Values
=>
[1]=>[1]=>0 [1,1]=>[1,1]=>0 [2]=>[2]=>0 [1,1,1]=>[1,1,1]=>0 [1,2]=>[1,2]=>0 [2,1]=>[2,1]=>1 [3]=>[3]=>0 [1,1,1,1]=>[1,1,1,1]=>0 [1,1,2]=>[1,1,2]=>0 [1,2,1]=>[2,1,1]=>2 [1,3]=>[1,3]=>0 [2,1,1]=>[1,2,1]=>1 [2,2]=>[2,2]=>0 [3,1]=>[3,1]=>1 [4]=>[4]=>0 [1,1,1,1,1]=>[1,1,1,1,1]=>0 [1,1,1,2]=>[1,1,1,2]=>0 [1,1,2,1]=>[2,1,1,1]=>3 [1,1,3]=>[1,1,3]=>0 [1,2,1,1]=>[1,2,1,1]=>2 [1,2,2]=>[1,2,2]=>0 [1,3,1]=>[3,1,1]=>2 [1,4]=>[1,4]=>0 [2,1,1,1]=>[1,1,2,1]=>1 [2,1,2]=>[2,1,2]=>1 [2,2,1]=>[2,2,1]=>2 [2,3]=>[2,3]=>0 [3,1,1]=>[1,3,1]=>1 [3,2]=>[3,2]=>1 [4,1]=>[4,1]=>1 [5]=>[5]=>0 [1,1,1,1,1,1]=>[1,1,1,1,1,1]=>0 [1,1,1,1,2]=>[1,1,1,1,2]=>0 [1,1,1,2,1]=>[2,1,1,1,1]=>4 [1,1,1,3]=>[1,1,1,3]=>0 [1,1,2,1,1]=>[1,2,1,1,1]=>3 [1,1,2,2]=>[1,1,2,2]=>0 [1,1,3,1]=>[3,1,1,1]=>3 [1,1,4]=>[1,1,4]=>0 [1,2,1,1,1]=>[1,1,2,1,1]=>2 [1,2,1,2]=>[2,1,1,2]=>2 [1,2,2,1]=>[2,1,2,1]=>3 [1,2,3]=>[1,2,3]=>0 [1,3,1,1]=>[1,3,1,1]=>2 [1,3,2]=>[3,1,2]=>2 [1,4,1]=>[4,1,1]=>2 [1,5]=>[1,5]=>0 [2,1,1,1,1]=>[1,1,1,2,1]=>1 [2,1,1,2]=>[1,2,1,2]=>1 [2,1,2,1]=>[2,2,1,1]=>4 [2,1,3]=>[2,1,3]=>1 [2,2,1,1]=>[1,2,2,1]=>2 [2,2,2]=>[2,2,2]=>0 [2,3,1]=>[2,3,1]=>2 [2,4]=>[2,4]=>0 [3,1,1,1]=>[1,1,3,1]=>1 [3,1,2]=>[1,3,2]=>1 [3,2,1]=>[3,2,1]=>3 [3,3]=>[3,3]=>0 [4,1,1]=>[1,4,1]=>1 [4,2]=>[4,2]=>1 [5,1]=>[5,1]=>1 [6]=>[6]=>0 [1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1]=>0 [1,1,1,1,1,2]=>[1,1,1,1,1,2]=>0 [1,1,1,1,2,1]=>[2,1,1,1,1,1]=>5 [1,1,1,1,3]=>[1,1,1,1,3]=>0 [1,1,1,2,1,1]=>[1,2,1,1,1,1]=>4 [1,1,1,2,2]=>[1,1,1,2,2]=>0 [1,1,1,3,1]=>[3,1,1,1,1]=>4 [1,1,1,4]=>[1,1,1,4]=>0 [1,1,2,1,1,1]=>[1,1,2,1,1,1]=>3 [1,1,2,1,2]=>[2,1,1,1,2]=>3 [1,1,2,2,1]=>[2,1,1,2,1]=>4 [1,1,2,3]=>[1,1,2,3]=>0 [1,1,3,1,1]=>[1,3,1,1,1]=>3 [1,1,3,2]=>[3,1,1,2]=>3 [1,1,4,1]=>[4,1,1,1]=>3 [1,1,5]=>[1,1,5]=>0 [1,2,1,1,1,1]=>[1,1,1,2,1,1]=>2 [1,2,1,1,2]=>[1,2,1,1,2]=>2 [1,2,1,2,1]=>[2,2,1,1,1]=>6 [1,2,1,3]=>[2,1,1,3]=>2 [1,2,2,1,1]=>[1,2,1,2,1]=>3 [1,2,2,2]=>[1,2,2,2]=>0 [1,2,3,1]=>[2,1,3,1]=>3 [1,2,4]=>[1,2,4]=>0 [1,3,1,1,1]=>[1,1,3,1,1]=>2 [1,3,1,2]=>[1,3,1,2]=>2 [1,3,2,1]=>[3,2,1,1]=>5 [1,3,3]=>[1,3,3]=>0 [1,4,1,1]=>[1,4,1,1]=>2 [1,4,2]=>[4,1,2]=>2 [1,5,1]=>[5,1,1]=>2 [1,6]=>[1,6]=>0 [2,1,1,1,1,1]=>[1,1,1,1,2,1]=>1 [2,1,1,1,2]=>[1,1,2,1,2]=>1 [2,1,1,2,1]=>[2,1,2,1,1]=>5 [2,1,1,3]=>[1,2,1,3]=>1 [2,1,2,1,1]=>[1,2,2,1,1]=>4 [2,1,2,2]=>[2,1,2,2]=>1 [2,1,3,1]=>[2,3,1,1]=>4 [2,1,4]=>[2,1,4]=>1 [2,2,1,1,1]=>[1,1,2,2,1]=>2 [2,2,1,2]=>[2,2,1,2]=>2 [2,2,2,1]=>[2,2,2,1]=>3 [2,2,3]=>[2,2,3]=>0 [2,3,1,1]=>[1,2,3,1]=>2 [2,3,2]=>[3,2,2]=>2 [2,4,1]=>[2,4,1]=>2 [2,5]=>[2,5]=>0 [3,1,1,1,1]=>[1,1,1,3,1]=>1 [3,1,1,2]=>[1,1,3,2]=>1 [3,1,2,1]=>[3,1,2,1]=>4 [3,1,3]=>[3,1,3]=>1 [3,2,1,1]=>[1,3,2,1]=>3 [3,2,2]=>[2,3,2]=>1 [3,3,1]=>[3,3,1]=>2 [3,4]=>[3,4]=>0 [4,1,1,1]=>[1,1,4,1]=>1 [4,1,2]=>[1,4,2]=>1 [4,2,1]=>[4,2,1]=>3 [4,3]=>[4,3]=>1 [5,1,1]=>[1,5,1]=>1 [5,2]=>[5,2]=>1 [6,1]=>[6,1]=>1 [7]=>[7]=>0 [1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1]=>0 [1,1,1,1,1,1,2]=>[1,1,1,1,1,1,2]=>0 [1,1,1,1,1,2,1]=>[2,1,1,1,1,1,1]=>6 [1,1,1,1,1,3]=>[1,1,1,1,1,3]=>0 [1,1,1,1,2,1,1]=>[1,2,1,1,1,1,1]=>5 [1,1,1,1,2,2]=>[1,1,1,1,2,2]=>0 [1,1,1,1,3,1]=>[3,1,1,1,1,1]=>5 [1,1,1,1,4]=>[1,1,1,1,4]=>0 [1,1,1,2,1,1,1]=>[1,1,2,1,1,1,1]=>4 [1,1,1,2,1,2]=>[2,1,1,1,1,2]=>4 [1,1,1,2,2,1]=>[2,1,1,1,2,1]=>5 [1,1,1,2,3]=>[1,1,1,2,3]=>0 [1,1,1,3,1,1]=>[1,3,1,1,1,1]=>4 [1,1,1,3,2]=>[3,1,1,1,2]=>4 [1,1,1,4,1]=>[4,1,1,1,1]=>4 [1,1,1,5]=>[1,1,1,5]=>0 [1,1,2,1,1,1,1]=>[1,1,1,2,1,1,1]=>3 [1,1,2,1,1,2]=>[1,2,1,1,1,2]=>3 [1,1,2,1,2,1]=>[2,2,1,1,1,1]=>8 [1,1,2,1,3]=>[2,1,1,1,3]=>3 [1,1,2,2,1,1]=>[1,2,1,1,2,1]=>4 [1,1,2,2,2]=>[1,1,2,2,2]=>0 [1,1,2,3,1]=>[2,1,1,3,1]=>4 [1,1,2,4]=>[1,1,2,4]=>0 [1,1,3,1,1,1]=>[1,1,3,1,1,1]=>3 [1,1,3,1,2]=>[1,3,1,1,2]=>3 [1,1,3,2,1]=>[3,2,1,1,1]=>7 [1,1,3,3]=>[1,1,3,3]=>0 [1,1,4,1,1]=>[1,4,1,1,1]=>3 [1,1,4,2]=>[4,1,1,2]=>3 [1,1,5,1]=>[5,1,1,1]=>3 [1,1,6]=>[1,1,6]=>0 [1,2,1,1,1,1,1]=>[1,1,1,1,2,1,1]=>2 [1,2,1,1,1,2]=>[1,1,2,1,1,2]=>2 [1,2,1,1,2,1]=>[2,1,2,1,1,1]=>7 [1,2,1,1,3]=>[1,2,1,1,3]=>2 [1,2,1,2,1,1]=>[1,2,2,1,1,1]=>6 [1,2,1,2,2]=>[2,1,1,2,2]=>2 [1,2,1,3,1]=>[2,3,1,1,1]=>6 [1,2,1,4]=>[2,1,1,4]=>2 [1,2,2,1,1,1]=>[1,1,2,1,2,1]=>3 [1,2,2,1,2]=>[2,1,2,1,2]=>3 [1,2,2,2,1]=>[2,1,2,2,1]=>4 [1,2,2,3]=>[1,2,2,3]=>0 [1,2,3,1,1]=>[1,2,1,3,1]=>3 [1,2,3,2]=>[3,1,2,2]=>3 [1,2,4,1]=>[2,1,4,1]=>3 [1,2,5]=>[1,2,5]=>0 [1,3,1,1,1,1]=>[1,1,1,3,1,1]=>2 [1,3,1,1,2]=>[1,1,3,1,2]=>2 [1,3,1,2,1]=>[3,1,2,1,1]=>6 [1,3,1,3]=>[3,1,1,3]=>2 [1,3,2,1,1]=>[1,3,2,1,1]=>5 [1,3,2,2]=>[1,3,2,2]=>2 [1,3,3,1]=>[3,1,3,1]=>3 [1,3,4]=>[1,3,4]=>0 [1,4,1,1,1]=>[1,1,4,1,1]=>2 [1,4,1,2]=>[1,4,1,2]=>2 [1,4,2,1]=>[4,2,1,1]=>5 [1,4,3]=>[4,1,3]=>2 [1,5,1,1]=>[1,5,1,1]=>2 [1,5,2]=>[5,1,2]=>2 [1,6,1]=>[6,1,1]=>2 [1,7]=>[1,7]=>0 [2,1,1,1,1,1,1]=>[1,1,1,1,1,2,1]=>1 [2,1,1,1,1,2]=>[1,1,1,2,1,2]=>1 [2,1,1,1,2,1]=>[2,1,1,2,1,1]=>6 [2,1,1,1,3]=>[1,1,2,1,3]=>1 [2,1,1,2,1,1]=>[1,2,1,2,1,1]=>5 [2,1,1,2,2]=>[1,2,1,2,2]=>1 [2,1,1,3,1]=>[2,1,3,1,1]=>5 [2,1,1,4]=>[1,2,1,4]=>1 [2,1,2,1,1,1]=>[1,1,2,2,1,1]=>4 [2,1,2,1,2]=>[2,2,1,1,2]=>4 [2,1,2,2,1]=>[2,2,1,2,1]=>5 [2,1,2,3]=>[2,1,2,3]=>1 [2,1,3,1,1]=>[1,2,3,1,1]=>4 [2,1,3,2]=>[3,2,1,2]=>4 [2,1,4,1]=>[2,4,1,1]=>4 [2,1,5]=>[2,1,5]=>1 [2,2,1,1,1,1]=>[1,1,1,2,2,1]=>2 [2,2,1,1,2]=>[1,2,2,1,2]=>2 [2,2,1,2,1]=>[2,2,2,1,1]=>6 [2,2,1,3]=>[2,2,1,3]=>2 [2,2,2,1,1]=>[1,2,2,2,1]=>3 [2,2,2,2]=>[2,2,2,2]=>0 [2,2,3,1]=>[2,2,3,1]=>3 [2,2,4]=>[2,2,4]=>0 [2,3,1,1,1]=>[1,1,2,3,1]=>2 [2,3,1,2]=>[2,1,3,2]=>2 [2,3,2,1]=>[3,2,2,1]=>5 [2,3,3]=>[2,3,3]=>0 [2,4,1,1]=>[1,2,4,1]=>2 [2,4,2]=>[4,2,2]=>2 [2,5,1]=>[2,5,1]=>2 [2,6]=>[2,6]=>0 [3,1,1,1,1,1]=>[1,1,1,1,3,1]=>1 [3,1,1,1,2]=>[1,1,1,3,2]=>1 [3,1,1,2,1]=>[3,1,1,2,1]=>5 [3,1,1,3]=>[1,3,1,3]=>1 [3,1,2,1,1]=>[1,3,1,2,1]=>4 [3,1,2,2]=>[1,2,3,2]=>1 [3,1,3,1]=>[3,3,1,1]=>4 [3,1,4]=>[3,1,4]=>1 [3,2,1,1,1]=>[1,1,3,2,1]=>3 [3,2,1,2]=>[2,3,1,2]=>3 [3,2,2,1]=>[2,3,2,1]=>4 [3,2,3]=>[3,2,3]=>1 [3,3,1,1]=>[1,3,3,1]=>2 [3,3,2]=>[3,3,2]=>2 [3,4,1]=>[3,4,1]=>2 [3,5]=>[3,5]=>0 [4,1,1,1,1]=>[1,1,1,4,1]=>1 [4,1,1,2]=>[1,1,4,2]=>1 [4,1,2,1]=>[4,1,2,1]=>4 [4,1,3]=>[1,4,3]=>1 [4,2,1,1]=>[1,4,2,1]=>3 [4,2,2]=>[2,4,2]=>1 [4,3,1]=>[4,3,1]=>3 [4,4]=>[4,4]=>0 [5,1,1,1]=>[1,1,5,1]=>1 [5,1,2]=>[1,5,2]=>1 [5,2,1]=>[5,2,1]=>3 [5,3]=>[5,3]=>1 [6,1,1]=>[1,6,1]=>1 [6,2]=>[6,2]=>1 [7,1]=>[7,1]=>1 [8]=>[8]=>0 [1,1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1,1]=>0 [1,1,1,1,1,1,1,2]=>[1,1,1,1,1,1,1,2]=>0 [1,1,1,1,1,1,2,1]=>[2,1,1,1,1,1,1,1]=>7 [1,1,1,1,1,1,3]=>[1,1,1,1,1,1,3]=>0 [1,1,1,1,1,2,1,1]=>[1,2,1,1,1,1,1,1]=>6 [1,1,1,1,1,2,2]=>[1,1,1,1,1,2,2]=>0 [1,1,1,1,1,3,1]=>[3,1,1,1,1,1,1]=>6 [1,1,1,1,1,4]=>[1,1,1,1,1,4]=>0 [1,1,1,1,2,1,1,1]=>[1,1,2,1,1,1,1,1]=>5 [1,1,1,1,2,1,2]=>[2,1,1,1,1,1,2]=>5 [1,1,1,1,2,2,1]=>[2,1,1,1,1,2,1]=>6 [1,1,1,1,2,3]=>[1,1,1,1,2,3]=>0 [1,1,1,1,3,1,1]=>[1,3,1,1,1,1,1]=>5 [1,1,1,1,3,2]=>[3,1,1,1,1,2]=>5 [1,1,1,1,4,1]=>[4,1,1,1,1,1]=>5 [1,1,1,1,5]=>[1,1,1,1,5]=>0 [1,1,1,2,1,1,1,1]=>[1,1,1,2,1,1,1,1]=>4 [1,1,1,2,1,1,2]=>[1,2,1,1,1,1,2]=>4 [1,1,1,2,1,2,1]=>[2,2,1,1,1,1,1]=>10 [1,1,1,2,1,3]=>[2,1,1,1,1,3]=>4 [1,1,1,2,2,1,1]=>[1,2,1,1,1,2,1]=>5 [1,1,1,2,2,2]=>[1,1,1,2,2,2]=>0 [1,1,1,2,3,1]=>[2,1,1,1,3,1]=>5 [1,1,1,2,4]=>[1,1,1,2,4]=>0 [1,1,1,3,1,1,1]=>[1,1,3,1,1,1,1]=>4 [1,1,1,3,1,2]=>[1,3,1,1,1,2]=>4 [1,1,1,3,2,1]=>[3,2,1,1,1,1]=>9 [1,1,1,3,3]=>[1,1,1,3,3]=>0 [1,1,1,4,1,1]=>[1,4,1,1,1,1]=>4 [1,1,1,4,2]=>[4,1,1,1,2]=>4 [1,1,1,5,1]=>[5,1,1,1,1]=>4 [1,1,1,6]=>[1,1,1,6]=>0 [1,1,2,1,1,1,1,1]=>[1,1,1,1,2,1,1,1]=>3 [1,1,2,1,1,1,2]=>[1,1,2,1,1,1,2]=>3 [1,1,2,1,1,2,1]=>[2,1,2,1,1,1,1]=>9 [1,1,2,1,1,3]=>[1,2,1,1,1,3]=>3 [1,1,2,1,2,1,1]=>[1,2,2,1,1,1,1]=>8 [1,1,2,1,2,2]=>[2,1,1,1,2,2]=>3 [1,1,2,1,3,1]=>[2,3,1,1,1,1]=>8 [1,1,2,1,4]=>[2,1,1,1,4]=>3 [1,1,2,2,1,1,1]=>[1,1,2,1,1,2,1]=>4 [1,1,2,2,1,2]=>[2,1,1,2,1,2]=>4 [1,1,2,2,2,1]=>[2,1,1,2,2,1]=>5 [1,1,2,2,3]=>[1,1,2,2,3]=>0 [1,1,2,3,1,1]=>[1,2,1,1,3,1]=>4 [1,1,2,3,2]=>[3,1,1,2,2]=>4 [1,1,2,4,1]=>[2,1,1,4,1]=>4 [1,1,2,5]=>[1,1,2,5]=>0 [1,1,3,1,1,1,1]=>[1,1,1,3,1,1,1]=>3 [1,1,3,1,1,2]=>[1,1,3,1,1,2]=>3 [1,1,3,1,2,1]=>[3,1,2,1,1,1]=>8 [1,1,3,1,3]=>[3,1,1,1,3]=>3 [1,1,3,2,1,1]=>[1,3,2,1,1,1]=>7 [1,1,3,2,2]=>[1,3,1,2,2]=>3 [1,1,3,3,1]=>[3,1,1,3,1]=>4 [1,1,3,4]=>[1,1,3,4]=>0 [1,1,4,1,1,1]=>[1,1,4,1,1,1]=>3 [1,1,4,1,2]=>[1,4,1,1,2]=>3 [1,1,4,2,1]=>[4,2,1,1,1]=>7 [1,1,4,3]=>[4,1,1,3]=>3 [1,1,5,1,1]=>[1,5,1,1,1]=>3 [1,1,5,2]=>[5,1,1,2]=>3 [1,1,6,1]=>[6,1,1,1]=>3 [1,1,7]=>[1,1,7]=>0 [1,2,1,1,1,1,1,1]=>[1,1,1,1,1,2,1,1]=>2 [1,2,1,1,1,1,2]=>[1,1,1,2,1,1,2]=>2 [1,2,1,1,1,2,1]=>[2,1,1,2,1,1,1]=>8 [1,2,1,1,1,3]=>[1,1,2,1,1,3]=>2 [1,2,1,1,2,1,1]=>[1,2,1,2,1,1,1]=>7 [1,2,1,1,2,2]=>[1,2,1,1,2,2]=>2 [1,2,1,1,3,1]=>[2,1,3,1,1,1]=>7 [1,2,1,1,4]=>[1,2,1,1,4]=>2 [1,2,1,2,1,1,1]=>[1,1,2,2,1,1,1]=>6 [1,2,1,2,1,2]=>[2,2,1,1,1,2]=>6 [1,2,1,2,2,1]=>[2,2,1,1,2,1]=>7 [1,2,1,2,3]=>[2,1,1,2,3]=>2 [1,2,1,3,1,1]=>[1,2,3,1,1,1]=>6 [1,2,1,3,2]=>[3,2,1,1,2]=>6 [1,2,1,4,1]=>[2,4,1,1,1]=>6 [1,2,1,5]=>[2,1,1,5]=>2 [1,2,2,1,1,1,1]=>[1,1,1,2,1,2,1]=>3 [1,2,2,1,1,2]=>[1,2,1,2,1,2]=>3 [1,2,2,1,2,1]=>[2,2,1,2,1,1]=>8 [1,2,2,1,3]=>[2,1,2,1,3]=>3 [1,2,2,2,1,1]=>[1,2,1,2,2,1]=>4 [1,2,2,2,2]=>[1,2,2,2,2]=>0 [1,2,2,3,1]=>[2,1,2,3,1]=>4 [1,2,2,4]=>[1,2,2,4]=>0 [1,2,3,1,1,1]=>[1,1,2,1,3,1]=>3 [1,2,3,1,2]=>[2,1,1,3,2]=>3 [1,2,3,2,1]=>[3,2,1,2,1]=>7 [1,2,3,3]=>[1,2,3,3]=>0 [1,2,4,1,1]=>[1,2,1,4,1]=>3 [1,2,4,2]=>[4,1,2,2]=>3 [1,2,5,1]=>[2,1,5,1]=>3 [1,2,6]=>[1,2,6]=>0 [1,3,1,1,1,1,1]=>[1,1,1,1,3,1,1]=>2 [1,3,1,1,1,2]=>[1,1,1,3,1,2]=>2 [1,3,1,1,2,1]=>[3,1,1,2,1,1]=>7 [1,3,1,1,3]=>[1,3,1,1,3]=>2 [1,3,1,2,1,1]=>[1,3,1,2,1,1]=>6 [1,3,1,2,2]=>[1,1,3,2,2]=>2 [1,3,1,3,1]=>[3,3,1,1,1]=>6 [1,3,1,4]=>[3,1,1,4]=>2 [1,3,2,1,1,1]=>[1,1,3,2,1,1]=>5 [1,3,2,1,2]=>[2,3,1,1,2]=>5 [1,3,2,2,1]=>[3,1,2,2,1]=>6 [1,3,2,3]=>[3,1,2,3]=>2 [1,3,3,1,1]=>[1,3,1,3,1]=>3 [1,3,3,2]=>[3,1,3,2]=>3 [1,3,4,1]=>[3,1,4,1]=>3 [1,3,5]=>[1,3,5]=>0 [1,4,1,1,1,1]=>[1,1,1,4,1,1]=>2 [1,4,1,1,2]=>[1,1,4,1,2]=>2 [1,4,1,2,1]=>[4,1,2,1,1]=>6 [1,4,1,3]=>[1,4,1,3]=>2 [1,4,2,1,1]=>[1,4,2,1,1]=>5 [1,4,2,2]=>[1,4,2,2]=>2 [1,4,3,1]=>[4,3,1,1]=>5 [1,4,4]=>[1,4,4]=>0 [1,5,1,1,1]=>[1,1,5,1,1]=>2 [1,5,1,2]=>[1,5,1,2]=>2 [1,5,2,1]=>[5,2,1,1]=>5 [1,5,3]=>[5,1,3]=>2 [1,6,1,1]=>[1,6,1,1]=>2 [1,6,2]=>[6,1,2]=>2 [1,7,1]=>[7,1,1]=>2 [1,8]=>[1,8]=>0 [2,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,2,1]=>1 [2,1,1,1,1,1,2]=>[1,1,1,1,2,1,2]=>1 [2,1,1,1,1,2,1]=>[2,1,1,1,2,1,1]=>7 [2,1,1,1,1,3]=>[1,1,1,2,1,3]=>1 [2,1,1,1,2,1,1]=>[1,2,1,1,2,1,1]=>6 [2,1,1,1,2,2]=>[1,1,2,1,2,2]=>1 [2,1,1,1,3,1]=>[2,1,1,3,1,1]=>6 [2,1,1,1,4]=>[1,1,2,1,4]=>1 [2,1,1,2,1,1,1]=>[1,1,2,1,2,1,1]=>5 [2,1,1,2,1,2]=>[2,1,2,1,1,2]=>5 [2,1,1,2,2,1]=>[2,1,2,1,2,1]=>6 [2,1,1,2,3]=>[1,2,1,2,3]=>1 [2,1,1,3,1,1]=>[1,2,1,3,1,1]=>5 [2,1,1,3,2]=>[3,1,2,1,2]=>5 [2,1,1,4,1]=>[2,1,4,1,1]=>5 [2,1,1,5]=>[1,2,1,5]=>1 [2,1,2,1,1,1,1]=>[1,1,1,2,2,1,1]=>4 [2,1,2,1,1,2]=>[1,2,2,1,1,2]=>4 [2,1,2,1,2,1]=>[2,2,2,1,1,1]=>9 [2,1,2,1,3]=>[2,2,1,1,3]=>4 [2,1,2,2,1,1]=>[1,2,2,1,2,1]=>5 [2,1,2,2,2]=>[2,1,2,2,2]=>1 [2,1,2,3,1]=>[2,2,1,3,1]=>5 [2,1,2,4]=>[2,1,2,4]=>1 [2,1,3,1,1,1]=>[1,1,2,3,1,1]=>4 [2,1,3,1,2]=>[2,1,3,1,2]=>4 [2,1,3,2,1]=>[3,2,2,1,1]=>8 [2,1,3,3]=>[2,1,3,3]=>1 [2,1,4,1,1]=>[1,2,4,1,1]=>4 [2,1,4,2]=>[4,2,1,2]=>4 [2,1,5,1]=>[2,5,1,1]=>4 [2,1,6]=>[2,1,6]=>1 [2,2,1,1,1,1,1]=>[1,1,1,1,2,2,1]=>2 [2,2,1,1,1,2]=>[1,1,2,2,1,2]=>2 [2,2,1,1,2,1]=>[2,1,2,2,1,1]=>7 [2,2,1,1,3]=>[1,2,2,1,3]=>2 [2,2,1,2,1,1]=>[1,2,2,2,1,1]=>6 [2,2,1,2,2]=>[2,2,1,2,2]=>2 [2,2,1,3,1]=>[2,2,3,1,1]=>6 [2,2,1,4]=>[2,2,1,4]=>2 [2,2,2,1,1,1]=>[1,1,2,2,2,1]=>3 [2,2,2,1,2]=>[2,2,2,1,2]=>3 [2,2,2,2,1]=>[2,2,2,2,1]=>4 [2,2,2,3]=>[2,2,2,3]=>0 [2,2,3,1,1]=>[1,2,2,3,1]=>3 [2,2,3,2]=>[3,2,2,2]=>3 [2,2,4,1]=>[2,2,4,1]=>3 [2,2,5]=>[2,2,5]=>0 [2,3,1,1,1,1]=>[1,1,1,2,3,1]=>2 [2,3,1,1,2]=>[1,2,1,3,2]=>2 [2,3,1,2,1]=>[2,3,1,2,1]=>6 [2,3,1,3]=>[2,3,1,3]=>2 [2,3,2,1,1]=>[1,3,2,2,1]=>5 [2,3,2,2]=>[2,3,2,2]=>2 [2,3,3,1]=>[2,3,3,1]=>3 [2,3,4]=>[2,3,4]=>0 [2,4,1,1,1]=>[1,1,2,4,1]=>2 [2,4,1,2]=>[2,1,4,2]=>2 [2,4,2,1]=>[4,2,2,1]=>5 [2,4,3]=>[4,2,3]=>2 [2,5,1,1]=>[1,2,5,1]=>2 [2,5,2]=>[5,2,2]=>2 [2,6,1]=>[2,6,1]=>2 [2,7]=>[2,7]=>0 [3,1,1,1,1,1,1]=>[1,1,1,1,1,3,1]=>1 [3,1,1,1,1,2]=>[1,1,1,1,3,2]=>1 [3,1,1,1,2,1]=>[3,1,1,1,2,1]=>6 [3,1,1,1,3]=>[1,1,3,1,3]=>1 [3,1,1,2,1,1]=>[1,3,1,1,2,1]=>5 [3,1,1,2,2]=>[1,1,2,3,2]=>1 [3,1,1,3,1]=>[3,1,3,1,1]=>5 [3,1,1,4]=>[1,3,1,4]=>1 [3,1,2,1,1,1]=>[1,1,3,1,2,1]=>4 [3,1,2,1,2]=>[1,3,2,1,2]=>4 [3,1,2,2,1]=>[2,1,3,2,1]=>5 [3,1,2,3]=>[1,3,2,3]=>1 [3,1,3,1,1]=>[1,3,3,1,1]=>4 [3,1,3,2]=>[3,3,1,2]=>4 [3,1,4,1]=>[3,4,1,1]=>4 [3,1,5]=>[3,1,5]=>1 [3,2,1,1,1,1]=>[1,1,1,3,2,1]=>3 [3,2,1,1,2]=>[1,2,3,1,2]=>3 [3,2,1,2,1]=>[2,3,2,1,1]=>7 [3,2,1,3]=>[3,2,1,3]=>3 [3,2,2,1,1]=>[1,2,3,2,1]=>4 [3,2,2,2]=>[2,2,3,2]=>1 [3,2,3,1]=>[3,2,3,1]=>4 [3,2,4]=>[3,2,4]=>1 [3,3,1,1,1]=>[1,1,3,3,1]=>2 [3,3,1,2]=>[1,3,3,2]=>2 [3,3,2,1]=>[3,3,2,1]=>5 [3,3,3]=>[3,3,3]=>0 [3,4,1,1]=>[1,3,4,1]=>2 [3,4,2]=>[3,4,2]=>2 [3,5,1]=>[3,5,1]=>2 [3,6]=>[3,6]=>0 [4,1,1,1,1,1]=>[1,1,1,1,4,1]=>1 [4,1,1,1,2]=>[1,1,1,4,2]=>1 [4,1,1,2,1]=>[4,1,1,2,1]=>5 [4,1,1,3]=>[1,1,4,3]=>1 [4,1,2,1,1]=>[1,4,1,2,1]=>4 [4,1,2,2]=>[1,2,4,2]=>1 [4,1,3,1]=>[4,1,3,1]=>4 [4,1,4]=>[4,1,4]=>1 [4,2,1,1,1]=>[1,1,4,2,1]=>3 [4,2,1,2]=>[2,4,1,2]=>3 [4,2,2,1]=>[2,4,2,1]=>4 [4,2,3]=>[2,4,3]=>1 [4,3,1,1]=>[1,4,3,1]=>3 [4,3,2]=>[4,3,2]=>3 [4,4,1]=>[4,4,1]=>2 [4,5]=>[4,5]=>0 [5,1,1,1,1]=>[1,1,1,5,1]=>1 [5,1,1,2]=>[1,1,5,2]=>1 [5,1,2,1]=>[5,1,2,1]=>4 [5,1,3]=>[1,5,3]=>1 [5,2,1,1]=>[1,5,2,1]=>3 [5,2,2]=>[2,5,2]=>1 [5,3,1]=>[5,3,1]=>3 [5,4]=>[5,4]=>1 [6,1,1,1]=>[1,1,6,1]=>1 [6,1,2]=>[1,6,2]=>1 [6,2,1]=>[6,2,1]=>3 [6,3]=>[6,3]=>1 [7,1,1]=>[1,7,1]=>1 [7,2]=>[7,2]=>1 [8,1]=>[8,1]=>1 [9]=>[9]=>0 [1,1,1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1,1,1]=>0 [1,1,1,1,1,1,2,2]=>[1,1,1,1,1,1,2,2]=>0 [1,1,1,1,1,2,1,2]=>[2,1,1,1,1,1,1,2]=>6 [1,1,1,1,2,1,2,1]=>[2,2,1,1,1,1,1,1]=>12 [1,1,1,1,3,3]=>[1,1,1,1,3,3]=>0 [1,1,1,2,1,1,1,2]=>[1,1,2,1,1,1,1,2]=>4 [1,1,1,3,1,3]=>[3,1,1,1,1,3]=>4 [1,1,1,5,1,1]=>[1,5,1,1,1,1]=>4 [1,1,2,1,1,1,2,1]=>[2,1,1,2,1,1,1,1]=>10 [1,1,2,1,2,1,1,1]=>[1,1,2,2,1,1,1,1]=>8 [1,1,2,1,2,3]=>[2,1,1,1,2,3]=>3 [1,1,2,1,3,2]=>[3,2,1,1,1,2]=>8 [1,1,2,2,2,2]=>[1,1,2,2,2,2]=>0 [1,1,2,4,1,1]=>[1,2,1,1,4,1]=>4 [1,1,3,1,3,1]=>[3,3,1,1,1,1]=>8 [1,1,3,3,1,1]=>[1,3,1,1,3,1]=>4 [1,1,4,4]=>[1,1,4,4]=>0 [1,2,1,1,1,1,1,2]=>[1,1,1,1,2,1,1,2]=>2 [1,2,1,2,2,2]=>[2,1,1,2,2,2]=>2 [1,2,2,2,1,2]=>[2,1,2,2,1,2]=>4 [1,2,6,1]=>[2,1,6,1]=>3 [1,3,1,1,1,3]=>[1,1,3,1,1,3]=>2 [1,3,2,4]=>[3,1,2,4]=>2 [1,4,1,4]=>[4,1,1,4]=>2 [1,4,3,2]=>[4,3,1,2]=>5 [1,6,1,2]=>[1,6,1,2]=>2 [1,7,1,1]=>[1,7,1,1]=>2 [1,8,1]=>[8,1,1]=>2 [1,9]=>[1,9]=>0 [2,1,1,1,1,1,2,1]=>[2,1,1,1,1,2,1,1]=>8 [2,1,1,1,2,1,1,1]=>[1,1,2,1,1,2,1,1]=>6 [2,1,1,1,2,3]=>[1,1,2,1,2,3]=>1 [2,1,1,1,3,2]=>[3,1,1,2,1,2]=>6 [2,1,1,2,1,3]=>[2,1,2,1,1,3]=>5 [2,1,1,3,2,1]=>[3,2,1,2,1,1]=>10 [2,1,2,1,1,1,1,1]=>[1,1,1,1,2,2,1,1]=>4 [2,1,2,1,2,2]=>[2,2,1,1,2,2]=>4 [2,1,3,4]=>[2,1,3,4]=>1 [2,1,4,3]=>[4,2,1,3]=>4 [2,2,1,1,3,1]=>[2,1,2,3,1,1]=>7 [2,2,1,2,1,2]=>[2,2,2,1,1,2]=>6 [2,2,2,1,2,1]=>[2,2,2,2,1,1]=>8 [2,2,3,3]=>[2,2,3,3]=>0 [2,3,2,3]=>[3,2,2,3]=>2 [2,6,1,1]=>[1,2,6,1]=>2 [3,1,1,1,3,1]=>[3,1,1,3,1,1]=>6 [3,1,2,1,1,2]=>[1,1,3,2,1,2]=>4 [3,1,3,1,1,1]=>[1,1,3,3,1,1]=>4 [3,2,3,2]=>[3,3,2,2]=>4 [3,5,1,1]=>[1,3,5,1]=>2 [4,1,1,2,1,1]=>[1,4,1,1,2,1]=>5 [4,1,4,1]=>[4,4,1,1]=>4 [4,4,1,1]=>[1,4,4,1]=>2 [5,1,1,1,1,1]=>[1,1,1,1,5,1]=>1 [5,3,1,1]=>[1,5,3,1]=>3 [5,5]=>[5,5]=>0 [6,2,1,1]=>[1,6,2,1]=>3 [7,1,1,1]=>[1,1,7,1]=>1 [8,1,1]=>[1,8,1]=>1 [9,1]=>[9,1]=>1
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Description
The number of inversions of an integer composition.
This is the number of pairs $(i,j)$ such that $i < j$ and $c_i > c_j$.
Map
Foata bijection
Description
The Foata bijection for compositions.
The Foata bijection $\phi$ is a bijection on the set of words whose letters are positive integers. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
  • If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
  • If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
  • $1$
  • $|1|4 \to 14$
  • $|14|2 \to 412$
  • $|4|1|2|5 \to 4125$
  • $|4|125|3 \to 45123.$
In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
This bijection sends the major index St000769The major index of a composition regarded as a word. to the number of inversions St000766The number of inversions of an integer composition..