Identifier
Values
=>
[1]=>0
[-1]=>1
[1,2]=>0
[1,-2]=>1
[-1,2]=>1
[-1,-2]=>2
[2,1]=>2
[2,-1]=>3
[-2,1]=>1
[-2,-1]=>2
[1,2,3]=>0
[1,2,-3]=>1
[1,-2,3]=>1
[1,-2,-3]=>2
[-1,2,3]=>1
[-1,2,-3]=>2
[-1,-2,3]=>2
[-1,-2,-3]=>3
[1,3,2]=>2
[1,3,-2]=>3
[1,-3,2]=>1
[1,-3,-2]=>2
[-1,3,2]=>3
[-1,3,-2]=>4
[-1,-3,2]=>2
[-1,-3,-2]=>3
[2,1,3]=>2
[2,1,-3]=>3
[2,-1,3]=>3
[2,-1,-3]=>4
[-2,1,3]=>1
[-2,1,-3]=>2
[-2,-1,3]=>2
[-2,-1,-3]=>3
[2,3,1]=>4
[2,3,-1]=>5
[2,-3,1]=>3
[2,-3,-1]=>4
[-2,3,1]=>3
[-2,3,-1]=>4
[-2,-3,1]=>2
[-2,-3,-1]=>3
[3,1,2]=>2
[3,1,-2]=>3
[3,-1,2]=>3
[3,-1,-2]=>4
[-3,1,2]=>1
[-3,1,-2]=>2
[-3,-1,2]=>2
[-3,-1,-2]=>3
[3,2,1]=>2
[3,2,-1]=>3
[3,-2,1]=>3
[3,-2,-1]=>4
[-3,2,1]=>1
[-3,2,-1]=>2
[-3,-2,1]=>2
[-3,-2,-1]=>3
[1,2,3,4]=>0
[1,2,3,-4]=>1
[1,2,-3,4]=>1
[1,2,-3,-4]=>2
[1,-2,3,4]=>1
[1,-2,3,-4]=>2
[1,-2,-3,4]=>2
[1,-2,-3,-4]=>3
[-1,2,3,4]=>1
[-1,2,3,-4]=>2
[-1,2,-3,4]=>2
[-1,2,-3,-4]=>3
[-1,-2,3,4]=>2
[-1,-2,3,-4]=>3
[-1,-2,-3,4]=>3
[-1,-2,-3,-4]=>4
[1,2,4,3]=>2
[1,2,4,-3]=>3
[1,2,-4,3]=>1
[1,2,-4,-3]=>2
[1,-2,4,3]=>3
[1,-2,4,-3]=>4
[1,-2,-4,3]=>2
[1,-2,-4,-3]=>3
[-1,2,4,3]=>3
[-1,2,4,-3]=>4
[-1,2,-4,3]=>2
[-1,2,-4,-3]=>3
[-1,-2,4,3]=>4
[-1,-2,4,-3]=>5
[-1,-2,-4,3]=>3
[-1,-2,-4,-3]=>4
[1,3,2,4]=>2
[1,3,2,-4]=>3
[1,3,-2,4]=>3
[1,3,-2,-4]=>4
[1,-3,2,4]=>1
[1,-3,2,-4]=>2
[1,-3,-2,4]=>2
[1,-3,-2,-4]=>3
[-1,3,2,4]=>3
[-1,3,2,-4]=>4
[-1,3,-2,4]=>4
[-1,3,-2,-4]=>5
[-1,-3,2,4]=>2
[-1,-3,2,-4]=>3
[-1,-3,-2,4]=>3
[-1,-3,-2,-4]=>4
[1,3,4,2]=>4
[1,3,4,-2]=>5
[1,3,-4,2]=>3
[1,3,-4,-2]=>4
[1,-3,4,2]=>3
[1,-3,4,-2]=>4
[1,-3,-4,2]=>2
[1,-3,-4,-2]=>3
[-1,3,4,2]=>5
[-1,3,4,-2]=>6
[-1,3,-4,2]=>4
[-1,3,-4,-2]=>5
[-1,-3,4,2]=>4
[-1,-3,4,-2]=>5
[-1,-3,-4,2]=>3
[-1,-3,-4,-2]=>4
[1,4,2,3]=>2
[1,4,2,-3]=>3
[1,4,-2,3]=>3
[1,4,-2,-3]=>4
[1,-4,2,3]=>1
[1,-4,2,-3]=>2
[1,-4,-2,3]=>2
[1,-4,-2,-3]=>3
[-1,4,2,3]=>3
[-1,4,2,-3]=>4
[-1,4,-2,3]=>4
[-1,4,-2,-3]=>5
[-1,-4,2,3]=>2
[-1,-4,2,-3]=>3
[-1,-4,-2,3]=>3
[-1,-4,-2,-3]=>4
[1,4,3,2]=>2
[1,4,3,-2]=>3
[1,4,-3,2]=>3
[1,4,-3,-2]=>4
[1,-4,3,2]=>1
[1,-4,3,-2]=>2
[1,-4,-3,2]=>2
[1,-4,-3,-2]=>3
[-1,4,3,2]=>3
[-1,4,3,-2]=>4
[-1,4,-3,2]=>4
[-1,4,-3,-2]=>5
[-1,-4,3,2]=>2
[-1,-4,3,-2]=>3
[-1,-4,-3,2]=>3
[-1,-4,-3,-2]=>4
[2,1,3,4]=>2
[2,1,3,-4]=>3
[2,1,-3,4]=>3
[2,1,-3,-4]=>4
[2,-1,3,4]=>3
[2,-1,3,-4]=>4
[2,-1,-3,4]=>4
[2,-1,-3,-4]=>5
[-2,1,3,4]=>1
[-2,1,3,-4]=>2
[-2,1,-3,4]=>2
[-2,1,-3,-4]=>3
[-2,-1,3,4]=>2
[-2,-1,3,-4]=>3
[-2,-1,-3,4]=>3
[-2,-1,-3,-4]=>4
[2,1,4,3]=>4
[2,1,4,-3]=>5
[2,1,-4,3]=>3
[2,1,-4,-3]=>4
[2,-1,4,3]=>5
[2,-1,4,-3]=>6
[2,-1,-4,3]=>4
[2,-1,-4,-3]=>5
[-2,1,4,3]=>3
[-2,1,4,-3]=>4
[-2,1,-4,3]=>2
[-2,1,-4,-3]=>3
[-2,-1,4,3]=>4
[-2,-1,4,-3]=>5
[-2,-1,-4,3]=>3
[-2,-1,-4,-3]=>4
[2,3,1,4]=>4
[2,3,1,-4]=>5
[2,3,-1,4]=>5
[2,3,-1,-4]=>6
[2,-3,1,4]=>3
[2,-3,1,-4]=>4
[2,-3,-1,4]=>4
[2,-3,-1,-4]=>5
[-2,3,1,4]=>3
[-2,3,1,-4]=>4
[-2,3,-1,4]=>4
[-2,3,-1,-4]=>5
[-2,-3,1,4]=>2
[-2,-3,1,-4]=>3
[-2,-3,-1,4]=>3
[-2,-3,-1,-4]=>4
[2,3,4,1]=>6
[2,3,4,-1]=>7
[2,3,-4,1]=>5
[2,3,-4,-1]=>6
[2,-3,4,1]=>5
[2,-3,4,-1]=>6
[2,-3,-4,1]=>4
[2,-3,-4,-1]=>5
[-2,3,4,1]=>5
[-2,3,4,-1]=>6
[-2,3,-4,1]=>4
[-2,3,-4,-1]=>5
[-2,-3,4,1]=>4
[-2,-3,4,-1]=>5
[-2,-3,-4,1]=>3
[-2,-3,-4,-1]=>4
[2,4,1,3]=>4
[2,4,1,-3]=>5
[2,4,-1,3]=>5
[2,4,-1,-3]=>6
[2,-4,1,3]=>3
[2,-4,1,-3]=>4
[2,-4,-1,3]=>4
[2,-4,-1,-3]=>5
[-2,4,1,3]=>3
[-2,4,1,-3]=>4
[-2,4,-1,3]=>4
[-2,4,-1,-3]=>5
[-2,-4,1,3]=>2
[-2,-4,1,-3]=>3
[-2,-4,-1,3]=>3
[-2,-4,-1,-3]=>4
[2,4,3,1]=>4
[2,4,3,-1]=>5
[2,4,-3,1]=>5
[2,4,-3,-1]=>6
[2,-4,3,1]=>3
[2,-4,3,-1]=>4
[2,-4,-3,1]=>4
[2,-4,-3,-1]=>5
[-2,4,3,1]=>3
[-2,4,3,-1]=>4
[-2,4,-3,1]=>4
[-2,4,-3,-1]=>5
[-2,-4,3,1]=>2
[-2,-4,3,-1]=>3
[-2,-4,-3,1]=>3
[-2,-4,-3,-1]=>4
[3,1,2,4]=>2
[3,1,2,-4]=>3
[3,1,-2,4]=>3
[3,1,-2,-4]=>4
[3,-1,2,4]=>3
[3,-1,2,-4]=>4
[3,-1,-2,4]=>4
[3,-1,-2,-4]=>5
[-3,1,2,4]=>1
[-3,1,2,-4]=>2
[-3,1,-2,4]=>2
[-3,1,-2,-4]=>3
[-3,-1,2,4]=>2
[-3,-1,2,-4]=>3
[-3,-1,-2,4]=>3
[-3,-1,-2,-4]=>4
[3,1,4,2]=>4
[3,1,4,-2]=>5
[3,1,-4,2]=>3
[3,1,-4,-2]=>4
[3,-1,4,2]=>5
[3,-1,4,-2]=>6
[3,-1,-4,2]=>4
[3,-1,-4,-2]=>5
[-3,1,4,2]=>3
[-3,1,4,-2]=>4
[-3,1,-4,2]=>2
[-3,1,-4,-2]=>3
[-3,-1,4,2]=>4
[-3,-1,4,-2]=>5
[-3,-1,-4,2]=>3
[-3,-1,-4,-2]=>4
[3,2,1,4]=>2
[3,2,1,-4]=>3
[3,2,-1,4]=>3
[3,2,-1,-4]=>4
[3,-2,1,4]=>3
[3,-2,1,-4]=>4
[3,-2,-1,4]=>4
[3,-2,-1,-4]=>5
[-3,2,1,4]=>1
[-3,2,1,-4]=>2
[-3,2,-1,4]=>2
[-3,2,-1,-4]=>3
[-3,-2,1,4]=>2
[-3,-2,1,-4]=>3
[-3,-2,-1,4]=>3
[-3,-2,-1,-4]=>4
[3,2,4,1]=>4
[3,2,4,-1]=>5
[3,2,-4,1]=>3
[3,2,-4,-1]=>4
[3,-2,4,1]=>5
[3,-2,4,-1]=>6
[3,-2,-4,1]=>4
[3,-2,-4,-1]=>5
[-3,2,4,1]=>3
[-3,2,4,-1]=>4
[-3,2,-4,1]=>2
[-3,2,-4,-1]=>3
[-3,-2,4,1]=>4
[-3,-2,4,-1]=>5
[-3,-2,-4,1]=>3
[-3,-2,-4,-1]=>4
[3,4,1,2]=>4
[3,4,1,-2]=>5
[3,4,-1,2]=>5
[3,4,-1,-2]=>6
[3,-4,1,2]=>3
[3,-4,1,-2]=>4
[3,-4,-1,2]=>4
[3,-4,-1,-2]=>5
[-3,4,1,2]=>3
[-3,4,1,-2]=>4
[-3,4,-1,2]=>4
[-3,4,-1,-2]=>5
[-3,-4,1,2]=>2
[-3,-4,1,-2]=>3
[-3,-4,-1,2]=>3
[-3,-4,-1,-2]=>4
[3,4,2,1]=>4
[3,4,2,-1]=>5
[3,4,-2,1]=>5
[3,4,-2,-1]=>6
[3,-4,2,1]=>3
[3,-4,2,-1]=>4
[3,-4,-2,1]=>4
[3,-4,-2,-1]=>5
[-3,4,2,1]=>3
[-3,4,2,-1]=>4
[-3,4,-2,1]=>4
[-3,4,-2,-1]=>5
[-3,-4,2,1]=>2
[-3,-4,2,-1]=>3
[-3,-4,-2,1]=>3
[-3,-4,-2,-1]=>4
[4,1,2,3]=>2
[4,1,2,-3]=>3
[4,1,-2,3]=>3
[4,1,-2,-3]=>4
[4,-1,2,3]=>3
[4,-1,2,-3]=>4
[4,-1,-2,3]=>4
[4,-1,-2,-3]=>5
[-4,1,2,3]=>1
[-4,1,2,-3]=>2
[-4,1,-2,3]=>2
[-4,1,-2,-3]=>3
[-4,-1,2,3]=>2
[-4,-1,2,-3]=>3
[-4,-1,-2,3]=>3
[-4,-1,-2,-3]=>4
[4,1,3,2]=>2
[4,1,3,-2]=>3
[4,1,-3,2]=>3
[4,1,-3,-2]=>4
[4,-1,3,2]=>3
[4,-1,3,-2]=>4
[4,-1,-3,2]=>4
[4,-1,-3,-2]=>5
[-4,1,3,2]=>1
[-4,1,3,-2]=>2
[-4,1,-3,2]=>2
[-4,1,-3,-2]=>3
[-4,-1,3,2]=>2
[-4,-1,3,-2]=>3
[-4,-1,-3,2]=>3
[-4,-1,-3,-2]=>4
[4,2,1,3]=>2
[4,2,1,-3]=>3
[4,2,-1,3]=>3
[4,2,-1,-3]=>4
[4,-2,1,3]=>3
[4,-2,1,-3]=>4
[4,-2,-1,3]=>4
[4,-2,-1,-3]=>5
[-4,2,1,3]=>1
[-4,2,1,-3]=>2
[-4,2,-1,3]=>2
[-4,2,-1,-3]=>3
[-4,-2,1,3]=>2
[-4,-2,1,-3]=>3
[-4,-2,-1,3]=>3
[-4,-2,-1,-3]=>4
[4,2,3,1]=>2
[4,2,3,-1]=>3
[4,2,-3,1]=>3
[4,2,-3,-1]=>4
[4,-2,3,1]=>3
[4,-2,3,-1]=>4
[4,-2,-3,1]=>4
[4,-2,-3,-1]=>5
[-4,2,3,1]=>1
[-4,2,3,-1]=>2
[-4,2,-3,1]=>2
[-4,2,-3,-1]=>3
[-4,-2,3,1]=>2
[-4,-2,3,-1]=>3
[-4,-2,-3,1]=>3
[-4,-2,-3,-1]=>4
[4,3,1,2]=>4
[4,3,1,-2]=>5
[4,3,-1,2]=>5
[4,3,-1,-2]=>6
[4,-3,1,2]=>3
[4,-3,1,-2]=>4
[4,-3,-1,2]=>4
[4,-3,-1,-2]=>5
[-4,3,1,2]=>3
[-4,3,1,-2]=>4
[-4,3,-1,2]=>4
[-4,3,-1,-2]=>5
[-4,-3,1,2]=>2
[-4,-3,1,-2]=>3
[-4,-3,-1,2]=>3
[-4,-3,-1,-2]=>4
[4,3,2,1]=>4
[4,3,2,-1]=>5
[4,3,-2,1]=>5
[4,3,-2,-1]=>6
[4,-3,2,1]=>3
[4,-3,2,-1]=>4
[4,-3,-2,1]=>4
[4,-3,-2,-1]=>5
[-4,3,2,1]=>3
[-4,3,2,-1]=>4
[-4,3,-2,1]=>4
[-4,3,-2,-1]=>5
[-4,-3,2,1]=>2
[-4,-3,2,-1]=>3
[-4,-3,-2,1]=>3
[-4,-3,-2,-1]=>4
[1,2,3,4,5]=>0
[1,2,3,4,-5]=>1
[1,2,3,-4,5]=>1
[1,2,3,-4,-5]=>2
[1,2,-3,4,5]=>1
[1,2,-3,4,-5]=>2
[1,2,-3,-4,5]=>2
[1,2,-3,-4,-5]=>3
[1,-2,3,4,5]=>1
[1,-2,3,4,-5]=>2
[1,-2,3,-4,5]=>2
[1,-2,3,-4,-5]=>3
[1,-2,-3,4,5]=>2
[1,-2,-3,4,-5]=>3
[1,-2,-3,-4,5]=>3
[1,-2,-3,-4,-5]=>4
[-1,2,3,4,5]=>1
[-1,2,3,4,-5]=>2
[-1,2,3,-4,5]=>2
[-1,2,3,-4,-5]=>3
[-1,2,-3,4,5]=>2
[-1,2,-3,4,-5]=>3
[-1,2,-3,-4,5]=>3
[-1,2,-3,-4,-5]=>4
[-1,-2,3,4,5]=>2
[-1,-2,3,4,-5]=>3
[-1,-2,3,-4,5]=>3
[-1,-2,3,-4,-5]=>4
[-1,-2,-3,4,5]=>3
[-1,-2,-3,4,-5]=>4
[-1,-2,-3,-4,5]=>4
[-1,-2,-3,-4,-5]=>5
[1,2,3,5,4]=>2
[1,2,3,5,-4]=>3
[1,2,3,-5,4]=>1
[1,2,3,-5,-4]=>2
[1,2,-3,5,4]=>3
[1,2,-3,5,-4]=>4
[1,2,-3,-5,4]=>2
[1,2,-3,-5,-4]=>3
[1,-2,3,5,4]=>3
[1,-2,3,5,-4]=>4
[1,-2,3,-5,4]=>2
[1,-2,3,-5,-4]=>3
[1,-2,-3,5,4]=>4
[1,-2,-3,5,-4]=>5
[1,-2,-3,-5,4]=>3
[1,-2,-3,-5,-4]=>4
[-1,2,3,5,4]=>3
[-1,2,3,5,-4]=>4
[-1,2,3,-5,4]=>2
[-1,2,3,-5,-4]=>3
[-1,2,-3,5,4]=>4
[-1,2,-3,5,-4]=>5
[-1,2,-3,-5,4]=>3
[-1,2,-3,-5,-4]=>4
[-1,-2,3,5,4]=>4
[-1,-2,3,5,-4]=>5
[-1,-2,3,-5,4]=>3
[-1,-2,3,-5,-4]=>4
[-1,-2,-3,5,4]=>5
[-1,-2,-3,5,-4]=>6
[-1,-2,-3,-5,4]=>4
[-1,-2,-3,-5,-4]=>5
[1,2,4,3,5]=>2
[1,2,4,3,-5]=>3
[1,2,4,-3,5]=>3
[1,2,4,-3,-5]=>4
[1,2,-4,3,5]=>1
[1,2,-4,3,-5]=>2
[1,2,-4,-3,5]=>2
[1,2,-4,-3,-5]=>3
[1,-2,4,3,5]=>3
[1,-2,4,3,-5]=>4
[1,-2,4,-3,5]=>4
[1,-2,4,-3,-5]=>5
[1,-2,-4,3,5]=>2
[1,-2,-4,3,-5]=>3
[1,-2,-4,-3,5]=>3
[1,-2,-4,-3,-5]=>4
[-1,2,4,3,5]=>3
[-1,2,4,3,-5]=>4
[-1,2,4,-3,5]=>4
[-1,2,4,-3,-5]=>5
[-1,2,-4,3,5]=>2
[-1,2,-4,3,-5]=>3
[-1,2,-4,-3,5]=>3
[-1,2,-4,-3,-5]=>4
[-1,-2,4,3,5]=>4
[-1,-2,4,3,-5]=>5
[-1,-2,4,-3,5]=>5
[-1,-2,4,-3,-5]=>6
[-1,-2,-4,3,5]=>3
[-1,-2,-4,3,-5]=>4
[-1,-2,-4,-3,5]=>4
[-1,-2,-4,-3,-5]=>5
[1,2,4,5,3]=>4
[1,2,4,5,-3]=>5
[1,2,4,-5,3]=>3
[1,2,4,-5,-3]=>4
[1,2,-4,5,3]=>3
[1,2,-4,5,-3]=>4
[1,2,-4,-5,3]=>2
[1,2,-4,-5,-3]=>3
[1,-2,4,5,3]=>5
[1,-2,4,5,-3]=>6
[1,-2,4,-5,3]=>4
[1,-2,4,-5,-3]=>5
[1,-2,-4,5,3]=>4
[1,-2,-4,5,-3]=>5
[1,-2,-4,-5,3]=>3
[1,-2,-4,-5,-3]=>4
[-1,2,4,5,3]=>5
[-1,2,4,5,-3]=>6
[-1,2,4,-5,3]=>4
[-1,2,4,-5,-3]=>5
[-1,2,-4,5,3]=>4
[-1,2,-4,5,-3]=>5
[-1,2,-4,-5,3]=>3
[-1,2,-4,-5,-3]=>4
[-1,-2,4,5,3]=>6
[-1,-2,4,5,-3]=>7
[-1,-2,4,-5,3]=>5
[-1,-2,4,-5,-3]=>6
[-1,-2,-4,5,3]=>5
[-1,-2,-4,5,-3]=>6
[-1,-2,-4,-5,3]=>4
[-1,-2,-4,-5,-3]=>5
[1,2,5,3,4]=>2
[1,2,5,3,-4]=>3
[1,2,5,-3,4]=>3
[1,2,5,-3,-4]=>4
[1,2,-5,3,4]=>1
[1,2,-5,3,-4]=>2
[1,2,-5,-3,4]=>2
[1,2,-5,-3,-4]=>3
[1,-2,5,3,4]=>3
[1,-2,5,3,-4]=>4
[1,-2,5,-3,4]=>4
[1,-2,5,-3,-4]=>5
[1,-2,-5,3,4]=>2
[1,-2,-5,3,-4]=>3
[1,-2,-5,-3,4]=>3
[1,-2,-5,-3,-4]=>4
[-1,2,5,3,4]=>3
[-1,2,5,3,-4]=>4
[-1,2,5,-3,4]=>4
[-1,2,5,-3,-4]=>5
[-1,2,-5,3,4]=>2
[-1,2,-5,3,-4]=>3
[-1,2,-5,-3,4]=>3
[-1,2,-5,-3,-4]=>4
[-1,-2,5,3,4]=>4
[-1,-2,5,3,-4]=>5
[-1,-2,5,-3,4]=>5
[-1,-2,5,-3,-4]=>6
[-1,-2,-5,3,4]=>3
[-1,-2,-5,3,-4]=>4
[-1,-2,-5,-3,4]=>4
[-1,-2,-5,-3,-4]=>5
[1,2,5,4,3]=>2
[1,2,5,4,-3]=>3
[1,2,5,-4,3]=>3
[1,2,5,-4,-3]=>4
[1,2,-5,4,3]=>1
[1,2,-5,4,-3]=>2
[1,2,-5,-4,3]=>2
[1,2,-5,-4,-3]=>3
[1,-2,5,4,3]=>3
[1,-2,5,4,-3]=>4
[1,-2,5,-4,3]=>4
[1,-2,5,-4,-3]=>5
[1,-2,-5,4,3]=>2
[1,-2,-5,4,-3]=>3
[1,-2,-5,-4,3]=>3
[1,-2,-5,-4,-3]=>4
[-1,2,5,4,3]=>3
[-1,2,5,4,-3]=>4
[-1,2,5,-4,3]=>4
[-1,2,5,-4,-3]=>5
[-1,2,-5,4,3]=>2
[-1,2,-5,4,-3]=>3
[-1,2,-5,-4,3]=>3
[-1,2,-5,-4,-3]=>4
[-1,-2,5,4,3]=>4
[-1,-2,5,4,-3]=>5
[-1,-2,5,-4,3]=>5
[-1,-2,5,-4,-3]=>6
[-1,-2,-5,4,3]=>3
[-1,-2,-5,4,-3]=>4
[-1,-2,-5,-4,3]=>4
[-1,-2,-5,-4,-3]=>5
[1,3,2,4,5]=>2
[1,3,2,4,-5]=>3
[1,3,2,-4,5]=>3
[1,3,2,-4,-5]=>4
[1,3,-2,4,5]=>3
[1,3,-2,4,-5]=>4
[1,3,-2,-4,5]=>4
[1,3,-2,-4,-5]=>5
[1,-3,2,4,5]=>1
[1,-3,2,4,-5]=>2
[1,-3,2,-4,5]=>2
[1,-3,2,-4,-5]=>3
[1,-3,-2,4,5]=>2
[1,-3,-2,4,-5]=>3
[1,-3,-2,-4,5]=>3
[1,-3,-2,-4,-5]=>4
[-1,3,2,4,5]=>3
[-1,3,2,4,-5]=>4
[-1,3,2,-4,5]=>4
[-1,3,2,-4,-5]=>5
[-1,3,-2,4,5]=>4
[-1,3,-2,4,-5]=>5
[-1,3,-2,-4,5]=>5
[-1,3,-2,-4,-5]=>6
[-1,-3,2,4,5]=>2
[-1,-3,2,4,-5]=>3
[-1,-3,2,-4,5]=>3
[-1,-3,2,-4,-5]=>4
[-1,-3,-2,4,5]=>3
[-1,-3,-2,4,-5]=>4
[-1,-3,-2,-4,5]=>4
[-1,-3,-2,-4,-5]=>5
[1,3,2,5,4]=>4
[1,3,2,5,-4]=>5
[1,3,2,-5,4]=>3
[1,3,2,-5,-4]=>4
[1,3,-2,5,4]=>5
[1,3,-2,5,-4]=>6
[1,3,-2,-5,4]=>4
[1,3,-2,-5,-4]=>5
[1,-3,2,5,4]=>3
[1,-3,2,5,-4]=>4
[1,-3,2,-5,4]=>2
[1,-3,2,-5,-4]=>3
[1,-3,-2,5,4]=>4
[1,-3,-2,5,-4]=>5
[1,-3,-2,-5,4]=>3
[1,-3,-2,-5,-4]=>4
[-1,3,2,5,4]=>5
[-1,3,2,5,-4]=>6
[-1,3,2,-5,4]=>4
[-1,3,2,-5,-4]=>5
[-1,3,-2,5,4]=>6
[-1,3,-2,5,-4]=>7
[-1,3,-2,-5,4]=>5
[-1,3,-2,-5,-4]=>6
[-1,-3,2,5,4]=>4
[-1,-3,2,5,-4]=>5
[-1,-3,2,-5,4]=>3
[-1,-3,2,-5,-4]=>4
[-1,-3,-2,5,4]=>5
[-1,-3,-2,5,-4]=>6
[-1,-3,-2,-5,4]=>4
[-1,-3,-2,-5,-4]=>5
[1,3,4,2,5]=>4
[1,3,4,2,-5]=>5
[1,3,4,-2,5]=>5
[1,3,4,-2,-5]=>6
[1,3,-4,2,5]=>3
[1,3,-4,2,-5]=>4
[1,3,-4,-2,5]=>4
[1,3,-4,-2,-5]=>5
[1,-3,4,2,5]=>3
[1,-3,4,2,-5]=>4
[1,-3,4,-2,5]=>4
[1,-3,4,-2,-5]=>5
[1,-3,-4,2,5]=>2
[1,-3,-4,2,-5]=>3
[1,-3,-4,-2,5]=>3
[1,-3,-4,-2,-5]=>4
[-1,3,4,2,5]=>5
[-1,3,4,2,-5]=>6
[-1,3,4,-2,5]=>6
[-1,3,4,-2,-5]=>7
[-1,3,-4,2,5]=>4
[-1,3,-4,2,-5]=>5
[-1,3,-4,-2,5]=>5
[-1,3,-4,-2,-5]=>6
[-1,-3,4,2,5]=>4
[-1,-3,4,2,-5]=>5
[-1,-3,4,-2,5]=>5
[-1,-3,4,-2,-5]=>6
[-1,-3,-4,2,5]=>3
[-1,-3,-4,2,-5]=>4
[-1,-3,-4,-2,5]=>4
[-1,-3,-4,-2,-5]=>5
[1,3,4,5,2]=>6
[1,3,4,5,-2]=>7
[1,3,4,-5,2]=>5
[1,3,4,-5,-2]=>6
[1,3,-4,5,2]=>5
[1,3,-4,5,-2]=>6
[1,3,-4,-5,2]=>4
[1,3,-4,-5,-2]=>5
[1,-3,4,5,2]=>5
[1,-3,4,5,-2]=>6
[1,-3,4,-5,2]=>4
[1,-3,4,-5,-2]=>5
[1,-3,-4,5,2]=>4
[1,-3,-4,5,-2]=>5
[1,-3,-4,-5,2]=>3
[1,-3,-4,-5,-2]=>4
[-1,3,4,5,2]=>7
[-1,3,4,5,-2]=>8
[-1,3,4,-5,2]=>6
[-1,3,4,-5,-2]=>7
[-1,3,-4,5,2]=>6
[-1,3,-4,5,-2]=>7
[-1,3,-4,-5,2]=>5
[-1,3,-4,-5,-2]=>6
[-1,-3,4,5,2]=>6
[-1,-3,4,5,-2]=>7
[-1,-3,4,-5,2]=>5
[-1,-3,4,-5,-2]=>6
[-1,-3,-4,5,2]=>5
[-1,-3,-4,5,-2]=>6
[-1,-3,-4,-5,2]=>4
[-1,-3,-4,-5,-2]=>5
[1,3,5,2,4]=>4
[1,3,5,2,-4]=>5
[1,3,5,-2,4]=>5
[1,3,5,-2,-4]=>6
[1,3,-5,2,4]=>3
[1,3,-5,2,-4]=>4
[1,3,-5,-2,4]=>4
[1,3,-5,-2,-4]=>5
[1,-3,5,2,4]=>3
[1,-3,5,2,-4]=>4
[1,-3,5,-2,4]=>4
[1,-3,5,-2,-4]=>5
[1,-3,-5,2,4]=>2
[1,-3,-5,2,-4]=>3
[1,-3,-5,-2,4]=>3
[1,-3,-5,-2,-4]=>4
[-1,3,5,2,4]=>5
[-1,3,5,2,-4]=>6
[-1,3,5,-2,4]=>6
[-1,3,5,-2,-4]=>7
[-1,3,-5,2,4]=>4
[-1,3,-5,2,-4]=>5
[-1,3,-5,-2,4]=>5
[-1,3,-5,-2,-4]=>6
[-1,-3,5,2,4]=>4
[-1,-3,5,2,-4]=>5
[-1,-3,5,-2,4]=>5
[-1,-3,5,-2,-4]=>6
[-1,-3,-5,2,4]=>3
[-1,-3,-5,2,-4]=>4
[-1,-3,-5,-2,4]=>4
[-1,-3,-5,-2,-4]=>5
[1,3,5,4,2]=>4
[1,3,5,4,-2]=>5
[1,3,5,-4,2]=>5
[1,3,5,-4,-2]=>6
[1,3,-5,4,2]=>3
[1,3,-5,4,-2]=>4
[1,3,-5,-4,2]=>4
[1,3,-5,-4,-2]=>5
[1,-3,5,4,2]=>3
[1,-3,5,4,-2]=>4
[1,-3,5,-4,2]=>4
[1,-3,5,-4,-2]=>5
[1,-3,-5,4,2]=>2
[1,-3,-5,4,-2]=>3
[1,-3,-5,-4,2]=>3
[1,-3,-5,-4,-2]=>4
[-1,3,5,4,2]=>5
[-1,3,5,4,-2]=>6
[-1,3,5,-4,2]=>6
[-1,3,5,-4,-2]=>7
[-1,3,-5,4,2]=>4
[-1,3,-5,4,-2]=>5
[-1,3,-5,-4,2]=>5
[-1,3,-5,-4,-2]=>6
[-1,-3,5,4,2]=>4
[-1,-3,5,4,-2]=>5
[-1,-3,5,-4,2]=>5
[-1,-3,5,-4,-2]=>6
[-1,-3,-5,4,2]=>3
[-1,-3,-5,4,-2]=>4
[-1,-3,-5,-4,2]=>4
[-1,-3,-5,-4,-2]=>5
[1,4,2,3,5]=>2
[1,4,2,3,-5]=>3
[1,4,2,-3,5]=>3
[1,4,2,-3,-5]=>4
[1,4,-2,3,5]=>3
[1,4,-2,3,-5]=>4
[1,4,-2,-3,5]=>4
[1,4,-2,-3,-5]=>5
[1,-4,2,3,5]=>1
[1,-4,2,3,-5]=>2
[1,-4,2,-3,5]=>2
[1,-4,2,-3,-5]=>3
[1,-4,-2,3,5]=>2
[1,-4,-2,3,-5]=>3
[1,-4,-2,-3,5]=>3
[1,-4,-2,-3,-5]=>4
[-1,4,2,3,5]=>3
[-1,4,2,3,-5]=>4
[-1,4,2,-3,5]=>4
[-1,4,2,-3,-5]=>5
[-1,4,-2,3,5]=>4
[-1,4,-2,3,-5]=>5
[-1,4,-2,-3,5]=>5
[-1,4,-2,-3,-5]=>6
[-1,-4,2,3,5]=>2
[-1,-4,2,3,-5]=>3
[-1,-4,2,-3,5]=>3
[-1,-4,2,-3,-5]=>4
[-1,-4,-2,3,5]=>3
[-1,-4,-2,3,-5]=>4
[-1,-4,-2,-3,5]=>4
[-1,-4,-2,-3,-5]=>5
[1,4,2,5,3]=>4
[1,4,2,5,-3]=>5
[1,4,2,-5,3]=>3
[1,4,2,-5,-3]=>4
[1,4,-2,5,3]=>5
[1,4,-2,5,-3]=>6
[1,4,-2,-5,3]=>4
[1,4,-2,-5,-3]=>5
[1,-4,2,5,3]=>3
[1,-4,2,5,-3]=>4
[1,-4,2,-5,3]=>2
[1,-4,2,-5,-3]=>3
[1,-4,-2,5,3]=>4
[1,-4,-2,5,-3]=>5
[1,-4,-2,-5,3]=>3
[1,-4,-2,-5,-3]=>4
[-1,4,2,5,3]=>5
[-1,4,2,5,-3]=>6
[-1,4,2,-5,3]=>4
[-1,4,2,-5,-3]=>5
[-1,4,-2,5,3]=>6
[-1,4,-2,5,-3]=>7
[-1,4,-2,-5,3]=>5
[-1,4,-2,-5,-3]=>6
[-1,-4,2,5,3]=>4
[-1,-4,2,5,-3]=>5
[-1,-4,2,-5,3]=>3
[-1,-4,2,-5,-3]=>4
[-1,-4,-2,5,3]=>5
[-1,-4,-2,5,-3]=>6
[-1,-4,-2,-5,3]=>4
[-1,-4,-2,-5,-3]=>5
[1,4,3,2,5]=>2
[1,4,3,2,-5]=>3
[1,4,3,-2,5]=>3
[1,4,3,-2,-5]=>4
[1,4,-3,2,5]=>3
[1,4,-3,2,-5]=>4
[1,4,-3,-2,5]=>4
[1,4,-3,-2,-5]=>5
[1,-4,3,2,5]=>1
[1,-4,3,2,-5]=>2
[1,-4,3,-2,5]=>2
[1,-4,3,-2,-5]=>3
[1,-4,-3,2,5]=>2
[1,-4,-3,2,-5]=>3
[1,-4,-3,-2,5]=>3
[1,-4,-3,-2,-5]=>4
[-1,4,3,2,5]=>3
[-1,4,3,2,-5]=>4
[-1,4,3,-2,5]=>4
[-1,4,3,-2,-5]=>5
[-1,4,-3,2,5]=>4
[-1,4,-3,2,-5]=>5
[-1,4,-3,-2,5]=>5
[-1,4,-3,-2,-5]=>6
[-1,-4,3,2,5]=>2
[-1,-4,3,2,-5]=>3
[-1,-4,3,-2,5]=>3
[-1,-4,3,-2,-5]=>4
[-1,-4,-3,2,5]=>3
[-1,-4,-3,2,-5]=>4
[-1,-4,-3,-2,5]=>4
[-1,-4,-3,-2,-5]=>5
[1,4,3,5,2]=>4
[1,4,3,5,-2]=>5
[1,4,3,-5,2]=>3
[1,4,3,-5,-2]=>4
[1,4,-3,5,2]=>5
[1,4,-3,5,-2]=>6
[1,4,-3,-5,2]=>4
[1,4,-3,-5,-2]=>5
[1,-4,3,5,2]=>3
[1,-4,3,5,-2]=>4
[1,-4,3,-5,2]=>2
[1,-4,3,-5,-2]=>3
[1,-4,-3,5,2]=>4
[1,-4,-3,5,-2]=>5
[1,-4,-3,-5,2]=>3
[1,-4,-3,-5,-2]=>4
[-1,4,3,5,2]=>5
[-1,4,3,5,-2]=>6
[-1,4,3,-5,2]=>4
[-1,4,3,-5,-2]=>5
[-1,4,-3,5,2]=>6
[-1,4,-3,5,-2]=>7
[-1,4,-3,-5,2]=>5
[-1,4,-3,-5,-2]=>6
[-1,-4,3,5,2]=>4
[-1,-4,3,5,-2]=>5
[-1,-4,3,-5,2]=>3
[-1,-4,3,-5,-2]=>4
[-1,-4,-3,5,2]=>5
[-1,-4,-3,5,-2]=>6
[-1,-4,-3,-5,2]=>4
[-1,-4,-3,-5,-2]=>5
[1,4,5,2,3]=>4
[1,4,5,2,-3]=>5
[1,4,5,-2,3]=>5
[1,4,5,-2,-3]=>6
[1,4,-5,2,3]=>3
[1,4,-5,2,-3]=>4
[1,4,-5,-2,3]=>4
[1,4,-5,-2,-3]=>5
[1,-4,5,2,3]=>3
[1,-4,5,2,-3]=>4
[1,-4,5,-2,3]=>4
[1,-4,5,-2,-3]=>5
[1,-4,-5,2,3]=>2
[1,-4,-5,2,-3]=>3
[1,-4,-5,-2,3]=>3
[1,-4,-5,-2,-3]=>4
[-1,4,5,2,3]=>5
[-1,4,5,2,-3]=>6
[-1,4,5,-2,3]=>6
[-1,4,5,-2,-3]=>7
[-1,4,-5,2,3]=>4
[-1,4,-5,2,-3]=>5
[-1,4,-5,-2,3]=>5
[-1,4,-5,-2,-3]=>6
[-1,-4,5,2,3]=>4
[-1,-4,5,2,-3]=>5
[-1,-4,5,-2,3]=>5
[-1,-4,5,-2,-3]=>6
[-1,-4,-5,2,3]=>3
[-1,-4,-5,2,-3]=>4
[-1,-4,-5,-2,3]=>4
[-1,-4,-5,-2,-3]=>5
[1,4,5,3,2]=>4
[1,4,5,3,-2]=>5
[1,4,5,-3,2]=>5
[1,4,5,-3,-2]=>6
[1,4,-5,3,2]=>3
[1,4,-5,3,-2]=>4
[1,4,-5,-3,2]=>4
[1,4,-5,-3,-2]=>5
[1,-4,5,3,2]=>3
[1,-4,5,3,-2]=>4
[1,-4,5,-3,2]=>4
[1,-4,5,-3,-2]=>5
[1,-4,-5,3,2]=>2
[1,-4,-5,3,-2]=>3
[1,-4,-5,-3,2]=>3
[1,-4,-5,-3,-2]=>4
[-1,4,5,3,2]=>5
[-1,4,5,3,-2]=>6
[-1,4,5,-3,2]=>6
[-1,4,5,-3,-2]=>7
[-1,4,-5,3,2]=>4
[-1,4,-5,3,-2]=>5
[-1,4,-5,-3,2]=>5
[-1,4,-5,-3,-2]=>6
[-1,-4,5,3,2]=>4
[-1,-4,5,3,-2]=>5
[-1,-4,5,-3,2]=>5
[-1,-4,5,-3,-2]=>6
[-1,-4,-5,3,2]=>3
[-1,-4,-5,3,-2]=>4
[-1,-4,-5,-3,2]=>4
[-1,-4,-5,-3,-2]=>5
[1,5,2,3,4]=>2
[1,5,2,3,-4]=>3
[1,5,2,-3,4]=>3
[1,5,2,-3,-4]=>4
[1,5,-2,3,4]=>3
[1,5,-2,3,-4]=>4
[1,5,-2,-3,4]=>4
[1,5,-2,-3,-4]=>5
[1,-5,2,3,4]=>1
[1,-5,2,3,-4]=>2
[1,-5,2,-3,4]=>2
[1,-5,2,-3,-4]=>3
[1,-5,-2,3,4]=>2
[1,-5,-2,3,-4]=>3
[1,-5,-2,-3,4]=>3
[1,-5,-2,-3,-4]=>4
[-1,5,2,3,4]=>3
[-1,5,2,3,-4]=>4
[-1,5,2,-3,4]=>4
[-1,5,2,-3,-4]=>5
[-1,5,-2,3,4]=>4
[-1,5,-2,3,-4]=>5
[-1,5,-2,-3,4]=>5
[-1,5,-2,-3,-4]=>6
[-1,-5,2,3,4]=>2
[-1,-5,2,3,-4]=>3
[-1,-5,2,-3,4]=>3
[-1,-5,2,-3,-4]=>4
[-1,-5,-2,3,4]=>3
[-1,-5,-2,3,-4]=>4
[-1,-5,-2,-3,4]=>4
[-1,-5,-2,-3,-4]=>5
[1,5,2,4,3]=>2
[1,5,2,4,-3]=>3
[1,5,2,-4,3]=>3
[1,5,2,-4,-3]=>4
[1,5,-2,4,3]=>3
[1,5,-2,4,-3]=>4
[1,5,-2,-4,3]=>4
[1,5,-2,-4,-3]=>5
[1,-5,2,4,3]=>1
[1,-5,2,4,-3]=>2
[1,-5,2,-4,3]=>2
[1,-5,2,-4,-3]=>3
[1,-5,-2,4,3]=>2
[1,-5,-2,4,-3]=>3
[1,-5,-2,-4,3]=>3
[1,-5,-2,-4,-3]=>4
[-1,5,2,4,3]=>3
[-1,5,2,4,-3]=>4
[-1,5,2,-4,3]=>4
[-1,5,2,-4,-3]=>5
[-1,5,-2,4,3]=>4
[-1,5,-2,4,-3]=>5
[-1,5,-2,-4,3]=>5
[-1,5,-2,-4,-3]=>6
[-1,-5,2,4,3]=>2
[-1,-5,2,4,-3]=>3
[-1,-5,2,-4,3]=>3
[-1,-5,2,-4,-3]=>4
[-1,-5,-2,4,3]=>3
[-1,-5,-2,4,-3]=>4
[-1,-5,-2,-4,3]=>4
[-1,-5,-2,-4,-3]=>5
[1,5,3,2,4]=>2
[1,5,3,2,-4]=>3
[1,5,3,-2,4]=>3
[1,5,3,-2,-4]=>4
[1,5,-3,2,4]=>3
[1,5,-3,2,-4]=>4
[1,5,-3,-2,4]=>4
[1,5,-3,-2,-4]=>5
[1,-5,3,2,4]=>1
[1,-5,3,2,-4]=>2
[1,-5,3,-2,4]=>2
[1,-5,3,-2,-4]=>3
[1,-5,-3,2,4]=>2
[1,-5,-3,2,-4]=>3
[1,-5,-3,-2,4]=>3
[1,-5,-3,-2,-4]=>4
[-1,5,3,2,4]=>3
[-1,5,3,2,-4]=>4
[-1,5,3,-2,4]=>4
[-1,5,3,-2,-4]=>5
[-1,5,-3,2,4]=>4
[-1,5,-3,2,-4]=>5
[-1,5,-3,-2,4]=>5
[-1,5,-3,-2,-4]=>6
[-1,-5,3,2,4]=>2
[-1,-5,3,2,-4]=>3
[-1,-5,3,-2,4]=>3
[-1,-5,3,-2,-4]=>4
[-1,-5,-3,2,4]=>3
[-1,-5,-3,2,-4]=>4
[-1,-5,-3,-2,4]=>4
[-1,-5,-3,-2,-4]=>5
[1,5,3,4,2]=>2
[1,5,3,4,-2]=>3
[1,5,3,-4,2]=>3
[1,5,3,-4,-2]=>4
[1,5,-3,4,2]=>3
[1,5,-3,4,-2]=>4
[1,5,-3,-4,2]=>4
[1,5,-3,-4,-2]=>5
[1,-5,3,4,2]=>1
[1,-5,3,4,-2]=>2
[1,-5,3,-4,2]=>2
[1,-5,3,-4,-2]=>3
[1,-5,-3,4,2]=>2
[1,-5,-3,4,-2]=>3
[1,-5,-3,-4,2]=>3
[1,-5,-3,-4,-2]=>4
[-1,5,3,4,2]=>3
[-1,5,3,4,-2]=>4
[-1,5,3,-4,2]=>4
[-1,5,3,-4,-2]=>5
[-1,5,-3,4,2]=>4
[-1,5,-3,4,-2]=>5
[-1,5,-3,-4,2]=>5
[-1,5,-3,-4,-2]=>6
[-1,-5,3,4,2]=>2
[-1,-5,3,4,-2]=>3
[-1,-5,3,-4,2]=>3
[-1,-5,3,-4,-2]=>4
[-1,-5,-3,4,2]=>3
[-1,-5,-3,4,-2]=>4
[-1,-5,-3,-4,2]=>4
[-1,-5,-3,-4,-2]=>5
[1,5,4,2,3]=>4
[1,5,4,2,-3]=>5
[1,5,4,-2,3]=>5
[1,5,4,-2,-3]=>6
[1,5,-4,2,3]=>3
[1,5,-4,2,-3]=>4
[1,5,-4,-2,3]=>4
[1,5,-4,-2,-3]=>5
[1,-5,4,2,3]=>3
[1,-5,4,2,-3]=>4
[1,-5,4,-2,3]=>4
[1,-5,4,-2,-3]=>5
[1,-5,-4,2,3]=>2
[1,-5,-4,2,-3]=>3
[1,-5,-4,-2,3]=>3
[1,-5,-4,-2,-3]=>4
[-1,5,4,2,3]=>5
[-1,5,4,2,-3]=>6
[-1,5,4,-2,3]=>6
[-1,5,4,-2,-3]=>7
[-1,5,-4,2,3]=>4
[-1,5,-4,2,-3]=>5
[-1,5,-4,-2,3]=>5
[-1,5,-4,-2,-3]=>6
[-1,-5,4,2,3]=>4
[-1,-5,4,2,-3]=>5
[-1,-5,4,-2,3]=>5
[-1,-5,4,-2,-3]=>6
[-1,-5,-4,2,3]=>3
[-1,-5,-4,2,-3]=>4
[-1,-5,-4,-2,3]=>4
[-1,-5,-4,-2,-3]=>5
[1,5,4,3,2]=>4
[1,5,4,3,-2]=>5
[1,5,4,-3,2]=>5
[1,5,4,-3,-2]=>6
[1,5,-4,3,2]=>3
[1,5,-4,3,-2]=>4
[1,5,-4,-3,2]=>4
[1,5,-4,-3,-2]=>5
[1,-5,4,3,2]=>3
[1,-5,4,3,-2]=>4
[1,-5,4,-3,2]=>4
[1,-5,4,-3,-2]=>5
[1,-5,-4,3,2]=>2
[1,-5,-4,3,-2]=>3
[1,-5,-4,-3,2]=>3
[1,-5,-4,-3,-2]=>4
[-1,5,4,3,2]=>5
[-1,5,4,3,-2]=>6
[-1,5,4,-3,2]=>6
[-1,5,4,-3,-2]=>7
[-1,5,-4,3,2]=>4
[-1,5,-4,3,-2]=>5
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Description
The flag excedance statistic of a signed permutation.
This is the number of negative entries plus twice the number of excedances of the signed permutation. That is,
$$fexc(\sigma) = 2exc(\sigma) + neg(\sigma),$$
where
$$exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}|$$
$$neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}|$$
It has the same distribution as the flag descent statistic.
This is the number of negative entries plus twice the number of excedances of the signed permutation. That is,
$$fexc(\sigma) = 2exc(\sigma) + neg(\sigma),$$
where
$$exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}|$$
$$neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}|$$
It has the same distribution as the flag descent statistic.
References
[1] Bastidas, J. The polytope algebra of generalized permutahedra MathSciNet:4339357
[2] Foata, D., Han, G.-N. Signed words and permutations. V. A sextuple distribution MathSciNet:2501235
[2] Foata, D., Han, G.-N. Signed words and permutations. V. A sextuple distribution MathSciNet:2501235
Code
def statistic(w):
l = list(w)
return 2*len([i for i in range(len(l)) if l[i]> i+1]) + (len([i for i in range(len(l)) if l[i] < 0]))
Created
Mar 31, 2023 at 09:34 by Jose Bastidas
Updated
Jul 05, 2023 at 17:38 by Jose Bastidas
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