- FindStat

edit this statistic or download as text // json

Identifier

St000577: Set partitions ⟶ ℤ

Values

=>
                            Cc0009;cc-rep
                            
                            
                            

                        {{1,2}}=>0
{{1},{2}}=>1
{{1,2,3}}=>0
{{1,2},{3}}=>1
{{1,3},{2}}=>1
{{1},{2,3}}=>2
{{1},{2},{3}}=>3
{{1,2,3,4}}=>0
{{1,2,3},{4}}=>1
{{1,2,4},{3}}=>1
{{1,2},{3,4}}=>2
{{1,2},{3},{4}}=>3
{{1,3,4},{2}}=>2
{{1,3},{2,4}}=>1
{{1,3},{2},{4}}=>3
{{1,4},{2,3}}=>1
{{1},{2,3,4}}=>3
{{1},{2,3},{4}}=>4
{{1,4},{2},{3}}=>3
{{1},{2,4},{3}}=>4
{{1},{2},{3,4}}=>5
{{1},{2},{3},{4}}=>6
{{1,2,3,4,5}}=>0
{{1,2,3,4},{5}}=>1
{{1,2,3,5},{4}}=>1
{{1,2,3},{4,5}}=>2
{{1,2,3},{4},{5}}=>3
{{1,2,4,5},{3}}=>2
{{1,2,4},{3,5}}=>1
{{1,2,4},{3},{5}}=>3
{{1,2,5},{3,4}}=>1
{{1,2},{3,4,5}}=>3
{{1,2},{3,4},{5}}=>4
{{1,2,5},{3},{4}}=>3
{{1,2},{3,5},{4}}=>4
{{1,2},{3},{4,5}}=>5
{{1,2},{3},{4},{5}}=>6
{{1,3,4,5},{2}}=>3
{{1,3,4},{2,5}}=>1
{{1,3,4},{2},{5}}=>4
{{1,3,5},{2,4}}=>1
{{1,3},{2,4,5}}=>2
{{1,3},{2,4},{5}}=>3
{{1,3,5},{2},{4}}=>4
{{1,3},{2,5},{4}}=>3
{{1,3},{2},{4,5}}=>5
{{1,3},{2},{4},{5}}=>6
{{1,4,5},{2,3}}=>2
{{1,4},{2,3,5}}=>1
{{1,4},{2,3},{5}}=>3
{{1,5},{2,3,4}}=>1
{{1},{2,3,4,5}}=>4
{{1},{2,3,4},{5}}=>5
{{1,5},{2,3},{4}}=>3
{{1},{2,3,5},{4}}=>5
{{1},{2,3},{4,5}}=>6
{{1},{2,3},{4},{5}}=>7
{{1,4,5},{2},{3}}=>5
{{1,4},{2,5},{3}}=>3
{{1,4},{2},{3,5}}=>4
{{1,4},{2},{3},{5}}=>6
{{1,5},{2,4},{3}}=>3
{{1},{2,4,5},{3}}=>6
{{1},{2,4},{3,5}}=>5
{{1},{2,4},{3},{5}}=>7
{{1,5},{2},{3,4}}=>4
{{1},{2,5},{3,4}}=>5
{{1},{2},{3,4,5}}=>7
{{1},{2},{3,4},{5}}=>8
{{1,5},{2},{3},{4}}=>6
{{1},{2,5},{3},{4}}=>7
{{1},{2},{3,5},{4}}=>8
{{1},{2},{3},{4,5}}=>9
{{1},{2},{3},{4},{5}}=>10
{{1,2,3,4,5,6}}=>0
{{1,2,3,4,5},{6}}=>1
{{1,2,3,4,6},{5}}=>1
{{1,2,3,4},{5,6}}=>2
{{1,2,3,4},{5},{6}}=>3
{{1,2,3,5,6},{4}}=>2
{{1,2,3,5},{4,6}}=>1
{{1,2,3,5},{4},{6}}=>3
{{1,2,3,6},{4,5}}=>1
{{1,2,3},{4,5,6}}=>3
{{1,2,3},{4,5},{6}}=>4
{{1,2,3,6},{4},{5}}=>3
{{1,2,3},{4,6},{5}}=>4
{{1,2,3},{4},{5,6}}=>5
{{1,2,3},{4},{5},{6}}=>6
{{1,2,4,5,6},{3}}=>3
{{1,2,4,5},{3,6}}=>1
{{1,2,4,5},{3},{6}}=>4
{{1,2,4,6},{3,5}}=>1
{{1,2,4},{3,5,6}}=>2
{{1,2,4},{3,5},{6}}=>3
{{1,2,4,6},{3},{5}}=>4
{{1,2,4},{3,6},{5}}=>3
{{1,2,4},{3},{5,6}}=>5
{{1,2,4},{3},{5},{6}}=>6
{{1,2,5,6},{3,4}}=>2
{{1,2,5},{3,4,6}}=>1
{{1,2,5},{3,4},{6}}=>3
{{1,2,6},{3,4,5}}=>1
{{1,2},{3,4,5,6}}=>4
{{1,2},{3,4,5},{6}}=>5
{{1,2,6},{3,4},{5}}=>3
{{1,2},{3,4,6},{5}}=>5
{{1,2},{3,4},{5,6}}=>6
{{1,2},{3,4},{5},{6}}=>7
{{1,2,5,6},{3},{4}}=>5
{{1,2,5},{3,6},{4}}=>3
{{1,2,5},{3},{4,6}}=>4
{{1,2,5},{3},{4},{6}}=>6
{{1,2,6},{3,5},{4}}=>3
{{1,2},{3,5,6},{4}}=>6
{{1,2},{3,5},{4,6}}=>5
{{1,2},{3,5},{4},{6}}=>7
{{1,2,6},{3},{4,5}}=>4
{{1,2},{3,6},{4,5}}=>5
{{1,2},{3},{4,5,6}}=>7
{{1,2},{3},{4,5},{6}}=>8
{{1,2,6},{3},{4},{5}}=>6
{{1,2},{3,6},{4},{5}}=>7
{{1,2},{3},{4,6},{5}}=>8
{{1,2},{3},{4},{5,6}}=>9
{{1,2},{3},{4},{5},{6}}=>10
{{1,3,4,5,6},{2}}=>4
{{1,3,4,5},{2,6}}=>1
{{1,3,4,5},{2},{6}}=>5
{{1,3,4,6},{2,5}}=>1
{{1,3,4},{2,5,6}}=>2
{{1,3,4},{2,5},{6}}=>3
{{1,3,4,6},{2},{5}}=>5
{{1,3,4},{2,6},{5}}=>3
{{1,3,4},{2},{5,6}}=>6
{{1,3,4},{2},{5},{6}}=>7
{{1,3,5,6},{2,4}}=>2
{{1,3,5},{2,4,6}}=>1
{{1,3,5},{2,4},{6}}=>3
{{1,3,6},{2,4,5}}=>1
{{1,3},{2,4,5,6}}=>3
{{1,3},{2,4,5},{6}}=>4
{{1,3,6},{2,4},{5}}=>3
{{1,3},{2,4,6},{5}}=>4
{{1,3},{2,4},{5,6}}=>5
{{1,3},{2,4},{5},{6}}=>6
{{1,3,5,6},{2},{4}}=>6
{{1,3,5},{2,6},{4}}=>3
{{1,3,5},{2},{4,6}}=>5
{{1,3,5},{2},{4},{6}}=>7
{{1,3,6},{2,5},{4}}=>3
{{1,3},{2,5,6},{4}}=>5
{{1,3},{2,5},{4,6}}=>4
{{1,3},{2,5},{4},{6}}=>6
{{1,3,6},{2},{4,5}}=>5
{{1,3},{2,6},{4,5}}=>4
{{1,3},{2},{4,5,6}}=>7
{{1,3},{2},{4,5},{6}}=>8
{{1,3,6},{2},{4},{5}}=>7
{{1,3},{2,6},{4},{5}}=>6
{{1,3},{2},{4,6},{5}}=>8
{{1,3},{2},{4},{5,6}}=>9
{{1,3},{2},{4},{5},{6}}=>10
{{1,4,5,6},{2,3}}=>3
{{1,4,5},{2,3,6}}=>1
{{1,4,5},{2,3},{6}}=>4
{{1,4,6},{2,3,5}}=>1
{{1,4},{2,3,5,6}}=>2
{{1,4},{2,3,5},{6}}=>3
{{1,4,6},{2,3},{5}}=>4
{{1,4},{2,3,6},{5}}=>3
{{1,4},{2,3},{5,6}}=>5
{{1,4},{2,3},{5},{6}}=>6
{{1,5,6},{2,3,4}}=>2
{{1,5},{2,3,4,6}}=>1
{{1,5},{2,3,4},{6}}=>3
{{1,6},{2,3,4,5}}=>1
{{1},{2,3,4,5,6}}=>5
{{1},{2,3,4,5},{6}}=>6
{{1,6},{2,3,4},{5}}=>3
{{1},{2,3,4,6},{5}}=>6
{{1},{2,3,4},{5,6}}=>7
{{1},{2,3,4},{5},{6}}=>8
{{1,5,6},{2,3},{4}}=>5
{{1,5},{2,3,6},{4}}=>3
{{1,5},{2,3},{4,6}}=>4
{{1,5},{2,3},{4},{6}}=>6
{{1,6},{2,3,5},{4}}=>3
{{1},{2,3,5,6},{4}}=>7
{{1},{2,3,5},{4,6}}=>6
{{1},{2,3,5},{4},{6}}=>8
{{1,6},{2,3},{4,5}}=>4
{{1},{2,3,6},{4,5}}=>6
{{1},{2,3},{4,5,6}}=>8
{{1},{2,3},{4,5},{6}}=>9
{{1,6},{2,3},{4},{5}}=>6
{{1},{2,3,6},{4},{5}}=>8
{{1},{2,3},{4,6},{5}}=>9
{{1},{2,3},{4},{5,6}}=>10
{{1},{2,3},{4},{5},{6}}=>11
{{1,4,5,6},{2},{3}}=>7
{{1,4,5},{2,6},{3}}=>4
{{1,4,5},{2},{3,6}}=>5
{{1,4,5},{2},{3},{6}}=>8
{{1,4,6},{2,5},{3}}=>4
{{1,4},{2,5,6},{3}}=>5
{{1,4},{2,5},{3,6}}=>3
{{1,4},{2,5},{3},{6}}=>6
{{1,4,6},{2},{3,5}}=>5
{{1,4},{2,6},{3,5}}=>3
{{1,4},{2},{3,5,6}}=>6
{{1,4},{2},{3,5},{6}}=>7
{{1,4,6},{2},{3},{5}}=>8
{{1,4},{2,6},{3},{5}}=>6
{{1,4},{2},{3,6},{5}}=>7
{{1,4},{2},{3},{5,6}}=>9
{{1,4},{2},{3},{5},{6}}=>10
{{1,5,6},{2,4},{3}}=>5
{{1,5},{2,4,6},{3}}=>4
{{1,5},{2,4},{3,6}}=>3
{{1,5},{2,4},{3},{6}}=>6
{{1,6},{2,4,5},{3}}=>4
{{1},{2,4,5,6},{3}}=>8
{{1},{2,4,5},{3,6}}=>6
{{1},{2,4,5},{3},{6}}=>9
{{1,6},{2,4},{3,5}}=>3
{{1},{2,4,6},{3,5}}=>6
{{1},{2,4},{3,5,6}}=>7
{{1},{2,4},{3,5},{6}}=>8
{{1,6},{2,4},{3},{5}}=>6
{{1},{2,4,6},{3},{5}}=>9
{{1},{2,4},{3,6},{5}}=>8
{{1},{2,4},{3},{5,6}}=>10
{{1},{2,4},{3},{5},{6}}=>11
{{1,5,6},{2},{3,4}}=>6
{{1,5},{2,6},{3,4}}=>3
{{1,5},{2},{3,4,6}}=>5
{{1,5},{2},{3,4},{6}}=>7
{{1,6},{2,5},{3,4}}=>3
{{1},{2,5,6},{3,4}}=>7
{{1},{2,5},{3,4,6}}=>6
{{1},{2,5},{3,4},{6}}=>8
{{1,6},{2},{3,4,5}}=>5
{{1},{2,6},{3,4,5}}=>6
{{1},{2},{3,4,5,6}}=>9
{{1},{2},{3,4,5},{6}}=>10
{{1,6},{2},{3,4},{5}}=>7
{{1},{2,6},{3,4},{5}}=>8
{{1},{2},{3,4,6},{5}}=>10
{{1},{2},{3,4},{5,6}}=>11
{{1},{2},{3,4},{5},{6}}=>12
{{1,5,6},{2},{3},{4}}=>9
{{1,5},{2,6},{3},{4}}=>6
{{1,5},{2},{3,6},{4}}=>7
{{1,5},{2},{3},{4,6}}=>8
{{1,5},{2},{3},{4},{6}}=>10
{{1,6},{2,5},{3},{4}}=>6
{{1},{2,5,6},{3},{4}}=>10
{{1},{2,5},{3,6},{4}}=>8
{{1},{2,5},{3},{4,6}}=>9
{{1},{2,5},{3},{4},{6}}=>11
{{1,6},{2},{3,5},{4}}=>7
{{1},{2,6},{3,5},{4}}=>8
{{1},{2},{3,5,6},{4}}=>11
{{1},{2},{3,5},{4,6}}=>10
{{1},{2},{3,5},{4},{6}}=>12
{{1,6},{2},{3},{4,5}}=>8
{{1},{2,6},{3},{4,5}}=>9
{{1},{2},{3,6},{4,5}}=>10
{{1},{2},{3},{4,5,6}}=>12
{{1},{2},{3},{4,5},{6}}=>13
{{1,6},{2},{3},{4},{5}}=>10
{{1},{2,6},{3},{4},{5}}=>11
{{1},{2},{3,6},{4},{5}}=>12
{{1},{2},{3},{4,6},{5}}=>13
{{1},{2},{3},{4},{5,6}}=>14
{{1},{2},{3},{4},{5},{6}}=>15
{{1,2,3,4,5,6,7}}=>0
{{1,2,3,4,5,6},{7}}=>1
{{1,2,3,4,5,7},{6}}=>1
{{1,2,3,4,5},{6,7}}=>2
{{1,2,3,4,5},{6},{7}}=>3
{{1,2,3,4,6,7},{5}}=>2
{{1,2,3,4,6},{5,7}}=>1
{{1,2,3,4,6},{5},{7}}=>3
{{1,2,3,4,7},{5,6}}=>1
{{1,2,3,4},{5,6,7}}=>3
{{1,2,3,4},{5,6},{7}}=>4
{{1,2,3,4,7},{5},{6}}=>3
{{1,2,3,4},{5,7},{6}}=>4
{{1,2,3,4},{5},{6,7}}=>5
{{1,2,3,4},{5},{6},{7}}=>6
{{1,2,3,5,6,7},{4}}=>3
{{1,2,3,5,6},{4,7}}=>1
{{1,2,3,5,6},{4},{7}}=>4
{{1,2,3,5,7},{4,6}}=>1
{{1,2,3,5},{4,6,7}}=>2
{{1,2,3,5},{4,6},{7}}=>3
{{1,2,3,5,7},{4},{6}}=>4
{{1,2,3,5},{4,7},{6}}=>3
{{1,2,3,5},{4},{6,7}}=>5
{{1,2,3,5},{4},{6},{7}}=>6
{{1,2,3,6,7},{4,5}}=>2
{{1,2,3,6},{4,5,7}}=>1
{{1,2,3,6},{4,5},{7}}=>3
{{1,2,3,7},{4,5,6}}=>1
{{1,2,3},{4,5,6,7}}=>4
{{1,2,3},{4,5,6},{7}}=>5
{{1,2,3,7},{4,5},{6}}=>3
{{1,2,3},{4,5,7},{6}}=>5
{{1,2,3},{4,5},{6,7}}=>6
{{1,2,3},{4,5},{6},{7}}=>7
{{1,2,3,6,7},{4},{5}}=>5
{{1,2,3,6},{4,7},{5}}=>3
{{1,2,3,6},{4},{5,7}}=>4
{{1,2,3,6},{4},{5},{7}}=>6
{{1,2,3,7},{4,6},{5}}=>3
{{1,2,3},{4,6,7},{5}}=>6
{{1,2,3},{4,6},{5,7}}=>5
{{1,2,3},{4,6},{5},{7}}=>7
{{1,2,3,7},{4},{5,6}}=>4
{{1,2,3},{4,7},{5,6}}=>5
{{1,2,3},{4},{5,6,7}}=>7
{{1,2,3},{4},{5,6},{7}}=>8
{{1,2,3,7},{4},{5},{6}}=>6
{{1,2,3},{4,7},{5},{6}}=>7
{{1,2,3},{4},{5,7},{6}}=>8
{{1,2,3},{4},{5},{6,7}}=>9
{{1,2,3},{4},{5},{6},{7}}=>10
{{1,2,4,5,6,7},{3}}=>4
{{1,2,4,5,6},{3,7}}=>1
{{1,2,4,5,6},{3},{7}}=>5
{{1,2,4,5,7},{3,6}}=>1
{{1,2,4,5},{3,6,7}}=>2
{{1,2,4,5},{3,6},{7}}=>3
{{1,2,4,5,7},{3},{6}}=>5
{{1,2,4,5},{3,7},{6}}=>3
{{1,2,4,5},{3},{6,7}}=>6
{{1,2,4,5},{3},{6},{7}}=>7
{{1,2,4,6,7},{3,5}}=>2
{{1,2,4,6},{3,5,7}}=>1
{{1,2,4,6},{3,5},{7}}=>3
{{1,2,4,7},{3,5,6}}=>1
{{1,2,4},{3,5,6,7}}=>3
{{1,2,4},{3,5,6},{7}}=>4
{{1,2,4,7},{3,5},{6}}=>3
{{1,2,4},{3,5,7},{6}}=>4
{{1,2,4},{3,5},{6,7}}=>5
{{1,2,4},{3,5},{6},{7}}=>6
{{1,2,4,6,7},{3},{5}}=>6
{{1,2,4,6},{3,7},{5}}=>3
{{1,2,4,6},{3},{5,7}}=>5
{{1,2,4,6},{3},{5},{7}}=>7
{{1,2,4,7},{3,6},{5}}=>3
{{1,2,4},{3,6,7},{5}}=>5
{{1,2,4},{3,6},{5,7}}=>4
{{1,2,4},{3,6},{5},{7}}=>6
{{1,2,4,7},{3},{5,6}}=>5
{{1,2,4},{3,7},{5,6}}=>4
{{1,2,4},{3},{5,6,7}}=>7
{{1,2,4},{3},{5,6},{7}}=>8
{{1,2,4,7},{3},{5},{6}}=>7
{{1,2,4},{3,7},{5},{6}}=>6
{{1,2,4},{3},{5,7},{6}}=>8
{{1,2,4},{3},{5},{6,7}}=>9
{{1,2,4},{3},{5},{6},{7}}=>10
{{1,2,5,6,7},{3,4}}=>3
{{1,2,5,6},{3,4,7}}=>1
{{1,2,5,6},{3,4},{7}}=>4
{{1,2,5,7},{3,4,6}}=>1
{{1,2,5},{3,4,6,7}}=>2
{{1,2,5},{3,4,6},{7}}=>3
{{1,2,5,7},{3,4},{6}}=>4
{{1,2,5},{3,4,7},{6}}=>3
{{1,2,5},{3,4},{6,7}}=>5
{{1,2,5},{3,4},{6},{7}}=>6
{{1,2,6,7},{3,4,5}}=>2
{{1,2,6},{3,4,5,7}}=>1
{{1,2,6},{3,4,5},{7}}=>3
{{1,2,7},{3,4,5,6}}=>1
{{1,2},{3,4,5,6,7}}=>5
{{1,2},{3,4,5,6},{7}}=>6
{{1,2,7},{3,4,5},{6}}=>3
{{1,2},{3,4,5,7},{6}}=>6
{{1,2},{3,4,5},{6,7}}=>7
{{1,2},{3,4,5},{6},{7}}=>8
{{1,2,6,7},{3,4},{5}}=>5
{{1,2,6},{3,4,7},{5}}=>3
{{1,2,6},{3,4},{5,7}}=>4
{{1,2,6},{3,4},{5},{7}}=>6
{{1,2,7},{3,4,6},{5}}=>3
{{1,2},{3,4,6,7},{5}}=>7
{{1,2},{3,4,6},{5,7}}=>6
{{1,2},{3,4,6},{5},{7}}=>8
{{1,2,7},{3,4},{5,6}}=>4
{{1,2},{3,4,7},{5,6}}=>6
{{1,2},{3,4},{5,6,7}}=>8
{{1,2},{3,4},{5,6},{7}}=>9
{{1,2,7},{3,4},{5},{6}}=>6
{{1,2},{3,4,7},{5},{6}}=>8
{{1,2},{3,4},{5,7},{6}}=>9
{{1,2},{3,4},{5},{6,7}}=>10
{{1,2},{3,4},{5},{6},{7}}=>11
{{1,2,5,6,7},{3},{4}}=>7
{{1,2,5,6},{3,7},{4}}=>4
{{1,2,5,6},{3},{4,7}}=>5
{{1,2,5,6},{3},{4},{7}}=>8
{{1,2,5,7},{3,6},{4}}=>4
{{1,2,5},{3,6,7},{4}}=>5
{{1,2,5},{3,6},{4,7}}=>3
{{1,2,5},{3,6},{4},{7}}=>6
{{1,2,5,7},{3},{4,6}}=>5
{{1,2,5},{3,7},{4,6}}=>3
{{1,2,5},{3},{4,6,7}}=>6
{{1,2,5},{3},{4,6},{7}}=>7
{{1,2,5,7},{3},{4},{6}}=>8
{{1,2,5},{3,7},{4},{6}}=>6
{{1,2,5},{3},{4,7},{6}}=>7
{{1,2,5},{3},{4},{6,7}}=>9
{{1,2,5},{3},{4},{6},{7}}=>10
{{1,2,6,7},{3,5},{4}}=>5
{{1,2,6},{3,5,7},{4}}=>4
{{1,2,6},{3,5},{4,7}}=>3
{{1,2,6},{3,5},{4},{7}}=>6
{{1,2,7},{3,5,6},{4}}=>4
{{1,2},{3,5,6,7},{4}}=>8
{{1,2},{3,5,6},{4,7}}=>6
{{1,2},{3,5,6},{4},{7}}=>9
{{1,2,7},{3,5},{4,6}}=>3
{{1,2},{3,5,7},{4,6}}=>6
{{1,2},{3,5},{4,6,7}}=>7
{{1,2},{3,5},{4,6},{7}}=>8
{{1,2,7},{3,5},{4},{6}}=>6
{{1,2},{3,5,7},{4},{6}}=>9
{{1,2},{3,5},{4,7},{6}}=>8
{{1,2},{3,5},{4},{6,7}}=>10
{{1,2},{3,5},{4},{6},{7}}=>11
{{1,2,6,7},{3},{4,5}}=>6
{{1,2,6},{3,7},{4,5}}=>3
{{1,2,6},{3},{4,5,7}}=>5
{{1,2,6},{3},{4,5},{7}}=>7
{{1,2,7},{3,6},{4,5}}=>3
{{1,2},{3,6,7},{4,5}}=>7
{{1,2},{3,6},{4,5,7}}=>6
{{1,2},{3,6},{4,5},{7}}=>8
{{1,2,7},{3},{4,5,6}}=>5
{{1,2},{3,7},{4,5,6}}=>6
{{1,2},{3},{4,5,6,7}}=>9
{{1,2},{3},{4,5,6},{7}}=>10
{{1,2,7},{3},{4,5},{6}}=>7
{{1,2},{3,7},{4,5},{6}}=>8
{{1,2},{3},{4,5,7},{6}}=>10
{{1,2},{3},{4,5},{6,7}}=>11
{{1,2},{3},{4,5},{6},{7}}=>12
{{1,2,6,7},{3},{4},{5}}=>9
{{1,2,6},{3,7},{4},{5}}=>6
{{1,2,6},{3},{4,7},{5}}=>7
{{1,2,6},{3},{4},{5,7}}=>8
{{1,2,6},{3},{4},{5},{7}}=>10
{{1,2,7},{3,6},{4},{5}}=>6
{{1,2},{3,6,7},{4},{5}}=>10
{{1,2},{3,6},{4,7},{5}}=>8
{{1,2},{3,6},{4},{5,7}}=>9
{{1,2},{3,6},{4},{5},{7}}=>11
{{1,2,7},{3},{4,6},{5}}=>7
{{1,2},{3,7},{4,6},{5}}=>8
{{1,2},{3},{4,6,7},{5}}=>11
{{1,2},{3},{4,6},{5,7}}=>10
{{1,2},{3},{4,6},{5},{7}}=>12
{{1,2,7},{3},{4},{5,6}}=>8
{{1,2},{3,7},{4},{5,6}}=>9
{{1,2},{3},{4,7},{5,6}}=>10
{{1,2},{3},{4},{5,6,7}}=>12
{{1,2},{3},{4},{5,6},{7}}=>13
{{1,2,7},{3},{4},{5},{6}}=>10
{{1,2},{3,7},{4},{5},{6}}=>11
{{1,2},{3},{4,7},{5},{6}}=>12
{{1,2},{3},{4},{5,7},{6}}=>13
{{1,2},{3},{4},{5},{6,7}}=>14
{{1,2},{3},{4},{5},{6},{7}}=>15
{{1,3,4,5,6,7},{2}}=>5
{{1,3,4,5,6},{2,7}}=>1
{{1,3,4,5,6},{2},{7}}=>6
{{1,3,4,5,7},{2,6}}=>1
{{1,3,4,5},{2,6,7}}=>2
{{1,3,4,5},{2,6},{7}}=>3
{{1,3,4,5,7},{2},{6}}=>6
{{1,3,4,5},{2,7},{6}}=>3
{{1,3,4,5},{2},{6,7}}=>7
{{1,3,4,5},{2},{6},{7}}=>8
{{1,3,4,6,7},{2,5}}=>2
{{1,3,4,6},{2,5,7}}=>1
{{1,3,4,6},{2,5},{7}}=>3
{{1,3,4,7},{2,5,6}}=>1
{{1,3,4},{2,5,6,7}}=>3
{{1,3,4},{2,5,6},{7}}=>4
{{1,3,4,7},{2,5},{6}}=>3
{{1,3,4},{2,5,7},{6}}=>4
{{1,3,4},{2,5},{6,7}}=>5
{{1,3,4},{2,5},{6},{7}}=>6
{{1,3,4,6,7},{2},{5}}=>7
{{1,3,4,6},{2,7},{5}}=>3
{{1,3,4,6},{2},{5,7}}=>6
{{1,3,4,6},{2},{5},{7}}=>8
{{1,3,4,7},{2,6},{5}}=>3
{{1,3,4},{2,6,7},{5}}=>5
{{1,3,4},{2,6},{5,7}}=>4
{{1,3,4},{2,6},{5},{7}}=>6
{{1,3,4,7},{2},{5,6}}=>6
{{1,3,4},{2,7},{5,6}}=>4
{{1,3,4},{2},{5,6,7}}=>8
{{1,3,4},{2},{5,6},{7}}=>9
{{1,3,4,7},{2},{5},{6}}=>8
{{1,3,4},{2,7},{5},{6}}=>6
{{1,3,4},{2},{5,7},{6}}=>9
{{1,3,4},{2},{5},{6,7}}=>10
{{1,3,4},{2},{5},{6},{7}}=>11
{{1,3,5,6,7},{2,4}}=>3
{{1,3,5,6},{2,4,7}}=>1
{{1,3,5,6},{2,4},{7}}=>4
{{1,3,5,7},{2,4,6}}=>1
{{1,3,5},{2,4,6,7}}=>2
{{1,3,5},{2,4,6},{7}}=>3
{{1,3,5,7},{2,4},{6}}=>4
{{1,3,5},{2,4,7},{6}}=>3
{{1,3,5},{2,4},{6,7}}=>5
{{1,3,5},{2,4},{6},{7}}=>6
{{1,3,6,7},{2,4,5}}=>2
{{1,3,6},{2,4,5,7}}=>1
{{1,3,6},{2,4,5},{7}}=>3
{{1,3,7},{2,4,5,6}}=>1
{{1,3},{2,4,5,6,7}}=>4
{{1,3},{2,4,5,6},{7}}=>5
{{1,3,7},{2,4,5},{6}}=>3
{{1,3},{2,4,5,7},{6}}=>5
{{1,3},{2,4,5},{6,7}}=>6
{{1,3},{2,4,5},{6},{7}}=>7
{{1,3,6,7},{2,4},{5}}=>5
{{1,3,6},{2,4,7},{5}}=>3
{{1,3,6},{2,4},{5,7}}=>4
{{1,3,6},{2,4},{5},{7}}=>6
{{1,3,7},{2,4,6},{5}}=>3
{{1,3},{2,4,6,7},{5}}=>6
{{1,3},{2,4,6},{5,7}}=>5
{{1,3},{2,4,6},{5},{7}}=>7
{{1,3,7},{2,4},{5,6}}=>4
{{1,3},{2,4,7},{5,6}}=>5
{{1,3},{2,4},{5,6,7}}=>7
{{1,3},{2,4},{5,6},{7}}=>8
{{1,3,7},{2,4},{5},{6}}=>6
{{1,3},{2,4,7},{5},{6}}=>7
{{1,3},{2,4},{5,7},{6}}=>8
{{1,3},{2,4},{5},{6,7}}=>9
{{1,3},{2,4},{5},{6},{7}}=>10
{{1,3,5,6,7},{2},{4}}=>8
{{1,3,5,6},{2,7},{4}}=>4
{{1,3,5,6},{2},{4,7}}=>6
{{1,3,5,6},{2},{4},{7}}=>9
{{1,3,5,7},{2,6},{4}}=>4
{{1,3,5},{2,6,7},{4}}=>5
{{1,3,5},{2,6},{4,7}}=>3
{{1,3,5},{2,6},{4},{7}}=>6
{{1,3,5,7},{2},{4,6}}=>6
{{1,3,5},{2,7},{4,6}}=>3
{{1,3,5},{2},{4,6,7}}=>7
{{1,3,5},{2},{4,6},{7}}=>8
{{1,3,5,7},{2},{4},{6}}=>9
{{1,3,5},{2,7},{4},{6}}=>6
{{1,3,5},{2},{4,7},{6}}=>8
{{1,3,5},{2},{4},{6,7}}=>10
{{1,3,5},{2},{4},{6},{7}}=>11
{{1,3,6,7},{2,5},{4}}=>5
{{1,3,6},{2,5,7},{4}}=>4
{{1,3,6},{2,5},{4,7}}=>3
{{1,3,6},{2,5},{4},{7}}=>6
{{1,3,7},{2,5,6},{4}}=>4
{{1,3},{2,5,6,7},{4}}=>7
{{1,3},{2,5,6},{4,7}}=>5
{{1,3},{2,5,6},{4},{7}}=>8
{{1,3,7},{2,5},{4,6}}=>3
{{1,3},{2,5,7},{4,6}}=>5
{{1,3},{2,5},{4,6,7}}=>6
{{1,3},{2,5},{4,6},{7}}=>7
{{1,3,7},{2,5},{4},{6}}=>6
{{1,3},{2,5,7},{4},{6}}=>8
{{1,3},{2,5},{4,7},{6}}=>7
{{1,3},{2,5},{4},{6,7}}=>9
{{1,3},{2,5},{4},{6},{7}}=>10
{{1,3,6,7},{2},{4,5}}=>7
{{1,3,6},{2,7},{4,5}}=>3
{{1,3,6},{2},{4,5,7}}=>6
{{1,3,6},{2},{4,5},{7}}=>8
{{1,3,7},{2,6},{4,5}}=>3
{{1,3},{2,6,7},{4,5}}=>6
{{1,3},{2,6},{4,5,7}}=>5
{{1,3},{2,6},{4,5},{7}}=>7
{{1,3,7},{2},{4,5,6}}=>6
{{1,3},{2,7},{4,5,6}}=>5
{{1,3},{2},{4,5,6,7}}=>9
{{1,3},{2},{4,5,6},{7}}=>10
{{1,3,7},{2},{4,5},{6}}=>8
{{1,3},{2,7},{4,5},{6}}=>7
{{1,3},{2},{4,5,7},{6}}=>10
{{1,3},{2},{4,5},{6,7}}=>11
{{1,3},{2},{4,5},{6},{7}}=>12
{{1,3,6,7},{2},{4},{5}}=>10
{{1,3,6},{2,7},{4},{5}}=>6
{{1,3,6},{2},{4,7},{5}}=>8
{{1,3,6},{2},{4},{5,7}}=>9
{{1,3,6},{2},{4},{5},{7}}=>11
{{1,3,7},{2,6},{4},{5}}=>6
{{1,3},{2,6,7},{4},{5}}=>9
{{1,3},{2,6},{4,7},{5}}=>7
{{1,3},{2,6},{4},{5,7}}=>8
{{1,3},{2,6},{4},{5},{7}}=>10
{{1,3,7},{2},{4,6},{5}}=>8
{{1,3},{2,7},{4,6},{5}}=>7
{{1,3},{2},{4,6,7},{5}}=>11
{{1,3},{2},{4,6},{5,7}}=>10
{{1,3},{2},{4,6},{5},{7}}=>12
{{1,3,7},{2},{4},{5,6}}=>9
{{1,3},{2,7},{4},{5,6}}=>8
{{1,3},{2},{4,7},{5,6}}=>10
{{1,3},{2},{4},{5,6,7}}=>12
{{1,3},{2},{4},{5,6},{7}}=>13
{{1,3,7},{2},{4},{5},{6}}=>11
{{1,3},{2,7},{4},{5},{6}}=>10
{{1,3},{2},{4,7},{5},{6}}=>12
{{1,3},{2},{4},{5,7},{6}}=>13
{{1,3},{2},{4},{5},{6,7}}=>14
{{1,3},{2},{4},{5},{6},{7}}=>15
{{1,4,5,6,7},{2,3}}=>4
{{1,4,5,6},{2,3,7}}=>1
{{1,4,5,6},{2,3},{7}}=>5
{{1,4,5,7},{2,3,6}}=>1
{{1,4,5},{2,3,6,7}}=>2
{{1,4,5},{2,3,6},{7}}=>3
{{1,4,5,7},{2,3},{6}}=>5
{{1,4,5},{2,3,7},{6}}=>3
{{1,4,5},{2,3},{6,7}}=>6
{{1,4,5},{2,3},{6},{7}}=>7
{{1,4,6,7},{2,3,5}}=>2
{{1,4,6},{2,3,5,7}}=>1
{{1,4,6},{2,3,5},{7}}=>3
{{1,4,7},{2,3,5,6}}=>1
{{1,4},{2,3,5,6,7}}=>3
{{1,4},{2,3,5,6},{7}}=>4
{{1,4,7},{2,3,5},{6}}=>3
{{1,4},{2,3,5,7},{6}}=>4
{{1,4},{2,3,5},{6,7}}=>5
{{1,4},{2,3,5},{6},{7}}=>6
{{1,4,6,7},{2,3},{5}}=>6
{{1,4,6},{2,3,7},{5}}=>3
{{1,4,6},{2,3},{5,7}}=>5
{{1,4,6},{2,3},{5},{7}}=>7
{{1,4,7},{2,3,6},{5}}=>3
{{1,4},{2,3,6,7},{5}}=>5
{{1,4},{2,3,6},{5,7}}=>4
{{1,4},{2,3,6},{5},{7}}=>6
{{1,4,7},{2,3},{5,6}}=>5
{{1,4},{2,3,7},{5,6}}=>4
{{1,4},{2,3},{5,6,7}}=>7
{{1,4},{2,3},{5,6},{7}}=>8
{{1,4,7},{2,3},{5},{6}}=>7
{{1,4},{2,3,7},{5},{6}}=>6
{{1,4},{2,3},{5,7},{6}}=>8
{{1,4},{2,3},{5},{6,7}}=>9
{{1,4},{2,3},{5},{6},{7}}=>10
{{1,5,6,7},{2,3,4}}=>3
{{1,5,6},{2,3,4,7}}=>1
{{1,5,6},{2,3,4},{7}}=>4
{{1,5,7},{2,3,4,6}}=>1
{{1,5},{2,3,4,6,7}}=>2
{{1,5},{2,3,4,6},{7}}=>3
{{1,5,7},{2,3,4},{6}}=>4
{{1,5},{2,3,4,7},{6}}=>3
{{1,5},{2,3,4},{6,7}}=>5
{{1,5},{2,3,4},{6},{7}}=>6
{{1,6,7},{2,3,4,5}}=>2
{{1,6},{2,3,4,5,7}}=>1
{{1,6},{2,3,4,5},{7}}=>3
{{1,7},{2,3,4,5,6}}=>1
{{1},{2,3,4,5,6,7}}=>6
{{1},{2,3,4,5,6},{7}}=>7
{{1,7},{2,3,4,5},{6}}=>3
{{1},{2,3,4,5,7},{6}}=>7
{{1},{2,3,4,5},{6,7}}=>8
{{1},{2,3,4,5},{6},{7}}=>9
{{1,6,7},{2,3,4},{5}}=>5
{{1,6},{2,3,4,7},{5}}=>3
{{1,6},{2,3,4},{5,7}}=>4
{{1,6},{2,3,4},{5},{7}}=>6
{{1,7},{2,3,4,6},{5}}=>3
{{1},{2,3,4,6,7},{5}}=>8
{{1},{2,3,4,6},{5,7}}=>7
{{1},{2,3,4,6},{5},{7}}=>9
{{1,7},{2,3,4},{5,6}}=>4
{{1},{2,3,4,7},{5,6}}=>7
{{1},{2,3,4},{5,6,7}}=>9
{{1},{2,3,4},{5,6},{7}}=>10
{{1,7},{2,3,4},{5},{6}}=>6
{{1},{2,3,4,7},{5},{6}}=>9
{{1},{2,3,4},{5,7},{6}}=>10
{{1},{2,3,4},{5},{6,7}}=>11
{{1},{2,3,4},{5},{6},{7}}=>12
{{1,5,6,7},{2,3},{4}}=>7
{{1,5,6},{2,3,7},{4}}=>4
{{1,5,6},{2,3},{4,7}}=>5
{{1,5,6},{2,3},{4},{7}}=>8
{{1,5,7},{2,3,6},{4}}=>4
{{1,5},{2,3,6,7},{4}}=>5
{{1,5},{2,3,6},{4,7}}=>3
{{1,5},{2,3,6},{4},{7}}=>6
{{1,5,7},{2,3},{4,6}}=>5
{{1,5},{2,3,7},{4,6}}=>3
{{1,5},{2,3},{4,6,7}}=>6
{{1,5},{2,3},{4,6},{7}}=>7
{{1,5,7},{2,3},{4},{6}}=>8
{{1,5},{2,3,7},{4},{6}}=>6
{{1,5},{2,3},{4,7},{6}}=>7
{{1,5},{2,3},{4},{6,7}}=>9
{{1,5},{2,3},{4},{6},{7}}=>10
{{1,6,7},{2,3,5},{4}}=>5
{{1,6},{2,3,5,7},{4}}=>4
{{1,6},{2,3,5},{4,7}}=>3
{{1,6},{2,3,5},{4},{7}}=>6
{{1,7},{2,3,5,6},{4}}=>4
{{1},{2,3,5,6,7},{4}}=>9
{{1},{2,3,5,6},{4,7}}=>7
{{1},{2,3,5,6},{4},{7}}=>10
{{1,7},{2,3,5},{4,6}}=>3
{{1},{2,3,5,7},{4,6}}=>7
{{1},{2,3,5},{4,6,7}}=>8
{{1},{2,3,5},{4,6},{7}}=>9
{{1,7},{2,3,5},{4},{6}}=>6
{{1},{2,3,5,7},{4},{6}}=>10
{{1},{2,3,5},{4,7},{6}}=>9
{{1},{2,3,5},{4},{6,7}}=>11
{{1},{2,3,5},{4},{6},{7}}=>12
{{1,6,7},{2,3},{4,5}}=>6
{{1,6},{2,3,7},{4,5}}=>3
{{1,6},{2,3},{4,5,7}}=>5
{{1,6},{2,3},{4,5},{7}}=>7
{{1,7},{2,3,6},{4,5}}=>3
{{1},{2,3,6,7},{4,5}}=>8
{{1},{2,3,6},{4,5,7}}=>7
{{1},{2,3,6},{4,5},{7}}=>9
{{1,7},{2,3},{4,5,6}}=>5
{{1},{2,3,7},{4,5,6}}=>7
{{1},{2,3},{4,5,6,7}}=>10
{{1},{2,3},{4,5,6},{7}}=>11
{{1,7},{2,3},{4,5},{6}}=>7
{{1},{2,3,7},{4,5},{6}}=>9
{{1},{2,3},{4,5,7},{6}}=>11
{{1},{2,3},{4,5},{6,7}}=>12
{{1},{2,3},{4,5},{6},{7}}=>13
{{1,6,7},{2,3},{4},{5}}=>9
{{1,6},{2,3,7},{4},{5}}=>6
{{1,6},{2,3},{4,7},{5}}=>7
{{1,6},{2,3},{4},{5,7}}=>8
{{1,6},{2,3},{4},{5},{7}}=>10
{{1,7},{2,3,6},{4},{5}}=>6
{{1},{2,3,6,7},{4},{5}}=>11
{{1},{2,3,6},{4,7},{5}}=>9
{{1},{2,3,6},{4},{5,7}}=>10
{{1},{2,3,6},{4},{5},{7}}=>12
{{1,7},{2,3},{4,6},{5}}=>7
{{1},{2,3,7},{4,6},{5}}=>9
{{1},{2,3},{4,6,7},{5}}=>12
{{1},{2,3},{4,6},{5,7}}=>11
{{1},{2,3},{4,6},{5},{7}}=>13
{{1,7},{2,3},{4},{5,6}}=>8
{{1},{2,3,7},{4},{5,6}}=>10
{{1},{2,3},{4,7},{5,6}}=>11
{{1},{2,3},{4},{5,6,7}}=>13
{{1},{2,3},{4},{5,6},{7}}=>14
{{1,7},{2,3},{4},{5},{6}}=>10
{{1},{2,3,7},{4},{5},{6}}=>12
{{1},{2,3},{4,7},{5},{6}}=>13
{{1},{2,3},{4},{5,7},{6}}=>14
{{1},{2,3},{4},{5},{6,7}}=>15
{{1},{2,3},{4},{5},{6},{7}}=>16
{{1,4,5,6,7},{2},{3}}=>9
{{1,4,5,6},{2,7},{3}}=>5
{{1,4,5,6},{2},{3,7}}=>6
{{1,4,5,6},{2},{3},{7}}=>10
{{1,4,5,7},{2,6},{3}}=>5
{{1,4,5},{2,6,7},{3}}=>6
{{1,4,5},{2,6},{3,7}}=>3
{{1,4,5},{2,6},{3},{7}}=>7
{{1,4,5,7},{2},{3,6}}=>6
{{1,4,5},{2,7},{3,6}}=>3
{{1,4,5},{2},{3,6,7}}=>7
{{1,4,5},{2},{3,6},{7}}=>8
{{1,4,5,7},{2},{3},{6}}=>10
{{1,4,5},{2,7},{3},{6}}=>7
{{1,4,5},{2},{3,7},{6}}=>8
{{1,4,5},{2},{3},{6,7}}=>11
{{1,4,5},{2},{3},{6},{7}}=>12
{{1,4,6,7},{2,5},{3}}=>6
{{1,4,6},{2,5,7},{3}}=>5
{{1,4,6},{2,5},{3,7}}=>3
{{1,4,6},{2,5},{3},{7}}=>7
{{1,4,7},{2,5,6},{3}}=>5
{{1,4},{2,5,6,7},{3}}=>7
{{1,4},{2,5,6},{3,7}}=>4
{{1,4},{2,5,6},{3},{7}}=>8
{{1,4,7},{2,5},{3,6}}=>3
{{1,4},{2,5,7},{3,6}}=>4
{{1,4},{2,5},{3,6,7}}=>5
{{1,4},{2,5},{3,6},{7}}=>6
{{1,4,7},{2,5},{3},{6}}=>7
{{1,4},{2,5,7},{3},{6}}=>8
{{1,4},{2,5},{3,7},{6}}=>6
{{1,4},{2,5},{3},{6,7}}=>9
{{1,4},{2,5},{3},{6},{7}}=>10
{{1,4,6,7},{2},{3,5}}=>7
{{1,4,6},{2,7},{3,5}}=>3
{{1,4,6},{2},{3,5,7}}=>6
{{1,4,6},{2},{3,5},{7}}=>8
{{1,4,7},{2,6},{3,5}}=>3
{{1,4},{2,6,7},{3,5}}=>5
{{1,4},{2,6},{3,5,7}}=>4
{{1,4},{2,6},{3,5},{7}}=>6
{{1,4,7},{2},{3,5,6}}=>6
{{1,4},{2,7},{3,5,6}}=>4
{{1,4},{2},{3,5,6,7}}=>8
{{1,4},{2},{3,5,6},{7}}=>9
{{1,4,7},{2},{3,5},{6}}=>8
{{1,4},{2,7},{3,5},{6}}=>6
{{1,4},{2},{3,5,7},{6}}=>9
{{1,4},{2},{3,5},{6,7}}=>10
{{1,4},{2},{3,5},{6},{7}}=>11
{{1,4,6,7},{2},{3},{5}}=>11
{{1,4,6},{2,7},{3},{5}}=>7
{{1,4,6},{2},{3,7},{5}}=>8
{{1,4,6},{2},{3},{5,7}}=>10
{{1,4,6},{2},{3},{5},{7}}=>12
{{1,4,7},{2,6},{3},{5}}=>7
{{1,4},{2,6,7},{3},{5}}=>9
{{1,4},{2,6},{3,7},{5}}=>6
{{1,4},{2,6},{3},{5,7}}=>8
{{1,4},{2,6},{3},{5},{7}}=>10
{{1,4,7},{2},{3,6},{5}}=>8
{{1,4},{2,7},{3,6},{5}}=>6
{{1,4},{2},{3,6,7},{5}}=>10
{{1,4},{2},{3,6},{5,7}}=>9
{{1,4},{2},{3,6},{5},{7}}=>11
{{1,4,7},{2},{3},{5,6}}=>10
{{1,4},{2,7},{3},{5,6}}=>8
{{1,4},{2},{3,7},{5,6}}=>9
{{1,4},{2},{3},{5,6,7}}=>12
{{1,4},{2},{3},{5,6},{7}}=>13
{{1,4,7},{2},{3},{5},{6}}=>12
{{1,4},{2,7},{3},{5},{6}}=>10
{{1,4},{2},{3,7},{5},{6}}=>11
{{1,4},{2},{3},{5,7},{6}}=>13
{{1,4},{2},{3},{5},{6,7}}=>14
{{1,4},{2},{3},{5},{6},{7}}=>15
{{1,5,6,7},{2,4},{3}}=>7
{{1,5,6},{2,4,7},{3}}=>5
{{1,5,6},{2,4},{3,7}}=>4
{{1,5,6},{2,4},{3},{7}}=>8
{{1,5,7},{2,4,6},{3}}=>5
{{1,5},{2,4,6,7},{3}}=>6
{{1,5},{2,4,6},{3,7}}=>3
{{1,5},{2,4,6},{3},{7}}=>7
{{1,5,7},{2,4},{3,6}}=>4
{{1,5},{2,4,7},{3,6}}=>3
{{1,5},{2,4},{3,6,7}}=>5
{{1,5},{2,4},{3,6},{7}}=>6
{{1,5,7},{2,4},{3},{6}}=>8
{{1,5},{2,4,7},{3},{6}}=>7
{{1,5},{2,4},{3,7},{6}}=>6
{{1,5},{2,4},{3},{6,7}}=>9
{{1,5},{2,4},{3},{6},{7}}=>10
{{1,6,7},{2,4,5},{3}}=>6
{{1,6},{2,4,5,7},{3}}=>5
{{1,6},{2,4,5},{3,7}}=>3
{{1,6},{2,4,5},{3},{7}}=>7
{{1,7},{2,4,5,6},{3}}=>5
{{1},{2,4,5,6,7},{3}}=>10
{{1},{2,4,5,6},{3,7}}=>7
{{1},{2,4,5,6},{3},{7}}=>11
{{1,7},{2,4,5},{3,6}}=>3
{{1},{2,4,5,7},{3,6}}=>7
{{1},{2,4,5},{3,6,7}}=>8
{{1},{2,4,5},{3,6},{7}}=>9
{{1,7},{2,4,5},{3},{6}}=>7
{{1},{2,4,5,7},{3},{6}}=>11
{{1},{2,4,5},{3,7},{6}}=>9
{{1},{2,4,5},{3},{6,7}}=>12
{{1},{2,4,5},{3},{6},{7}}=>13
{{1,6,7},{2,4},{3,5}}=>5
{{1,6},{2,4,7},{3,5}}=>3
{{1,6},{2,4},{3,5,7}}=>4
{{1,6},{2,4},{3,5},{7}}=>6
{{1,7},{2,4,6},{3,5}}=>3
{{1},{2,4,6,7},{3,5}}=>8
{{1},{2,4,6},{3,5,7}}=>7
{{1},{2,4,6},{3,5},{7}}=>9
{{1,7},{2,4},{3,5,6}}=>4
{{1},{2,4,7},{3,5,6}}=>7
{{1},{2,4},{3,5,6,7}}=>9
{{1},{2,4},{3,5,6},{7}}=>10
{{1,7},{2,4},{3,5},{6}}=>6
{{1},{2,4,7},{3,5},{6}}=>9
{{1},{2,4},{3,5,7},{6}}=>10
{{1},{2,4},{3,5},{6,7}}=>11
{{1},{2,4},{3,5},{6},{7}}=>12
{{1,6,7},{2,4},{3},{5}}=>9
{{1,6},{2,4,7},{3},{5}}=>7
{{1,6},{2,4},{3,7},{5}}=>6
{{1,6},{2,4},{3},{5,7}}=>8
{{1,6},{2,4},{3},{5},{7}}=>10
{{1,7},{2,4,6},{3},{5}}=>7
{{1},{2,4,6,7},{3},{5}}=>12
{{1},{2,4,6},{3,7},{5}}=>9
{{1},{2,4,6},{3},{5,7}}=>11
{{1},{2,4,6},{3},{5},{7}}=>13
{{1,7},{2,4},{3,6},{5}}=>6
{{1},{2,4,7},{3,6},{5}}=>9
{{1},{2,4},{3,6,7},{5}}=>11
{{1},{2,4},{3,6},{5,7}}=>10
{{1},{2,4},{3,6},{5},{7}}=>12
{{1,7},{2,4},{3},{5,6}}=>8
{{1},{2,4,7},{3},{5,6}}=>11
{{1},{2,4},{3,7},{5,6}}=>10
{{1},{2,4},{3},{5,6,7}}=>13
{{1},{2,4},{3},{5,6},{7}}=>14
{{1,7},{2,4},{3},{5},{6}}=>10
{{1},{2,4,7},{3},{5},{6}}=>13
{{1},{2,4},{3,7},{5},{6}}=>12
{{1},{2,4},{3},{5,7},{6}}=>14
{{1},{2,4},{3},{5},{6,7}}=>15
{{1},{2,4},{3},{5},{6},{7}}=>16
{{1,5,6,7},{2},{3,4}}=>8
{{1,5,6},{2,7},{3,4}}=>4
{{1,5,6},{2},{3,4,7}}=>6
{{1,5,6},{2},{3,4},{7}}=>9
{{1,5,7},{2,6},{3,4}}=>4
{{1,5},{2,6,7},{3,4}}=>5
{{1,5},{2,6},{3,4,7}}=>3
{{1,5},{2,6},{3,4},{7}}=>6
{{1,5,7},{2},{3,4,6}}=>6
{{1,5},{2,7},{3,4,6}}=>3
{{1,5},{2},{3,4,6,7}}=>7
{{1,5},{2},{3,4,6},{7}}=>8
{{1,5,7},{2},{3,4},{6}}=>9
{{1,5},{2,7},{3,4},{6}}=>6
{{1,5},{2},{3,4,7},{6}}=>8
{{1,5},{2},{3,4},{6,7}}=>10
{{1,5},{2},{3,4},{6},{7}}=>11
{{1,6,7},{2,5},{3,4}}=>5
{{1,6},{2,5,7},{3,4}}=>4
{{1,6},{2,5},{3,4,7}}=>3
{{1,6},{2,5},{3,4},{7}}=>6
{{1,7},{2,5,6},{3,4}}=>4
{{1},{2,5,6,7},{3,4}}=>9
{{1},{2,5,6},{3,4,7}}=>7
{{1},{2,5,6},{3,4},{7}}=>10
{{1,7},{2,5},{3,4,6}}=>3
{{1},{2,5,7},{3,4,6}}=>7
{{1},{2,5},{3,4,6,7}}=>8
{{1},{2,5},{3,4,6},{7}}=>9
{{1,7},{2,5},{3,4},{6}}=>6
{{1},{2,5,7},{3,4},{6}}=>10
{{1},{2,5},{3,4,7},{6}}=>9
{{1},{2,5},{3,4},{6,7}}=>11
{{1},{2,5},{3,4},{6},{7}}=>12
{{1,6,7},{2},{3,4,5}}=>7
{{1,6},{2,7},{3,4,5}}=>3
{{1,6},{2},{3,4,5,7}}=>6
{{1,6},{2},{3,4,5},{7}}=>8
{{1,7},{2,6},{3,4,5}}=>3
{{1},{2,6,7},{3,4,5}}=>8
{{1},{2,6},{3,4,5,7}}=>7
{{1},{2,6},{3,4,5},{7}}=>9
{{1,7},{2},{3,4,5,6}}=>6
{{1},{2,7},{3,4,5,6}}=>7
{{1},{2},{3,4,5,6,7}}=>11
{{1},{2},{3,4,5,6},{7}}=>12
{{1,7},{2},{3,4,5},{6}}=>8
{{1},{2,7},{3,4,5},{6}}=>9
{{1},{2},{3,4,5,7},{6}}=>12
{{1},{2},{3,4,5},{6,7}}=>13
{{1},{2},{3,4,5},{6},{7}}=>14
{{1,6,7},{2},{3,4},{5}}=>10
{{1,6},{2,7},{3,4},{5}}=>6
{{1,6},{2},{3,4,7},{5}}=>8
{{1,6},{2},{3,4},{5,7}}=>9
{{1,6},{2},{3,4},{5},{7}}=>11
{{1,7},{2,6},{3,4},{5}}=>6
{{1},{2,6,7},{3,4},{5}}=>11
{{1},{2,6},{3,4,7},{5}}=>9
{{1},{2,6},{3,4},{5,7}}=>10
{{1},{2,6},{3,4},{5},{7}}=>12
{{1,7},{2},{3,4,6},{5}}=>8
{{1},{2,7},{3,4,6},{5}}=>9
{{1},{2},{3,4,6,7},{5}}=>13
{{1},{2},{3,4,6},{5,7}}=>12
{{1},{2},{3,4,6},{5},{7}}=>14
{{1,7},{2},{3,4},{5,6}}=>9
{{1},{2,7},{3,4},{5,6}}=>10
{{1},{2},{3,4,7},{5,6}}=>12
{{1},{2},{3,4},{5,6,7}}=>14
{{1},{2},{3,4},{5,6},{7}}=>15
{{1,7},{2},{3,4},{5},{6}}=>11
{{1},{2,7},{3,4},{5},{6}}=>12
{{1},{2},{3,4,7},{5},{6}}=>14
{{1},{2},{3,4},{5,7},{6}}=>15
{{1},{2},{3,4},{5},{6,7}}=>16
{{1},{2},{3,4},{5},{6},{7}}=>17
{{1,5,6,7},{2},{3},{4}}=>12
{{1,5,6},{2,7},{3},{4}}=>8
{{1,5,6},{2},{3,7},{4}}=>9
{{1,5,6},{2},{3},{4,7}}=>10
{{1,5,6},{2},{3},{4},{7}}=>13
{{1,5,7},{2,6},{3},{4}}=>8
{{1,5},{2,6,7},{3},{4}}=>9
{{1,5},{2,6},{3,7},{4}}=>6
{{1,5},{2,6},{3},{4,7}}=>7
{{1,5},{2,6},{3},{4},{7}}=>10
{{1,5,7},{2},{3,6},{4}}=>9
{{1,5},{2,7},{3,6},{4}}=>6
{{1,5},{2},{3,6,7},{4}}=>10
{{1,5},{2},{3,6},{4,7}}=>8
{{1,5},{2},{3,6},{4},{7}}=>11
{{1,5,7},{2},{3},{4,6}}=>10
{{1,5},{2,7},{3},{4,6}}=>7
{{1,5},{2},{3,7},{4,6}}=>8
{{1,5},{2},{3},{4,6,7}}=>11
{{1,5},{2},{3},{4,6},{7}}=>12
{{1,5,7},{2},{3},{4},{6}}=>13
{{1,5},{2,7},{3},{4},{6}}=>10
{{1,5},{2},{3,7},{4},{6}}=>11
{{1,5},{2},{3},{4,7},{6}}=>12
{{1,5},{2},{3},{4},{6,7}}=>14
{{1,5},{2},{3},{4},{6},{7}}=>15
{{1,6,7},{2,5},{3},{4}}=>9
{{1,6},{2,5,7},{3},{4}}=>8
{{1,6},{2,5},{3,7},{4}}=>6
{{1,6},{2,5},{3},{4,7}}=>7
{{1,6},{2,5},{3},{4},{7}}=>10
{{1,7},{2,5,6},{3},{4}}=>8
{{1},{2,5,6,7},{3},{4}}=>13
{{1},{2,5,6},{3,7},{4}}=>10
{{1},{2,5,6},{3},{4,7}}=>11
{{1},{2,5,6},{3},{4},{7}}=>14
{{1,7},{2,5},{3,6},{4}}=>6
{{1},{2,5,7},{3,6},{4}}=>10
{{1},{2,5},{3,6,7},{4}}=>11
{{1},{2,5},{3,6},{4,7}}=>9
{{1},{2,5},{3,6},{4},{7}}=>12
{{1,7},{2,5},{3},{4,6}}=>7
{{1},{2,5,7},{3},{4,6}}=>11
{{1},{2,5},{3,7},{4,6}}=>9
{{1},{2,5},{3},{4,6,7}}=>12
{{1},{2,5},{3},{4,6},{7}}=>13
{{1,7},{2,5},{3},{4},{6}}=>10
{{1},{2,5,7},{3},{4},{6}}=>14
{{1},{2,5},{3,7},{4},{6}}=>12
{{1},{2,5},{3},{4,7},{6}}=>13
{{1},{2,5},{3},{4},{6,7}}=>15
{{1},{2,5},{3},{4},{6},{7}}=>16
{{1,6,7},{2},{3,5},{4}}=>10
{{1,6},{2,7},{3,5},{4}}=>6
{{1,6},{2},{3,5,7},{4}}=>9
{{1,6},{2},{3,5},{4,7}}=>8
{{1,6},{2},{3,5},{4},{7}}=>11
{{1,7},{2,6},{3,5},{4}}=>6
{{1},{2,6,7},{3,5},{4}}=>11
{{1},{2,6},{3,5,7},{4}}=>10
{{1},{2,6},{3,5},{4,7}}=>9
{{1},{2,6},{3,5},{4},{7}}=>12
{{1,7},{2},{3,5,6},{4}}=>9
{{1},{2,7},{3,5,6},{4}}=>10
{{1},{2},{3,5,6,7},{4}}=>14
{{1},{2},{3,5,6},{4,7}}=>12
{{1},{2},{3,5,6},{4},{7}}=>15
{{1,7},{2},{3,5},{4,6}}=>8
{{1},{2,7},{3,5},{4,6}}=>9
{{1},{2},{3,5,7},{4,6}}=>12
{{1},{2},{3,5},{4,6,7}}=>13
{{1},{2},{3,5},{4,6},{7}}=>14
{{1,7},{2},{3,5},{4},{6}}=>11
{{1},{2,7},{3,5},{4},{6}}=>12
{{1},{2},{3,5,7},{4},{6}}=>15
{{1},{2},{3,5},{4,7},{6}}=>14
{{1},{2},{3,5},{4},{6,7}}=>16
{{1},{2},{3,5},{4},{6},{7}}=>17
{{1,6,7},{2},{3},{4,5}}=>11
{{1,6},{2,7},{3},{4,5}}=>7
{{1,6},{2},{3,7},{4,5}}=>8
{{1,6},{2},{3},{4,5,7}}=>10
{{1,6},{2},{3},{4,5},{7}}=>12
{{1,7},{2,6},{3},{4,5}}=>7
{{1},{2,6,7},{3},{4,5}}=>12
{{1},{2,6},{3,7},{4,5}}=>9
{{1},{2,6},{3},{4,5,7}}=>11
{{1},{2,6},{3},{4,5},{7}}=>13
{{1,7},{2},{3,6},{4,5}}=>8
{{1},{2,7},{3,6},{4,5}}=>9
{{1},{2},{3,6,7},{4,5}}=>13
{{1},{2},{3,6},{4,5,7}}=>12
{{1},{2},{3,6},{4,5},{7}}=>14
{{1,7},{2},{3},{4,5,6}}=>10
{{1},{2,7},{3},{4,5,6}}=>11
{{1},{2},{3,7},{4,5,6}}=>12
{{1},{2},{3},{4,5,6,7}}=>15
{{1},{2},{3},{4,5,6},{7}}=>16
{{1,7},{2},{3},{4,5},{6}}=>12
{{1},{2,7},{3},{4,5},{6}}=>13
{{1},{2},{3,7},{4,5},{6}}=>14
{{1},{2},{3},{4,5,7},{6}}=>16
{{1},{2},{3},{4,5},{6,7}}=>17
{{1},{2},{3},{4,5},{6},{7}}=>18
{{1,6,7},{2},{3},{4},{5}}=>14
{{1,6},{2,7},{3},{4},{5}}=>10
{{1,6},{2},{3,7},{4},{5}}=>11
{{1,6},{2},{3},{4,7},{5}}=>12
{{1,6},{2},{3},{4},{5,7}}=>13
{{1,6},{2},{3},{4},{5},{7}}=>15
{{1,7},{2,6},{3},{4},{5}}=>10
{{1},{2,6,7},{3},{4},{5}}=>15
{{1},{2,6},{3,7},{4},{5}}=>12
{{1},{2,6},{3},{4,7},{5}}=>13
{{1},{2,6},{3},{4},{5,7}}=>14
{{1},{2,6},{3},{4},{5},{7}}=>16
{{1,7},{2},{3,6},{4},{5}}=>11
{{1},{2,7},{3,6},{4},{5}}=>12
{{1},{2},{3,6,7},{4},{5}}=>16
{{1},{2},{3,6},{4,7},{5}}=>14
{{1},{2},{3,6},{4},{5,7}}=>15
{{1},{2},{3,6},{4},{5},{7}}=>17
{{1,7},{2},{3},{4,6},{5}}=>12
{{1},{2,7},{3},{4,6},{5}}=>13
{{1},{2},{3,7},{4,6},{5}}=>14
{{1},{2},{3},{4,6,7},{5}}=>17
{{1},{2},{3},{4,6},{5,7}}=>16
{{1},{2},{3},{4,6},{5},{7}}=>18
{{1,7},{2},{3},{4},{5,6}}=>13
{{1},{2,7},{3},{4},{5,6}}=>14
{{1},{2},{3,7},{4},{5,6}}=>15
{{1},{2},{3},{4,7},{5,6}}=>16
{{1},{2},{3},{4},{5,6,7}}=>18
{{1},{2},{3},{4},{5,6},{7}}=>19
{{1,7},{2},{3},{4},{5},{6}}=>15
{{1},{2,7},{3},{4},{5},{6}}=>16
{{1},{2},{3,7},{4},{5},{6}}=>17
{{1},{2},{3},{4,7},{5},{6}}=>18
{{1},{2},{3},{4},{5,7},{6}}=>19
{{1},{2},{3},{4},{5},{6,7}}=>20
{{1},{2},{3},{4},{5},{6},{7}}=>21
{{1},{2},{3,4,5,6,7,8}}=>13
{{1},{2,4,5,6,7,8},{3}}=>12
{{1},{2,3,5,6,7,8},{4}}=>11
{{1},{2,3,4,6,7,8},{5}}=>10
{{1},{2,3,4,5,7,8},{6}}=>9
{{1},{2,3,4,5,6,7},{8}}=>13
{{1},{2,3,4,5,6,8},{7}}=>8
{{1},{2,3,4,5,6,7,8}}=>7
{{1,2},{3,4,5,6,7,8}}=>7
{{1,4,5,6,7,8},{2},{3}}=>11
{{1,3,5,6,7,8},{2},{4}}=>10
{{1,3,4,5,6,7,8},{2}}=>6
{{1,4,5,6,7,8},{2,3}}=>6
{{1,2,4,5,6,7,8},{3}}=>5
{{1,2,5,6,7,8},{3,4}}=>5
{{1,2,3,5,6,7,8},{4}}=>4
{{1,2,3,6,7,8},{4,5}}=>4
{{1,2,3,4,6,7,8},{5}}=>3
{{1,2,3,4,5,6},{7,8}}=>7
{{1,2,3,4,7,8},{5,6}}=>3
{{1,2,3,4,5,7,8},{6}}=>2
{{1,2,3,4,5,6,7},{8}}=>7
{{1,8},{2,3,4,5,6,7}}=>6
{{1,2,3,4,5,8},{6,7}}=>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}}=>5
{{1,3,4,6,7,8},{2,5}}=>4
{{1,2,4,6,7,8},{3,5}}=>4
{{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
{{1,3,4,5,6,8},{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
{{1,3,4,5,6,7},{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
{{1,3},{2,4,5,6,7,8}}=>6
{{1,4},{2,3,5,6,7,8}}=>5
{{1,5},{2,3,4,6,7,8}}=>4
{{1,6},{2,3,4,5,7,8}}=>3
{{1,7},{2,3,4,5,6,8}}=>2
                    

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Description

The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element.
This is the number of pairs $i\lt j$ in different blocks such that $i$ is the maximal element of a block.

References

[1] Chern, B., Diaconis, P., Kane, D. M., Rhoades, R. C. Closed expressions for averages of set partition statistics MathSciNet:3338726 arXiv:1304.4309

Code

def statistic(pi):
    return len(pattern_occurrences(pi, [[1],[2]], [], [1], [], []))

def pattern_occurrences(pi, P, First, Last, Arcs, Consecutives):
    """We assume that pi is a SetPartition of {1,2,...,n} and P is a
    SetPartition of {1,2,...,k}.
    """
    occurrences = []
    pi = SetPartition(pi)
    P = SetPartition(P)

    openers = [min(B) for B in pi]
    closers = [max(B) for B in pi]
    pi_sorted = sorted([sorted(b) for b in pi])
    edges = [(a,b) for B in pi_sorted for (a,b) in zip(B, B[1:])]

    for s in Subsets(pi.base_set(), P.size()):
        s = sorted(s)
        pi_r = pi.restriction(s)
        if pi_r.standardization() != P:
            continue

        X = pi_r.base_set()

        if any(X[i-1] not in openers for i in First):
            continue

        if any(X[i-1] not in closers for i in Last):
            continue

        if any((X[i-1], X[j-1]) not in edges for (i,j) in Arcs):
            continue

        if any(abs(X[i-1]-X[j-1]) != 1 for (i,j) in Consecutives):
            continue

        occurrences += [s]
    return occurrences

Created

Aug 10, 2016 at 23:16 by Martin Rubey

Updated

Aug 10, 2016 at 23:16 by Martin Rubey