1,2,3,6],[4,5,8],[7]]=>4
[[1,3,4,5],[2,6,8],[7]]=>7
[[1,2,4,5],[3,6,8],[7]]=>6
[[1,2,3,5],[4,6,8],[7]]=>5
[[1,2,3,4],[5,6,8],[7]]=>1
[[1,3,5,7],[2,4,6],[8]]=>12
[[1,2,5,7],[3,4,6],[8]]=>6
[[1,3,4,7],[2,5,6],[8]]=>8
[[1,2,4,7],[3,5,6],[8]]=>7
[[1,2,3,7],[4,5,6],[8]]=>2
[[1,3,5,6],[2,4,7],[8]]=>10
[[1,2,5,6],[3,4,7],[8]]=>4
[[1,3,4,6],[2,5,7],[8]]=>9
[[1,2,4,6],[3,5,7],[8]]=>8
[[1,2,3,6],[4,5,7],[8]]=>3
[[1,3,4,5],[2,6,7],[8]]=>6
[[1,2,4,5],[3,6,7],[8]]=>5
[[1,2,3,5],[4,6,7],[8]]=>4
[[1,2,3,4],[5,6,7],[8]]=>0
[[1,4,7,8],[2,5],[3,6]]=>7
[[1,3,7,8],[2,5],[4,6]]=>12
[[1,2,7,8],[3,5],[4,6]]=>6
[[1,3,7,8],[2,4],[5,6]]=>8
[[1,2,7,8],[3,4],[5,6]]=>2
[[1,4,6,8],[2,5],[3,7]]=>9
[[1,3,6,8],[2,5],[4,7]]=>14
[[1,2,6,8],[3,5],[4,7]]=>8
[[1,3,6,8],[2,4],[5,7]]=>10
[[1,2,6,8],[3,4],[5,7]]=>4
[[1,4,5,8],[2,6],[3,7]]=>6
[[1,3,5,8],[2,6],[4,7]]=>11
[[1,2,5,8],[3,6],[4,7]]=>5
[[1,3,4,8],[2,6],[5,7]]=>10
[[1,2,4,8],[3,6],[5,7]]=>9
[[1,2,3,8],[4,6],[5,7]]=>4
[[1,3,5,8],[2,4],[6,7]]=>11
[[1,2,5,8],[3,4],[6,7]]=>5
[[1,3,4,8],[2,5],[6,7]]=>7
[[1,2,4,8],[3,5],[6,7]]=>6
[[1,2,3,8],[4,5],[6,7]]=>1
[[1,4,6,7],[2,5],[3,8]]=>8
[[1,3,6,7],[2,5],[4,8]]=>13
[[1,2,6,7],[3,5],[4,8]]=>7
[[1,3,6,7],[2,4],[5,8]]=>9
[[1,2,6,7],[3,4],[5,8]]=>3
[[1,4,5,7],[2,6],[3,8]]=>7
[[1,3,5,7],[2,6],[4,8]]=>12
[[1,2,5,7],[3,6],[4,8]]=>6
[[1,3,4,7],[2,6],[5,8]]=>11
[[1,2,4,7],[3,6],[5,8]]=>10
[[1,2,3,7],[4,6],[5,8]]=>5
[[1,3,5,7],[2,4],[6,8]]=>12
[[1,2,5,7],[3,4],[6,8]]=>6
[[1,3,4,7],[2,5],[6,8]]=>8
[[1,2,4,7],[3,5],[6,8]]=>7
[[1,2,3,7],[4,5],[6,8]]=>2
[[1,4,5,6],[2,7],[3,8]]=>5
[[1,3,5,6],[2,7],[4,8]]=>10
[[1,2,5,6],[3,7],[4,8]]=>4
[[1,3,4,6],[2,7],[5,8]]=>9
[[1,2,4,6],[3,7],[5,8]]=>8
[[1,2,3,6],[4,7],[5,8]]=>3
[[1,3,4,5],[2,7],[6,8]]=>8
[[1,2,4,5],[3,7],[6,8]]=>7
[[1,2,3,5],[4,7],[6,8]]=>6
[[1,2,3,4],[5,7],[6,8]]=>2
[[1,3,5,6],[2,4],[7,8]]=>10
[[1,2,5,6],[3,4],[7,8]]=>4
[[1,3,4,6],[2,5],[7,8]]=>9
[[1,2,4,6],[3,5],[7,8]]=>8
[[1,2,3,6],[4,5],[7,8]]=>3
[[1,3,4,5],[2,6],[7,8]]=>6
[[1,2,4,5],[3,6],[7,8]]=>5
[[1,2,3,5],[4,6],[7,8]]=>4
[[1,2,3,4],[5,6],[7,8]]=>0
[[1,5,7,8],[2,6],[3],[4]]=>6
[[1,4,7,8],[2,6],[3],[5]]=>10
[[1,3,7,8],[2,6],[4],[5]]=>11
[[1,2,7,8],[3,6],[4],[5]]=>5
[[1,4,7,8],[2,5],[3],[6]]=>7
[[1,3,7,8],[2,5],[4],[6]]=>12
[[1,2,7,8],[3,5],[4],[6]]=>6
[[1,3,7,8],[2,4],[5],[6]]=>8
[[1,2,7,8],[3,4],[5],[6]]=>2
[[1,5,6,8],[2,7],[3],[4]]=>5
[[1,4,6,8],[2,7],[3],[5]]=>9
[[1,3,6,8],[2,7],[4],[5]]=>10
[[1,2,6,8],[3,7],[4],[5]]=>4
[[1,4,5,8],[2,7],[3],[6]]=>8
[[1,3,5,8],[2,7],[4],[6]]=>13
[[1,2,5,8],[3,7],[4],[6]]=>7
[[1,3,4,8],[2,7],[5],[6]]=>9
[[1,2,4,8],[3,7],[5],[6]]=>8
[[1,2,3,8],[4,7],[5],[6]]=>3
[[1,4,6,8],[2,5],[3],[7]]=>9
[[1,3,6,8],[2,5],[4],[7]]=>14
[[1,2,6,8],[3,5],[4],[7]]=>8
[[1,3,6,8],[2,4],[5],[7]]=>10
[[1,2,6,8],[3,4],[5],[7]]=>4
[[1,4,5,8],[2,6],[3],[7]]=>6
[[1,3,5,8],[2,6],[4],[7]]=>11
[[1,2,5,8],[3,6],[4],[7]]=>5
[[1,3,4,8],[2,6],[5],[7]]=>10
[[1,2,4,8],[3,6],[5],[7]]=>9
[[1,2,3,8],[4,6],[5],[7]]=>4
[[1,3,5,8],[2,4],[6],[7]]=>11
[[1,2,5,8],[3,4],[6],[7]]=>5
[[1,3,4,8],[2,5],[6],[7]]=>7
[[1,2,4,8],[3,5],[6],[7]]=>6
[[1,2,3,8],[4,5],[6],[7]]=>1
[[1,5,6,7],[2,8],[3],[4]]=>4
[[1,4,6,7],[2,8],[3],[5]]=>8
[[1,3,6,7],[2,8],[4],[5]]=>9
[[1,2,6,7],[3,8],[4],[5]]=>3
[[1,4,5,7],[2,8],[3],[6]]=>7
[[1,3,5,7],[2,8],[4],[6]]=>12
[[1,2,5,7],[3,8],[4],[6]]=>6
[[1,3,4,7],[2,8],[5],[6]]=>8
[[1,2,4,7],[3,8],[5],[6]]=>7
[[1,2,3,7],[4,8],[5],[6]]=>2
[[1,4,5,6],[2,8],[3],[7]]=>6
[[1,3,5,6],[2,8],[4],[7]]=>11
[[1,2,5,6],[3,8],[4],[7]]=>5
[[1,3,4,6],[2,8],[5],[7]]=>10
[[1,2,4,6],[3,8],[5],[7]]=>9
[[1,2,3,6],[4,8],[5],[7]]=>4
[[1,3,4,5],[2,8],[6],[7]]=>7
[[1,2,4,5],[3,8],[6],[7]]=>6
[[1,2,3,5],[4,8],[6],[7]]=>5
[[1,2,3,4],[5,8],[6],[7]]=>1
[[1,4,6,7],[2,5],[3],[8]]=>8
[[1,3,6,7],[2,5],[4],[8]]=>13
[[1,2,6,7],[3,5],[4],[8]]=>7
[[1,3,6,7],[2,4],[5],[8]]=>9
[[1,2,6,7],[3,4],[5],[8]]=>3
[[1,4,5,7],[2,6],[3],[8]]=>7
[[1,3,5,7],[2,6],[4],[8]]=>12
[[1,2,5,7],[3,6],[4],[8]]=>6
[[1,3,4,7],[2,6],[5],[8]]=>11
[[1,2,4,7],[3,6],[5],[8]]=>10
[[1,2,3,7],[4,6],[5],[8]]=>5
[[1,3,5,7],[2,4],[6],[8]]=>12
[[1,2,5,7],[3,4],[6],[8]]=>6
[[1,3,4,7],[2,5],[6],[8]]=>8
[[1,2,4,7],[3,5],[6],[8]]=>7
[[1,2,3,7],[4,5],[6],[8]]=>2
[[1,4,5,6],[2,7],[3],[8]]=>5
[[1,3,5,6],[2,7],[4],[8]]=>10
[[1,2,5,6],[3,7],[4],[8]]=>4
[[1,3,4,6],[2,7],[5],[8]]=>9
[[1,2,4,6],[3,7],[5],[8]]=>8
[[1,2,3,6],[4,7],[5],[8]]=>3
[[1,3,4,5],[2,7],[6],[8]]=>8
[[1,2,4,5],[3,7],[6],[8]]=>7
[[1,2,3,5],[4,7],[6],[8]]=>6
[[1,2,3,4],[5,7],[6],[8]]=>2
[[1,3,5,6],[2,4],[7],[8]]=>10
[[1,2,5,6],[3,4],[7],[8]]=>4
[[1,3,4,6],[2,5],[7],[8]]=>9
[[1,2,4,6],[3,5],[7],[8]]=>8
[[1,2,3,6],[4,5],[7],[8]]=>3
[[1,3,4,5],[2,6],[7],[8]]=>6
[[1,2,4,5],[3,6],[7],[8]]=>5
[[1,2,3,5],[4,6],[7],[8]]=>4
[[1,2,3,4],[5,6],[7],[8]]=>0
[[1,6,7,8],[2],[3],[4],[5]]=>3
[[1,5,7,8],[2],[3],[4],[6]]=>6
[[1,4,7,8],[2],[3],[5],[6]]=>7
[[1,3,7,8],[2],[4],[5],[6]]=>8
[[1,2,7,8],[3],[4],[5],[6]]=>2
[[1,5,6,8],[2],[3],[4],[7]]=>5
[[1,4,6,8],[2],[3],[5],[7]]=>9
[[1,3,6,8],[2],[4],[5],[7]]=>10
[[1,2,6,8],[3],[4],[5],[7]]=>4
[[1,4,5,8],[2],[3],[6],[7]]=>6
[[1,3,5,8],[2],[4],[6],[7]]=>11
[[1,2,5,8],[3],[4],[6],[7]]=>5
[[1,3,4,8],[2],[5],[6],[7]]=>7
[[1,2,4,8],[3],[5],[6],[7]]=>6
[[1,2,3,8],[4],[5],[6],[7]]=>1
[[1,5,6,7],[2],[3],[4],[8]]=>4
[[1,4,6,7],[2],[3],[5],[8]]=>8
[[1,3,6,7],[2],[4],[5],[8]]=>9
[[1,2,6,7],[3],[4],[5],[8]]=>3
[[1,4,5,7],[2],[3],[6],[8]]=>7
[[1,3,5,7],[2],[4],[6],[8]]=>12
[[1,2,5,7],[3],[4],[6],[8]]=>6
[[1,3,4,7],[2],[5],[6],[8]]=>8
[[1,2,4,7],[3],[5],[6],[8]]=>7
[[1,2,3,7],[4],[5],[6],[8]]=>2
[[1,4,5,6],[2],[3],[7],[8]]=>5
[[1,3,5,6],[2],[4],[7],[8]]=>10
[[1,2,5,6],[3],[4],[7],[8]]=>4
[[1,3,4,6],[2],[5],[7],[8]]=>9
[[1,2,4,6],[3],[5],[7],[8]]=>8
[[1,2,3,6],[4],[5],[7],[8]]=>3
[[1,3,4,5],[2],[6],[7],[8]]=>6
[[1,2,4,5],[3],[6],[7],[8]]=>5
[[1,2,3,5],[4],[6],[7],[8]]=>4
[[1,2,3,4],[5],[6],[7],[8]]=>0
[[1,4,7],[2,5,8],[3,6]]=>7
[[1,3,7],[2,5,8],[4,6]]=>12
[[1,2,7],[3,5,8],[4,6]]=>6
[[1,3,7],[2,4,8],[5,6]]=>8
[[1,2,7],[3,4,8],[5,6]]=>2
[[1,4,6],[2,5,8],[3,7]]=>9
[[1,3,6],[2,5,8],[4,7]]=>14
[[1,2,6],[3,5,8],[4,7]]=>8
[[1,3,6],[2,4,8],[5,7]]=>10
[[1,2,6],[3,4,8],[5,7]]=>4
[[1,4,5],[2,6,8],[3,7]]=>6
[[1,3,5],[2,6,8],[4,7]]=>11
[[1,2,5],[3,6,8],[4,7]]=>5
[[1,3,4],[2,6,8],[5,7]]=>10
[[1,2,4],[3,6,8],[5,7]]=>9
[[1,2,3],[4,6,8],[5,7]]=>4
[[1,3,5],[2,4,8],[6,7]]=>11
[[1,2,5],[3,4,8],[6,7]]=>5
[[1,3,4],[2,5,8],[6,7]]=>7
[[1,2,4],[3,5,8],[6,7]]=>6
[[1,2,3],[4,5,8],[6,7]]=>1
[[1,4,6],[2,5,7],[3,8]]=>8
[[1,3,6],[2,5,7],[4,8]]=>13
[[1,2,6],[3,5,7],[4,8]]=>7
[[1,3,6],[2,4,7],[5,8]]=>9
[[1,2,6],[3,4,7],[5,8]]=>3
[[1,4,5],[2,6,7],[3,8]]=>5
[[1,3,5],[2,6,7],[4,8]]=>10
[[1,2,5],[3,6,7],[4,8]]=>4
[[1,3,4],[2,6,7],[5,8]]=>9
[[1,2,4],[3,6,7],[5,8]]=>8
[[1,2,3],[4,6,7],[5,8]]=>3
[[1,3,5],[2,4,7],[6,8]]=>12
[[1,2,5],[3,4,7],[6,8]]=>6
[[1,3,4],[2,5,7],[6,8]]=>8
[[1,2,4],[3,5,7],[6,8]]=>7
[[1,2,3],[4,5,7],[6,8]]=>2
[[1,3,5],[2,4,6],[7,8]]=>10
[[1,2,5],[3,4,6],[7,8]]=>4
[[1,3,4],[2,5,6],[7,8]]=>6
[[1,2,4],[3,5,6],[7,8]]=>5
[[1,2,3],[4,5,6],[7,8]]=>0
[[1,5,7],[2,6,8],[3],[4]]=>6
[[1,4,7],[2,6,8],[3],[5]]=>10
[[1,3,7],[2,6,8],[4],[5]]=>11
[[1,2,7],[3,6,8],[4],[5]]=>5
[[1,4,7],[2,5,8],[3],[6]]=>7
[[1,3,7],[2,5,8],[4],[6]]=>12
[[1,2,7],[3,5,8],[4],[6]]=>6
[[1,3,7],[2,4,8],[5],[6]]=>8
[[1,2,7],[3,4,8],[5],[6]]=>2
[[1,5,6],[2,7,8],[3],[4]]=>4
[[1,4,6],[2,7,8],[3],[5]]=>8
[[1,3,6],[2,7,8],[4],[5]]=>9
[[1,2,6],[3,7,8],[4],[5]]=>3
[[1,4,5],[2,7,8],[3],[6]]=>7
[[1,3,5],[2,7,8],[4],[6]]=>12
[[1,2,5],[3,7,8],[4],[6]]=>6
[[1,3,4],[2,7,8],[5],[6]]=>8
[[1,2,4],[3,7,8],[5],[6]]=>7
[[1,2,3],[4,7,8],[5],[6]]=>2
[[1,4,6],[2,5,8],[3],[7]]=>9
[[1,3,6],[2,5,8],[4],[7]]=>14
[[1,2,6],[3,5,8],[4],[7]]=>8
[[1,3,6],[2,4,8],[5],[7]]=>10
[[1,2,6],[3,4,8],[5],[7]]=>4
[[1,4,5],[2,6,8],[3],[7]]=>6
[[1,3,5],[2,6,8],[4],[7]]=>11
[[1,2,5],[3,6,8],[4],[7]]=>5
[[1,3,4],[2,6,8],[5],[7]]=>10
[[1,2,4],[3,6,8],[5],[7]]=>9
[[1,2,3],[4,6,8],[5],[7]]=>4
[[1,3,5],[2,4,8],[6],[7]]=>11
[[1,2,5],[3,4,8],[6],[7]]=>5
[[1,3,4],[2,5,8],[6],[7]]=>7
[[1,2,4],[3,5,8],[6],[7]]=>6
[[1,2,3],[4,5,8],[6],[7]]=>1
[[1,4,6],[2,5,7],[3],[8]]=>8
[[1,3,6],[2,5,7],[4],[8]]=>13
[[1,2,6],[3,5,7],[4],[8]]=>7
[[1,3,6],[2,4,7],[5],[8]]=>9
[[1,2,6],[3,4,7],[5],[8]]=>3
[[1,4,5],[2,6,7],[3],[8]]=>5
[[1,3,5],[2,6,7],[4],[8]]=>10
[[1,2,5],[3,6,7],[4],[8]]=>4
[[1,3,4],[2,6,7],[5],[8]]=>9
[[1,2,4],[3,6,7],[5],[8]]=>8
[[1,2,3],[4,6,7],[5],[8]]=>3
[[1,3,5],[2,4,7],[6],[8]]=>12
[[1,2,5],[3,4,7],[6],[8]]=>6
[[1,3,4],[2,5,7],[6],[8]]=>8
[[1,2,4],[3,5,7],[6],[8]]=>7
[[1,2,3],[4,5,7],[6],[8]]=>2
[[1,3,5],[2,4,6],[7],[8]]=>10
[[1,2,5],[3,4,6],[7],[8]]=>4
[[1,3,4],[2,5,6],[7],[8]]=>6
[[1,2,4],[3,5,6],[7],[8]]=>5
[[1,2,3],[4,5,6],[7],[8]]=>0
[[1,5,8],[2,6],[3,7],[4]]=>5
[[1,4,8],[2,6],[3,7],[5]]=>9
[[1,3,8],[2,6],[4,7],[5]]=>10
[[1,2,8],[3,6],[4,7],[5]]=>4
[[1,4,8],[2,5],[3,7],[6]]=>8
[[1,3,8],[2,5],[4,7],[6]]=>13
[[1,2,8],[3,5],[4,7],[6]]=>7
[[1,3,8],[2,4],[5,7],[6]]=>9
[[1,2,8],[3,4],[5,7],[6]]=>3
[[1,4,8],[2,5],[3,6],[7]]=>6
[[1,3,8],[2,5],[4,6],[7]]=>11
[[1,2,8],[3,5],[4,6],[7]]=>5
[[1,3,8],[2,4],[5,6],[7]]=>7
[[1,2,8],[3,4],[5,6],[7]]=>1
[[1,5,7],[2,6],[3,8],[4]]=>6
[[1,4,7],[2,6],[3,8],[5]]=>10
[[1,3,7],[2,6],[4,8],[5]]=>11
[[1,2,7],[3,6],[4,8],[5]]=>5
[[1,4,7],[2,5],[3,8],[6]]=>7
[[1,3,7],[2,5],[4,8],[6]]=>12
[[1,2,7],[3,5],[4,8],[6]]=>6
[[1,3,7],[2,4],[5,8],[6]]=>8
[[1,2,7],[3,4],[5,8],[6]]=>2
[[1,5,6],[2,7],[3,8],[4]]=>4
[[1,4,6],[2,7],[3,8],[5]]=>8
[[1,3,6],[2,7],[4,8],[5]]=>9
[[1,2,6],[3,7],[4,8],[5]]=>3
[[1,4,5],[2,7],[3,8],[6]]=>7
[[1,3,5],[2,7],[4,8],[6]]=>12
[[1,2,5],[3,7],[4,8],[6]]=>6
[[1,3,4],[2,7],[5,8],[6]]=>8
[[1,2,4],[3,7],[5,8],[6]]=>7
[[1,2,3],[4,7],[5,8],[6]]=>2
[[1,4,6],[2,5],[3,8],[7]]=>9
[[1,3,6],[2,5],[4,8],[7]]=>14
[[1,2,6],[3,5],[4,8],[7]]=>8
[[1,3,6],[2,4],[5,8],[7]]=>10
[[1,2,6],[3,4],[5,8],[7]]=>4
[[1,4,5],[2,6],[3,8],[7]]=>6
[[1,3,5],[2,6],[4,8],[7]]=>11
[[1,2,5],[3,6],[4,8],[7]]=>5
[[1,3,4],[2,6],[5,8],[7]]=>10
[[1,2,4],[3,6],[5,8],[7]]=>9
[[1,2,3],[4,6],[5,8],[7]]=>4
[[1,3,5],[2,4],[6,8],[7]]=>11
[[1,2,5],[3,4],[6,8],[7]]=>5
[[1,3,4],[2,5],[6,8],[7]]=>7
[[1,2,4],[3,5],[6,8],[7]]=>6
[[1,2,3],[4,5],[6,8],[7]]=>1
[[1,4,7],[2,5],[3,6],[8]]=>7
[[1,3,7],[2,5],[4,6],[8]]=>12
[[1,2,7],[3,5],[4,6],[8]]=>6
[[1,3,7],[2,4],[5,6],[8]]=>8
[[1,2,7],[3,4],[5,6],[8]]=>2
[[1,4,6],[2,5],[3,7],[8]]=>8
[[1,3,6],[2,5],[4,7],[8]]=>13
[[1,2,6],[3,5],[4,7],[8]]=>7
[[1,3,6],[2,4],[5,7],[8]]=>9
[[1,2,6],[3,4],[5,7],[8]]=>3
[[1,4,5],[2,6],[3,7],[8]]=>5
[[1,3,5],[2,6],[4,7],[8]]=>10
[[1,2,5],[3,6],[4,7],[8]]=>4
[[1,3,4],[2,6],[5,7],[8]]=>9
[[1,2,4],[3,6],[5,7],[8]]=>8
[[1,2,3],[4,6],[5,7],[8]]=>3
[[1,3,5],[2,4],[6,7],[8]]=>10
[[1,2,5],[3,4],[6,7],[8]]=>4
[[1,3,4],[2,5],[6,7],[8]]=>6
[[1,2,4],[3,5],[6,7],[8]]=>5
[[1,2,3],[4,5],[6,7],[8]]=>0
[[1,6,8],[2,7],[3],[4],[5]]=>4
[[1,5,8],[2,7],[3],[4],[6]]=>7
[[1,4,8],[2,7],[3],[5],[6]]=>8
[[1,3,8],[2,7],[4],[5],[6]]=>9
[[1,2,8],[3,7],[4],[5],[6]]=>3
[[1,5,8],[2,6],[3],[4],[7]]=>5
[[1,4,8],[2,6],[3],[5],[7]]=>9
[[1,3,8],[2,6],[4],[5],[7]]=>10
[[1,2,8],[3,6],[4],[5],[7]]=>4
[[1,4,8],[2,5],[3],[6],[7]]=>6
[[1,3,8],[2,5],[4],[6],[7]]=>11
[[1,2,8],[3,5],[4],[6],[7]]=>5
[[1,3,8],[2,4],[5],[6],[7]]=>7
[[1,2,8],[3,4],[5],[6],[7]]=>1
[[1,6,7],[2,8],[3],[4],[5]]=>3
[[1,5,7],[2,8],[3],[4],[6]]=>6
[[1,4,7],[2,8],[3],[5],[6]]=>7
[[1,3,7],[2,8],[4],[5],[6]]=>8
[[1,2,7],[3,8],[4],[5],[6]]=>2
[[1,5,6],[2,8],[3],[4],[7]]=>5
[[1,4,6],[2,8],[3],[5],[7]]=>9
[[1,3,6],[2,8],[4],[5],[7]]=>10
[[1,2,6],[3,8],[4],[5],[7]]=>4
[[1,4,5],[2,8],[3],[6],[7]]=>6
[[1,3,5],[2,8],[4],[6],[7]]=>11
[[1,2,5],[3,8],[4],[6],[7]]=>5
[[1,3,4],[2,8],[5],[6],[7]]=>7
[[1,2,4],[3,8],[5],[6],[7]]=>6
[[1,2,3],[4,8],[5],[6],[7]]=>1
[[1,5,7],[2,6],[3],[4],[8]]=>6
[[1,4,7],[2,6],[3],[5],[8]]=>10
[[1,3,7],[2,6],[4],[5],[8]]=>11
[[1,2,7],[3,6],[4],[5],[8]]=>5
[[1,4,7],[2,5],[3],[6],[8]]=>7
[[1,3,7],[2,5],[4],[6],[8]]=>12
[[1,2,7],[3,5],[4],[6],[8]]=>6
[[1,3,7],[2,4],[5],[6],[8]]=>8
[[1,2,7],[3,4],[5],[6],[8]]=>2
[[1,5,6],[2,7],[3],[4],[8]]=>4
[[1,4,6],[2,7],[3],[5],[8]]=>8
[[1,3,6],[2,7],[4],[5],[8]]=>9
[[1,2,6],[3,7],[4],[5],[8]]=>3
[[1,4,5],[2,7],[3],[6],[8]]=>7
[[1,3,5],[2,7],[4],[6],[8]]=>12
[[1,2,5],[3,7],[4],[6],[8]]=>6
[[1,3,4],[2,7],[5],[6],[8]]=>8
[[1,2,4],[3,7],[5],[6],[8]]=>7
[[1,2,3],[4,7],[5],[6],[8]]=>2
[[1,4,6],[2,5],[3],[7],[8]]=>8
[[1,3,6],[2,5],[4],[7],[8]]=>13
[[1,2,6],[3,5],[4],[7],[8]]=>7
[[1,3,6],[2,4],[5],[7],[8]]=>9
[[1,2,6],[3,4],[5],[7],[8]]=>3
[[1,4,5],[2,6],[3],[7],[8]]=>5
[[1,3,5],[2,6],[4],[7],[8]]=>10
[[1,2,5],[3,6],[4],[7],[8]]=>4
[[1,3,4],[2,6],[5],[7],[8]]=>9
[[1,2,4],[3,6],[5],[7],[8]]=>8
[[1,2,3],[4,6],[5],[7],[8]]=>3
[[1,3,5],[2,4],[6],[7],[8]]=>10
[[1,2,5],[3,4],[6],[7],[8]]=>4
[[1,3,4],[2,5],[6],[7],[8]]=>6
[[1,2,4],[3,5],[6],[7],[8]]=>5
[[1,2,3],[4,5],[6],[7],[8]]=>0
[[1,7,8],[2],[3],[4],[5],[6]]=>2
[[1,6,8],[2],[3],[4],[5],[7]]=>4
[[1,5,8],[2],[3],[4],[6],[7]]=>5
[[1,4,8],[2],[3],[5],[6],[7]]=>6
[[1,3,8],[2],[4],[5],[6],[7]]=>7
[[1,2,8],[3],[4],[5],[6],[7]]=>1
[[1,6,7],[2],[3],[4],[5],[8]]=>3
[[1,5,7],[2],[3],[4],[6],[8]]=>6
[[1,4,7],[2],[3],[5],[6],[8]]=>7
[[1,3,7],[2],[4],[5],[6],[8]]=>8
[[1,2,7],[3],[4],[5],[6],[8]]=>2
[[1,5,6],[2],[3],[4],[7],[8]]=>4
[[1,4,6],[2],[3],[5],[7],[8]]=>8
[[1,3,6],[2],[4],[5],[7],[8]]=>9
[[1,2,6],[3],[4],[5],[7],[8]]=>3
[[1,4,5],[2],[3],[6],[7],[8]]=>5
[[1,3,5],[2],[4],[6],[7],[8]]=>10
[[1,2,5],[3],[4],[6],[7],[8]]=>4
[[1,3,4],[2],[5],[6],[7],[8]]=>6
[[1,2,4],[3],[5],[6],[7],[8]]=>5
[[1,2,3],[4],[5],[6],[7],[8]]=>0
[[1,5],[2,6],[3,7],[4,8]]=>4
[[1,4],[2,6],[3,7],[5,8]]=>8
[[1,3],[2,6],[4,7],[5,8]]=>9
[[1,2],[3,6],[4,7],[5,8]]=>3
[[1,4],[2,5],[3,7],[6,8]]=>7
[[1,3],[2,5],[4,7],[6,8]]=>12
[[1,2],[3,5],[4,7],[6,8]]=>6
[[1,3],[2,4],[5,7],[6,8]]=>8
[[1,2],[3,4],[5,7],[6,8]]=>2
[[1,4],[2,5],[3,6],[7,8]]=>5
[[1,3],[2,5],[4,6],[7,8]]=>10
[[1,2],[3,5],[4,6],[7,8]]=>4
[[1,3],[2,4],[5,6],[7,8]]=>6
[[1,2],[3,4],[5,6],[7,8]]=>0
[[1,6],[2,7],[3,8],[4],[5]]=>3
[[1,5],[2,7],[3,8],[4],[6]]=>6
[[1,4],[2,7],[3,8],[5],[6]]=>7
[[1,3],[2,7],[4,8],[5],[6]]=>8
[[1,2],[3,7],[4,8],[5],[6]]=>2
[[1,5],[2,6],[3,8],[4],[7]]=>5
[[1,4],[2,6],[3,8],[5],[7]]=>9
[[1,3],[2,6],[4,8],[5],[7]]=>10
[[1,2],[3,6],[4,8],[5],[7]]=>4
[[1,4],[2,5],[3,8],[6],[7]]=>6
[[1,3],[2,5],[4,8],[6],[7]]=>11
[[1,2],[3,5],[4,8],[6],[7]]=>5
[[1,3],[2,4],[5,8],[6],[7]]=>7
[[1,2],[3,4],[5,8],[6],[7]]=>1
[[1,5],[2,6],[3,7],[4],[8]]=>4
[[1,4],[2,6],[3,7],[5],[8]]=>8
[[1,3],[2,6],[4,7],[5],[8]]=>9
[[1,2],[3,6],[4,7],[5],[8]]=>3
[[1,4],[2,5],[3,7],[6],[8]]=>7
[[1,3],[2,5],[4,7],[6],[8]]=>12
[[1,2],[3,5],[4,7],[6],[8]]=>6
[[1,3],[2,4],[5,7],[6],[8]]=>8
[[1,2],[3,4],[5,7],[6],[8]]=>2
[[1,4],[2,5],[3,6],[7],[8]]=>5
[[1,3],[2,5],[4,6],[7],[8]]=>10
[[1,2],[3,5],[4,6],[7],[8]]=>4
[[1,3],[2,4],[5,6],[7],[8]]=>6
[[1,2],[3,4],[5,6],[7],[8]]=>0
[[1,7],[2,8],[3],[4],[5],[6]]=>2
[[1,6],[2,8],[3],[4],[5],[7]]=>4
[[1,5],[2,8],[3],[4],[6],[7]]=>5
[[1,4],[2,8],[3],[5],[6],[7]]=>6
[[1,3],[2,8],[4],[5],[6],[7]]=>7
[[1,2],[3,8],[4],[5],[6],[7]]=>1
[[1,6],[2,7],[3],[4],[5],[8]]=>3
[[1,5],[2,7],[3],[4],[6],[8]]=>6
[[1,4],[2,7],[3],[5],[6],[8]]=>7
[[1,3],[2,7],[4],[5],[6],[8]]=>8
[[1,2],[3,7],[4],[5],[6],[8]]=>2
[[1,5],[2,6],[3],[4],[7],[8]]=>4
[[1,4],[2,6],[3],[5],[7],[8]]=>8
[[1,3],[2,6],[4],[5],[7],[8]]=>9
[[1,2],[3,6],[4],[5],[7],[8]]=>3
[[1,4],[2,5],[3],[6],[7],[8]]=>5
[[1,3],[2,5],[4],[6],[7],[8]]=>10
[[1,2],[3,5],[4],[6],[7],[8]]=>4
[[1,3],[2,4],[5],[6],[7],[8]]=>6
[[1,2],[3,4],[5],[6],[7],[8]]=>0
[[1,8],[2],[3],[4],[5],[6],[7]]=>1
[[1,7],[2],[3],[4],[5],[6],[8]]=>2
[[1,6],[2],[3],[4],[5],[7],[8]]=>3
[[1,5],[2],[3],[4],[6],[7],[8]]=>4
[[1,4],[2],[3],[5],[6],[7],[8]]=>5
[[1,3],[2],[4],[5],[6],[7],[8]]=>6
[[1,2],[3],[4],[5],[6],[7],[8]]=>0
[[1],[2],[3],[4],[5],[6],[7],[8]]=>0
                    

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Description

The natural comajor index of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
The natural comajor index of a tableau of size $n$ with natural descent set $D$ is then $\sum_{d\in D} n-d$.

References

[1] Hopkins, S. Two majs for standard Young tableaux? MathOverflow:385374

Code

def natural_descents(T):
    n = T.size()
    D = []
    for i in range(1, n):
        for row in T:
            if row.count(i):
                break
            if row.count(i+1):
                D.append(i)
                break
    return D

def statistic(T):
    n = T.size()
    D = natural_descents(T)
    return sum(n-d for d in D)

Created

Mar 04, 2021 at 08:46 by Martin Rubey

Updated

Mar 04, 2021 at 08:46 by Martin Rubey