Identifier
Values
=>
[1]=>1
[-1]=>1
[1,2]=>1
[1,-2]=>1
[-1,2]=>1
[-1,-2]=>3
[2,1]=>1
[2,-1]=>0
[-2,1]=>0
[-2,-1]=>1
[1,2,3]=>1
[1,2,-3]=>1
[1,-2,3]=>1
[1,-2,-3]=>3
[-1,2,3]=>1
[-1,2,-3]=>3
[-1,-2,3]=>5
[-1,-2,-3]=>13
[1,3,2]=>1
[1,3,-2]=>0
[1,-3,2]=>0
[1,-3,-2]=>1
[-1,3,2]=>1
[-1,3,-2]=>0
[-1,-3,2]=>0
[-1,-3,-2]=>1
[2,1,3]=>1
[2,1,-3]=>1
[2,-1,3]=>0
[2,-1,-3]=>0
[-2,1,3]=>0
[-2,1,-3]=>0
[-2,-1,3]=>3
[-2,-1,-3]=>3
[2,3,1]=>0
[2,3,-1]=>0
[2,-3,1]=>0
[2,-3,-1]=>0
[-2,3,1]=>0
[-2,3,-1]=>0
[-2,-3,1]=>0
[-2,-3,-1]=>0
[3,1,2]=>0
[3,1,-2]=>0
[3,-1,2]=>0
[3,-1,-2]=>0
[-3,1,2]=>0
[-3,1,-2]=>0
[-3,-1,2]=>0
[-3,-1,-2]=>0
[3,2,1]=>3
[3,2,-1]=>0
[3,-2,1]=>1
[3,-2,-1]=>0
[-3,2,1]=>0
[-3,2,-1]=>3
[-3,-2,1]=>0
[-3,-2,-1]=>1
[1,2,3,4]=>1
[1,2,3,-4]=>1
[1,2,-3,4]=>1
[1,2,-3,-4]=>3
[1,-2,3,4]=>1
[1,-2,3,-4]=>3
[1,-2,-3,4]=>5
[1,-2,-3,-4]=>13
[-1,2,3,4]=>1
[-1,2,3,-4]=>3
[-1,2,-3,4]=>5
[-1,2,-3,-4]=>13
[-1,-2,3,4]=>7
[-1,-2,3,-4]=>15
[-1,-2,-3,4]=>19
[-1,-2,-3,-4]=>85
[1,2,4,3]=>1
[1,2,4,-3]=>0
[1,2,-4,3]=>0
[1,2,-4,-3]=>1
[1,-2,4,3]=>1
[1,-2,4,-3]=>0
[1,-2,-4,3]=>0
[1,-2,-4,-3]=>1
[-1,2,4,3]=>1
[-1,2,4,-3]=>0
[-1,2,-4,3]=>0
[-1,2,-4,-3]=>1
[-1,-2,4,3]=>9
[-1,-2,4,-3]=>0
[-1,-2,-4,3]=>0
[-1,-2,-4,-3]=>9
[1,3,2,4]=>1
[1,3,2,-4]=>1
[1,3,-2,4]=>0
[1,3,-2,-4]=>0
[1,-3,2,4]=>0
[1,-3,2,-4]=>0
[1,-3,-2,4]=>3
[1,-3,-2,-4]=>3
[-1,3,2,4]=>1
[-1,3,2,-4]=>3
[-1,3,-2,4]=>0
[-1,3,-2,-4]=>0
[-1,-3,2,4]=>0
[-1,-3,2,-4]=>0
[-1,-3,-2,4]=>3
[-1,-3,-2,-4]=>5
[1,3,4,2]=>0
[1,3,4,-2]=>0
[1,3,-4,2]=>0
[1,3,-4,-2]=>0
[1,-3,4,2]=>0
[1,-3,4,-2]=>0
[1,-3,-4,2]=>0
[1,-3,-4,-2]=>0
[-1,3,4,2]=>0
[-1,3,4,-2]=>0
[-1,3,-4,2]=>0
[-1,3,-4,-2]=>0
[-1,-3,4,2]=>0
[-1,-3,4,-2]=>0
[-1,-3,-4,2]=>0
[-1,-3,-4,-2]=>0
[1,4,2,3]=>0
[1,4,2,-3]=>0
[1,4,-2,3]=>0
[1,4,-2,-3]=>0
[1,-4,2,3]=>0
[1,-4,2,-3]=>0
[1,-4,-2,3]=>0
[1,-4,-2,-3]=>0
[-1,4,2,3]=>0
[-1,4,2,-3]=>0
[-1,4,-2,3]=>0
[-1,4,-2,-3]=>0
[-1,-4,2,3]=>0
[-1,-4,2,-3]=>0
[-1,-4,-2,3]=>0
[-1,-4,-2,-3]=>0
[1,4,3,2]=>3
[1,4,3,-2]=>0
[1,4,-3,2]=>1
[1,4,-3,-2]=>0
[1,-4,3,2]=>0
[1,-4,3,-2]=>3
[1,-4,-3,2]=>0
[1,-4,-3,-2]=>1
[-1,4,3,2]=>3
[-1,4,3,-2]=>0
[-1,4,-3,2]=>3
[-1,4,-3,-2]=>0
[-1,-4,3,2]=>0
[-1,-4,3,-2]=>3
[-1,-4,-3,2]=>0
[-1,-4,-3,-2]=>3
[2,1,3,4]=>1
[2,1,3,-4]=>1
[2,1,-3,4]=>1
[2,1,-3,-4]=>3
[2,-1,3,4]=>0
[2,-1,3,-4]=>0
[2,-1,-3,4]=>0
[2,-1,-3,-4]=>0
[-2,1,3,4]=>0
[-2,1,3,-4]=>0
[-2,1,-3,4]=>0
[-2,1,-3,-4]=>0
[-2,-1,3,4]=>5
[-2,-1,3,-4]=>5
[-2,-1,-3,4]=>5
[-2,-1,-3,-4]=>19
[2,1,4,3]=>1
[2,1,4,-3]=>0
[2,1,-4,3]=>0
[2,1,-4,-3]=>1
[2,-1,4,3]=>0
[2,-1,4,-3]=>0
[2,-1,-4,3]=>0
[2,-1,-4,-3]=>0
[-2,1,4,3]=>0
[-2,1,4,-3]=>0
[-2,1,-4,3]=>0
[-2,1,-4,-3]=>0
[-2,-1,4,3]=>7
[-2,-1,4,-3]=>0
[-2,-1,-4,3]=>0
[-2,-1,-4,-3]=>7
[2,3,1,4]=>0
[2,3,1,-4]=>0
[2,3,-1,4]=>0
[2,3,-1,-4]=>0
[2,-3,1,4]=>0
[2,-3,1,-4]=>0
[2,-3,-1,4]=>0
[2,-3,-1,-4]=>0
[-2,3,1,4]=>0
[-2,3,1,-4]=>0
[-2,3,-1,4]=>0
[-2,3,-1,-4]=>0
[-2,-3,1,4]=>0
[-2,-3,1,-4]=>0
[-2,-3,-1,4]=>0
[-2,-3,-1,-4]=>0
[2,3,4,1]=>0
[2,3,4,-1]=>0
[2,3,-4,1]=>0
[2,3,-4,-1]=>0
[2,-3,4,1]=>0
[2,-3,4,-1]=>0
[2,-3,-4,1]=>0
[2,-3,-4,-1]=>0
[-2,3,4,1]=>0
[-2,3,4,-1]=>0
[-2,3,-4,1]=>0
[-2,3,-4,-1]=>0
[-2,-3,4,1]=>0
[-2,-3,4,-1]=>0
[-2,-3,-4,1]=>0
[-2,-3,-4,-1]=>0
[2,4,1,3]=>0
[2,4,1,-3]=>0
[2,4,-1,3]=>0
[2,4,-1,-3]=>0
[2,-4,1,3]=>0
[2,-4,1,-3]=>0
[2,-4,-1,3]=>0
[2,-4,-1,-3]=>0
[-2,4,1,3]=>0
[-2,4,1,-3]=>0
[-2,4,-1,3]=>0
[-2,4,-1,-3]=>0
[-2,-4,1,3]=>0
[-2,-4,1,-3]=>0
[-2,-4,-1,3]=>0
[-2,-4,-1,-3]=>0
[2,4,3,1]=>0
[2,4,3,-1]=>0
[2,4,-3,1]=>0
[2,4,-3,-1]=>0
[2,-4,3,1]=>0
[2,-4,3,-1]=>0
[2,-4,-3,1]=>0
[2,-4,-3,-1]=>0
[-2,4,3,1]=>0
[-2,4,3,-1]=>0
[-2,4,-3,1]=>0
[-2,4,-3,-1]=>0
[-2,-4,3,1]=>0
[-2,-4,3,-1]=>0
[-2,-4,-3,1]=>0
[-2,-4,-3,-1]=>0
[3,1,2,4]=>0
[3,1,2,-4]=>0
[3,1,-2,4]=>0
[3,1,-2,-4]=>0
[3,-1,2,4]=>0
[3,-1,2,-4]=>0
[3,-1,-2,4]=>0
[3,-1,-2,-4]=>0
[-3,1,2,4]=>0
[-3,1,2,-4]=>0
[-3,1,-2,4]=>0
[-3,1,-2,-4]=>0
[-3,-1,2,4]=>0
[-3,-1,2,-4]=>0
[-3,-1,-2,4]=>0
[-3,-1,-2,-4]=>0
[3,1,4,2]=>0
[3,1,4,-2]=>0
[3,1,-4,2]=>0
[3,1,-4,-2]=>0
[3,-1,4,2]=>0
[3,-1,4,-2]=>0
[3,-1,-4,2]=>0
[3,-1,-4,-2]=>0
[-3,1,4,2]=>0
[-3,1,4,-2]=>0
[-3,1,-4,2]=>0
[-3,1,-4,-2]=>0
[-3,-1,4,2]=>0
[-3,-1,4,-2]=>0
[-3,-1,-4,2]=>0
[-3,-1,-4,-2]=>0
[3,2,1,4]=>3
[3,2,1,-4]=>3
[3,2,-1,4]=>0
[3,2,-1,-4]=>0
[3,-2,1,4]=>1
[3,-2,1,-4]=>3
[3,-2,-1,4]=>0
[3,-2,-1,-4]=>0
[-3,2,1,4]=>0
[-3,2,1,-4]=>0
[-3,2,-1,4]=>5
[-3,2,-1,-4]=>5
[-3,-2,1,4]=>0
[-3,-2,1,-4]=>0
[-3,-2,-1,4]=>3
[-3,-2,-1,-4]=>5
[3,2,4,1]=>0
[3,2,4,-1]=>0
[3,2,-4,1]=>0
[3,2,-4,-1]=>0
[3,-2,4,1]=>0
[3,-2,4,-1]=>0
[3,-2,-4,1]=>0
[3,-2,-4,-1]=>0
[-3,2,4,1]=>0
[-3,2,4,-1]=>0
[-3,2,-4,1]=>0
[-3,2,-4,-1]=>0
[-3,-2,4,1]=>0
[-3,-2,4,-1]=>0
[-3,-2,-4,1]=>0
[-3,-2,-4,-1]=>0
[3,4,1,2]=>1
[3,4,1,-2]=>0
[3,4,-1,2]=>0
[3,4,-1,-2]=>0
[3,-4,1,2]=>0
[3,-4,1,-2]=>1
[3,-4,-1,2]=>0
[3,-4,-1,-2]=>0
[-3,4,1,2]=>0
[-3,4,1,-2]=>0
[-3,4,-1,2]=>7
[-3,4,-1,-2]=>0
[-3,-4,1,2]=>0
[-3,-4,1,-2]=>0
[-3,-4,-1,2]=>0
[-3,-4,-1,-2]=>7
[3,4,2,1]=>0
[3,4,2,-1]=>0
[3,4,-2,1]=>0
[3,4,-2,-1]=>0
[3,-4,2,1]=>0
[3,-4,2,-1]=>0
[3,-4,-2,1]=>0
[3,-4,-2,-1]=>0
[-3,4,2,1]=>0
[-3,4,2,-1]=>0
[-3,4,-2,1]=>0
[-3,4,-2,-1]=>0
[-3,-4,2,1]=>0
[-3,-4,2,-1]=>0
[-3,-4,-2,1]=>0
[-3,-4,-2,-1]=>0
[4,1,2,3]=>0
[4,1,2,-3]=>0
[4,1,-2,3]=>0
[4,1,-2,-3]=>0
[4,-1,2,3]=>0
[4,-1,2,-3]=>0
[4,-1,-2,3]=>0
[4,-1,-2,-3]=>0
[-4,1,2,3]=>0
[-4,1,2,-3]=>0
[-4,1,-2,3]=>0
[-4,1,-2,-3]=>0
[-4,-1,2,3]=>0
[-4,-1,2,-3]=>0
[-4,-1,-2,3]=>0
[-4,-1,-2,-3]=>0
[4,1,3,2]=>0
[4,1,3,-2]=>0
[4,1,-3,2]=>0
[4,1,-3,-2]=>0
[4,-1,3,2]=>0
[4,-1,3,-2]=>0
[4,-1,-3,2]=>0
[4,-1,-3,-2]=>0
[-4,1,3,2]=>0
[-4,1,3,-2]=>0
[-4,1,-3,2]=>0
[-4,1,-3,-2]=>0
[-4,-1,3,2]=>0
[-4,-1,3,-2]=>0
[-4,-1,-3,2]=>0
[-4,-1,-3,-2]=>0
[4,2,1,3]=>0
[4,2,1,-3]=>0
[4,2,-1,3]=>0
[4,2,-1,-3]=>0
[4,-2,1,3]=>0
[4,-2,1,-3]=>0
[4,-2,-1,3]=>0
[4,-2,-1,-3]=>0
[-4,2,1,3]=>0
[-4,2,1,-3]=>0
[-4,2,-1,3]=>0
[-4,2,-1,-3]=>0
[-4,-2,1,3]=>0
[-4,-2,1,-3]=>0
[-4,-2,-1,3]=>0
[-4,-2,-1,-3]=>0
[4,2,3,1]=>5
[4,2,3,-1]=>0
[4,2,-3,1]=>3
[4,2,-3,-1]=>0
[4,-2,3,1]=>3
[4,-2,3,-1]=>0
[4,-2,-3,1]=>3
[4,-2,-3,-1]=>0
[-4,2,3,1]=>0
[-4,2,3,-1]=>5
[-4,2,-3,1]=>0
[-4,2,-3,-1]=>3
[-4,-2,3,1]=>0
[-4,-2,3,-1]=>3
[-4,-2,-3,1]=>0
[-4,-2,-3,-1]=>3
[4,3,1,2]=>0
[4,3,1,-2]=>0
[4,3,-1,2]=>0
[4,3,-1,-2]=>0
[4,-3,1,2]=>0
[4,-3,1,-2]=>0
[4,-3,-1,2]=>0
[4,-3,-1,-2]=>0
[-4,3,1,2]=>0
[-4,3,1,-2]=>0
[-4,3,-1,2]=>0
[-4,3,-1,-2]=>0
[-4,-3,1,2]=>0
[-4,-3,1,-2]=>0
[-4,-3,-1,2]=>0
[-4,-3,-1,-2]=>0
[4,3,2,1]=>7
[4,3,2,-1]=>0
[4,3,-2,1]=>0
[4,3,-2,-1]=>0
[4,-3,2,1]=>0
[4,-3,2,-1]=>0
[4,-3,-2,1]=>1
[4,-3,-2,-1]=>0
[-4,3,2,1]=>0
[-4,3,2,-1]=>7
[-4,3,-2,1]=>0
[-4,3,-2,-1]=>0
[-4,-3,2,1]=>0
[-4,-3,2,-1]=>0
[-4,-3,-2,1]=>0
[-4,-3,-2,-1]=>1
[1,2,3,4,5]=>1
[1,2,3,4,-5]=>1
[1,2,3,-4,5]=>1
[1,2,3,-4,-5]=>3
[1,2,-3,4,5]=>1
[1,2,-3,4,-5]=>3
[1,2,-3,-4,5]=>5
[1,2,-3,-4,-5]=>13
[1,-2,3,4,5]=>1
[1,-2,3,4,-5]=>3
[1,-2,3,-4,5]=>5
[1,-2,3,-4,-5]=>13
[1,-2,-3,4,5]=>7
[1,-2,-3,4,-5]=>15
[1,-2,-3,-4,5]=>19
[1,-2,-3,-4,-5]=>85
[-1,2,3,4,5]=>1
[-1,2,3,4,-5]=>3
[-1,2,3,-4,5]=>5
[-1,2,3,-4,-5]=>13
[-1,2,-3,4,5]=>7
[-1,2,-3,4,-5]=>15
[-1,2,-3,-4,5]=>19
[-1,2,-3,-4,-5]=>85
[-1,-2,3,4,5]=>9
[-1,-2,3,4,-5]=>17
[-1,-2,3,-4,5]=>21
[-1,-2,3,-4,-5]=>91
[-1,-2,-3,4,5]=>25
[-1,-2,-3,4,-5]=>103
[-1,-2,-3,-4,5]=>169
[-1,-2,-3,-4,-5]=>681
[1,2,3,5,4]=>1
[1,2,3,5,-4]=>0
[1,2,3,-5,4]=>0
[1,2,3,-5,-4]=>1
[1,2,-3,5,4]=>1
[1,2,-3,5,-4]=>0
[1,2,-3,-5,4]=>0
[1,2,-3,-5,-4]=>1
[1,-2,3,5,4]=>1
[1,-2,3,5,-4]=>0
[1,-2,3,-5,4]=>0
[1,-2,3,-5,-4]=>1
[1,-2,-3,5,4]=>9
[1,-2,-3,5,-4]=>0
[1,-2,-3,-5,4]=>0
[1,-2,-3,-5,-4]=>9
[-1,2,3,5,4]=>1
[-1,2,3,5,-4]=>0
[-1,2,3,-5,4]=>0
[-1,2,3,-5,-4]=>1
[-1,2,-3,5,4]=>9
[-1,2,-3,5,-4]=>0
[-1,2,-3,-5,4]=>0
[-1,2,-3,-5,-4]=>9
[-1,-2,3,5,4]=>11
[-1,-2,3,5,-4]=>0
[-1,-2,3,-5,4]=>0
[-1,-2,3,-5,-4]=>11
[-1,-2,-3,5,4]=>31
[-1,-2,-3,5,-4]=>0
[-1,-2,-3,-5,4]=>0
[-1,-2,-3,-5,-4]=>31
[1,2,4,3,5]=>1
[1,2,4,3,-5]=>1
[1,2,4,-3,5]=>0
[1,2,4,-3,-5]=>0
[1,2,-4,3,5]=>0
[1,2,-4,3,-5]=>0
[1,2,-4,-3,5]=>3
[1,2,-4,-3,-5]=>3
[1,-2,4,3,5]=>1
[1,-2,4,3,-5]=>3
[1,-2,4,-3,5]=>0
[1,-2,4,-3,-5]=>0
[1,-2,-4,3,5]=>0
[1,-2,-4,3,-5]=>0
[1,-2,-4,-3,5]=>3
[1,-2,-4,-3,-5]=>5
[-1,2,4,3,5]=>1
[-1,2,4,3,-5]=>3
[-1,2,4,-3,5]=>0
[-1,2,4,-3,-5]=>0
[-1,2,-4,3,5]=>0
[-1,2,-4,3,-5]=>0
[-1,2,-4,-3,5]=>3
[-1,2,-4,-3,-5]=>5
[-1,-2,4,3,5]=>11
[-1,-2,4,3,-5]=>19
[-1,-2,4,-3,5]=>0
[-1,-2,4,-3,-5]=>0
[-1,-2,-4,3,5]=>0
[-1,-2,-4,3,-5]=>0
[-1,-2,-4,-3,5]=>41
[-1,-2,-4,-3,-5]=>49
[1,2,4,5,3]=>0
[1,2,4,5,-3]=>0
[1,2,4,-5,3]=>0
[1,2,4,-5,-3]=>0
[1,2,-4,5,3]=>0
[1,2,-4,5,-3]=>0
[1,2,-4,-5,3]=>0
[1,2,-4,-5,-3]=>0
[1,-2,4,5,3]=>0
[1,-2,4,5,-3]=>0
[1,-2,4,-5,3]=>0
[1,-2,4,-5,-3]=>0
[1,-2,-4,5,3]=>0
[1,-2,-4,5,-3]=>0
[1,-2,-4,-5,3]=>0
[1,-2,-4,-5,-3]=>0
[-1,2,4,5,3]=>0
[-1,2,4,5,-3]=>0
[-1,2,4,-5,3]=>0
[-1,2,4,-5,-3]=>0
[-1,2,-4,5,3]=>0
[-1,2,-4,5,-3]=>0
[-1,2,-4,-5,3]=>0
[-1,2,-4,-5,-3]=>0
[-1,-2,4,5,3]=>0
[-1,-2,4,5,-3]=>0
[-1,-2,4,-5,3]=>0
[-1,-2,4,-5,-3]=>0
[-1,-2,-4,5,3]=>0
[-1,-2,-4,5,-3]=>0
[-1,-2,-4,-5,3]=>0
[-1,-2,-4,-5,-3]=>0
[1,2,5,3,4]=>0
[1,2,5,3,-4]=>0
[1,2,5,-3,4]=>0
[1,2,5,-3,-4]=>0
[1,2,-5,3,4]=>0
[1,2,-5,3,-4]=>0
[1,2,-5,-3,4]=>0
[1,2,-5,-3,-4]=>0
[1,-2,5,3,4]=>0
[1,-2,5,3,-4]=>0
[1,-2,5,-3,4]=>0
[1,-2,5,-3,-4]=>0
[1,-2,-5,3,4]=>0
[1,-2,-5,3,-4]=>0
[1,-2,-5,-3,4]=>0
[1,-2,-5,-3,-4]=>0
[-1,2,5,3,4]=>0
[-1,2,5,3,-4]=>0
[-1,2,5,-3,4]=>0
[-1,2,5,-3,-4]=>0
[-1,2,-5,3,4]=>0
[-1,2,-5,3,-4]=>0
[-1,2,-5,-3,4]=>0
[-1,2,-5,-3,-4]=>0
[-1,-2,5,3,4]=>0
[-1,-2,5,3,-4]=>0
[-1,-2,5,-3,4]=>0
[-1,-2,5,-3,-4]=>0
[-1,-2,-5,3,4]=>0
[-1,-2,-5,3,-4]=>0
[-1,-2,-5,-3,4]=>0
[-1,-2,-5,-3,-4]=>0
[1,2,5,4,3]=>3
[1,2,5,4,-3]=>0
[1,2,5,-4,3]=>1
[1,2,5,-4,-3]=>0
[1,2,-5,4,3]=>0
[1,2,-5,4,-3]=>3
[1,2,-5,-4,3]=>0
[1,2,-5,-4,-3]=>1
[1,-2,5,4,3]=>3
[1,-2,5,4,-3]=>0
[1,-2,5,-4,3]=>3
[1,-2,5,-4,-3]=>0
[1,-2,-5,4,3]=>0
[1,-2,-5,4,-3]=>3
[1,-2,-5,-4,3]=>0
[1,-2,-5,-4,-3]=>3
[-1,2,5,4,3]=>3
[-1,2,5,4,-3]=>0
[-1,2,5,-4,3]=>3
[-1,2,5,-4,-3]=>0
[-1,2,-5,4,3]=>0
[-1,2,-5,4,-3]=>3
[-1,2,-5,-4,3]=>0
[-1,2,-5,-4,-3]=>3
[-1,-2,5,4,3]=>41
[-1,-2,5,4,-3]=>0
[-1,-2,5,-4,3]=>19
[-1,-2,5,-4,-3]=>0
[-1,-2,-5,4,3]=>0
[-1,-2,-5,4,-3]=>41
[-1,-2,-5,-4,3]=>0
[-1,-2,-5,-4,-3]=>19
[1,3,2,4,5]=>1
[1,3,2,4,-5]=>1
[1,3,2,-4,5]=>1
[1,3,2,-4,-5]=>3
[1,3,-2,4,5]=>0
[1,3,-2,4,-5]=>0
[1,3,-2,-4,5]=>0
[1,3,-2,-4,-5]=>0
[1,-3,2,4,5]=>0
[1,-3,2,4,-5]=>0
[1,-3,2,-4,5]=>0
[1,-3,2,-4,-5]=>0
[1,-3,-2,4,5]=>5
[1,-3,-2,4,-5]=>5
[1,-3,-2,-4,5]=>5
[1,-3,-2,-4,-5]=>19
[-1,3,2,4,5]=>1
[-1,3,2,4,-5]=>3
[-1,3,2,-4,5]=>5
[-1,3,2,-4,-5]=>13
[-1,3,-2,4,5]=>0
[-1,3,-2,4,-5]=>0
[-1,3,-2,-4,5]=>0
[-1,3,-2,-4,-5]=>0
[-1,-3,2,4,5]=>0
[-1,-3,2,4,-5]=>0
[-1,-3,2,-4,5]=>0
[-1,-3,2,-4,-5]=>0
[-1,-3,-2,4,5]=>5
[-1,-3,-2,4,-5]=>11
[-1,-3,-2,-4,5]=>13
[-1,-3,-2,-4,-5]=>37
[1,3,2,5,4]=>1
[1,3,2,5,-4]=>0
[1,3,2,-5,4]=>0
[1,3,2,-5,-4]=>1
[1,3,-2,5,4]=>0
[1,3,-2,5,-4]=>0
[1,3,-2,-5,4]=>0
[1,3,-2,-5,-4]=>0
[1,-3,2,5,4]=>0
[1,-3,2,5,-4]=>0
[1,-3,2,-5,4]=>0
[1,-3,2,-5,-4]=>0
[1,-3,-2,5,4]=>7
[1,-3,-2,5,-4]=>0
[1,-3,-2,-5,4]=>0
[1,-3,-2,-5,-4]=>7
[-1,3,2,5,4]=>1
[-1,3,2,5,-4]=>0
[-1,3,2,-5,4]=>0
[-1,3,2,-5,-4]=>1
[-1,3,-2,5,4]=>0
[-1,3,-2,5,-4]=>0
[-1,3,-2,-5,4]=>0
[-1,3,-2,-5,-4]=>0
[-1,-3,2,5,4]=>0
[-1,-3,2,5,-4]=>0
[-1,-3,2,-5,4]=>0
[-1,-3,2,-5,-4]=>0
[-1,-3,-2,5,4]=>7
[-1,-3,-2,5,-4]=>0
[-1,-3,-2,-5,4]=>0
[-1,-3,-2,-5,-4]=>7
[1,3,4,2,5]=>0
[1,3,4,2,-5]=>0
[1,3,4,-2,5]=>0
[1,3,4,-2,-5]=>0
[1,3,-4,2,5]=>0
[1,3,-4,2,-5]=>0
[1,3,-4,-2,5]=>0
[1,3,-4,-2,-5]=>0
[1,-3,4,2,5]=>0
[1,-3,4,2,-5]=>0
[1,-3,4,-2,5]=>0
[1,-3,4,-2,-5]=>0
[1,-3,-4,2,5]=>0
[1,-3,-4,2,-5]=>0
[1,-3,-4,-2,5]=>0
[1,-3,-4,-2,-5]=>0
[-1,3,4,2,5]=>0
[-1,3,4,2,-5]=>0
[-1,3,4,-2,5]=>0
[-1,3,4,-2,-5]=>0
[-1,3,-4,2,5]=>0
[-1,3,-4,2,-5]=>0
[-1,3,-4,-2,5]=>0
[-1,3,-4,-2,-5]=>0
[-1,-3,4,2,5]=>0
[-1,-3,4,2,-5]=>0
[-1,-3,4,-2,5]=>0
[-1,-3,4,-2,-5]=>0
[-1,-3,-4,2,5]=>0
[-1,-3,-4,2,-5]=>0
[-1,-3,-4,-2,5]=>0
[-1,-3,-4,-2,-5]=>0
[1,3,4,5,2]=>0
[1,3,4,5,-2]=>0
[1,3,4,-5,2]=>0
[1,3,4,-5,-2]=>0
[1,3,-4,5,2]=>0
[1,3,-4,5,-2]=>0
[1,3,-4,-5,2]=>0
[1,3,-4,-5,-2]=>0
[1,-3,4,5,2]=>0
[1,-3,4,5,-2]=>0
[1,-3,4,-5,2]=>0
[1,-3,4,-5,-2]=>0
[1,-3,-4,5,2]=>0
[1,-3,-4,5,-2]=>0
[1,-3,-4,-5,2]=>0
[1,-3,-4,-5,-2]=>0
[-1,3,4,5,2]=>0
[-1,3,4,5,-2]=>0
[-1,3,4,-5,2]=>0
[-1,3,4,-5,-2]=>0
[-1,3,-4,5,2]=>0
[-1,3,-4,5,-2]=>0
[-1,3,-4,-5,2]=>0
[-1,3,-4,-5,-2]=>0
[-1,-3,4,5,2]=>0
[-1,-3,4,5,-2]=>0
[-1,-3,4,-5,2]=>0
[-1,-3,4,-5,-2]=>0
[-1,-3,-4,5,2]=>0
[-1,-3,-4,5,-2]=>0
[-1,-3,-4,-5,2]=>0
[-1,-3,-4,-5,-2]=>0
[1,3,5,2,4]=>0
[1,3,5,2,-4]=>0
[1,3,5,-2,4]=>0
[1,3,5,-2,-4]=>0
[1,3,-5,2,4]=>0
[1,3,-5,2,-4]=>0
[1,3,-5,-2,4]=>0
[1,3,-5,-2,-4]=>0
[1,-3,5,2,4]=>0
[1,-3,5,2,-4]=>0
[1,-3,5,-2,4]=>0
[1,-3,5,-2,-4]=>0
[1,-3,-5,2,4]=>0
[1,-3,-5,2,-4]=>0
[1,-3,-5,-2,4]=>0
[1,-3,-5,-2,-4]=>0
[-1,3,5,2,4]=>0
[-1,3,5,2,-4]=>0
[-1,3,5,-2,4]=>0
[-1,3,5,-2,-4]=>0
[-1,3,-5,2,4]=>0
[-1,3,-5,2,-4]=>0
[-1,3,-5,-2,4]=>0
[-1,3,-5,-2,-4]=>0
[-1,-3,5,2,4]=>0
[-1,-3,5,2,-4]=>0
[-1,-3,5,-2,4]=>0
[-1,-3,5,-2,-4]=>0
[-1,-3,-5,2,4]=>0
[-1,-3,-5,2,-4]=>0
[-1,-3,-5,-2,4]=>0
[-1,-3,-5,-2,-4]=>0
[1,3,5,4,2]=>0
[1,3,5,4,-2]=>0
[1,3,5,-4,2]=>0
[1,3,5,-4,-2]=>0
[1,3,-5,4,2]=>0
[1,3,-5,4,-2]=>0
[1,3,-5,-4,2]=>0
[1,3,-5,-4,-2]=>0
[1,-3,5,4,2]=>0
[1,-3,5,4,-2]=>0
[1,-3,5,-4,2]=>0
[1,-3,5,-4,-2]=>0
[1,-3,-5,4,2]=>0
[1,-3,-5,4,-2]=>0
[1,-3,-5,-4,2]=>0
[1,-3,-5,-4,-2]=>0
[-1,3,5,4,2]=>0
[-1,3,5,4,-2]=>0
[-1,3,5,-4,2]=>0
[-1,3,5,-4,-2]=>0
[-1,3,-5,4,2]=>0
[-1,3,-5,4,-2]=>0
[-1,3,-5,-4,2]=>0
[-1,3,-5,-4,-2]=>0
[-1,-3,5,4,2]=>0
[-1,-3,5,4,-2]=>0
[-1,-3,5,-4,2]=>0
[-1,-3,5,-4,-2]=>0
[-1,-3,-5,4,2]=>0
[-1,-3,-5,4,-2]=>0
[-1,-3,-5,-4,2]=>0
[-1,-3,-5,-4,-2]=>0
[1,4,2,3,5]=>0
[1,4,2,3,-5]=>0
[1,4,2,-3,5]=>0
[1,4,2,-3,-5]=>0
[1,4,-2,3,5]=>0
[1,4,-2,3,-5]=>0
[1,4,-2,-3,5]=>0
[1,4,-2,-3,-5]=>0
[1,-4,2,3,5]=>0
[1,-4,2,3,-5]=>0
[1,-4,2,-3,5]=>0
[1,-4,2,-3,-5]=>0
[1,-4,-2,3,5]=>0
[1,-4,-2,3,-5]=>0
[1,-4,-2,-3,5]=>0
[1,-4,-2,-3,-5]=>0
[-1,4,2,3,5]=>0
[-1,4,2,3,-5]=>0
[-1,4,2,-3,5]=>0
[-1,4,2,-3,-5]=>0
[-1,4,-2,3,5]=>0
[-1,4,-2,3,-5]=>0
[-1,4,-2,-3,5]=>0
[-1,4,-2,-3,-5]=>0
[-1,-4,2,3,5]=>0
[-1,-4,2,3,-5]=>0
[-1,-4,2,-3,5]=>0
[-1,-4,2,-3,-5]=>0
[-1,-4,-2,3,5]=>0
[-1,-4,-2,3,-5]=>0
[-1,-4,-2,-3,5]=>0
[-1,-4,-2,-3,-5]=>0
[1,4,2,5,3]=>0
[1,4,2,5,-3]=>0
[1,4,2,-5,3]=>0
[1,4,2,-5,-3]=>0
[1,4,-2,5,3]=>0
[1,4,-2,5,-3]=>0
[1,4,-2,-5,3]=>0
[1,4,-2,-5,-3]=>0
[1,-4,2,5,3]=>0
[1,-4,2,5,-3]=>0
[1,-4,2,-5,3]=>0
[1,-4,2,-5,-3]=>0
[1,-4,-2,5,3]=>0
[1,-4,-2,5,-3]=>0
[1,-4,-2,-5,3]=>0
[1,-4,-2,-5,-3]=>0
[-1,4,2,5,3]=>0
[-1,4,2,5,-3]=>0
[-1,4,2,-5,3]=>0
[-1,4,2,-5,-3]=>0
[-1,4,-2,5,3]=>0
[-1,4,-2,5,-3]=>0
[-1,4,-2,-5,3]=>0
[-1,4,-2,-5,-3]=>0
[-1,-4,2,5,3]=>0
[-1,-4,2,5,-3]=>0
[-1,-4,2,-5,3]=>0
[-1,-4,2,-5,-3]=>0
[-1,-4,-2,5,3]=>0
[-1,-4,-2,5,-3]=>0
[-1,-4,-2,-5,3]=>0
[-1,-4,-2,-5,-3]=>0
[1,4,3,2,5]=>3
[1,4,3,2,-5]=>3
[1,4,3,-2,5]=>0
[1,4,3,-2,-5]=>0
[1,4,-3,2,5]=>1
[1,4,-3,2,-5]=>3
[1,4,-3,-2,5]=>0
[1,4,-3,-2,-5]=>0
[1,-4,3,2,5]=>0
[1,-4,3,2,-5]=>0
[1,-4,3,-2,5]=>5
[1,-4,3,-2,-5]=>5
[1,-4,-3,2,5]=>0
[1,-4,-3,2,-5]=>0
[1,-4,-3,-2,5]=>3
[1,-4,-3,-2,-5]=>5
[-1,4,3,2,5]=>3
[-1,4,3,2,-5]=>9
[-1,4,3,-2,5]=>0
[-1,4,3,-2,-5]=>0
[-1,4,-3,2,5]=>5
[-1,4,-3,2,-5]=>13
[-1,4,-3,-2,5]=>0
[-1,4,-3,-2,-5]=>0
[-1,-4,3,2,5]=>0
[-1,-4,3,2,-5]=>0
[-1,-4,3,-2,5]=>5
[-1,-4,3,-2,-5]=>11
[-1,-4,-3,2,5]=>0
[-1,-4,-3,2,-5]=>0
[-1,-4,-3,-2,5]=>11
[-1,-4,-3,-2,-5]=>19
[1,4,3,5,2]=>0
[1,4,3,5,-2]=>0
[1,4,3,-5,2]=>0
[1,4,3,-5,-2]=>0
[1,4,-3,5,2]=>0
[1,4,-3,5,-2]=>0
[1,4,-3,-5,2]=>0
[1,4,-3,-5,-2]=>0
[1,-4,3,5,2]=>0
[1,-4,3,5,-2]=>0
[1,-4,3,-5,2]=>0
[1,-4,3,-5,-2]=>0
[1,-4,-3,5,2]=>0
[1,-4,-3,5,-2]=>0
[1,-4,-3,-5,2]=>0
[1,-4,-3,-5,-2]=>0
[-1,4,3,5,2]=>0
[-1,4,3,5,-2]=>0
[-1,4,3,-5,2]=>0
[-1,4,3,-5,-2]=>0
[-1,4,-3,5,2]=>0
[-1,4,-3,5,-2]=>0
[-1,4,-3,-5,2]=>0
[-1,4,-3,-5,-2]=>0
[-1,-4,3,5,2]=>0
[-1,-4,3,5,-2]=>0
[-1,-4,3,-5,2]=>0
[-1,-4,3,-5,-2]=>0
[-1,-4,-3,5,2]=>0
[-1,-4,-3,5,-2]=>0
[-1,-4,-3,-5,2]=>0
[-1,-4,-3,-5,-2]=>0
[1,4,5,2,3]=>1
[1,4,5,2,-3]=>0
[1,4,5,-2,3]=>0
[1,4,5,-2,-3]=>0
[1,4,-5,2,3]=>0
[1,4,-5,2,-3]=>1
[1,4,-5,-2,3]=>0
[1,4,-5,-2,-3]=>0
[1,-4,5,2,3]=>0
[1,-4,5,2,-3]=>0
[1,-4,5,-2,3]=>7
[1,-4,5,-2,-3]=>0
[1,-4,-5,2,3]=>0
[1,-4,-5,2,-3]=>0
[1,-4,-5,-2,3]=>0
[1,-4,-5,-2,-3]=>7
[-1,4,5,2,3]=>1
[-1,4,5,2,-3]=>0
[-1,4,5,-2,3]=>0
[-1,4,5,-2,-3]=>0
[-1,4,-5,2,3]=>0
[-1,4,-5,2,-3]=>1
[-1,4,-5,-2,3]=>0
[-1,4,-5,-2,-3]=>0
[-1,-4,5,2,3]=>0
[-1,-4,5,2,-3]=>0
[-1,-4,5,-2,3]=>7
[-1,-4,5,-2,-3]=>0
[-1,-4,-5,2,3]=>0
[-1,-4,-5,2,-3]=>0
[-1,-4,-5,-2,3]=>0
[-1,-4,-5,-2,-3]=>7
[1,4,5,3,2]=>0
[1,4,5,3,-2]=>0
[1,4,5,-3,2]=>0
[1,4,5,-3,-2]=>0
[1,4,-5,3,2]=>0
[1,4,-5,3,-2]=>0
[1,4,-5,-3,2]=>0
[1,4,-5,-3,-2]=>0
[1,-4,5,3,2]=>0
[1,-4,5,3,-2]=>0
[1,-4,5,-3,2]=>0
[1,-4,5,-3,-2]=>0
[1,-4,-5,3,2]=>0
[1,-4,-5,3,-2]=>0
[1,-4,-5,-3,2]=>0
[1,-4,-5,-3,-2]=>0
[-1,4,5,3,2]=>0
[-1,4,5,3,-2]=>0
[-1,4,5,-3,2]=>0
[-1,4,5,-3,-2]=>0
[-1,4,-5,3,2]=>0
[-1,4,-5,3,-2]=>0
[-1,4,-5,-3,2]=>0
[-1,4,-5,-3,-2]=>0
[-1,-4,5,3,2]=>0
[-1,-4,5,3,-2]=>0
[-1,-4,5,-3,2]=>0
[-1,-4,5,-3,-2]=>0
[-1,-4,-5,3,2]=>0
[-1,-4,-5,3,-2]=>0
[-1,-4,-5,-3,2]=>0
[-1,-4,-5,-3,-2]=>0
[1,5,2,3,4]=>0
[1,5,2,3,-4]=>0
[1,5,2,-3,4]=>0
[1,5,2,-3,-4]=>0
[1,5,-2,3,4]=>0
[1,5,-2,3,-4]=>0
[1,5,-2,-3,4]=>0
[1,5,-2,-3,-4]=>0
[1,-5,2,3,4]=>0
[1,-5,2,3,-4]=>0
[1,-5,2,-3,4]=>0
[1,-5,2,-3,-4]=>0
[1,-5,-2,3,4]=>0
[1,-5,-2,3,-4]=>0
[1,-5,-2,-3,4]=>0
[1,-5,-2,-3,-4]=>0
[-1,5,2,3,4]=>0
[-1,5,2,3,-4]=>0
[-1,5,2,-3,4]=>0
[-1,5,2,-3,-4]=>0
[-1,5,-2,3,4]=>0
[-1,5,-2,3,-4]=>0
[-1,5,-2,-3,4]=>0
[-1,5,-2,-3,-4]=>0
[-1,-5,2,3,4]=>0
[-1,-5,2,3,-4]=>0
[-1,-5,2,-3,4]=>0
[-1,-5,2,-3,-4]=>0
[-1,-5,-2,3,4]=>0
[-1,-5,-2,3,-4]=>0
[-1,-5,-2,-3,4]=>0
[-1,-5,-2,-3,-4]=>0
[1,5,2,4,3]=>0
[1,5,2,4,-3]=>0
[1,5,2,-4,3]=>0
[1,5,2,-4,-3]=>0
[1,5,-2,4,3]=>0
[1,5,-2,4,-3]=>0
[1,5,-2,-4,3]=>0
[1,5,-2,-4,-3]=>0
[1,-5,2,4,3]=>0
[1,-5,2,4,-3]=>0
[1,-5,2,-4,3]=>0
[1,-5,2,-4,-3]=>0
[1,-5,-2,4,3]=>0
[1,-5,-2,4,-3]=>0
[1,-5,-2,-4,3]=>0
[1,-5,-2,-4,-3]=>0
[-1,5,2,4,3]=>0
[-1,5,2,4,-3]=>0
[-1,5,2,-4,3]=>0
[-1,5,2,-4,-3]=>0
[-1,5,-2,4,3]=>0
[-1,5,-2,4,-3]=>0
[-1,5,-2,-4,3]=>0
[-1,5,-2,-4,-3]=>0
[-1,-5,2,4,3]=>0
[-1,-5,2,4,-3]=>0
[-1,-5,2,-4,3]=>0
[-1,-5,2,-4,-3]=>0
[-1,-5,-2,4,3]=>0
[-1,-5,-2,4,-3]=>0
[-1,-5,-2,-4,3]=>0
[-1,-5,-2,-4,-3]=>0
[1,5,3,2,4]=>0
[1,5,3,2,-4]=>0
[1,5,3,-2,4]=>0
[1,5,3,-2,-4]=>0
[1,5,-3,2,4]=>0
[1,5,-3,2,-4]=>0
[1,5,-3,-2,4]=>0
[1,5,-3,-2,-4]=>0
[1,-5,3,2,4]=>0
[1,-5,3,2,-4]=>0
[1,-5,3,-2,4]=>0
[1,-5,3,-2,-4]=>0
[1,-5,-3,2,4]=>0
[1,-5,-3,2,-4]=>0
[1,-5,-3,-2,4]=>0
[1,-5,-3,-2,-4]=>0
[-1,5,3,2,4]=>0
[-1,5,3,2,-4]=>0
[-1,5,3,-2,4]=>0
[-1,5,3,-2,-4]=>0
[-1,5,-3,2,4]=>0
[-1,5,-3,2,-4]=>0
[-1,5,-3,-2,4]=>0
[-1,5,-3,-2,-4]=>0
[-1,-5,3,2,4]=>0
[-1,-5,3,2,-4]=>0
[-1,-5,3,-2,4]=>0
[-1,-5,3,-2,-4]=>0
[-1,-5,-3,2,4]=>0
[-1,-5,-3,2,-4]=>0
[-1,-5,-3,-2,4]=>0
[-1,-5,-3,-2,-4]=>0
[1,5,3,4,2]=>5
[1,5,3,4,-2]=>0
[1,5,3,-4,2]=>3
[1,5,3,-4,-2]=>0
[1,5,-3,4,2]=>3
[1,5,-3,4,-2]=>0
[1,5,-3,-4,2]=>3
[1,5,-3,-4,-2]=>0
[1,-5,3,4,2]=>0
[1,-5,3,4,-2]=>5
[1,-5,3,-4,2]=>0
[1,-5,3,-4,-2]=>3
[1,-5,-3,4,2]=>0
[1,-5,-3,4,-2]=>3
[1,-5,-3,-4,2]=>0
[1,-5,-3,-4,-2]=>3
[-1,5,3,4,2]=>5
[-1,5,3,4,-2]=>0
[-1,5,3,-4,2]=>9
[-1,5,3,-4,-2]=>0
[-1,5,-3,4,2]=>11
[-1,5,-3,4,-2]=>0
[-1,5,-3,-4,2]=>13
[-1,5,-3,-4,-2]=>0
[-1,-5,3,4,2]=>0
[-1,-5,3,4,-2]=>5
[-1,-5,3,-4,2]=>0
[-1,-5,3,-4,-2]=>9
[-1,-5,-3,4,2]=>0
[-1,-5,-3,4,-2]=>11
[-1,-5,-3,-4,2]=>0
[-1,-5,-3,-4,-2]=>13
[1,5,4,2,3]=>0
[1,5,4,2,-3]=>0
[1,5,4,-2,3]=>0
[1,5,4,-2,-3]=>0
[1,5,-4,2,3]=>0
[1,5,-4,2,-3]=>0
[1,5,-4,-2,3]=>0
[1,5,-4,-2,-3]=>0
[1,-5,4,2,3]=>0
[1,-5,4,2,-3]=>0
[1,-5,4,-2,3]=>0
[1,-5,4,-2,-3]=>0
[1,-5,-4,2,3]=>0
[1,-5,-4,2,-3]=>0
[1,-5,-4,-2,3]=>0
[1,-5,-4,-2,-3]=>0
[-1,5,4,2,3]=>0
[-1,5,4,2,-3]=>0
[-1,5,4,-2,3]=>0
[-1,5,4,-2,-3]=>0
[-1,5,-4,2,3]=>0
[-1,5,-4,2,-3]=>0
[-1,5,-4,-2,3]=>0
[-1,5,-4,-2,-3]=>0
[-1,-5,4,2,3]=>0
[-1,-5,4,2,-3]=>0
[-1,-5,4,-2,3]=>0
[-1,-5,4,-2,-3]=>0
[-1,-5,-4,2,3]=>0
[-1,-5,-4,2,-3]=>0
[-1,-5,-4,-2,3]=>0
[-1,-5,-4,-2,-3]=>0
[1,5,4,3,2]=>7
[1,5,4,3,-2]=>0
[1,5,4,-3,2]=>0
[1,5,4,-3,-2]=>0
[1,5,-4,3,2]=>0
[1,5,-4,3,-2]=>0
[1,5,-4,-3,2]=>1
[1,5,-4,-3,-2]=>0
[1,-5,4,3,2]=>0
[1,-5,4,3,-2]=>7
[1,-5,4,-3,2]=>0
[1,-5,4,-3,-2]=>0
[1,-5,-4,3,2]=>0
[1,-5,-4,3,-2]=>0
[1,-5,-4,-3,2]=>0
[1,-5,-4,-3,-2]=>1
[-1,5,4,3,2]=>7
[-1,5,4,3,-2]=>0
[-1,5,4,-3,2]=>0
[-1,5,4,-3,-2]=>0
[-1,5,-4,3,2]=>0
[-1,5,-4,3,-2]=>0
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Description
The number of Hecke atoms of a signed permutation.
For a signed permutation $z\in\mathfrak H_n$, this is the cardinality of the set
$$ \{ w\in\mathfrak H_n | w^{-1} \star w = z\}, $$
where $\star$ denotes the Demazure product. Note that $w\mapsto w^{-1}\star w$ is a surjection onto the set of involutions.
For a signed permutation $z\in\mathfrak H_n$, this is the cardinality of the set
$$ \{ w\in\mathfrak H_n | w^{-1} \star w = z\}, $$
where $\star$ denotes the Demazure product. Note that $w\mapsto w^{-1}\star w$ is a surjection onto the set of involutions.
References
[1] Hamaker, Z., Marberg, E. Atoms for signed permutations arXiv:1802.09805
Code
def statistic(z):
return len(set(w for w in SignedPermutations(len(z)) if w.inverse().apply_demazure_product(w) == z))
Created
Nov 26, 2022 at 12:09 by Martin Rubey
Updated
Nov 26, 2022 at 12:09 by Martin Rubey
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