- FindStat

edit this statistic or download as text // json

Identifier

St001804: Standard tableaux ⟶ ℤ

Values

=>
                            Cc0007;cc-rep
                            
                            
                            

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

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Description

The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.

Code

def statistic(T):
    if not T:
        return 0
    w = len(T[0])
    l = len(T)
    d = 0
    while d < l and len(T[d]) == w and T[0][w-1] < T[l-1-d][0]:
        d += 1
    return len(T)-d

Created

Jun 06, 2022 at 19:33 by Martin Rubey

Updated

Jun 06, 2022 at 19:33 by Martin Rubey