- FindStat

edit this statistic or download as text // json

Identifier

St001895: Signed permutations ⟶ ℤ

Values

=>
                            
                            
                            
                            

                        [1]=>0
[-1]=>1
[1,2]=>0
[1,-2]=>1
[-1,2]=>1
[-1,-2]=>2
[2,1]=>0
[2,-1]=>1
[-2,1]=>1
[-2,-1]=>0
[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]=>0
[1,3,-2]=>1
[1,-3,2]=>1
[1,-3,-2]=>0
[-1,3,2]=>1
[-1,3,-2]=>2
[-1,-3,2]=>2
[-1,-3,-2]=>1
[2,1,3]=>0
[2,1,-3]=>1
[2,-1,3]=>1
[2,-1,-3]=>2
[-2,1,3]=>1
[-2,1,-3]=>2
[-2,-1,3]=>0
[-2,-1,-3]=>1
[2,3,1]=>0
[2,3,-1]=>1
[2,-3,1]=>1
[2,-3,-1]=>0
[-2,3,1]=>1
[-2,3,-1]=>0
[-2,-3,1]=>0
[-2,-3,-1]=>1
[3,1,2]=>0
[3,1,-2]=>1
[3,-1,2]=>1
[3,-1,-2]=>0
[-3,1,2]=>1
[-3,1,-2]=>0
[-3,-1,2]=>0
[-3,-1,-2]=>1
[3,2,1]=>0
[3,2,-1]=>1
[3,-2,1]=>1
[3,-2,-1]=>0
[-3,2,1]=>1
[-3,2,-1]=>0
[-3,-2,1]=>0
[-3,-2,-1]=>1
[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]=>0
[1,2,4,-3]=>1
[1,2,-4,3]=>1
[1,2,-4,-3]=>0
[1,-2,4,3]=>1
[1,-2,4,-3]=>2
[1,-2,-4,3]=>2
[1,-2,-4,-3]=>1
[-1,2,4,3]=>1
[-1,2,4,-3]=>2
[-1,2,-4,3]=>2
[-1,2,-4,-3]=>1
[-1,-2,4,3]=>2
[-1,-2,4,-3]=>3
[-1,-2,-4,3]=>3
[-1,-2,-4,-3]=>2
[1,3,2,4]=>0
[1,3,2,-4]=>1
[1,3,-2,4]=>1
[1,3,-2,-4]=>2
[1,-3,2,4]=>1
[1,-3,2,-4]=>2
[1,-3,-2,4]=>0
[1,-3,-2,-4]=>1
[-1,3,2,4]=>1
[-1,3,2,-4]=>2
[-1,3,-2,4]=>2
[-1,3,-2,-4]=>3
[-1,-3,2,4]=>2
[-1,-3,2,-4]=>3
[-1,-3,-2,4]=>1
[-1,-3,-2,-4]=>2
[1,3,4,2]=>0
[1,3,4,-2]=>1
[1,3,-4,2]=>1
[1,3,-4,-2]=>0
[1,-3,4,2]=>1
[1,-3,4,-2]=>0
[1,-3,-4,2]=>0
[1,-3,-4,-2]=>1
[-1,3,4,2]=>1
[-1,3,4,-2]=>2
[-1,3,-4,2]=>2
[-1,3,-4,-2]=>1
[-1,-3,4,2]=>2
[-1,-3,4,-2]=>1
[-1,-3,-4,2]=>1
[-1,-3,-4,-2]=>2
[1,4,2,3]=>0
[1,4,2,-3]=>1
[1,4,-2,3]=>1
[1,4,-2,-3]=>0
[1,-4,2,3]=>1
[1,-4,2,-3]=>0
[1,-4,-2,3]=>0
[1,-4,-2,-3]=>1
[-1,4,2,3]=>1
[-1,4,2,-3]=>2
[-1,4,-2,3]=>2
[-1,4,-2,-3]=>1
[-1,-4,2,3]=>2
[-1,-4,2,-3]=>1
[-1,-4,-2,3]=>1
[-1,-4,-2,-3]=>2
[1,4,3,2]=>0
[1,4,3,-2]=>1
[1,4,-3,2]=>1
[1,4,-3,-2]=>0
[1,-4,3,2]=>1
[1,-4,3,-2]=>0
[1,-4,-3,2]=>0
[1,-4,-3,-2]=>1
[-1,4,3,2]=>1
[-1,4,3,-2]=>2
[-1,4,-3,2]=>2
[-1,4,-3,-2]=>1
[-1,-4,3,2]=>2
[-1,-4,3,-2]=>1
[-1,-4,-3,2]=>1
[-1,-4,-3,-2]=>2
[2,1,3,4]=>0
[2,1,3,-4]=>1
[2,1,-3,4]=>1
[2,1,-3,-4]=>2
[2,-1,3,4]=>1
[2,-1,3,-4]=>2
[2,-1,-3,4]=>2
[2,-1,-3,-4]=>3
[-2,1,3,4]=>1
[-2,1,3,-4]=>2
[-2,1,-3,4]=>2
[-2,1,-3,-4]=>3
[-2,-1,3,4]=>0
[-2,-1,3,-4]=>1
[-2,-1,-3,4]=>1
[-2,-1,-3,-4]=>2
[2,1,4,3]=>0
[2,1,4,-3]=>1
[2,1,-4,3]=>1
[2,1,-4,-3]=>0
[2,-1,4,3]=>1
[2,-1,4,-3]=>2
[2,-1,-4,3]=>2
[2,-1,-4,-3]=>1
[-2,1,4,3]=>1
[-2,1,4,-3]=>2
[-2,1,-4,3]=>2
[-2,1,-4,-3]=>1
[-2,-1,4,3]=>0
[-2,-1,4,-3]=>1
[-2,-1,-4,3]=>1
[-2,-1,-4,-3]=>0
[2,3,1,4]=>0
[2,3,1,-4]=>1
[2,3,-1,4]=>1
[2,3,-1,-4]=>2
[2,-3,1,4]=>1
[2,-3,1,-4]=>2
[2,-3,-1,4]=>0
[2,-3,-1,-4]=>1
[-2,3,1,4]=>1
[-2,3,1,-4]=>2
[-2,3,-1,4]=>0
[-2,3,-1,-4]=>1
[-2,-3,1,4]=>0
[-2,-3,1,-4]=>1
[-2,-3,-1,4]=>1
[-2,-3,-1,-4]=>2
[2,3,4,1]=>0
[2,3,4,-1]=>1
[2,3,-4,1]=>1
[2,3,-4,-1]=>0
[2,-3,4,1]=>1
[2,-3,4,-1]=>0
[2,-3,-4,1]=>0
[2,-3,-4,-1]=>1
[-2,3,4,1]=>1
[-2,3,4,-1]=>0
[-2,3,-4,1]=>0
[-2,3,-4,-1]=>1
[-2,-3,4,1]=>0
[-2,-3,4,-1]=>1
[-2,-3,-4,1]=>1
[-2,-3,-4,-1]=>0
[2,4,1,3]=>0
[2,4,1,-3]=>1
[2,4,-1,3]=>1
[2,4,-1,-3]=>0
[2,-4,1,3]=>1
[2,-4,1,-3]=>0
[2,-4,-1,3]=>0
[2,-4,-1,-3]=>1
[-2,4,1,3]=>1
[-2,4,1,-3]=>0
[-2,4,-1,3]=>0
[-2,4,-1,-3]=>1
[-2,-4,1,3]=>0
[-2,-4,1,-3]=>1
[-2,-4,-1,3]=>1
[-2,-4,-1,-3]=>0
[2,4,3,1]=>0
[2,4,3,-1]=>1
[2,4,-3,1]=>1
[2,4,-3,-1]=>0
[2,-4,3,1]=>1
[2,-4,3,-1]=>0
[2,-4,-3,1]=>0
[2,-4,-3,-1]=>1
[-2,4,3,1]=>1
[-2,4,3,-1]=>0
[-2,4,-3,1]=>0
[-2,4,-3,-1]=>1
[-2,-4,3,1]=>0
[-2,-4,3,-1]=>1
[-2,-4,-3,1]=>1
[-2,-4,-3,-1]=>0
[3,1,2,4]=>0
[3,1,2,-4]=>1
[3,1,-2,4]=>1
[3,1,-2,-4]=>2
[3,-1,2,4]=>1
[3,-1,2,-4]=>2
[3,-1,-2,4]=>0
[3,-1,-2,-4]=>1
[-3,1,2,4]=>1
[-3,1,2,-4]=>2
[-3,1,-2,4]=>0
[-3,1,-2,-4]=>1
[-3,-1,2,4]=>0
[-3,-1,2,-4]=>1
[-3,-1,-2,4]=>1
[-3,-1,-2,-4]=>2
[3,1,4,2]=>0
[3,1,4,-2]=>1
[3,1,-4,2]=>1
[3,1,-4,-2]=>0
[3,-1,4,2]=>1
[3,-1,4,-2]=>0
[3,-1,-4,2]=>0
[3,-1,-4,-2]=>1
[-3,1,4,2]=>1
[-3,1,4,-2]=>0
[-3,1,-4,2]=>0
[-3,1,-4,-2]=>1
[-3,-1,4,2]=>0
[-3,-1,4,-2]=>1
[-3,-1,-4,2]=>1
[-3,-1,-4,-2]=>0
[3,2,1,4]=>0
[3,2,1,-4]=>1
[3,2,-1,4]=>1
[3,2,-1,-4]=>2
[3,-2,1,4]=>1
[3,-2,1,-4]=>2
[3,-2,-1,4]=>0
[3,-2,-1,-4]=>1
[-3,2,1,4]=>1
[-3,2,1,-4]=>2
[-3,2,-1,4]=>0
[-3,2,-1,-4]=>1
[-3,-2,1,4]=>0
[-3,-2,1,-4]=>1
[-3,-2,-1,4]=>1
[-3,-2,-1,-4]=>2
[3,2,4,1]=>0
[3,2,4,-1]=>1
[3,2,-4,1]=>1
[3,2,-4,-1]=>0
[3,-2,4,1]=>1
[3,-2,4,-1]=>0
[3,-2,-4,1]=>0
[3,-2,-4,-1]=>1
[-3,2,4,1]=>1
[-3,2,4,-1]=>0
[-3,2,-4,1]=>0
[-3,2,-4,-1]=>1
[-3,-2,4,1]=>0
[-3,-2,4,-1]=>1
[-3,-2,-4,1]=>1
[-3,-2,-4,-1]=>0
[3,4,1,2]=>0
[3,4,1,-2]=>1
[3,4,-1,2]=>1
[3,4,-1,-2]=>0
[3,-4,1,2]=>1
[3,-4,1,-2]=>0
[3,-4,-1,2]=>0
[3,-4,-1,-2]=>1
[-3,4,1,2]=>1
[-3,4,1,-2]=>0
[-3,4,-1,2]=>0
[-3,4,-1,-2]=>1
[-3,-4,1,2]=>0
[-3,-4,1,-2]=>1
[-3,-4,-1,2]=>1
[-3,-4,-1,-2]=>0
[3,4,2,1]=>0
[3,4,2,-1]=>1
[3,4,-2,1]=>1
[3,4,-2,-1]=>0
[3,-4,2,1]=>1
[3,-4,2,-1]=>0
[3,-4,-2,1]=>0
[3,-4,-2,-1]=>1
[-3,4,2,1]=>1
[-3,4,2,-1]=>0
[-3,4,-2,1]=>0
[-3,4,-2,-1]=>1
[-3,-4,2,1]=>0
[-3,-4,2,-1]=>1
[-3,-4,-2,1]=>1
[-3,-4,-2,-1]=>0
[4,1,2,3]=>0
[4,1,2,-3]=>1
[4,1,-2,3]=>1
[4,1,-2,-3]=>0
[4,-1,2,3]=>1
[4,-1,2,-3]=>0
[4,-1,-2,3]=>0
[4,-1,-2,-3]=>1
[-4,1,2,3]=>1
[-4,1,2,-3]=>0
[-4,1,-2,3]=>0
[-4,1,-2,-3]=>1
[-4,-1,2,3]=>0
[-4,-1,2,-3]=>1
[-4,-1,-2,3]=>1
[-4,-1,-2,-3]=>0
[4,1,3,2]=>0
[4,1,3,-2]=>1
[4,1,-3,2]=>1
[4,1,-3,-2]=>0
[4,-1,3,2]=>1
[4,-1,3,-2]=>0
[4,-1,-3,2]=>0
[4,-1,-3,-2]=>1
[-4,1,3,2]=>1
[-4,1,3,-2]=>0
[-4,1,-3,2]=>0
[-4,1,-3,-2]=>1
[-4,-1,3,2]=>0
[-4,-1,3,-2]=>1
[-4,-1,-3,2]=>1
[-4,-1,-3,-2]=>0
[4,2,1,3]=>0
[4,2,1,-3]=>1
[4,2,-1,3]=>1
[4,2,-1,-3]=>0
[4,-2,1,3]=>1
[4,-2,1,-3]=>0
[4,-2,-1,3]=>0
[4,-2,-1,-3]=>1
[-4,2,1,3]=>1
[-4,2,1,-3]=>0
[-4,2,-1,3]=>0
[-4,2,-1,-3]=>1
[-4,-2,1,3]=>0
[-4,-2,1,-3]=>1
[-4,-2,-1,3]=>1
[-4,-2,-1,-3]=>0
[4,2,3,1]=>0
[4,2,3,-1]=>1
[4,2,-3,1]=>1
[4,2,-3,-1]=>0
[4,-2,3,1]=>1
[4,-2,3,-1]=>0
[4,-2,-3,1]=>0
[4,-2,-3,-1]=>1
[-4,2,3,1]=>1
[-4,2,3,-1]=>0
[-4,2,-3,1]=>0
[-4,2,-3,-1]=>1
[-4,-2,3,1]=>0
[-4,-2,3,-1]=>1
[-4,-2,-3,1]=>1
[-4,-2,-3,-1]=>0
[4,3,1,2]=>0
[4,3,1,-2]=>1
[4,3,-1,2]=>1
[4,3,-1,-2]=>0
[4,-3,1,2]=>1
[4,-3,1,-2]=>0
[4,-3,-1,2]=>0
[4,-3,-1,-2]=>1
[-4,3,1,2]=>1
[-4,3,1,-2]=>0
[-4,3,-1,2]=>0
[-4,3,-1,-2]=>1
[-4,-3,1,2]=>0
[-4,-3,1,-2]=>1
[-4,-3,-1,2]=>1
[-4,-3,-1,-2]=>0
[4,3,2,1]=>0
[4,3,2,-1]=>1
[4,3,-2,1]=>1
[4,3,-2,-1]=>0
[4,-3,2,1]=>1
[4,-3,2,-1]=>0
[4,-3,-2,1]=>0
[4,-3,-2,-1]=>1
[-4,3,2,1]=>1
[-4,3,2,-1]=>0
[-4,3,-2,1]=>0
[-4,3,-2,-1]=>1
[-4,-3,2,1]=>0
[-4,-3,2,-1]=>1
[-4,-3,-2,1]=>1
[-4,-3,-2,-1]=>0
[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]=>0
[1,2,3,5,-4]=>1
[1,2,3,-5,4]=>1
[1,2,3,-5,-4]=>0
[1,2,-3,5,4]=>1
[1,2,-3,5,-4]=>2
[1,2,-3,-5,4]=>2
[1,2,-3,-5,-4]=>1
[1,-2,3,5,4]=>1
[1,-2,3,5,-4]=>2
[1,-2,3,-5,4]=>2
[1,-2,3,-5,-4]=>1
[1,-2,-3,5,4]=>2
[1,-2,-3,5,-4]=>3
[1,-2,-3,-5,4]=>3
[1,-2,-3,-5,-4]=>2
[-1,2,3,5,4]=>1
[-1,2,3,5,-4]=>2
[-1,2,3,-5,4]=>2
[-1,2,3,-5,-4]=>1
[-1,2,-3,5,4]=>2
[-1,2,-3,5,-4]=>3
[-1,2,-3,-5,4]=>3
[-1,2,-3,-5,-4]=>2
[-1,-2,3,5,4]=>2
[-1,-2,3,5,-4]=>3
[-1,-2,3,-5,4]=>3
[-1,-2,3,-5,-4]=>2
[-1,-2,-3,5,4]=>3
[-1,-2,-3,5,-4]=>4
[-1,-2,-3,-5,4]=>4
[-1,-2,-3,-5,-4]=>3
[1,2,4,3,5]=>0
[1,2,4,3,-5]=>1
[1,2,4,-3,5]=>1
[1,2,4,-3,-5]=>2
[1,2,-4,3,5]=>1
[1,2,-4,3,-5]=>2
[1,2,-4,-3,5]=>0
[1,2,-4,-3,-5]=>1
[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]=>2
[1,-2,-4,3,-5]=>3
[1,-2,-4,-3,5]=>1
[1,-2,-4,-3,-5]=>2
[-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]=>2
[-1,2,-4,3,-5]=>3
[-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]=>3
[-1,-2,-4,3,-5]=>4
[-1,-2,-4,-3,5]=>2
[-1,-2,-4,-3,-5]=>3
[1,2,4,5,3]=>0
[1,2,4,5,-3]=>1
[1,2,4,-5,3]=>1
[1,2,4,-5,-3]=>0
[1,2,-4,5,3]=>1
[1,2,-4,5,-3]=>0
[1,2,-4,-5,3]=>0
[1,2,-4,-5,-3]=>1
[1,-2,4,5,3]=>1
[1,-2,4,5,-3]=>2
[1,-2,4,-5,3]=>2
[1,-2,4,-5,-3]=>1
[1,-2,-4,5,3]=>2
[1,-2,-4,5,-3]=>1
[1,-2,-4,-5,3]=>1
[1,-2,-4,-5,-3]=>2
[-1,2,4,5,3]=>1
[-1,2,4,5,-3]=>2
[-1,2,4,-5,3]=>2
[-1,2,4,-5,-3]=>1
[-1,2,-4,5,3]=>2
[-1,2,-4,5,-3]=>1
[-1,2,-4,-5,3]=>1
[-1,2,-4,-5,-3]=>2
[-1,-2,4,5,3]=>2
[-1,-2,4,5,-3]=>3
[-1,-2,4,-5,3]=>3
[-1,-2,4,-5,-3]=>2
[-1,-2,-4,5,3]=>3
[-1,-2,-4,5,-3]=>2
[-1,-2,-4,-5,3]=>2
[-1,-2,-4,-5,-3]=>3
[1,2,5,3,4]=>0
[1,2,5,3,-4]=>1
[1,2,5,-3,4]=>1
[1,2,5,-3,-4]=>0
[1,2,-5,3,4]=>1
[1,2,-5,3,-4]=>0
[1,2,-5,-3,4]=>0
[1,2,-5,-3,-4]=>1
[1,-2,5,3,4]=>1
[1,-2,5,3,-4]=>2
[1,-2,5,-3,4]=>2
[1,-2,5,-3,-4]=>1
[1,-2,-5,3,4]=>2
[1,-2,-5,3,-4]=>1
[1,-2,-5,-3,4]=>1
[1,-2,-5,-3,-4]=>2
[-1,2,5,3,4]=>1
[-1,2,5,3,-4]=>2
[-1,2,5,-3,4]=>2
[-1,2,5,-3,-4]=>1
[-1,2,-5,3,4]=>2
[-1,2,-5,3,-4]=>1
[-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]=>2
[-1,-2,-5,3,4]=>3
[-1,-2,-5,3,-4]=>2
[-1,-2,-5,-3,4]=>2
[-1,-2,-5,-3,-4]=>3
[1,2,5,4,3]=>0
[1,2,5,4,-3]=>1
[1,2,5,-4,3]=>1
[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]=>1
[1,-2,5,4,3]=>1
[1,-2,5,4,-3]=>2
[1,-2,5,-4,3]=>2
[1,-2,5,-4,-3]=>1
[1,-2,-5,4,3]=>2
[1,-2,-5,4,-3]=>1
[1,-2,-5,-4,3]=>1
[1,-2,-5,-4,-3]=>2
[-1,2,5,4,3]=>1
[-1,2,5,4,-3]=>2
[-1,2,5,-4,3]=>2
[-1,2,5,-4,-3]=>1
[-1,2,-5,4,3]=>2
[-1,2,-5,4,-3]=>1
[-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]=>2
[-1,-2,-5,4,3]=>3
[-1,-2,-5,4,-3]=>2
[-1,-2,-5,-4,3]=>2
[-1,-2,-5,-4,-3]=>3
[1,3,2,4,5]=>0
[1,3,2,4,-5]=>1
[1,3,2,-4,5]=>1
[1,3,2,-4,-5]=>2
[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]=>1
[1,-3,2,4,-5]=>2
[1,-3,2,-4,5]=>2
[1,-3,2,-4,-5]=>3
[1,-3,-2,4,5]=>0
[1,-3,-2,4,-5]=>1
[1,-3,-2,-4,5]=>1
[1,-3,-2,-4,-5]=>2
[-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]=>2
[-1,-3,2,4,-5]=>3
[-1,-3,2,-4,5]=>3
[-1,-3,2,-4,-5]=>4
[-1,-3,-2,4,5]=>1
[-1,-3,-2,4,-5]=>2
[-1,-3,-2,-4,5]=>2
[-1,-3,-2,-4,-5]=>3
[1,3,2,5,4]=>0
[1,3,2,5,-4]=>1
[1,3,2,-5,4]=>1
[1,3,2,-5,-4]=>0
[1,3,-2,5,4]=>1
[1,3,-2,5,-4]=>2
[1,3,-2,-5,4]=>2
[1,3,-2,-5,-4]=>1
[1,-3,2,5,4]=>1
[1,-3,2,5,-4]=>2
[1,-3,2,-5,4]=>2
[1,-3,2,-5,-4]=>1
[1,-3,-2,5,4]=>0
[1,-3,-2,5,-4]=>1
[1,-3,-2,-5,4]=>1
[1,-3,-2,-5,-4]=>0
[-1,3,2,5,4]=>1
[-1,3,2,5,-4]=>2
[-1,3,2,-5,4]=>2
[-1,3,2,-5,-4]=>1
[-1,3,-2,5,4]=>2
[-1,3,-2,5,-4]=>3
[-1,3,-2,-5,4]=>3
[-1,3,-2,-5,-4]=>2
[-1,-3,2,5,4]=>2
[-1,-3,2,5,-4]=>3
[-1,-3,2,-5,4]=>3
[-1,-3,2,-5,-4]=>2
[-1,-3,-2,5,4]=>1
[-1,-3,-2,5,-4]=>2
[-1,-3,-2,-5,4]=>2
[-1,-3,-2,-5,-4]=>1
[1,3,4,2,5]=>0
[1,3,4,2,-5]=>1
[1,3,4,-2,5]=>1
[1,3,4,-2,-5]=>2
[1,3,-4,2,5]=>1
[1,3,-4,2,-5]=>2
[1,3,-4,-2,5]=>0
[1,3,-4,-2,-5]=>1
[1,-3,4,2,5]=>1
[1,-3,4,2,-5]=>2
[1,-3,4,-2,5]=>0
[1,-3,4,-2,-5]=>1
[1,-3,-4,2,5]=>0
[1,-3,-4,2,-5]=>1
[1,-3,-4,-2,5]=>1
[1,-3,-4,-2,-5]=>2
[-1,3,4,2,5]=>1
[-1,3,4,2,-5]=>2
[-1,3,4,-2,5]=>2
[-1,3,4,-2,-5]=>3
[-1,3,-4,2,5]=>2
[-1,3,-4,2,-5]=>3
[-1,3,-4,-2,5]=>1
[-1,3,-4,-2,-5]=>2
[-1,-3,4,2,5]=>2
[-1,-3,4,2,-5]=>3
[-1,-3,4,-2,5]=>1
[-1,-3,4,-2,-5]=>2
[-1,-3,-4,2,5]=>1
[-1,-3,-4,2,-5]=>2
[-1,-3,-4,-2,5]=>2
[-1,-3,-4,-2,-5]=>3
[1,3,4,5,2]=>0
[1,3,4,5,-2]=>1
[1,3,4,-5,2]=>1
[1,3,4,-5,-2]=>0
[1,3,-4,5,2]=>1
[1,3,-4,5,-2]=>0
[1,3,-4,-5,2]=>0
[1,3,-4,-5,-2]=>1
[1,-3,4,5,2]=>1
[1,-3,4,5,-2]=>0
[1,-3,4,-5,2]=>0
[1,-3,4,-5,-2]=>1
[1,-3,-4,5,2]=>0
[1,-3,-4,5,-2]=>1
[1,-3,-4,-5,2]=>1
[1,-3,-4,-5,-2]=>0
[-1,3,4,5,2]=>1
[-1,3,4,5,-2]=>2
[-1,3,4,-5,2]=>2
[-1,3,4,-5,-2]=>1
[-1,3,-4,5,2]=>2
[-1,3,-4,5,-2]=>1
[-1,3,-4,-5,2]=>1
[-1,3,-4,-5,-2]=>2
[-1,-3,4,5,2]=>2
[-1,-3,4,5,-2]=>1
[-1,-3,4,-5,2]=>1
[-1,-3,4,-5,-2]=>2
[-1,-3,-4,5,2]=>1
[-1,-3,-4,5,-2]=>2
[-1,-3,-4,-5,2]=>2
[-1,-3,-4,-5,-2]=>1
[1,3,5,2,4]=>0
[1,3,5,2,-4]=>1
[1,3,5,-2,4]=>1
[1,3,5,-2,-4]=>0
[1,3,-5,2,4]=>1
[1,3,-5,2,-4]=>0
[1,3,-5,-2,4]=>0
[1,3,-5,-2,-4]=>1
[1,-3,5,2,4]=>1
[1,-3,5,2,-4]=>0
[1,-3,5,-2,4]=>0
[1,-3,5,-2,-4]=>1
[1,-3,-5,2,4]=>0
[1,-3,-5,2,-4]=>1
[1,-3,-5,-2,4]=>1
[1,-3,-5,-2,-4]=>0
[-1,3,5,2,4]=>1
[-1,3,5,2,-4]=>2
[-1,3,5,-2,4]=>2
[-1,3,5,-2,-4]=>1
[-1,3,-5,2,4]=>2
[-1,3,-5,2,-4]=>1
[-1,3,-5,-2,4]=>1
[-1,3,-5,-2,-4]=>2
[-1,-3,5,2,4]=>2
[-1,-3,5,2,-4]=>1
[-1,-3,5,-2,4]=>1
[-1,-3,5,-2,-4]=>2
[-1,-3,-5,2,4]=>1
[-1,-3,-5,2,-4]=>2
[-1,-3,-5,-2,4]=>2
[-1,-3,-5,-2,-4]=>1
[1,3,5,4,2]=>0
[1,3,5,4,-2]=>1
[1,3,5,-4,2]=>1
[1,3,5,-4,-2]=>0
[1,3,-5,4,2]=>1
[1,3,-5,4,-2]=>0
[1,3,-5,-4,2]=>0
[1,3,-5,-4,-2]=>1
[1,-3,5,4,2]=>1
[1,-3,5,4,-2]=>0
[1,-3,5,-4,2]=>0
[1,-3,5,-4,-2]=>1
[1,-3,-5,4,2]=>0
[1,-3,-5,4,-2]=>1
[1,-3,-5,-4,2]=>1
[1,-3,-5,-4,-2]=>0
[-1,3,5,4,2]=>1
[-1,3,5,4,-2]=>2
[-1,3,5,-4,2]=>2
[-1,3,5,-4,-2]=>1
[-1,3,-5,4,2]=>2
[-1,3,-5,4,-2]=>1
[-1,3,-5,-4,2]=>1
[-1,3,-5,-4,-2]=>2
[-1,-3,5,4,2]=>2
[-1,-3,5,4,-2]=>1
[-1,-3,5,-4,2]=>1
[-1,-3,5,-4,-2]=>2
[-1,-3,-5,4,2]=>1
[-1,-3,-5,4,-2]=>2
[-1,-3,-5,-4,2]=>2
[-1,-3,-5,-4,-2]=>1
[1,4,2,3,5]=>0
[1,4,2,3,-5]=>1
[1,4,2,-3,5]=>1
[1,4,2,-3,-5]=>2
[1,4,-2,3,5]=>1
[1,4,-2,3,-5]=>2
[1,4,-2,-3,5]=>0
[1,4,-2,-3,-5]=>1
[1,-4,2,3,5]=>1
[1,-4,2,3,-5]=>2
[1,-4,2,-3,5]=>0
[1,-4,2,-3,-5]=>1
[1,-4,-2,3,5]=>0
[1,-4,-2,3,-5]=>1
[1,-4,-2,-3,5]=>1
[1,-4,-2,-3,-5]=>2
[-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]=>1
[-1,4,-2,-3,-5]=>2
[-1,-4,2,3,5]=>2
[-1,-4,2,3,-5]=>3
[-1,-4,2,-3,5]=>1
[-1,-4,2,-3,-5]=>2
[-1,-4,-2,3,5]=>1
[-1,-4,-2,3,-5]=>2
[-1,-4,-2,-3,5]=>2
[-1,-4,-2,-3,-5]=>3
[1,4,2,5,3]=>0
[1,4,2,5,-3]=>1
[1,4,2,-5,3]=>1
[1,4,2,-5,-3]=>0
[1,4,-2,5,3]=>1
[1,4,-2,5,-3]=>0
[1,4,-2,-5,3]=>0
[1,4,-2,-5,-3]=>1
[1,-4,2,5,3]=>1
[1,-4,2,5,-3]=>0
[1,-4,2,-5,3]=>0
[1,-4,2,-5,-3]=>1
[1,-4,-2,5,3]=>0
[1,-4,-2,5,-3]=>1
[1,-4,-2,-5,3]=>1
[1,-4,-2,-5,-3]=>0
[-1,4,2,5,3]=>1
[-1,4,2,5,-3]=>2
[-1,4,2,-5,3]=>2
[-1,4,2,-5,-3]=>1
[-1,4,-2,5,3]=>2
[-1,4,-2,5,-3]=>1
[-1,4,-2,-5,3]=>1
[-1,4,-2,-5,-3]=>2
[-1,-4,2,5,3]=>2
[-1,-4,2,5,-3]=>1
[-1,-4,2,-5,3]=>1
[-1,-4,2,-5,-3]=>2
[-1,-4,-2,5,3]=>1
[-1,-4,-2,5,-3]=>2
[-1,-4,-2,-5,3]=>2
[-1,-4,-2,-5,-3]=>1
[1,4,3,2,5]=>0
[1,4,3,2,-5]=>1
[1,4,3,-2,5]=>1
[1,4,3,-2,-5]=>2
[1,4,-3,2,5]=>1
[1,4,-3,2,-5]=>2
[1,4,-3,-2,5]=>0
[1,4,-3,-2,-5]=>1
[1,-4,3,2,5]=>1
[1,-4,3,2,-5]=>2
[1,-4,3,-2,5]=>0
[1,-4,3,-2,-5]=>1
[1,-4,-3,2,5]=>0
[1,-4,-3,2,-5]=>1
[1,-4,-3,-2,5]=>1
[1,-4,-3,-2,-5]=>2
[-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]=>1
[-1,4,-3,-2,-5]=>2
[-1,-4,3,2,5]=>2
[-1,-4,3,2,-5]=>3
[-1,-4,3,-2,5]=>1
[-1,-4,3,-2,-5]=>2
[-1,-4,-3,2,5]=>1
[-1,-4,-3,2,-5]=>2
[-1,-4,-3,-2,5]=>2
[-1,-4,-3,-2,-5]=>3
[1,4,3,5,2]=>0
[1,4,3,5,-2]=>1
[1,4,3,-5,2]=>1
[1,4,3,-5,-2]=>0
[1,4,-3,5,2]=>1
[1,4,-3,5,-2]=>0
[1,4,-3,-5,2]=>0
[1,4,-3,-5,-2]=>1
[1,-4,3,5,2]=>1
[1,-4,3,5,-2]=>0
[1,-4,3,-5,2]=>0
[1,-4,3,-5,-2]=>1
[1,-4,-3,5,2]=>0
[1,-4,-3,5,-2]=>1
[1,-4,-3,-5,2]=>1
[1,-4,-3,-5,-2]=>0
[-1,4,3,5,2]=>1
[-1,4,3,5,-2]=>2
[-1,4,3,-5,2]=>2
[-1,4,3,-5,-2]=>1
[-1,4,-3,5,2]=>2
[-1,4,-3,5,-2]=>1
[-1,4,-3,-5,2]=>1
[-1,4,-3,-5,-2]=>2
[-1,-4,3,5,2]=>2
[-1,-4,3,5,-2]=>1
[-1,-4,3,-5,2]=>1
[-1,-4,3,-5,-2]=>2
[-1,-4,-3,5,2]=>1
[-1,-4,-3,5,-2]=>2
[-1,-4,-3,-5,2]=>2
[-1,-4,-3,-5,-2]=>1
[1,4,5,2,3]=>0
[1,4,5,2,-3]=>1
[1,4,5,-2,3]=>1
[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]=>1
[1,-4,5,2,3]=>1
[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]=>1
[1,-4,-5,-2,3]=>1
[1,-4,-5,-2,-3]=>0
[-1,4,5,2,3]=>1
[-1,4,5,2,-3]=>2
[-1,4,5,-2,3]=>2
[-1,4,5,-2,-3]=>1
[-1,4,-5,2,3]=>2
[-1,4,-5,2,-3]=>1
[-1,4,-5,-2,3]=>1
[-1,4,-5,-2,-3]=>2
[-1,-4,5,2,3]=>2
[-1,-4,5,2,-3]=>1
[-1,-4,5,-2,3]=>1
[-1,-4,5,-2,-3]=>2
[-1,-4,-5,2,3]=>1
[-1,-4,-5,2,-3]=>2
[-1,-4,-5,-2,3]=>2
[-1,-4,-5,-2,-3]=>1
[1,4,5,3,2]=>0
[1,4,5,3,-2]=>1
[1,4,5,-3,2]=>1
[1,4,5,-3,-2]=>0
[1,4,-5,3,2]=>1
[1,4,-5,3,-2]=>0
[1,4,-5,-3,2]=>0
[1,4,-5,-3,-2]=>1
[1,-4,5,3,2]=>1
[1,-4,5,3,-2]=>0
[1,-4,5,-3,2]=>0
[1,-4,5,-3,-2]=>1
[1,-4,-5,3,2]=>0
[1,-4,-5,3,-2]=>1
[1,-4,-5,-3,2]=>1
[1,-4,-5,-3,-2]=>0
[-1,4,5,3,2]=>1
[-1,4,5,3,-2]=>2
[-1,4,5,-3,2]=>2
[-1,4,5,-3,-2]=>1
[-1,4,-5,3,2]=>2
[-1,4,-5,3,-2]=>1
[-1,4,-5,-3,2]=>1
[-1,4,-5,-3,-2]=>2
[-1,-4,5,3,2]=>2
[-1,-4,5,3,-2]=>1
[-1,-4,5,-3,2]=>1
[-1,-4,5,-3,-2]=>2
[-1,-4,-5,3,2]=>1
[-1,-4,-5,3,-2]=>2
[-1,-4,-5,-3,2]=>2
[-1,-4,-5,-3,-2]=>1
[1,5,2,3,4]=>0
[1,5,2,3,-4]=>1
[1,5,2,-3,4]=>1
[1,5,2,-3,-4]=>0
[1,5,-2,3,4]=>1
[1,5,-2,3,-4]=>0
[1,5,-2,-3,4]=>0
[1,5,-2,-3,-4]=>1
[1,-5,2,3,4]=>1
[1,-5,2,3,-4]=>0
[1,-5,2,-3,4]=>0
[1,-5,2,-3,-4]=>1
[1,-5,-2,3,4]=>0
[1,-5,-2,3,-4]=>1
[1,-5,-2,-3,4]=>1
[1,-5,-2,-3,-4]=>0
[-1,5,2,3,4]=>1
[-1,5,2,3,-4]=>2
[-1,5,2,-3,4]=>2
[-1,5,2,-3,-4]=>1
[-1,5,-2,3,4]=>2
[-1,5,-2,3,-4]=>1
[-1,5,-2,-3,4]=>1
[-1,5,-2,-3,-4]=>2
[-1,-5,2,3,4]=>2
[-1,-5,2,3,-4]=>1
[-1,-5,2,-3,4]=>1
[-1,-5,2,-3,-4]=>2
[-1,-5,-2,3,4]=>1
[-1,-5,-2,3,-4]=>2
[-1,-5,-2,-3,4]=>2
[-1,-5,-2,-3,-4]=>1
[1,5,2,4,3]=>0
[1,5,2,4,-3]=>1
[1,5,2,-4,3]=>1
[1,5,2,-4,-3]=>0
[1,5,-2,4,3]=>1
[1,5,-2,4,-3]=>0
[1,5,-2,-4,3]=>0
[1,5,-2,-4,-3]=>1
[1,-5,2,4,3]=>1
[1,-5,2,4,-3]=>0
[1,-5,2,-4,3]=>0
[1,-5,2,-4,-3]=>1
[1,-5,-2,4,3]=>0
[1,-5,-2,4,-3]=>1
[1,-5,-2,-4,3]=>1
[1,-5,-2,-4,-3]=>0
[-1,5,2,4,3]=>1
[-1,5,2,4,-3]=>2
[-1,5,2,-4,3]=>2
[-1,5,2,-4,-3]=>1
[-1,5,-2,4,3]=>2
[-1,5,-2,4,-3]=>1
[-1,5,-2,-4,3]=>1
[-1,5,-2,-4,-3]=>2
[-1,-5,2,4,3]=>2
[-1,-5,2,4,-3]=>1
[-1,-5,2,-4,3]=>1
[-1,-5,2,-4,-3]=>2
[-1,-5,-2,4,3]=>1
[-1,-5,-2,4,-3]=>2
[-1,-5,-2,-4,3]=>2
[-1,-5,-2,-4,-3]=>1
[1,5,3,2,4]=>0
[1,5,3,2,-4]=>1
[1,5,3,-2,4]=>1
[1,5,3,-2,-4]=>0
[1,5,-3,2,4]=>1
[1,5,-3,2,-4]=>0
[1,5,-3,-2,4]=>0
[1,5,-3,-2,-4]=>1
[1,-5,3,2,4]=>1
[1,-5,3,2,-4]=>0
[1,-5,3,-2,4]=>0
[1,-5,3,-2,-4]=>1
[1,-5,-3,2,4]=>0
[1,-5,-3,2,-4]=>1
[1,-5,-3,-2,4]=>1
[1,-5,-3,-2,-4]=>0
[-1,5,3,2,4]=>1
[-1,5,3,2,-4]=>2
[-1,5,3,-2,4]=>2
[-1,5,3,-2,-4]=>1
[-1,5,-3,2,4]=>2
[-1,5,-3,2,-4]=>1
[-1,5,-3,-2,4]=>1
[-1,5,-3,-2,-4]=>2
[-1,-5,3,2,4]=>2
[-1,-5,3,2,-4]=>1
[-1,-5,3,-2,4]=>1
[-1,-5,3,-2,-4]=>2
[-1,-5,-3,2,4]=>1
[-1,-5,-3,2,-4]=>2
[-1,-5,-3,-2,4]=>2
[-1,-5,-3,-2,-4]=>1
[1,5,3,4,2]=>0
[1,5,3,4,-2]=>1
[1,5,3,-4,2]=>1
[1,5,3,-4,-2]=>0
[1,5,-3,4,2]=>1
[1,5,-3,4,-2]=>0
[1,5,-3,-4,2]=>0
[1,5,-3,-4,-2]=>1
[1,-5,3,4,2]=>1
[1,-5,3,4,-2]=>0
[1,-5,3,-4,2]=>0
[1,-5,3,-4,-2]=>1
[1,-5,-3,4,2]=>0
[1,-5,-3,4,-2]=>1
[1,-5,-3,-4,2]=>1
[1,-5,-3,-4,-2]=>0
[-1,5,3,4,2]=>1
[-1,5,3,4,-2]=>2
[-1,5,3,-4,2]=>2
[-1,5,3,-4,-2]=>1
[-1,5,-3,4,2]=>2
[-1,5,-3,4,-2]=>1
[-1,5,-3,-4,2]=>1
[-1,5,-3,-4,-2]=>2
[-1,-5,3,4,2]=>2
[-1,-5,3,4,-2]=>1
[-1,-5,3,-4,2]=>1
[-1,-5,3,-4,-2]=>2
[-1,-5,-3,4,2]=>1
[-1,-5,-3,4,-2]=>2
[-1,-5,-3,-4,2]=>2
[-1,-5,-3,-4,-2]=>1
[1,5,4,2,3]=>0
[1,5,4,2,-3]=>1
[1,5,4,-2,3]=>1
[1,5,4,-2,-3]=>0
[1,5,-4,2,3]=>1
[1,5,-4,2,-3]=>0
[1,5,-4,-2,3]=>0
[1,5,-4,-2,-3]=>1
[1,-5,4,2,3]=>1
[1,-5,4,2,-3]=>0
[1,-5,4,-2,3]=>0
[1,-5,4,-2,-3]=>1
[1,-5,-4,2,3]=>0
[1,-5,-4,2,-3]=>1
[1,-5,-4,-2,3]=>1
[1,-5,-4,-2,-3]=>0
[-1,5,4,2,3]=>1
[-1,5,4,2,-3]=>2
[-1,5,4,-2,3]=>2
[-1,5,4,-2,-3]=>1
[-1,5,-4,2,3]=>2
[-1,5,-4,2,-3]=>1
[-1,5,-4,-2,3]=>1
[-1,5,-4,-2,-3]=>2
[-1,-5,4,2,3]=>2
[-1,-5,4,2,-3]=>1
[-1,-5,4,-2,3]=>1
[-1,-5,4,-2,-3]=>2
[-1,-5,-4,2,3]=>1
[-1,-5,-4,2,-3]=>2
[-1,-5,-4,-2,3]=>2
[-1,-5,-4,-2,-3]=>1
[1,5,4,3,2]=>0
[1,5,4,3,-2]=>1
[1,5,4,-3,2]=>1
[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]=>1
[1,-5,4,3,2]=>1
[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]=>1
[1,-5,-4,-3,2]=>1
[1,-5,-4,-3,-2]=>0
[-1,5,4,3,2]=>1
[-1,5,4,3,-2]=>2
[-1,5,4,-3,2]=>2
[-1,5,4,-3,-2]=>1
[-1,5,-4,3,2]=>2
[-1,5,-4,3,-2]=>1
                    

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The oddness of a signed permutation.
The direct sum of two signed permutations $\sigma\in\mathfrak H_k$ and $\tau\in\mathfrak H_m$ is the signed permutation in $\mathfrak H_{k+m}$ obtained by concatenating $\sigma$ with the result of increasing the absolute value of every entry in $\tau$ by $k$.
This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.

References

[1] Bagno, E., Biagioli, R., Novick, M., Woo, A. Depth in classical Coexter groups arXiv:1507.01180

Code

def statistic(sigma):
    return sum(1 for b in decomposition(sigma)
               if is_odd(b.colors().count(-1)))

def decomposition(sigma):
    """
    sage: sigma = SignedPermutation([2,-4,3,-1,7,-5,6])
    sage: decomposition(sigma)
    [[2, -4, 3, -1], [3, -1, 2]]

    sage: sigma = SignedPermutation([2,-4,3,-6,7,-5,1])
    sage: decomposition(sigma)
    [[2, -4, 3, -6, 7, -5, 1]]
    """
    pi = sigma.permutation()
    summands = []
    j = 0
    for i in range(1, len(pi)):
        if max(pi[:i]) <= i:
            summand = SignedPermutation([e-j if e > 0 else e + j for e in sigma[j:i]])
            summands.append(summand)
            j = i
    summand = SignedPermutation([e-j if e > 0 else e + j for e in sigma[j:]])
    summands.append(summand)
    return summands

Created

Mar 31, 2023 at 21:09 by Martin Rubey

Updated

Mar 31, 2023 at 21:09 by Martin Rubey