Identifier
Values
=>
[1]=>0
[1,1]=>0
[2]=>0
[1,1,1]=>0
[1,2]=>0
[2,1]=>1
[3]=>0
[1,1,1,1]=>0
[1,1,2]=>0
[1,2,1]=>1
[1,3]=>0
[2,1,1]=>2
[2,2]=>0
[3,1]=>1
[4]=>0
[1,1,1,1,1]=>0
[1,1,1,2]=>0
[1,1,2,1]=>1
[1,1,3]=>0
[1,2,1,1]=>2
[1,2,2]=>0
[1,3,1]=>1
[1,4]=>0
[2,1,1,1]=>3
[2,1,2]=>1
[2,2,1]=>2
[2,3]=>0
[3,1,1]=>2
[3,2]=>1
[4,1]=>1
[5]=>0
[1,1,1,1,1,1]=>0
[1,1,1,1,2]=>0
[1,1,1,2,1]=>1
[1,1,1,3]=>0
[1,1,2,1,1]=>2
[1,1,2,2]=>0
[1,1,3,1]=>1
[1,1,4]=>0
[1,2,1,1,1]=>3
[1,2,1,2]=>1
[1,2,2,1]=>2
[1,2,3]=>0
[1,3,1,1]=>2
[1,3,2]=>1
[1,4,1]=>1
[1,5]=>0
[2,1,1,1,1]=>4
[2,1,1,2]=>2
[2,1,2,1]=>3
[2,1,3]=>1
[2,2,1,1]=>4
[2,2,2]=>0
[2,3,1]=>2
[2,4]=>0
[3,1,1,1]=>3
[3,1,2]=>2
[3,2,1]=>3
[3,3]=>0
[4,1,1]=>2
[4,2]=>1
[5,1]=>1
[6]=>0
[1,1,1,1,1,1,1]=>0
[1,1,1,1,1,2]=>0
[1,1,1,1,2,1]=>1
[1,1,1,1,3]=>0
[1,1,1,2,1,1]=>2
[1,1,1,2,2]=>0
[1,1,1,3,1]=>1
[1,1,1,4]=>0
[1,1,2,1,1,1]=>3
[1,1,2,1,2]=>1
[1,1,2,2,1]=>2
[1,1,2,3]=>0
[1,1,3,1,1]=>2
[1,1,3,2]=>1
[1,1,4,1]=>1
[1,1,5]=>0
[1,2,1,1,1,1]=>4
[1,2,1,1,2]=>2
[1,2,1,2,1]=>3
[1,2,1,3]=>1
[1,2,2,1,1]=>4
[1,2,2,2]=>0
[1,2,3,1]=>2
[1,2,4]=>0
[1,3,1,1,1]=>3
[1,3,1,2]=>2
[1,3,2,1]=>3
[1,3,3]=>0
[1,4,1,1]=>2
[1,4,2]=>1
[1,5,1]=>1
[1,6]=>0
[2,1,1,1,1,1]=>5
[2,1,1,1,2]=>3
[2,1,1,2,1]=>4
[2,1,1,3]=>2
[2,1,2,1,1]=>5
[2,1,2,2]=>1
[2,1,3,1]=>3
[2,1,4]=>1
[2,2,1,1,1]=>6
[2,2,1,2]=>2
[2,2,2,1]=>3
[2,2,3]=>0
[2,3,1,1]=>4
[2,3,2]=>1
[2,4,1]=>2
[2,5]=>0
[3,1,1,1,1]=>4
[3,1,1,2]=>3
[3,1,2,1]=>4
[3,1,3]=>1
[3,2,1,1]=>5
[3,2,2]=>2
[3,3,1]=>2
[3,4]=>0
[4,1,1,1]=>3
[4,1,2]=>2
[4,2,1]=>3
[4,3]=>1
[5,1,1]=>2
[5,2]=>1
[6,1]=>1
[7]=>0
[1,1,1,1,1,1,1,1]=>0
[1,1,1,1,1,1,2]=>0
[1,1,1,1,1,2,1]=>1
[1,1,1,1,1,3]=>0
[1,1,1,1,2,1,1]=>2
[1,1,1,1,2,2]=>0
[1,1,1,1,3,1]=>1
[1,1,1,1,4]=>0
[1,1,1,2,1,1,1]=>3
[1,1,1,2,1,2]=>1
[1,1,1,2,2,1]=>2
[1,1,1,2,3]=>0
[1,1,1,3,1,1]=>2
[1,1,1,3,2]=>1
[1,1,1,4,1]=>1
[1,1,1,5]=>0
[1,1,2,1,1,1,1]=>4
[1,1,2,1,1,2]=>2
[1,1,2,1,2,1]=>3
[1,1,2,1,3]=>1
[1,1,2,2,1,1]=>4
[1,1,2,2,2]=>0
[1,1,2,3,1]=>2
[1,1,2,4]=>0
[1,1,3,1,1,1]=>3
[1,1,3,1,2]=>2
[1,1,3,2,1]=>3
[1,1,3,3]=>0
[1,1,4,1,1]=>2
[1,1,4,2]=>1
[1,1,5,1]=>1
[1,1,6]=>0
[1,2,1,1,1,1,1]=>5
[1,2,1,1,1,2]=>3
[1,2,1,1,2,1]=>4
[1,2,1,1,3]=>2
[1,2,1,2,1,1]=>5
[1,2,1,2,2]=>1
[1,2,1,3,1]=>3
[1,2,1,4]=>1
[1,2,2,1,1,1]=>6
[1,2,2,1,2]=>2
[1,2,2,2,1]=>3
[1,2,2,3]=>0
[1,2,3,1,1]=>4
[1,2,3,2]=>1
[1,2,4,1]=>2
[1,2,5]=>0
[1,3,1,1,1,1]=>4
[1,3,1,1,2]=>3
[1,3,1,2,1]=>4
[1,3,1,3]=>1
[1,3,2,1,1]=>5
[1,3,2,2]=>2
[1,3,3,1]=>2
[1,3,4]=>0
[1,4,1,1,1]=>3
[1,4,1,2]=>2
[1,4,2,1]=>3
[1,4,3]=>1
[1,5,1,1]=>2
[1,5,2]=>1
[1,6,1]=>1
[1,7]=>0
[2,1,1,1,1,1,1]=>6
[2,1,1,1,1,2]=>4
[2,1,1,1,2,1]=>5
[2,1,1,1,3]=>3
[2,1,1,2,1,1]=>6
[2,1,1,2,2]=>2
[2,1,1,3,1]=>4
[2,1,1,4]=>2
[2,1,2,1,1,1]=>7
[2,1,2,1,2]=>3
[2,1,2,2,1]=>4
[2,1,2,3]=>1
[2,1,3,1,1]=>5
[2,1,3,2]=>2
[2,1,4,1]=>3
[2,1,5]=>1
[2,2,1,1,1,1]=>8
[2,2,1,1,2]=>4
[2,2,1,2,1]=>5
[2,2,1,3]=>2
[2,2,2,1,1]=>6
[2,2,2,2]=>0
[2,2,3,1]=>3
[2,2,4]=>0
[2,3,1,1,1]=>6
[2,3,1,2]=>3
[2,3,2,1]=>4
[2,3,3]=>0
[2,4,1,1]=>4
[2,4,2]=>1
[2,5,1]=>2
[2,6]=>0
[3,1,1,1,1,1]=>5
[3,1,1,1,2]=>4
[3,1,1,2,1]=>5
[3,1,1,3]=>2
[3,1,2,1,1]=>6
[3,1,2,2]=>3
[3,1,3,1]=>3
[3,1,4]=>1
[3,2,1,1,1]=>7
[3,2,1,2]=>4
[3,2,2,1]=>5
[3,2,3]=>1
[3,3,1,1]=>4
[3,3,2]=>2
[3,4,1]=>2
[3,5]=>0
[4,1,1,1,1]=>4
[4,1,1,2]=>3
[4,1,2,1]=>4
[4,1,3]=>2
[4,2,1,1]=>5
[4,2,2]=>2
[4,3,1]=>3
[4,4]=>0
[5,1,1,1]=>3
[5,1,2]=>2
[5,2,1]=>3
[5,3]=>1
[6,1,1]=>2
[6,2]=>1
[7,1]=>1
[8]=>0
[1,1,1,1,1,1,1,1,1]=>0
[1,1,1,1,1,1,1,2]=>0
[1,1,1,1,1,1,2,1]=>1
[1,1,1,1,1,1,3]=>0
[1,1,1,1,1,2,1,1]=>2
[1,1,1,1,1,2,2]=>0
[1,1,1,1,1,3,1]=>1
[1,1,1,1,1,4]=>0
[1,1,1,1,2,1,1,1]=>3
[1,1,1,1,2,1,2]=>1
[1,1,1,1,2,2,1]=>2
[1,1,1,1,2,3]=>0
[1,1,1,1,3,1,1]=>2
[1,1,1,1,3,2]=>1
[1,1,1,1,4,1]=>1
[1,1,1,1,5]=>0
[1,1,1,2,1,1,1,1]=>4
[1,1,1,2,1,1,2]=>2
[1,1,1,2,1,2,1]=>3
[1,1,1,2,1,3]=>1
[1,1,1,2,2,1,1]=>4
[1,1,1,2,2,2]=>0
[1,1,1,2,3,1]=>2
[1,1,1,2,4]=>0
[1,1,1,3,1,1,1]=>3
[1,1,1,3,1,2]=>2
[1,1,1,3,2,1]=>3
[1,1,1,3,3]=>0
[1,1,1,4,1,1]=>2
[1,1,1,4,2]=>1
[1,1,1,5,1]=>1
[1,1,1,6]=>0
[1,1,2,1,1,1,1,1]=>5
[1,1,2,1,1,1,2]=>3
[1,1,2,1,1,2,1]=>4
[1,1,2,1,1,3]=>2
[1,1,2,1,2,1,1]=>5
[1,1,2,1,2,2]=>1
[1,1,2,1,3,1]=>3
[1,1,2,1,4]=>1
[1,1,2,2,1,1,1]=>6
[1,1,2,2,1,2]=>2
[1,1,2,2,2,1]=>3
[1,1,2,2,3]=>0
[1,1,2,3,1,1]=>4
[1,1,2,3,2]=>1
[1,1,2,4,1]=>2
[1,1,2,5]=>0
[1,1,3,1,1,1,1]=>4
[1,1,3,1,1,2]=>3
[1,1,3,1,2,1]=>4
[1,1,3,1,3]=>1
[1,1,3,2,1,1]=>5
[1,1,3,2,2]=>2
[1,1,3,3,1]=>2
[1,1,3,4]=>0
[1,1,4,1,1,1]=>3
[1,1,4,1,2]=>2
[1,1,4,2,1]=>3
[1,1,4,3]=>1
[1,1,5,1,1]=>2
[1,1,5,2]=>1
[1,1,6,1]=>1
[1,1,7]=>0
[1,2,1,1,1,1,1,1]=>6
[1,2,1,1,1,1,2]=>4
[1,2,1,1,1,2,1]=>5
[1,2,1,1,1,3]=>3
[1,2,1,1,2,1,1]=>6
[1,2,1,1,2,2]=>2
[1,2,1,1,3,1]=>4
[1,2,1,1,4]=>2
[1,2,1,2,1,1,1]=>7
[1,2,1,2,1,2]=>3
[1,2,1,2,2,1]=>4
[1,2,1,2,3]=>1
[1,2,1,3,1,1]=>5
[1,2,1,3,2]=>2
[1,2,1,4,1]=>3
[1,2,1,5]=>1
[1,2,2,1,1,1,1]=>8
[1,2,2,1,1,2]=>4
[1,2,2,1,2,1]=>5
[1,2,2,1,3]=>2
[1,2,2,2,1,1]=>6
[1,2,2,2,2]=>0
[1,2,2,3,1]=>3
[1,2,2,4]=>0
[1,2,3,1,1,1]=>6
[1,2,3,1,2]=>3
[1,2,3,2,1]=>4
[1,2,3,3]=>0
[1,2,4,1,1]=>4
[1,2,4,2]=>1
[1,2,5,1]=>2
[1,2,6]=>0
[1,3,1,1,1,1,1]=>5
[1,3,1,1,1,2]=>4
[1,3,1,1,2,1]=>5
[1,3,1,1,3]=>2
[1,3,1,2,1,1]=>6
[1,3,1,2,2]=>3
[1,3,1,3,1]=>3
[1,3,1,4]=>1
[1,3,2,1,1,1]=>7
[1,3,2,1,2]=>4
[1,3,2,2,1]=>5
[1,3,2,3]=>1
[1,3,3,1,1]=>4
[1,3,3,2]=>2
[1,3,4,1]=>2
[1,3,5]=>0
[1,4,1,1,1,1]=>4
[1,4,1,1,2]=>3
[1,4,1,2,1]=>4
[1,4,1,3]=>2
[1,4,2,1,1]=>5
[1,4,2,2]=>2
[1,4,3,1]=>3
[1,4,4]=>0
[1,5,1,1,1]=>3
[1,5,1,2]=>2
[1,5,2,1]=>3
[1,5,3]=>1
[1,6,1,1]=>2
[1,6,2]=>1
[1,7,1]=>1
[1,8]=>0
[2,1,1,1,1,1,1,1]=>7
[2,1,1,1,1,1,2]=>5
[2,1,1,1,1,2,1]=>6
[2,1,1,1,1,3]=>4
[2,1,1,1,2,1,1]=>7
[2,1,1,1,2,2]=>3
[2,1,1,1,3,1]=>5
[2,1,1,1,4]=>3
[2,1,1,2,1,1,1]=>8
[2,1,1,2,1,2]=>4
[2,1,1,2,2,1]=>5
[2,1,1,2,3]=>2
[2,1,1,3,1,1]=>6
[2,1,1,3,2]=>3
[2,1,1,4,1]=>4
[2,1,1,5]=>2
[2,1,2,1,1,1,1]=>9
[2,1,2,1,1,2]=>5
[2,1,2,1,2,1]=>6
[2,1,2,1,3]=>3
[2,1,2,2,1,1]=>7
[2,1,2,2,2]=>1
[2,1,2,3,1]=>4
[2,1,2,4]=>1
[2,1,3,1,1,1]=>7
[2,1,3,1,2]=>4
[2,1,3,2,1]=>5
[2,1,3,3]=>1
[2,1,4,1,1]=>5
[2,1,4,2]=>2
[2,1,5,1]=>3
[2,1,6]=>1
[2,2,1,1,1,1,1]=>10
[2,2,1,1,1,2]=>6
[2,2,1,1,2,1]=>7
[2,2,1,1,3]=>4
[2,2,1,2,1,1]=>8
[2,2,1,2,2]=>2
[2,2,1,3,1]=>5
[2,2,1,4]=>2
[2,2,2,1,1,1]=>9
[2,2,2,1,2]=>3
[2,2,2,2,1]=>4
[2,2,2,3]=>0
[2,2,3,1,1]=>6
[2,2,3,2]=>1
[2,2,4,1]=>3
[2,2,5]=>0
[2,3,1,1,1,1]=>8
[2,3,1,1,2]=>5
[2,3,1,2,1]=>6
[2,3,1,3]=>2
[2,3,2,1,1]=>7
[2,3,2,2]=>2
[2,3,3,1]=>3
[2,3,4]=>0
[2,4,1,1,1]=>6
[2,4,1,2]=>3
[2,4,2,1]=>4
[2,4,3]=>1
[2,5,1,1]=>4
[2,5,2]=>1
[2,6,1]=>2
[2,7]=>0
[3,1,1,1,1,1,1]=>6
[3,1,1,1,1,2]=>5
[3,1,1,1,2,1]=>6
[3,1,1,1,3]=>3
[3,1,1,2,1,1]=>7
[3,1,1,2,2]=>4
[3,1,1,3,1]=>4
[3,1,1,4]=>2
[3,1,2,1,1,1]=>8
[3,1,2,1,2]=>5
[3,1,2,2,1]=>6
[3,1,2,3]=>2
[3,1,3,1,1]=>5
[3,1,3,2]=>3
[3,1,4,1]=>3
[3,1,5]=>1
[3,2,1,1,1,1]=>9
[3,2,1,1,2]=>6
[3,2,1,2,1]=>7
[3,2,1,3]=>3
[3,2,2,1,1]=>8
[3,2,2,2]=>3
[3,2,3,1]=>4
[3,2,4]=>1
[3,3,1,1,1]=>6
[3,3,1,2]=>4
[3,3,2,1]=>5
[3,3,3]=>0
[3,4,1,1]=>4
[3,4,2]=>2
[3,5,1]=>2
[3,6]=>0
[4,1,1,1,1,1]=>5
[4,1,1,1,2]=>4
[4,1,1,2,1]=>5
[4,1,1,3]=>3
[4,1,2,1,1]=>6
[4,1,2,2]=>3
[4,1,3,1]=>4
[4,1,4]=>1
[4,2,1,1,1]=>7
[4,2,1,2]=>4
[4,2,2,1]=>5
[4,2,3]=>2
[4,3,1,1]=>5
[4,3,2]=>3
[4,4,1]=>2
[4,5]=>0
[5,1,1,1,1]=>4
[5,1,1,2]=>3
[5,1,2,1]=>4
[5,1,3]=>2
[5,2,1,1]=>5
[5,2,2]=>2
[5,3,1]=>3
[5,4]=>1
[6,1,1,1]=>3
[6,1,2]=>2
[6,2,1]=>3
[6,3]=>1
[7,1,1]=>2
[7,2]=>1
[8,1]=>1
[9]=>0
[1,1,1,1,1,1,1,1,1,1]=>0
[1,1,1,1,1,1,2,2]=>0
[1,1,1,1,2,1,1,2]=>2
[1,1,1,1,2,2,1,1]=>4
[1,1,1,1,3,3]=>0
[1,1,1,1,5,1]=>1
[1,1,2,1,1,1,1,2]=>4
[1,1,2,1,1,2,1,1]=>6
[1,1,2,1,2,3]=>1
[1,1,2,2,1,1,1,1]=>8
[1,1,2,2,2,2]=>0
[1,1,3,1,1,3]=>2
[1,1,3,2,1,2]=>4
[1,1,3,3,1,1]=>4
[1,1,4,4]=>0
[1,1,7,1]=>1
[1,2,1,1,4,1]=>4
[1,2,6,1]=>2
[1,3,1,1,3,1]=>4
[1,3,5,1]=>2
[1,4,1,1,2,1]=>5
[1,4,4,1]=>2
[1,5,1,1,1,1]=>4
[1,5,3,1]=>3
[1,6,1,2]=>2
[1,6,2,1]=>3
[1,7,1,1]=>2
[1,8,1]=>1
[1,9]=>0
[2,1,1,1,1,1,1,2]=>6
[2,1,1,1,1,2,1,1]=>8
[2,1,1,1,2,3]=>3
[2,1,1,2,1,1,1,1]=>10
[2,1,1,2,2,2]=>2
[2,1,2,1,1,3]=>5
[2,1,2,2,1,2]=>4
[2,1,2,3,1,1]=>7
[2,1,3,4]=>1
[2,1,6,1]=>3
[2,2,1,1,1,1,1,1]=>12
[2,2,1,1,2,2]=>4
[2,2,2,1,1,2]=>6
[2,2,2,2,1,1]=>8
[2,2,3,3]=>0
[3,1,1,1,1,3]=>4
[3,1,1,2,1,2]=>6
[3,1,1,3,1,1]=>6
[3,1,2,4]=>2
[3,2,1,1,1,2]=>8
[3,2,1,2,1,1]=>10
[3,2,2,3]=>2
[3,3,1,1,1,1]=>8
[3,3,2,2]=>4
[4,1,1,4]=>2
[4,2,1,3]=>4
[4,3,1,2]=>5
[4,4,1,1]=>4
[5,5]=>0
[8,1,1]=>2
[9,1]=>1
[1,10]=>0
[1,8,1,1]=>2
[1,7,2,1]=>3
[1,6,3,1]=>3
[1,5,4,1]=>3
[1,4,5,1]=>2
[1,3,6,1]=>2
[1,2,7,1]=>2
[1,1,8,1]=>1
[10,1]=>1
[1,1,1,1,1,1,1,1,1,1,1,1]=>0
[1,1,1,1,1,1,1,1,2,2]=>0
[1,1,1,1,1,1,2,2,1,1]=>4
[1,1,1,1,1,1,2,1,1,2]=>2
[1,1,1,1,1,1,3,3]=>0
[1,1,1,1,2,2,1,1,1,1]=>8
[1,1,1,1,2,2,2,2]=>0
[1,1,1,1,2,1,1,2,1,1]=>6
[1,1,1,1,2,1,1,1,1,2]=>4
[1,1,1,1,2,1,2,3]=>1
[1,1,1,1,3,3,1,1]=>4
[1,1,1,1,3,2,1,2]=>4
[1,1,1,1,3,1,1,3]=>2
[1,1,1,1,4,4]=>0
[1,1,2,2,1,1,1,1,1,1]=>12
[1,1,2,2,1,1,2,2]=>4
[1,1,2,2,2,2,1,1]=>8
[1,1,2,2,2,1,1,2]=>6
[1,1,2,2,3,3]=>0
[1,1,2,1,1,2,1,1,1,1]=>10
[1,1,2,1,1,2,2,2]=>2
[1,1,2,1,1,1,1,2,1,1]=>8
[1,1,2,1,1,1,1,1,1,2]=>6
[1,1,2,1,1,1,2,3]=>3
[1,1,2,1,2,3,1,1]=>7
[1,1,2,1,2,2,1,2]=>4
[1,1,2,1,2,1,1,3]=>5
[1,1,2,1,3,4]=>1
[1,1,3,3,1,1,1,1]=>8
[1,1,3,3,2,2]=>4
[1,1,3,2,1,2,1,1]=>10
[1,1,3,2,1,1,1,2]=>8
[1,1,3,2,2,3]=>2
[1,1,3,1,1,3,1,1]=>6
[1,1,3,1,1,2,1,2]=>6
[1,1,3,1,1,1,1,3]=>4
[1,1,3,1,2,4]=>2
[1,1,4,4,1,1]=>4
[1,1,4,3,1,2]=>5
[1,1,4,2,1,3]=>4
[1,1,4,1,1,4]=>2
[1,1,5,5]=>0
[2,2,1,1,1,1,1,1,1,1]=>16
[2,2,1,1,1,1,2,2]=>8
[2,2,1,1,2,2,1,1]=>12
[2,2,1,1,2,1,1,2]=>10
[2,2,1,1,3,3]=>4
[2,2,2,2,1,1,1,1]=>16
[2,2,2,2,2,2]=>0
[2,2,2,1,1,2,1,1]=>14
[2,2,2,1,1,1,1,2]=>12
[2,2,2,1,2,3]=>3
[2,2,3,3,1,1]=>8
[2,2,3,2,1,2]=>6
[2,2,3,1,1,3]=>6
[2,2,4,4]=>0
[2,1,1,2,1,1,1,1,1,1]=>14
[2,1,1,2,1,1,2,2]=>6
[2,1,1,2,2,2,1,1]=>10
[2,1,1,2,2,1,1,2]=>8
[2,1,1,2,3,3]=>2
[2,1,1,1,1,2,1,1,1,1]=>12
[2,1,1,1,1,2,2,2]=>4
[2,1,1,1,1,1,1,2,1,1]=>10
[2,1,1,1,1,1,1,1,1,2]=>8
[2,1,1,1,1,1,2,3]=>5
[2,1,1,1,2,3,1,1]=>9
[2,1,1,1,2,2,1,2]=>6
[2,1,1,1,2,1,1,3]=>7
[2,1,1,1,3,4]=>3
[2,1,2,3,1,1,1,1]=>13
[2,1,2,3,2,2]=>3
[2,1,2,2,1,2,1,1]=>12
[2,1,2,2,1,1,1,2]=>10
[2,1,2,2,2,3]=>1
[2,1,2,1,1,3,1,1]=>11
[2,1,2,1,1,2,1,2]=>8
[2,1,2,1,1,1,1,3]=>9
[2,1,2,1,2,4]=>3
[2,1,3,4,1,1]=>7
[2,1,3,3,1,2]=>6
[2,1,3,2,1,3]=>5
[2,1,3,1,1,4]=>5
[2,1,4,5]=>1
[3,3,1,1,1,1,1,1]=>12
[3,3,1,1,2,2]=>8
[3,3,2,2,1,1]=>12
[3,3,2,1,1,2]=>10
[3,3,3,3]=>0
[3,2,1,2,1,1,1,1]=>16
[3,2,1,2,2,2]=>6
[3,2,1,1,1,2,1,1]=>14
[3,2,1,1,1,1,1,2]=>12
[3,2,1,1,2,3]=>6
[3,2,2,3,1,1]=>10
[3,2,2,2,1,2]=>8
[3,2,2,1,1,3]=>8
[3,2,3,4]=>1
[3,1,1,3,1,1,1,1]=>10
[3,1,1,3,2,2]=>6
[3,1,1,2,1,2,1,1]=>12
[3,1,1,2,1,1,1,2]=>10
[3,1,1,2,2,3]=>4
[3,1,1,1,1,3,1,1]=>8
[3,1,1,1,1,2,1,2]=>8
[3,1,1,1,1,1,1,3]=>6
[3,1,1,1,2,4]=>4
[3,1,2,4,1,1]=>8
[3,1,2,3,1,2]=>7
[3,1,2,2,1,3]=>6
[3,1,2,1,1,4]=>6
[3,1,3,5]=>1
[4,4,1,1,1,1]=>8
[4,4,2,2]=>4
[4,3,1,2,1,1]=>11
[4,3,1,1,1,2]=>9
[4,3,2,3]=>4
[4,2,1,3,1,1]=>10
[4,2,1,2,1,2]=>8
[4,2,1,1,1,3]=>8
[4,2,2,4]=>2
[4,1,1,4,1,1]=>6
[4,1,1,3,1,2]=>7
[4,1,1,2,1,3]=>6
[4,1,1,1,1,4]=>4
[4,1,2,5]=>2
[5,5,1,1]=>4
[5,4,1,2]=>5
[5,3,1,3]=>4
[5,2,1,4]=>4
[5,1,1,5]=>2
[6,6]=>0
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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 $c_i > c_j$.
This is the number of pairs $(i,j)$ such that $i < j$ and $c_i > c_j$.
References
[1] MacMahon, P. A. Two Applications of General Theorems in Combinatory Analysis: (1) To the Theory of Inversions of Permutations; (2) To the Ascertainment of the Numbers of Terms in the Development of a Determinant which has Amongst its Elements an Arbitrary Number of Zeros MathSciNet:1576566
Code
def statistic(w):
return len([(i,j) for i in range(len(w)) for j in range(i, len(w)) if w[i] > w[j]])
Created
Apr 09, 2017 at 14:37 by Martin Rubey
Updated
Jan 11, 2018 at 09:38 by Martin Rubey
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