- FindStat

edit this statistic or download as text // json

Identifier

St001993: Standard tableaux ⟶ ℤ

Values

=>
                            Cc0007;cc-rep
                            
                            
                            

                        [[1]]=>1
[[1,2]]=>1
[[1],[2]]=>1
[[1,2,3]]=>1
[[1,3],[2]]=>1
[[1,2],[3]]=>2
[[1],[2],[3]]=>1
[[1,2,3,4]]=>1
[[1,3,4],[2]]=>2
[[1,2,4],[3]]=>2
[[1,2,3],[4]]=>3
[[1,3],[2,4]]=>1
[[1,2],[3,4]]=>2
[[1,4],[2],[3]]=>1
[[1,3],[2],[4]]=>2
[[1,2],[3],[4]]=>2
[[1],[2],[3],[4]]=>1
[[1,2,3,4,5]]=>1
[[1,3,4,5],[2]]=>3
[[1,2,4,5],[3]]=>3
[[1,2,3,5],[4]]=>3
[[1,2,3,4],[5]]=>4
[[1,3,5],[2,4]]=>1
[[1,2,5],[3,4]]=>2
[[1,3,4],[2,5]]=>2
[[1,2,4],[3,5]]=>2
[[1,2,3],[4,5]]=>3
[[1,4,5],[2],[3]]=>2
[[1,3,5],[2],[4]]=>2
[[1,2,5],[3],[4]]=>2
[[1,3,4],[2],[5]]=>3
[[1,2,4],[3],[5]]=>3
[[1,2,3],[4],[5]]=>3
[[1,4],[2,5],[3]]=>1
[[1,3],[2,5],[4]]=>2
[[1,2],[3,5],[4]]=>2
[[1,3],[2,4],[5]]=>2
[[1,2],[3,4],[5]]=>2
[[1,5],[2],[3],[4]]=>1
[[1,4],[2],[3],[5]]=>2
[[1,3],[2],[4],[5]]=>2
[[1,2],[3],[4],[5]]=>2
[[1],[2],[3],[4],[5]]=>1
[[1,2,3,4,5,6]]=>1
[[1,3,4,5,6],[2]]=>4
[[1,2,4,5,6],[3]]=>4
[[1,2,3,5,6],[4]]=>4
[[1,2,3,4,6],[5]]=>4
[[1,2,3,4,5],[6]]=>5
[[1,3,5,6],[2,4]]=>2
[[1,2,5,6],[3,4]]=>2
[[1,3,4,6],[2,5]]=>2
[[1,2,4,6],[3,5]]=>2
[[1,2,3,6],[4,5]]=>3
[[1,3,4,5],[2,6]]=>3
[[1,2,4,5],[3,6]]=>3
[[1,2,3,5],[4,6]]=>3
[[1,2,3,4],[5,6]]=>4
[[1,4,5,6],[2],[3]]=>3
[[1,3,5,6],[2],[4]]=>3
[[1,2,5,6],[3],[4]]=>3
[[1,3,4,6],[2],[5]]=>3
[[1,2,4,6],[3],[5]]=>3
[[1,2,3,6],[4],[5]]=>3
[[1,3,4,5],[2],[6]]=>4
[[1,2,4,5],[3],[6]]=>4
[[1,2,3,5],[4],[6]]=>4
[[1,2,3,4],[5],[6]]=>4
[[1,3,5],[2,4,6]]=>1
[[1,2,5],[3,4,6]]=>2
[[1,3,4],[2,5,6]]=>2
[[1,2,4],[3,5,6]]=>2
[[1,2,3],[4,5,6]]=>3
[[1,4,6],[2,5],[3]]=>2
[[1,3,6],[2,5],[4]]=>2
[[1,2,6],[3,5],[4]]=>2
[[1,3,6],[2,4],[5]]=>2
[[1,2,6],[3,4],[5]]=>2
[[1,4,5],[2,6],[3]]=>2
[[1,3,5],[2,6],[4]]=>2
[[1,2,5],[3,6],[4]]=>2
[[1,3,4],[2,6],[5]]=>3
[[1,2,4],[3,6],[5]]=>3
[[1,2,3],[4,6],[5]]=>3
[[1,3,5],[2,4],[6]]=>3
[[1,2,5],[3,4],[6]]=>3
[[1,3,4],[2,5],[6]]=>3
[[1,2,4],[3,5],[6]]=>3
[[1,2,3],[4,5],[6]]=>3
[[1,5,6],[2],[3],[4]]=>2
[[1,4,6],[2],[3],[5]]=>2
[[1,3,6],[2],[4],[5]]=>2
[[1,2,6],[3],[4],[5]]=>2
[[1,4,5],[2],[3],[6]]=>3
[[1,3,5],[2],[4],[6]]=>3
[[1,2,5],[3],[4],[6]]=>3
[[1,3,4],[2],[5],[6]]=>3
[[1,2,4],[3],[5],[6]]=>3
[[1,2,3],[4],[5],[6]]=>3
[[1,4],[2,5],[3,6]]=>1
[[1,3],[2,5],[4,6]]=>2
[[1,2],[3,5],[4,6]]=>2
[[1,3],[2,4],[5,6]]=>2
[[1,2],[3,4],[5,6]]=>2
[[1,5],[2,6],[3],[4]]=>1
[[1,4],[2,6],[3],[5]]=>2
[[1,3],[2,6],[4],[5]]=>2
[[1,2],[3,6],[4],[5]]=>2
[[1,4],[2,5],[3],[6]]=>2
[[1,3],[2,5],[4],[6]]=>2
[[1,2],[3,5],[4],[6]]=>2
[[1,3],[2,4],[5],[6]]=>2
[[1,2],[3,4],[5],[6]]=>2
[[1,6],[2],[3],[4],[5]]=>1
[[1,5],[2],[3],[4],[6]]=>2
[[1,4],[2],[3],[5],[6]]=>2
[[1,3],[2],[4],[5],[6]]=>2
[[1,2],[3],[4],[5],[6]]=>2
[[1],[2],[3],[4],[5],[6]]=>1
[[1,2,3,4,5,6,7]]=>1
[[1,3,4,5,6,7],[2]]=>5
[[1,2,4,5,6,7],[3]]=>5
[[1,2,3,5,6,7],[4]]=>5
[[1,2,3,4,6,7],[5]]=>5
[[1,2,3,4,5,7],[6]]=>5
[[1,2,3,4,5,6],[7]]=>6
[[1,3,5,6,7],[2,4]]=>3
[[1,2,5,6,7],[3,4]]=>3
[[1,3,4,6,7],[2,5]]=>3
[[1,2,4,6,7],[3,5]]=>3
[[1,2,3,6,7],[4,5]]=>3
[[1,3,4,5,7],[2,6]]=>3
[[1,2,4,5,7],[3,6]]=>3
[[1,2,3,5,7],[4,6]]=>3
[[1,2,3,4,7],[5,6]]=>4
[[1,3,4,5,6],[2,7]]=>4
[[1,2,4,5,6],[3,7]]=>4
[[1,2,3,5,6],[4,7]]=>4
[[1,2,3,4,6],[5,7]]=>4
[[1,2,3,4,5],[6,7]]=>5
[[1,4,5,6,7],[2],[3]]=>4
[[1,3,5,6,7],[2],[4]]=>4
[[1,2,5,6,7],[3],[4]]=>4
[[1,3,4,6,7],[2],[5]]=>4
[[1,2,4,6,7],[3],[5]]=>4
[[1,2,3,6,7],[4],[5]]=>4
[[1,3,4,5,7],[2],[6]]=>4
[[1,2,4,5,7],[3],[6]]=>4
[[1,2,3,5,7],[4],[6]]=>4
[[1,2,3,4,7],[5],[6]]=>4
[[1,3,4,5,6],[2],[7]]=>5
[[1,2,4,5,6],[3],[7]]=>5
[[1,2,3,5,6],[4],[7]]=>5
[[1,2,3,4,6],[5],[7]]=>5
[[1,2,3,4,5],[6],[7]]=>5
[[1,3,5,7],[2,4,6]]=>1
[[1,2,5,7],[3,4,6]]=>2
[[1,3,4,7],[2,5,6]]=>2
[[1,2,4,7],[3,5,6]]=>2
[[1,2,3,7],[4,5,6]]=>3
[[1,3,5,6],[2,4,7]]=>2
[[1,2,5,6],[3,4,7]]=>2
[[1,3,4,6],[2,5,7]]=>2
[[1,2,4,6],[3,5,7]]=>2
[[1,2,3,6],[4,5,7]]=>3
[[1,3,4,5],[2,6,7]]=>3
[[1,2,4,5],[3,6,7]]=>3
[[1,2,3,5],[4,6,7]]=>3
[[1,2,3,4],[5,6,7]]=>4
[[1,4,6,7],[2,5],[3]]=>3
[[1,3,6,7],[2,5],[4]]=>3
[[1,2,6,7],[3,5],[4]]=>3
[[1,3,6,7],[2,4],[5]]=>3
[[1,2,6,7],[3,4],[5]]=>3
[[1,4,5,7],[2,6],[3]]=>3
[[1,3,5,7],[2,6],[4]]=>3
[[1,2,5,7],[3,6],[4]]=>3
[[1,3,4,7],[2,6],[5]]=>3
[[1,2,4,7],[3,6],[5]]=>3
[[1,2,3,7],[4,6],[5]]=>3
[[1,3,5,7],[2,4],[6]]=>3
[[1,2,5,7],[3,4],[6]]=>3
[[1,3,4,7],[2,5],[6]]=>3
[[1,2,4,7],[3,5],[6]]=>3
[[1,2,3,7],[4,5],[6]]=>3
[[1,4,5,6],[2,7],[3]]=>3
[[1,3,5,6],[2,7],[4]]=>3
[[1,2,5,6],[3,7],[4]]=>3
[[1,3,4,6],[2,7],[5]]=>3
[[1,2,4,6],[3,7],[5]]=>3
[[1,2,3,6],[4,7],[5]]=>3
[[1,3,4,5],[2,7],[6]]=>4
[[1,2,4,5],[3,7],[6]]=>4
[[1,2,3,5],[4,7],[6]]=>4
[[1,2,3,4],[5,7],[6]]=>4
[[1,3,5,6],[2,4],[7]]=>4
[[1,2,5,6],[3,4],[7]]=>4
[[1,3,4,6],[2,5],[7]]=>4
[[1,2,4,6],[3,5],[7]]=>4
[[1,2,3,6],[4,5],[7]]=>4
[[1,3,4,5],[2,6],[7]]=>4
[[1,2,4,5],[3,6],[7]]=>4
[[1,2,3,5],[4,6],[7]]=>4
[[1,2,3,4],[5,6],[7]]=>4
[[1,5,6,7],[2],[3],[4]]=>3
[[1,4,6,7],[2],[3],[5]]=>3
[[1,3,6,7],[2],[4],[5]]=>3
[[1,2,6,7],[3],[4],[5]]=>3
[[1,4,5,7],[2],[3],[6]]=>3
[[1,3,5,7],[2],[4],[6]]=>3
[[1,2,5,7],[3],[4],[6]]=>3
[[1,3,4,7],[2],[5],[6]]=>3
[[1,2,4,7],[3],[5],[6]]=>3
[[1,2,3,7],[4],[5],[6]]=>3
[[1,4,5,6],[2],[3],[7]]=>4
[[1,3,5,6],[2],[4],[7]]=>4
[[1,2,5,6],[3],[4],[7]]=>4
[[1,3,4,6],[2],[5],[7]]=>4
[[1,2,4,6],[3],[5],[7]]=>4
[[1,2,3,6],[4],[5],[7]]=>4
[[1,3,4,5],[2],[6],[7]]=>4
[[1,2,4,5],[3],[6],[7]]=>4
[[1,2,3,5],[4],[6],[7]]=>4
[[1,2,3,4],[5],[6],[7]]=>4
[[1,4,6],[2,5,7],[3]]=>2
[[1,3,6],[2,5,7],[4]]=>2
[[1,2,6],[3,5,7],[4]]=>2
[[1,3,6],[2,4,7],[5]]=>2
[[1,2,6],[3,4,7],[5]]=>2
[[1,4,5],[2,6,7],[3]]=>2
[[1,3,5],[2,6,7],[4]]=>2
[[1,2,5],[3,6,7],[4]]=>2
[[1,3,4],[2,6,7],[5]]=>3
[[1,2,4],[3,6,7],[5]]=>3
[[1,2,3],[4,6,7],[5]]=>3
[[1,3,5],[2,4,7],[6]]=>3
[[1,2,5],[3,4,7],[6]]=>3
[[1,3,4],[2,5,7],[6]]=>3
[[1,2,4],[3,5,7],[6]]=>3
[[1,2,3],[4,5,7],[6]]=>3
[[1,3,5],[2,4,6],[7]]=>3
[[1,2,5],[3,4,6],[7]]=>3
[[1,3,4],[2,5,6],[7]]=>3
[[1,2,4],[3,5,6],[7]]=>3
[[1,2,3],[4,5,6],[7]]=>3
[[1,4,7],[2,5],[3,6]]=>1
[[1,3,7],[2,5],[4,6]]=>2
[[1,2,7],[3,5],[4,6]]=>2
[[1,3,7],[2,4],[5,6]]=>2
[[1,2,7],[3,4],[5,6]]=>2
[[1,4,6],[2,5],[3,7]]=>2
[[1,3,6],[2,5],[4,7]]=>2
[[1,2,6],[3,5],[4,7]]=>2
[[1,3,6],[2,4],[5,7]]=>2
[[1,2,6],[3,4],[5,7]]=>2
[[1,4,5],[2,6],[3,7]]=>2
[[1,3,5],[2,6],[4,7]]=>2
[[1,2,5],[3,6],[4,7]]=>2
[[1,3,4],[2,6],[5,7]]=>3
[[1,2,4],[3,6],[5,7]]=>3
[[1,2,3],[4,6],[5,7]]=>3
[[1,3,5],[2,4],[6,7]]=>3
[[1,2,5],[3,4],[6,7]]=>3
[[1,3,4],[2,5],[6,7]]=>3
[[1,2,4],[3,5],[6,7]]=>3
[[1,2,3],[4,5],[6,7]]=>3
[[1,5,7],[2,6],[3],[4]]=>2
[[1,4,7],[2,6],[3],[5]]=>2
[[1,3,7],[2,6],[4],[5]]=>2
[[1,2,7],[3,6],[4],[5]]=>2
[[1,4,7],[2,5],[3],[6]]=>2
[[1,3,7],[2,5],[4],[6]]=>2
[[1,2,7],[3,5],[4],[6]]=>2
[[1,3,7],[2,4],[5],[6]]=>2
[[1,2,7],[3,4],[5],[6]]=>2
[[1,5,6],[2,7],[3],[4]]=>2
[[1,4,6],[2,7],[3],[5]]=>2
[[1,3,6],[2,7],[4],[5]]=>2
[[1,2,6],[3,7],[4],[5]]=>2
[[1,4,5],[2,7],[3],[6]]=>3
[[1,3,5],[2,7],[4],[6]]=>3
[[1,2,5],[3,7],[4],[6]]=>3
[[1,3,4],[2,7],[5],[6]]=>3
[[1,2,4],[3,7],[5],[6]]=>3
[[1,2,3],[4,7],[5],[6]]=>3
[[1,4,6],[2,5],[3],[7]]=>3
[[1,3,6],[2,5],[4],[7]]=>3
[[1,2,6],[3,5],[4],[7]]=>3
[[1,3,6],[2,4],[5],[7]]=>3
[[1,2,6],[3,4],[5],[7]]=>3
[[1,4,5],[2,6],[3],[7]]=>3
[[1,3,5],[2,6],[4],[7]]=>3
[[1,2,5],[3,6],[4],[7]]=>3
[[1,3,4],[2,6],[5],[7]]=>3
[[1,2,4],[3,6],[5],[7]]=>3
[[1,2,3],[4,6],[5],[7]]=>3
[[1,3,5],[2,4],[6],[7]]=>3
[[1,2,5],[3,4],[6],[7]]=>3
[[1,3,4],[2,5],[6],[7]]=>3
[[1,2,4],[3,5],[6],[7]]=>3
[[1,2,3],[4,5],[6],[7]]=>3
[[1,6,7],[2],[3],[4],[5]]=>2
[[1,5,7],[2],[3],[4],[6]]=>2
[[1,4,7],[2],[3],[5],[6]]=>2
[[1,3,7],[2],[4],[5],[6]]=>2
[[1,2,7],[3],[4],[5],[6]]=>2
[[1,5,6],[2],[3],[4],[7]]=>3
[[1,4,6],[2],[3],[5],[7]]=>3
[[1,3,6],[2],[4],[5],[7]]=>3
[[1,2,6],[3],[4],[5],[7]]=>3
[[1,4,5],[2],[3],[6],[7]]=>3
[[1,3,5],[2],[4],[6],[7]]=>3
[[1,2,5],[3],[4],[6],[7]]=>3
[[1,3,4],[2],[5],[6],[7]]=>3
[[1,2,4],[3],[5],[6],[7]]=>3
[[1,2,3],[4],[5],[6],[7]]=>3
[[1,5],[2,6],[3,7],[4]]=>1
[[1,4],[2,6],[3,7],[5]]=>2
[[1,3],[2,6],[4,7],[5]]=>2
[[1,2],[3,6],[4,7],[5]]=>2
[[1,4],[2,5],[3,7],[6]]=>2
[[1,3],[2,5],[4,7],[6]]=>2
[[1,2],[3,5],[4,7],[6]]=>2
[[1,3],[2,4],[5,7],[6]]=>2
[[1,2],[3,4],[5,7],[6]]=>2
[[1,4],[2,5],[3,6],[7]]=>2
[[1,3],[2,5],[4,6],[7]]=>2
[[1,2],[3,5],[4,6],[7]]=>2
[[1,3],[2,4],[5,6],[7]]=>2
[[1,2],[3,4],[5,6],[7]]=>2
[[1,6],[2,7],[3],[4],[5]]=>1
[[1,5],[2,7],[3],[4],[6]]=>2
[[1,4],[2,7],[3],[5],[6]]=>2
[[1,3],[2,7],[4],[5],[6]]=>2
[[1,2],[3,7],[4],[5],[6]]=>2
[[1,5],[2,6],[3],[4],[7]]=>2
[[1,4],[2,6],[3],[5],[7]]=>2
[[1,3],[2,6],[4],[5],[7]]=>2
[[1,2],[3,6],[4],[5],[7]]=>2
[[1,4],[2,5],[3],[6],[7]]=>2
[[1,3],[2,5],[4],[6],[7]]=>2
[[1,2],[3,5],[4],[6],[7]]=>2
[[1,3],[2,4],[5],[6],[7]]=>2
[[1,2],[3,4],[5],[6],[7]]=>2
[[1,7],[2],[3],[4],[5],[6]]=>1
[[1,6],[2],[3],[4],[5],[7]]=>2
[[1,5],[2],[3],[4],[6],[7]]=>2
[[1,4],[2],[3],[5],[6],[7]]=>2
[[1,3],[2],[4],[5],[6],[7]]=>2
[[1,2],[3],[4],[5],[6],[7]]=>2
[[1],[2],[3],[4],[5],[6],[7]]=>1
[[1,2,3,4,5,6,7,8]]=>1
[[1,3,4,5,6,7,8],[2]]=>6
[[1,2,4,5,6,7,8],[3]]=>6
[[1,2,3,5,6,7,8],[4]]=>6
[[1,2,3,4,6,7,8],[5]]=>6
[[1,2,3,4,5,7,8],[6]]=>6
[[1,2,3,4,5,6,8],[7]]=>6
[[1,2,3,4,5,6,7],[8]]=>7
[[1,3,5,6,7,8],[2,4]]=>4
[[1,2,5,6,7,8],[3,4]]=>4
[[1,3,4,6,7,8],[2,5]]=>4
[[1,2,4,6,7,8],[3,5]]=>4
[[1,2,3,6,7,8],[4,5]]=>4
[[1,3,4,5,7,8],[2,6]]=>4
[[1,2,4,5,7,8],[3,6]]=>4
[[1,2,3,5,7,8],[4,6]]=>4
[[1,2,3,4,7,8],[5,6]]=>4
[[1,3,4,5,6,8],[2,7]]=>4
[[1,2,4,5,6,8],[3,7]]=>4
[[1,2,3,5,6,8],[4,7]]=>4
[[1,2,3,4,6,8],[5,7]]=>4
[[1,2,3,4,5,8],[6,7]]=>5
[[1,3,4,5,6,7],[2,8]]=>5
[[1,2,4,5,6,7],[3,8]]=>5
[[1,2,3,5,6,7],[4,8]]=>5
[[1,2,3,4,6,7],[5,8]]=>5
[[1,2,3,4,5,7],[6,8]]=>5
[[1,2,3,4,5,6],[7,8]]=>6
[[1,4,5,6,7,8],[2],[3]]=>5
[[1,3,5,6,7,8],[2],[4]]=>5
[[1,2,5,6,7,8],[3],[4]]=>5
[[1,3,4,6,7,8],[2],[5]]=>5
[[1,2,4,6,7,8],[3],[5]]=>5
[[1,2,3,6,7,8],[4],[5]]=>5
[[1,3,4,5,7,8],[2],[6]]=>5
[[1,2,4,5,7,8],[3],[6]]=>5
[[1,2,3,5,7,8],[4],[6]]=>5
[[1,2,3,4,7,8],[5],[6]]=>5
[[1,3,4,5,6,8],[2],[7]]=>5
[[1,2,4,5,6,8],[3],[7]]=>5
[[1,2,3,5,6,8],[4],[7]]=>5
[[1,2,3,4,6,8],[5],[7]]=>5
[[1,2,3,4,5,8],[6],[7]]=>5
[[1,3,4,5,6,7],[2],[8]]=>6
[[1,2,4,5,6,7],[3],[8]]=>6
[[1,2,3,5,6,7],[4],[8]]=>6
[[1,2,3,4,6,7],[5],[8]]=>6
[[1,2,3,4,5,7],[6],[8]]=>6
[[1,2,3,4,5,6],[7],[8]]=>6
[[1,3,5,7,8],[2,4,6]]=>2
[[1,2,5,7,8],[3,4,6]]=>2
[[1,3,4,7,8],[2,5,6]]=>2
[[1,2,4,7,8],[3,5,6]]=>2
[[1,2,3,7,8],[4,5,6]]=>3
[[1,3,5,6,8],[2,4,7]]=>2
[[1,2,5,6,8],[3,4,7]]=>2
[[1,3,4,6,8],[2,5,7]]=>2
[[1,2,4,6,8],[3,5,7]]=>2
[[1,2,3,6,8],[4,5,7]]=>3
[[1,3,4,5,8],[2,6,7]]=>3
[[1,2,4,5,8],[3,6,7]]=>3
[[1,2,3,5,8],[4,6,7]]=>3
[[1,2,3,4,8],[5,6,7]]=>4
[[1,3,5,6,7],[2,4,8]]=>3
[[1,2,5,6,7],[3,4,8]]=>3
[[1,3,4,6,7],[2,5,8]]=>3
[[1,2,4,6,7],[3,5,8]]=>3
[[1,2,3,6,7],[4,5,8]]=>3
[[1,3,4,5,7],[2,6,8]]=>3
[[1,2,4,5,7],[3,6,8]]=>3
[[1,2,3,5,7],[4,6,8]]=>3
[[1,2,3,4,7],[5,6,8]]=>4
[[1,3,4,5,6],[2,7,8]]=>4
[[1,2,4,5,6],[3,7,8]]=>4
[[1,2,3,5,6],[4,7,8]]=>4
[[1,2,3,4,6],[5,7,8]]=>4
[[1,2,3,4,5],[6,7,8]]=>5
[[1,4,6,7,8],[2,5],[3]]=>4
[[1,3,6,7,8],[2,5],[4]]=>4
[[1,2,6,7,8],[3,5],[4]]=>4
[[1,3,6,7,8],[2,4],[5]]=>4
[[1,2,6,7,8],[3,4],[5]]=>4
[[1,4,5,7,8],[2,6],[3]]=>4
[[1,3,5,7,8],[2,6],[4]]=>4
[[1,2,5,7,8],[3,6],[4]]=>4
[[1,3,4,7,8],[2,6],[5]]=>4
[[1,2,4,7,8],[3,6],[5]]=>4
[[1,2,3,7,8],[4,6],[5]]=>4
[[1,3,5,7,8],[2,4],[6]]=>4
[[1,2,5,7,8],[3,4],[6]]=>4
[[1,3,4,7,8],[2,5],[6]]=>4
[[1,2,4,7,8],[3,5],[6]]=>4
[[1,2,3,7,8],[4,5],[6]]=>4
[[1,4,5,6,8],[2,7],[3]]=>4
[[1,3,5,6,8],[2,7],[4]]=>4
[[1,2,5,6,8],[3,7],[4]]=>4
[[1,3,4,6,8],[2,7],[5]]=>4
[[1,2,4,6,8],[3,7],[5]]=>4
[[1,2,3,6,8],[4,7],[5]]=>4
[[1,3,4,5,8],[2,7],[6]]=>4
[[1,2,4,5,8],[3,7],[6]]=>4
[[1,2,3,5,8],[4,7],[6]]=>4
[[1,2,3,4,8],[5,7],[6]]=>4
[[1,3,5,6,8],[2,4],[7]]=>4
[[1,2,5,6,8],[3,4],[7]]=>4
[[1,3,4,6,8],[2,5],[7]]=>4
[[1,2,4,6,8],[3,5],[7]]=>4
[[1,2,3,6,8],[4,5],[7]]=>4
[[1,3,4,5,8],[2,6],[7]]=>4
[[1,2,4,5,8],[3,6],[7]]=>4
[[1,2,3,5,8],[4,6],[7]]=>4
[[1,2,3,4,8],[5,6],[7]]=>4
[[1,4,5,6,7],[2,8],[3]]=>4
[[1,3,5,6,7],[2,8],[4]]=>4
[[1,2,5,6,7],[3,8],[4]]=>4
[[1,3,4,6,7],[2,8],[5]]=>4
[[1,2,4,6,7],[3,8],[5]]=>4
[[1,2,3,6,7],[4,8],[5]]=>4
[[1,3,4,5,7],[2,8],[6]]=>4
[[1,2,4,5,7],[3,8],[6]]=>4
[[1,2,3,5,7],[4,8],[6]]=>4
[[1,2,3,4,7],[5,8],[6]]=>4
[[1,3,4,5,6],[2,8],[7]]=>5
[[1,2,4,5,6],[3,8],[7]]=>5
[[1,2,3,5,6],[4,8],[7]]=>5
[[1,2,3,4,6],[5,8],[7]]=>5
[[1,2,3,4,5],[6,8],[7]]=>5
[[1,3,5,6,7],[2,4],[8]]=>5
[[1,2,5,6,7],[3,4],[8]]=>5
[[1,3,4,6,7],[2,5],[8]]=>5
[[1,2,4,6,7],[3,5],[8]]=>5
[[1,2,3,6,7],[4,5],[8]]=>5
[[1,3,4,5,7],[2,6],[8]]=>5
[[1,2,4,5,7],[3,6],[8]]=>5
[[1,2,3,5,7],[4,6],[8]]=>5
[[1,2,3,4,7],[5,6],[8]]=>5
[[1,3,4,5,6],[2,7],[8]]=>5
[[1,2,4,5,6],[3,7],[8]]=>5
[[1,2,3,5,6],[4,7],[8]]=>5
[[1,2,3,4,6],[5,7],[8]]=>5
[[1,2,3,4,5],[6,7],[8]]=>5
[[1,5,6,7,8],[2],[3],[4]]=>4
[[1,4,6,7,8],[2],[3],[5]]=>4
[[1,3,6,7,8],[2],[4],[5]]=>4
[[1,2,6,7,8],[3],[4],[5]]=>4
[[1,4,5,7,8],[2],[3],[6]]=>4
[[1,3,5,7,8],[2],[4],[6]]=>4
[[1,2,5,7,8],[3],[4],[6]]=>4
[[1,3,4,7,8],[2],[5],[6]]=>4
[[1,2,4,7,8],[3],[5],[6]]=>4
[[1,2,3,7,8],[4],[5],[6]]=>4
[[1,4,5,6,8],[2],[3],[7]]=>4
[[1,3,5,6,8],[2],[4],[7]]=>4
[[1,2,5,6,8],[3],[4],[7]]=>4
[[1,3,4,6,8],[2],[5],[7]]=>4
[[1,2,4,6,8],[3],[5],[7]]=>4
[[1,2,3,6,8],[4],[5],[7]]=>4
[[1,3,4,5,8],[2],[6],[7]]=>4
[[1,2,4,5,8],[3],[6],[7]]=>4
[[1,2,3,5,8],[4],[6],[7]]=>4
[[1,2,3,4,8],[5],[6],[7]]=>4
[[1,4,5,6,7],[2],[3],[8]]=>5
[[1,3,5,6,7],[2],[4],[8]]=>5
[[1,2,5,6,7],[3],[4],[8]]=>5
[[1,3,4,6,7],[2],[5],[8]]=>5
[[1,2,4,6,7],[3],[5],[8]]=>5
[[1,2,3,6,7],[4],[5],[8]]=>5
[[1,3,4,5,7],[2],[6],[8]]=>5
[[1,2,4,5,7],[3],[6],[8]]=>5
[[1,2,3,5,7],[4],[6],[8]]=>5
[[1,2,3,4,7],[5],[6],[8]]=>5
[[1,3,4,5,6],[2],[7],[8]]=>5
[[1,2,4,5,6],[3],[7],[8]]=>5
[[1,2,3,5,6],[4],[7],[8]]=>5
[[1,2,3,4,6],[5],[7],[8]]=>5
[[1,2,3,4,5],[6],[7],[8]]=>5
[[1,3,5,7],[2,4,6,8]]=>1
[[1,2,5,7],[3,4,6,8]]=>2
[[1,3,4,7],[2,5,6,8]]=>2
[[1,2,4,7],[3,5,6,8]]=>2
[[1,2,3,7],[4,5,6,8]]=>3
[[1,3,5,6],[2,4,7,8]]=>2
[[1,2,5,6],[3,4,7,8]]=>2
[[1,3,4,6],[2,5,7,8]]=>2
[[1,2,4,6],[3,5,7,8]]=>2
[[1,2,3,6],[4,5,7,8]]=>3
[[1,3,4,5],[2,6,7,8]]=>3
[[1,2,4,5],[3,6,7,8]]=>3
[[1,2,3,5],[4,6,7,8]]=>3
[[1,2,3,4],[5,6,7,8]]=>4
[[1,4,6,8],[2,5,7],[3]]=>3
[[1,3,6,8],[2,5,7],[4]]=>3
[[1,2,6,8],[3,5,7],[4]]=>3
[[1,3,6,8],[2,4,7],[5]]=>3
[[1,2,6,8],[3,4,7],[5]]=>3
[[1,4,5,8],[2,6,7],[3]]=>3
[[1,3,5,8],[2,6,7],[4]]=>3
[[1,2,5,8],[3,6,7],[4]]=>3
[[1,3,4,8],[2,6,7],[5]]=>3
[[1,2,4,8],[3,6,7],[5]]=>3
[[1,2,3,8],[4,6,7],[5]]=>3
[[1,3,5,8],[2,4,7],[6]]=>3
[[1,2,5,8],[3,4,7],[6]]=>3
[[1,3,4,8],[2,5,7],[6]]=>3
[[1,2,4,8],[3,5,7],[6]]=>3
[[1,2,3,8],[4,5,7],[6]]=>3
[[1,3,5,8],[2,4,6],[7]]=>3
[[1,2,5,8],[3,4,6],[7]]=>3
[[1,3,4,8],[2,5,6],[7]]=>3
[[1,2,4,8],[3,5,6],[7]]=>3
[[1,2,3,8],[4,5,6],[7]]=>3
[[1,4,6,7],[2,5,8],[3]]=>3
[[1,3,6,7],[2,5,8],[4]]=>3
[[1,2,6,7],[3,5,8],[4]]=>3
[[1,3,6,7],[2,4,8],[5]]=>3
[[1,2,6,7],[3,4,8],[5]]=>3
[[1,4,5,7],[2,6,8],[3]]=>3
[[1,3,5,7],[2,6,8],[4]]=>3
[[1,2,5,7],[3,6,8],[4]]=>3
[[1,3,4,7],[2,6,8],[5]]=>3
[[1,2,4,7],[3,6,8],[5]]=>3
[[1,2,3,7],[4,6,8],[5]]=>3
[[1,3,5,7],[2,4,8],[6]]=>3
[[1,2,5,7],[3,4,8],[6]]=>3
[[1,3,4,7],[2,5,8],[6]]=>3
[[1,2,4,7],[3,5,8],[6]]=>3
[[1,2,3,7],[4,5,8],[6]]=>3
[[1,4,5,6],[2,7,8],[3]]=>3
[[1,3,5,6],[2,7,8],[4]]=>3
[[1,2,5,6],[3,7,8],[4]]=>3
[[1,3,4,6],[2,7,8],[5]]=>3
[[1,2,4,6],[3,7,8],[5]]=>3
[[1,2,3,6],[4,7,8],[5]]=>3
[[1,3,4,5],[2,7,8],[6]]=>4
[[1,2,4,5],[3,7,8],[6]]=>4
[[1,2,3,5],[4,7,8],[6]]=>4
[[1,2,3,4],[5,7,8],[6]]=>4
[[1,3,5,6],[2,4,8],[7]]=>4
[[1,2,5,6],[3,4,8],[7]]=>4
[[1,3,4,6],[2,5,8],[7]]=>4
[[1,2,4,6],[3,5,8],[7]]=>4
[[1,2,3,6],[4,5,8],[7]]=>4
[[1,3,4,5],[2,6,8],[7]]=>4
[[1,2,4,5],[3,6,8],[7]]=>4
[[1,2,3,5],[4,6,8],[7]]=>4
[[1,2,3,4],[5,6,8],[7]]=>4
[[1,3,5,7],[2,4,6],[8]]=>4
[[1,2,5,7],[3,4,6],[8]]=>4
[[1,3,4,7],[2,5,6],[8]]=>4
[[1,2,4,7],[3,5,6],[8]]=>4
[[1,2,3,7],[4,5,6],[8]]=>4
[[1,3,5,6],[2,4,7],[8]]=>4
[[1,2,5,6],[3,4,7],[8]]=>4
[[1,3,4,6],[2,5,7],[8]]=>4
[[1,2,4,6],[3,5,7],[8]]=>4
[[1,2,3,6],[4,5,7],[8]]=>4
[[1,3,4,5],[2,6,7],[8]]=>4
[[1,2,4,5],[3,6,7],[8]]=>4
[[1,2,3,5],[4,6,7],[8]]=>4
[[1,2,3,4],[5,6,7],[8]]=>4
[[1,4,7,8],[2,5],[3,6]]=>2
[[1,3,7,8],[2,5],[4,6]]=>2
[[1,2,7,8],[3,5],[4,6]]=>2
[[1,3,7,8],[2,4],[5,6]]=>2
[[1,2,7,8],[3,4],[5,6]]=>2
[[1,4,6,8],[2,5],[3,7]]=>2
[[1,3,6,8],[2,5],[4,7]]=>2
[[1,2,6,8],[3,5],[4,7]]=>2
[[1,3,6,8],[2,4],[5,7]]=>2
[[1,2,6,8],[3,4],[5,7]]=>2
[[1,4,5,8],[2,6],[3,7]]=>2
[[1,3,5,8],[2,6],[4,7]]=>2
[[1,2,5,8],[3,6],[4,7]]=>2
[[1,3,4,8],[2,6],[5,7]]=>3
[[1,2,4,8],[3,6],[5,7]]=>3
[[1,2,3,8],[4,6],[5,7]]=>3
[[1,3,5,8],[2,4],[6,7]]=>3
[[1,2,5,8],[3,4],[6,7]]=>3
[[1,3,4,8],[2,5],[6,7]]=>3
[[1,2,4,8],[3,5],[6,7]]=>3
[[1,2,3,8],[4,5],[6,7]]=>3
[[1,4,6,7],[2,5],[3,8]]=>3
[[1,3,6,7],[2,5],[4,8]]=>3
[[1,2,6,7],[3,5],[4,8]]=>3
[[1,3,6,7],[2,4],[5,8]]=>3
[[1,2,6,7],[3,4],[5,8]]=>3
[[1,4,5,7],[2,6],[3,8]]=>3
[[1,3,5,7],[2,6],[4,8]]=>3
[[1,2,5,7],[3,6],[4,8]]=>3
[[1,3,4,7],[2,6],[5,8]]=>3
[[1,2,4,7],[3,6],[5,8]]=>3
[[1,2,3,7],[4,6],[5,8]]=>3
[[1,3,5,7],[2,4],[6,8]]=>3
[[1,2,5,7],[3,4],[6,8]]=>3
[[1,3,4,7],[2,5],[6,8]]=>3
[[1,2,4,7],[3,5],[6,8]]=>3
[[1,2,3,7],[4,5],[6,8]]=>3
[[1,4,5,6],[2,7],[3,8]]=>3
[[1,3,5,6],[2,7],[4,8]]=>3
[[1,2,5,6],[3,7],[4,8]]=>3
[[1,3,4,6],[2,7],[5,8]]=>3
[[1,2,4,6],[3,7],[5,8]]=>3
[[1,2,3,6],[4,7],[5,8]]=>3
[[1,3,4,5],[2,7],[6,8]]=>4
[[1,2,4,5],[3,7],[6,8]]=>4
[[1,2,3,5],[4,7],[6,8]]=>4
[[1,2,3,4],[5,7],[6,8]]=>4
[[1,3,5,6],[2,4],[7,8]]=>4
[[1,2,5,6],[3,4],[7,8]]=>4
[[1,3,4,6],[2,5],[7,8]]=>4
[[1,2,4,6],[3,5],[7,8]]=>4
[[1,2,3,6],[4,5],[7,8]]=>4
[[1,3,4,5],[2,6],[7,8]]=>4
[[1,2,4,5],[3,6],[7,8]]=>4
[[1,2,3,5],[4,6],[7,8]]=>4
[[1,2,3,4],[5,6],[7,8]]=>4
[[1,5,7,8],[2,6],[3],[4]]=>3
[[1,4,7,8],[2,6],[3],[5]]=>3
[[1,3,7,8],[2,6],[4],[5]]=>3
[[1,2,7,8],[3,6],[4],[5]]=>3
[[1,4,7,8],[2,5],[3],[6]]=>3
[[1,3,7,8],[2,5],[4],[6]]=>3
[[1,2,7,8],[3,5],[4],[6]]=>3
[[1,3,7,8],[2,4],[5],[6]]=>3
[[1,2,7,8],[3,4],[5],[6]]=>3
[[1,5,6,8],[2,7],[3],[4]]=>3
[[1,4,6,8],[2,7],[3],[5]]=>3
[[1,3,6,8],[2,7],[4],[5]]=>3
[[1,2,6,8],[3,7],[4],[5]]=>3
[[1,4,5,8],[2,7],[3],[6]]=>3
[[1,3,5,8],[2,7],[4],[6]]=>3
[[1,2,5,8],[3,7],[4],[6]]=>3
[[1,3,4,8],[2,7],[5],[6]]=>3
[[1,2,4,8],[3,7],[5],[6]]=>3
[[1,2,3,8],[4,7],[5],[6]]=>3
[[1,4,6,8],[2,5],[3],[7]]=>3
[[1,3,6,8],[2,5],[4],[7]]=>3
[[1,2,6,8],[3,5],[4],[7]]=>3
[[1,3,6,8],[2,4],[5],[7]]=>3
[[1,2,6,8],[3,4],[5],[7]]=>3
[[1,4,5,8],[2,6],[3],[7]]=>3
[[1,3,5,8],[2,6],[4],[7]]=>3
[[1,2,5,8],[3,6],[4],[7]]=>3
[[1,3,4,8],[2,6],[5],[7]]=>3
[[1,2,4,8],[3,6],[5],[7]]=>3
[[1,2,3,8],[4,6],[5],[7]]=>3
[[1,3,5,8],[2,4],[6],[7]]=>3
[[1,2,5,8],[3,4],[6],[7]]=>3
[[1,3,4,8],[2,5],[6],[7]]=>3
[[1,2,4,8],[3,5],[6],[7]]=>3
[[1,2,3,8],[4,5],[6],[7]]=>3
[[1,5,6,7],[2,8],[3],[4]]=>3
[[1,4,6,7],[2,8],[3],[5]]=>3
[[1,3,6,7],[2,8],[4],[5]]=>3
[[1,2,6,7],[3,8],[4],[5]]=>3
[[1,4,5,7],[2,8],[3],[6]]=>3
[[1,3,5,7],[2,8],[4],[6]]=>3
[[1,2,5,7],[3,8],[4],[6]]=>3
[[1,3,4,7],[2,8],[5],[6]]=>3
[[1,2,4,7],[3,8],[5],[6]]=>3
[[1,2,3,7],[4,8],[5],[6]]=>3
[[1,4,5,6],[2,8],[3],[7]]=>4
[[1,3,5,6],[2,8],[4],[7]]=>4
[[1,2,5,6],[3,8],[4],[7]]=>4
[[1,3,4,6],[2,8],[5],[7]]=>4
[[1,2,4,6],[3,8],[5],[7]]=>4
[[1,2,3,6],[4,8],[5],[7]]=>4
[[1,3,4,5],[2,8],[6],[7]]=>4
[[1,2,4,5],[3,8],[6],[7]]=>4
[[1,2,3,5],[4,8],[6],[7]]=>4
[[1,2,3,4],[5,8],[6],[7]]=>4
[[1,4,6,7],[2,5],[3],[8]]=>4
[[1,3,6,7],[2,5],[4],[8]]=>4
[[1,2,6,7],[3,5],[4],[8]]=>4
[[1,3,6,7],[2,4],[5],[8]]=>4
[[1,2,6,7],[3,4],[5],[8]]=>4
[[1,4,5,7],[2,6],[3],[8]]=>4
[[1,3,5,7],[2,6],[4],[8]]=>4
[[1,2,5,7],[3,6],[4],[8]]=>4
[[1,3,4,7],[2,6],[5],[8]]=>4
[[1,2,4,7],[3,6],[5],[8]]=>4
[[1,2,3,7],[4,6],[5],[8]]=>4
[[1,3,5,7],[2,4],[6],[8]]=>4
[[1,2,5,7],[3,4],[6],[8]]=>4
[[1,3,4,7],[2,5],[6],[8]]=>4
[[1,2,4,7],[3,5],[6],[8]]=>4
[[1,2,3,7],[4,5],[6],[8]]=>4
[[1,4,5,6],[2,7],[3],[8]]=>4
[[1,3,5,6],[2,7],[4],[8]]=>4
[[1,2,5,6],[3,7],[4],[8]]=>4
[[1,3,4,6],[2,7],[5],[8]]=>4
[[1,2,4,6],[3,7],[5],[8]]=>4
[[1,2,3,6],[4,7],[5],[8]]=>4
[[1,3,4,5],[2,7],[6],[8]]=>4
[[1,2,4,5],[3,7],[6],[8]]=>4
[[1,2,3,5],[4,7],[6],[8]]=>4
[[1,2,3,4],[5,7],[6],[8]]=>4
[[1,3,5,6],[2,4],[7],[8]]=>4
[[1,2,5,6],[3,4],[7],[8]]=>4
[[1,3,4,6],[2,5],[7],[8]]=>4
[[1,2,4,6],[3,5],[7],[8]]=>4
[[1,2,3,6],[4,5],[7],[8]]=>4
[[1,3,4,5],[2,6],[7],[8]]=>4
[[1,2,4,5],[3,6],[7],[8]]=>4
[[1,2,3,5],[4,6],[7],[8]]=>4
[[1,2,3,4],[5,6],[7],[8]]=>4
[[1,6,7,8],[2],[3],[4],[5]]=>3
[[1,5,7,8],[2],[3],[4],[6]]=>3
[[1,4,7,8],[2],[3],[5],[6]]=>3
[[1,3,7,8],[2],[4],[5],[6]]=>3
[[1,2,7,8],[3],[4],[5],[6]]=>3
[[1,5,6,8],[2],[3],[4],[7]]=>3
[[1,4,6,8],[2],[3],[5],[7]]=>3
[[1,3,6,8],[2],[4],[5],[7]]=>3
[[1,2,6,8],[3],[4],[5],[7]]=>3
[[1,4,5,8],[2],[3],[6],[7]]=>3
[[1,3,5,8],[2],[4],[6],[7]]=>3
[[1,2,5,8],[3],[4],[6],[7]]=>3
[[1,3,4,8],[2],[5],[6],[7]]=>3
[[1,2,4,8],[3],[5],[6],[7]]=>3
[[1,2,3,8],[4],[5],[6],[7]]=>3
[[1,5,6,7],[2],[3],[4],[8]]=>4
[[1,4,6,7],[2],[3],[5],[8]]=>4
[[1,3,6,7],[2],[4],[5],[8]]=>4
[[1,2,6,7],[3],[4],[5],[8]]=>4
[[1,4,5,7],[2],[3],[6],[8]]=>4
[[1,3,5,7],[2],[4],[6],[8]]=>4
[[1,2,5,7],[3],[4],[6],[8]]=>4
[[1,3,4,7],[2],[5],[6],[8]]=>4
[[1,2,4,7],[3],[5],[6],[8]]=>4
[[1,2,3,7],[4],[5],[6],[8]]=>4
[[1,4,5,6],[2],[3],[7],[8]]=>4
[[1,3,5,6],[2],[4],[7],[8]]=>4
[[1,2,5,6],[3],[4],[7],[8]]=>4
[[1,3,4,6],[2],[5],[7],[8]]=>4
[[1,2,4,6],[3],[5],[7],[8]]=>4
[[1,2,3,6],[4],[5],[7],[8]]=>4
[[1,3,4,5],[2],[6],[7],[8]]=>4
[[1,2,4,5],[3],[6],[7],[8]]=>4
[[1,2,3,5],[4],[6],[7],[8]]=>4
[[1,2,3,4],[5],[6],[7],[8]]=>4
[[1,4,7],[2,5,8],[3,6]]=>1
[[1,3,7],[2,5,8],[4,6]]=>2
[[1,2,7],[3,5,8],[4,6]]=>2
[[1,3,7],[2,4,8],[5,6]]=>2
[[1,2,7],[3,4,8],[5,6]]=>2
[[1,4,6],[2,5,8],[3,7]]=>2
[[1,3,6],[2,5,8],[4,7]]=>2
[[1,2,6],[3,5,8],[4,7]]=>2
[[1,3,6],[2,4,8],[5,7]]=>2
[[1,2,6],[3,4,8],[5,7]]=>2
[[1,4,5],[2,6,8],[3,7]]=>2
[[1,3,5],[2,6,8],[4,7]]=>2
[[1,2,5],[3,6,8],[4,7]]=>2
[[1,3,4],[2,6,8],[5,7]]=>3
[[1,2,4],[3,6,8],[5,7]]=>3
[[1,2,3],[4,6,8],[5,7]]=>3
[[1,3,5],[2,4,8],[6,7]]=>3
[[1,2,5],[3,4,8],[6,7]]=>3
[[1,3,4],[2,5,8],[6,7]]=>3
[[1,2,4],[3,5,8],[6,7]]=>3
[[1,2,3],[4,5,8],[6,7]]=>3
[[1,4,6],[2,5,7],[3,8]]=>2
[[1,3,6],[2,5,7],[4,8]]=>2
[[1,2,6],[3,5,7],[4,8]]=>2
[[1,3,6],[2,4,7],[5,8]]=>2
[[1,2,6],[3,4,7],[5,8]]=>2
[[1,4,5],[2,6,7],[3,8]]=>2
[[1,3,5],[2,6,7],[4,8]]=>2
[[1,2,5],[3,6,7],[4,8]]=>2
[[1,3,4],[2,6,7],[5,8]]=>3
[[1,2,4],[3,6,7],[5,8]]=>3
[[1,2,3],[4,6,7],[5,8]]=>3
[[1,3,5],[2,4,7],[6,8]]=>3
[[1,2,5],[3,4,7],[6,8]]=>3
[[1,3,4],[2,5,7],[6,8]]=>3
[[1,2,4],[3,5,7],[6,8]]=>3
[[1,2,3],[4,5,7],[6,8]]=>3
[[1,3,5],[2,4,6],[7,8]]=>3
[[1,2,5],[3,4,6],[7,8]]=>3
[[1,3,4],[2,5,6],[7,8]]=>3
[[1,2,4],[3,5,6],[7,8]]=>3
[[1,2,3],[4,5,6],[7,8]]=>3
[[1,5,7],[2,6,8],[3],[4]]=>2
[[1,4,7],[2,6,8],[3],[5]]=>2
[[1,3,7],[2,6,8],[4],[5]]=>2
[[1,2,7],[3,6,8],[4],[5]]=>2
[[1,4,7],[2,5,8],[3],[6]]=>2
[[1,3,7],[2,5,8],[4],[6]]=>2
[[1,2,7],[3,5,8],[4],[6]]=>2
[[1,3,7],[2,4,8],[5],[6]]=>2
[[1,2,7],[3,4,8],[5],[6]]=>2
[[1,5,6],[2,7,8],[3],[4]]=>2
[[1,4,6],[2,7,8],[3],[5]]=>2
[[1,3,6],[2,7,8],[4],[5]]=>2
[[1,2,6],[3,7,8],[4],[5]]=>2
[[1,4,5],[2,7,8],[3],[6]]=>3
[[1,3,5],[2,7,8],[4],[6]]=>3
[[1,2,5],[3,7,8],[4],[6]]=>3
[[1,3,4],[2,7,8],[5],[6]]=>3
[[1,2,4],[3,7,8],[5],[6]]=>3
[[1,2,3],[4,7,8],[5],[6]]=>3
[[1,4,6],[2,5,8],[3],[7]]=>3
[[1,3,6],[2,5,8],[4],[7]]=>3
[[1,2,6],[3,5,8],[4],[7]]=>3
[[1,3,6],[2,4,8],[5],[7]]=>3
[[1,2,6],[3,4,8],[5],[7]]=>3
[[1,4,5],[2,6,8],[3],[7]]=>3
[[1,3,5],[2,6,8],[4],[7]]=>3
[[1,2,5],[3,6,8],[4],[7]]=>3
[[1,3,4],[2,6,8],[5],[7]]=>3
[[1,2,4],[3,6,8],[5],[7]]=>3
[[1,2,3],[4,6,8],[5],[7]]=>3
[[1,3,5],[2,4,8],[6],[7]]=>3
[[1,2,5],[3,4,8],[6],[7]]=>3
[[1,3,4],[2,5,8],[6],[7]]=>3
[[1,2,4],[3,5,8],[6],[7]]=>3
[[1,2,3],[4,5,8],[6],[7]]=>3
[[1,4,6],[2,5,7],[3],[8]]=>3
[[1,3,6],[2,5,7],[4],[8]]=>3
[[1,2,6],[3,5,7],[4],[8]]=>3
[[1,3,6],[2,4,7],[5],[8]]=>3
[[1,2,6],[3,4,7],[5],[8]]=>3
[[1,4,5],[2,6,7],[3],[8]]=>3
[[1,3,5],[2,6,7],[4],[8]]=>3
[[1,2,5],[3,6,7],[4],[8]]=>3
[[1,3,4],[2,6,7],[5],[8]]=>3
[[1,2,4],[3,6,7],[5],[8]]=>3
[[1,2,3],[4,6,7],[5],[8]]=>3
[[1,3,5],[2,4,7],[6],[8]]=>3
[[1,2,5],[3,4,7],[6],[8]]=>3
[[1,3,4],[2,5,7],[6],[8]]=>3
[[1,2,4],[3,5,7],[6],[8]]=>3
[[1,2,3],[4,5,7],[6],[8]]=>3
[[1,3,5],[2,4,6],[7],[8]]=>3
[[1,2,5],[3,4,6],[7],[8]]=>3
[[1,3,4],[2,5,6],[7],[8]]=>3
[[1,2,4],[3,5,6],[7],[8]]=>3
[[1,2,3],[4,5,6],[7],[8]]=>3
[[1,5,8],[2,6],[3,7],[4]]=>2
[[1,4,8],[2,6],[3,7],[5]]=>2
[[1,3,8],[2,6],[4,7],[5]]=>2
[[1,2,8],[3,6],[4,7],[5]]=>2
[[1,4,8],[2,5],[3,7],[6]]=>2
[[1,3,8],[2,5],[4,7],[6]]=>2
[[1,2,8],[3,5],[4,7],[6]]=>2
[[1,3,8],[2,4],[5,7],[6]]=>2
[[1,2,8],[3,4],[5,7],[6]]=>2
[[1,4,8],[2,5],[3,6],[7]]=>2
[[1,3,8],[2,5],[4,6],[7]]=>2
[[1,2,8],[3,5],[4,6],[7]]=>2
[[1,3,8],[2,4],[5,6],[7]]=>2
[[1,2,8],[3,4],[5,6],[7]]=>2
[[1,5,7],[2,6],[3,8],[4]]=>2
[[1,4,7],[2,6],[3,8],[5]]=>2
[[1,3,7],[2,6],[4,8],[5]]=>2
[[1,2,7],[3,6],[4,8],[5]]=>2
[[1,4,7],[2,5],[3,8],[6]]=>2
[[1,3,7],[2,5],[4,8],[6]]=>2
[[1,2,7],[3,5],[4,8],[6]]=>2
[[1,3,7],[2,4],[5,8],[6]]=>2
[[1,2,7],[3,4],[5,8],[6]]=>2
[[1,5,6],[2,7],[3,8],[4]]=>2
[[1,4,6],[2,7],[3,8],[5]]=>2
[[1,3,6],[2,7],[4,8],[5]]=>2
[[1,2,6],[3,7],[4,8],[5]]=>2
[[1,4,5],[2,7],[3,8],[6]]=>3
[[1,3,5],[2,7],[4,8],[6]]=>3
[[1,2,5],[3,7],[4,8],[6]]=>3
[[1,3,4],[2,7],[5,8],[6]]=>3
[[1,2,4],[3,7],[5,8],[6]]=>3
[[1,2,3],[4,7],[5,8],[6]]=>3
[[1,4,6],[2,5],[3,8],[7]]=>3
[[1,3,6],[2,5],[4,8],[7]]=>3
[[1,2,6],[3,5],[4,8],[7]]=>3
[[1,3,6],[2,4],[5,8],[7]]=>3
[[1,2,6],[3,4],[5,8],[7]]=>3
[[1,4,5],[2,6],[3,8],[7]]=>3
[[1,3,5],[2,6],[4,8],[7]]=>3
[[1,2,5],[3,6],[4,8],[7]]=>3
[[1,3,4],[2,6],[5,8],[7]]=>3
[[1,2,4],[3,6],[5,8],[7]]=>3
[[1,2,3],[4,6],[5,8],[7]]=>3
[[1,3,5],[2,4],[6,8],[7]]=>3
[[1,2,5],[3,4],[6,8],[7]]=>3
[[1,3,4],[2,5],[6,8],[7]]=>3
[[1,2,4],[3,5],[6,8],[7]]=>3
[[1,2,3],[4,5],[6,8],[7]]=>3
[[1,4,7],[2,5],[3,6],[8]]=>3
[[1,3,7],[2,5],[4,6],[8]]=>3
[[1,2,7],[3,5],[4,6],[8]]=>3
[[1,3,7],[2,4],[5,6],[8]]=>3
[[1,2,7],[3,4],[5,6],[8]]=>3
[[1,4,6],[2,5],[3,7],[8]]=>3
[[1,3,6],[2,5],[4,7],[8]]=>3
[[1,2,6],[3,5],[4,7],[8]]=>3
[[1,3,6],[2,4],[5,7],[8]]=>3
[[1,2,6],[3,4],[5,7],[8]]=>3
[[1,4,5],[2,6],[3,7],[8]]=>3
[[1,3,5],[2,6],[4,7],[8]]=>3
[[1,2,5],[3,6],[4,7],[8]]=>3
[[1,3,4],[2,6],[5,7],[8]]=>3
[[1,2,4],[3,6],[5,7],[8]]=>3
[[1,2,3],[4,6],[5,7],[8]]=>3
[[1,3,5],[2,4],[6,7],[8]]=>3
[[1,2,5],[3,4],[6,7],[8]]=>3
[[1,3,4],[2,5],[6,7],[8]]=>3
[[1,2,4],[3,5],[6,7],[8]]=>3
[[1,2,3],[4,5],[6,7],[8]]=>3
[[1,6,8],[2,7],[3],[4],[5]]=>2
[[1,5,8],[2,7],[3],[4],[6]]=>2
[[1,4,8],[2,7],[3],[5],[6]]=>2
[[1,3,8],[2,7],[4],[5],[6]]=>2
[[1,2,8],[3,7],[4],[5],[6]]=>2
[[1,5,8],[2,6],[3],[4],[7]]=>2
[[1,4,8],[2,6],[3],[5],[7]]=>2
[[1,3,8],[2,6],[4],[5],[7]]=>2
[[1,2,8],[3,6],[4],[5],[7]]=>2
[[1,4,8],[2,5],[3],[6],[7]]=>2
[[1,3,8],[2,5],[4],[6],[7]]=>2
[[1,2,8],[3,5],[4],[6],[7]]=>2
[[1,3,8],[2,4],[5],[6],[7]]=>2
[[1,2,8],[3,4],[5],[6],[7]]=>2
[[1,6,7],[2,8],[3],[4],[5]]=>2
[[1,5,7],[2,8],[3],[4],[6]]=>2
[[1,4,7],[2,8],[3],[5],[6]]=>2
[[1,3,7],[2,8],[4],[5],[6]]=>2
[[1,2,7],[3,8],[4],[5],[6]]=>2
[[1,5,6],[2,8],[3],[4],[7]]=>3
[[1,4,6],[2,8],[3],[5],[7]]=>3
[[1,3,6],[2,8],[4],[5],[7]]=>3
[[1,2,6],[3,8],[4],[5],[7]]=>3
[[1,4,5],[2,8],[3],[6],[7]]=>3
[[1,3,5],[2,8],[4],[6],[7]]=>3
[[1,2,5],[3,8],[4],[6],[7]]=>3
[[1,3,4],[2,8],[5],[6],[7]]=>3
[[1,2,4],[3,8],[5],[6],[7]]=>3
[[1,2,3],[4,8],[5],[6],[7]]=>3
[[1,5,7],[2,6],[3],[4],[8]]=>3
[[1,4,7],[2,6],[3],[5],[8]]=>3
[[1,3,7],[2,6],[4],[5],[8]]=>3
[[1,2,7],[3,6],[4],[5],[8]]=>3
[[1,4,7],[2,5],[3],[6],[8]]=>3
[[1,3,7],[2,5],[4],[6],[8]]=>3
[[1,2,7],[3,5],[4],[6],[8]]=>3
[[1,3,7],[2,4],[5],[6],[8]]=>3
[[1,2,7],[3,4],[5],[6],[8]]=>3
[[1,5,6],[2,7],[3],[4],[8]]=>3
[[1,4,6],[2,7],[3],[5],[8]]=>3
[[1,3,6],[2,7],[4],[5],[8]]=>3
[[1,2,6],[3,7],[4],[5],[8]]=>3
[[1,4,5],[2,7],[3],[6],[8]]=>3
[[1,3,5],[2,7],[4],[6],[8]]=>3
[[1,2,5],[3,7],[4],[6],[8]]=>3
[[1,3,4],[2,7],[5],[6],[8]]=>3
[[1,2,4],[3,7],[5],[6],[8]]=>3
[[1,2,3],[4,7],[5],[6],[8]]=>3
[[1,4,6],[2,5],[3],[7],[8]]=>3
[[1,3,6],[2,5],[4],[7],[8]]=>3
[[1,2,6],[3,5],[4],[7],[8]]=>3
[[1,3,6],[2,4],[5],[7],[8]]=>3
[[1,2,6],[3,4],[5],[7],[8]]=>3
[[1,4,5],[2,6],[3],[7],[8]]=>3
[[1,3,5],[2,6],[4],[7],[8]]=>3
[[1,2,5],[3,6],[4],[7],[8]]=>3
[[1,3,4],[2,6],[5],[7],[8]]=>3
[[1,2,4],[3,6],[5],[7],[8]]=>3
[[1,2,3],[4,6],[5],[7],[8]]=>3
[[1,3,5],[2,4],[6],[7],[8]]=>3
[[1,2,5],[3,4],[6],[7],[8]]=>3
[[1,3,4],[2,5],[6],[7],[8]]=>3
[[1,2,4],[3,5],[6],[7],[8]]=>3
[[1,2,3],[4,5],[6],[7],[8]]=>3
[[1,7,8],[2],[3],[4],[5],[6]]=>2
[[1,6,8],[2],[3],[4],[5],[7]]=>2
[[1,5,8],[2],[3],[4],[6],[7]]=>2
[[1,4,8],[2],[3],[5],[6],[7]]=>2
[[1,3,8],[2],[4],[5],[6],[7]]=>2
[[1,2,8],[3],[4],[5],[6],[7]]=>2
[[1,6,7],[2],[3],[4],[5],[8]]=>3
[[1,5,7],[2],[3],[4],[6],[8]]=>3
[[1,4,7],[2],[3],[5],[6],[8]]=>3
[[1,3,7],[2],[4],[5],[6],[8]]=>3
[[1,2,7],[3],[4],[5],[6],[8]]=>3
[[1,5,6],[2],[3],[4],[7],[8]]=>3
[[1,4,6],[2],[3],[5],[7],[8]]=>3
[[1,3,6],[2],[4],[5],[7],[8]]=>3
[[1,2,6],[3],[4],[5],[7],[8]]=>3
[[1,4,5],[2],[3],[6],[7],[8]]=>3
[[1,3,5],[2],[4],[6],[7],[8]]=>3
[[1,2,5],[3],[4],[6],[7],[8]]=>3
[[1,3,4],[2],[5],[6],[7],[8]]=>3
[[1,2,4],[3],[5],[6],[7],[8]]=>3
[[1,2,3],[4],[5],[6],[7],[8]]=>3
[[1,5],[2,6],[3,7],[4,8]]=>1
[[1,4],[2,6],[3,7],[5,8]]=>2
[[1,3],[2,6],[4,7],[5,8]]=>2
[[1,2],[3,6],[4,7],[5,8]]=>2
[[1,4],[2,5],[3,7],[6,8]]=>2
[[1,3],[2,5],[4,7],[6,8]]=>2
[[1,2],[3,5],[4,7],[6,8]]=>2
[[1,3],[2,4],[5,7],[6,8]]=>2
[[1,2],[3,4],[5,7],[6,8]]=>2
[[1,4],[2,5],[3,6],[7,8]]=>2
[[1,3],[2,5],[4,6],[7,8]]=>2
[[1,2],[3,5],[4,6],[7,8]]=>2
[[1,3],[2,4],[5,6],[7,8]]=>2
[[1,2],[3,4],[5,6],[7,8]]=>2
[[1,6],[2,7],[3,8],[4],[5]]=>1
[[1,5],[2,7],[3,8],[4],[6]]=>2
[[1,4],[2,7],[3,8],[5],[6]]=>2
[[1,3],[2,7],[4,8],[5],[6]]=>2
[[1,2],[3,7],[4,8],[5],[6]]=>2
[[1,5],[2,6],[3,8],[4],[7]]=>2
[[1,4],[2,6],[3,8],[5],[7]]=>2
[[1,3],[2,6],[4,8],[5],[7]]=>2
[[1,2],[3,6],[4,8],[5],[7]]=>2
[[1,4],[2,5],[3,8],[6],[7]]=>2
[[1,3],[2,5],[4,8],[6],[7]]=>2
[[1,2],[3,5],[4,8],[6],[7]]=>2
[[1,3],[2,4],[5,8],[6],[7]]=>2
[[1,2],[3,4],[5,8],[6],[7]]=>2
[[1,5],[2,6],[3,7],[4],[8]]=>2
[[1,4],[2,6],[3,7],[5],[8]]=>2
[[1,3],[2,6],[4,7],[5],[8]]=>2
[[1,2],[3,6],[4,7],[5],[8]]=>2
[[1,4],[2,5],[3,7],[6],[8]]=>2
[[1,3],[2,5],[4,7],[6],[8]]=>2
[[1,2],[3,5],[4,7],[6],[8]]=>2
[[1,3],[2,4],[5,7],[6],[8]]=>2
[[1,2],[3,4],[5,7],[6],[8]]=>2
[[1,4],[2,5],[3,6],[7],[8]]=>2
[[1,3],[2,5],[4,6],[7],[8]]=>2
[[1,2],[3,5],[4,6],[7],[8]]=>2
[[1,3],[2,4],[5,6],[7],[8]]=>2
[[1,2],[3,4],[5,6],[7],[8]]=>2
[[1,7],[2,8],[3],[4],[5],[6]]=>1
[[1,6],[2,8],[3],[4],[5],[7]]=>2
[[1,5],[2,8],[3],[4],[6],[7]]=>2
[[1,4],[2,8],[3],[5],[6],[7]]=>2
[[1,3],[2,8],[4],[5],[6],[7]]=>2
[[1,2],[3,8],[4],[5],[6],[7]]=>2
[[1,6],[2,7],[3],[4],[5],[8]]=>2
[[1,5],[2,7],[3],[4],[6],[8]]=>2
[[1,4],[2,7],[3],[5],[6],[8]]=>2
[[1,3],[2,7],[4],[5],[6],[8]]=>2
[[1,2],[3,7],[4],[5],[6],[8]]=>2
[[1,5],[2,6],[3],[4],[7],[8]]=>2
[[1,4],[2,6],[3],[5],[7],[8]]=>2
[[1,3],[2,6],[4],[5],[7],[8]]=>2
[[1,2],[3,6],[4],[5],[7],[8]]=>2
[[1,4],[2,5],[3],[6],[7],[8]]=>2
[[1,3],[2,5],[4],[6],[7],[8]]=>2
[[1,2],[3,5],[4],[6],[7],[8]]=>2
[[1,3],[2,4],[5],[6],[7],[8]]=>2
[[1,2],[3,4],[5],[6],[7],[8]]=>2
[[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]]=>2
[[1,5],[2],[3],[4],[6],[7],[8]]=>2
[[1,4],[2],[3],[5],[6],[7],[8]]=>2
[[1,3],[2],[4],[5],[6],[7],[8]]=>2
[[1,2],[3],[4],[5],[6],[7],[8]]=>2
[[1],[2],[3],[4],[5],[6],[7],[8]]=>1
                    

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Description

The smallest shift such that the standard tableau is cylindric.
A standard tableau is $(d, L)$-cylindric, if it has $d$ rows and the entry in column $i$ of the last row is strictly smaller than the entry in column $i+L$ of the first row.

References

[1] Gessel, I. M., Krattenthaler, C. Cylindric partitions MathSciNet:1389777

Code

def statistic(T):
    if not T:
        return 0
    for i in range(len(T[0])-len(T[-1]), len(T[0])+1):
        if all(T[0][i+j] is not None
               and T[-1][j] < T[0][i+j] for j in range(min(len(T[-1]), len(T[0])-i))):
            return i

Created

Mar 11, 2026 at 16:00 by Martin Rubey

Updated

Mar 11, 2026 at 16:33 by Martin Rubey