- FindStat

Identifier

St001731: Permutations ⟶ ℤ

Values

=>
                            
                            
                            
                            

                        [1,2]=>0
[2,1]=>0
[1,2,3]=>0
[1,3,2]=>0
[2,1,3]=>0
[2,3,1]=>1
[3,1,2]=>1
[3,2,1]=>0
[1,2,3,4]=>0
[1,2,4,3]=>0
[1,3,2,4]=>0
[1,3,4,2]=>1
[1,4,2,3]=>1
[1,4,3,2]=>0
[2,1,3,4]=>0
[2,1,4,3]=>1
[2,3,1,4]=>1
[2,3,4,1]=>6
[2,4,1,3]=>6
[2,4,3,1]=>1
[3,1,2,4]=>1
[3,1,4,2]=>6
[3,2,1,4]=>0
[3,2,4,1]=>1
[3,4,1,2]=>1
[3,4,2,1]=>6
[4,1,2,3]=>6
[4,1,3,2]=>1
[4,2,1,3]=>1
[4,2,3,1]=>0
[4,3,1,2]=>6
[4,3,2,1]=>1
[1,2,3,4,5]=>0
[1,2,3,5,4]=>0
[1,2,4,3,5]=>0
[1,2,4,5,3]=>1
[1,2,5,3,4]=>1
[1,2,5,4,3]=>0
[1,3,2,4,5]=>0
[1,3,2,5,4]=>1
[1,3,4,2,5]=>1
[1,3,4,5,2]=>6
[1,3,5,2,4]=>6
[1,3,5,4,2]=>1
[1,4,2,3,5]=>1
[1,4,2,5,3]=>6
[1,4,3,2,5]=>0
[1,4,3,5,2]=>1
[1,4,5,2,3]=>1
[1,4,5,3,2]=>6
[1,5,2,3,4]=>6
[1,5,2,4,3]=>1
[1,5,3,2,4]=>1
[1,5,3,4,2]=>0
[1,5,4,2,3]=>6
[1,5,4,3,2]=>1
[2,1,3,4,5]=>0
[2,1,3,5,4]=>1
[2,1,4,3,5]=>1
[2,1,4,5,3]=>4
[2,1,5,3,4]=>4
[2,1,5,4,3]=>1
[2,3,1,4,5]=>1
[2,3,1,5,4]=>4
[2,3,4,1,5]=>6
[2,3,4,5,1]=>20
[2,3,5,1,4]=>20
[2,3,5,4,1]=>6
[2,4,1,3,5]=>6
[2,4,1,5,3]=>20
[2,4,3,1,5]=>1
[2,4,3,5,1]=>6
[2,4,5,1,3]=>4
[2,4,5,3,1]=>20
[2,5,1,3,4]=>20
[2,5,1,4,3]=>6
[2,5,3,1,4]=>6
[2,5,3,4,1]=>1
[2,5,4,1,3]=>20
[2,5,4,3,1]=>4
[3,1,2,4,5]=>1
[3,1,2,5,4]=>4
[3,1,4,2,5]=>6
[3,1,4,5,2]=>20
[3,1,5,2,4]=>20
[3,1,5,4,2]=>6
[3,2,1,4,5]=>0
[3,2,1,5,4]=>1
[3,2,4,1,5]=>1
[3,2,4,5,1]=>6
[3,2,5,1,4]=>6
[3,2,5,4,1]=>1
[3,4,1,2,5]=>1
[3,4,1,5,2]=>4
[3,4,2,1,5]=>6
[3,4,2,5,1]=>20
[3,4,5,1,2]=>20
[3,4,5,2,1]=>4
[3,5,1,2,4]=>4
[3,5,1,4,2]=>1
[3,5,2,1,4]=>20
[3,5,2,4,1]=>6
[3,5,4,1,2]=>4
[3,5,4,2,1]=>20
[4,1,2,3,5]=>6
[4,1,2,5,3]=>20
[4,1,3,2,5]=>1
[4,1,3,5,2]=>6
[4,1,5,2,3]=>4
[4,1,5,3,2]=>20
[4,2,1,3,5]=>1
[4,2,1,5,3]=>6
[4,2,3,1,5]=>0
[4,2,3,5,1]=>1
[4,2,5,1,3]=>1
[4,2,5,3,1]=>6
[4,3,1,2,5]=>6
[4,3,1,5,2]=>20
[4,3,2,1,5]=>1
[4,3,2,5,1]=>4
[4,3,5,1,2]=>4
[4,3,5,2,1]=>20
[4,5,1,2,3]=>20
[4,5,1,3,2]=>4
[4,5,2,1,3]=>4
[4,5,2,3,1]=>20
[4,5,3,1,2]=>1
[4,5,3,2,1]=>6
[5,1,2,3,4]=>20
[5,1,2,4,3]=>6
[5,1,3,2,4]=>6
[5,1,3,4,2]=>1
[5,1,4,2,3]=>20
[5,1,4,3,2]=>4
[5,2,1,3,4]=>6
[5,2,1,4,3]=>1
[5,2,3,1,4]=>1
[5,2,3,4,1]=>0
[5,2,4,1,3]=>6
[5,2,4,3,1]=>1
[5,3,1,2,4]=>20
[5,3,1,4,2]=>6
[5,3,2,1,4]=>4
[5,3,2,4,1]=>1
[5,3,4,1,2]=>20
[5,3,4,2,1]=>4
[5,4,1,2,3]=>4
[5,4,1,3,2]=>20
[5,4,2,1,3]=>20
[5,4,2,3,1]=>4
[5,4,3,1,2]=>6
[5,4,3,2,1]=>1
[1,2,3,4,5,6]=>0
[1,2,3,4,6,5]=>0
[1,2,3,5,4,6]=>0
[1,2,3,5,6,4]=>1
[1,2,3,6,4,5]=>1
[1,2,3,6,5,4]=>0
[1,2,4,3,5,6]=>0
[1,2,4,3,6,5]=>1
[1,2,4,5,3,6]=>1
[1,2,4,5,6,3]=>6
[1,2,4,6,3,5]=>6
[1,2,4,6,5,3]=>1
[1,2,5,3,4,6]=>1
[1,2,5,3,6,4]=>6
[1,2,5,4,3,6]=>0
[1,2,5,4,6,3]=>1
[1,2,5,6,3,4]=>1
[1,2,5,6,4,3]=>6
[1,2,6,3,4,5]=>6
[1,2,6,3,5,4]=>1
[1,2,6,4,3,5]=>1
[1,2,6,4,5,3]=>0
[1,2,6,5,3,4]=>6
[1,2,6,5,4,3]=>1
[1,3,2,4,5,6]=>0
[1,3,2,4,6,5]=>1
[1,3,2,5,4,6]=>1
[1,3,2,5,6,4]=>4
[1,3,2,6,4,5]=>4
[1,3,2,6,5,4]=>1
[1,3,4,2,5,6]=>1
[1,3,4,2,6,5]=>4
[1,3,4,5,2,6]=>6
[1,3,4,5,6,2]=>20
[1,3,4,6,2,5]=>20
[1,3,4,6,5,2]=>6
[1,3,5,2,4,6]=>6
[1,3,5,2,6,4]=>20
[1,3,5,4,2,6]=>1
[1,3,5,4,6,2]=>6
[1,3,5,6,2,4]=>4
[1,3,5,6,4,2]=>20
[1,3,6,2,4,5]=>20
[1,3,6,2,5,4]=>6
[1,3,6,4,2,5]=>6
[1,3,6,4,5,2]=>1
[1,3,6,5,2,4]=>20
[1,3,6,5,4,2]=>4
[1,4,2,3,5,6]=>1
[1,4,2,3,6,5]=>4
[1,4,2,5,3,6]=>6
[1,4,2,5,6,3]=>20
[1,4,2,6,3,5]=>20
[1,4,2,6,5,3]=>6
[1,4,3,2,5,6]=>0
[1,4,3,2,6,5]=>1
[1,4,3,5,2,6]=>1
[1,4,3,5,6,2]=>6
[1,4,3,6,2,5]=>6
[1,4,3,6,5,2]=>1
[1,4,5,2,3,6]=>1
[1,4,5,2,6,3]=>4
[1,4,5,3,2,6]=>6
[1,4,5,3,6,2]=>20
[1,4,5,6,2,3]=>20
[1,4,5,6,3,2]=>4
[1,4,6,2,3,5]=>4
[1,4,6,2,5,3]=>1
[1,4,6,3,2,5]=>20
[1,4,6,3,5,2]=>6
[1,4,6,5,2,3]=>4
[1,4,6,5,3,2]=>20
[1,5,2,3,4,6]=>6
[1,5,2,3,6,4]=>20
[1,5,2,4,3,6]=>1
[1,5,2,4,6,3]=>6
[1,5,2,6,3,4]=>4
[1,5,2,6,4,3]=>20
[1,5,3,2,4,6]=>1
[1,5,3,2,6,4]=>6
[1,5,3,4,2,6]=>0
[1,5,3,4,6,2]=>1
[1,5,3,6,2,4]=>1
[1,5,3,6,4,2]=>6
[1,5,4,2,3,6]=>6
[1,5,4,2,6,3]=>20
[1,5,4,3,2,6]=>1
[1,5,4,3,6,2]=>4
[1,5,4,6,2,3]=>4
[1,5,4,6,3,2]=>20
[1,5,6,2,3,4]=>20
[1,5,6,2,4,3]=>4
[1,5,6,3,2,4]=>4
[1,5,6,3,4,2]=>20
[1,5,6,4,2,3]=>1
[1,5,6,4,3,2]=>6
[1,6,2,3,4,5]=>20
[1,6,2,3,5,4]=>6
[1,6,2,4,3,5]=>6
[1,6,2,4,5,3]=>1
[1,6,2,5,3,4]=>20
[1,6,2,5,4,3]=>4
[1,6,3,2,4,5]=>6
[1,6,3,2,5,4]=>1
[1,6,3,4,2,5]=>1
[1,6,3,4,5,2]=>0
[1,6,3,5,2,4]=>6
[1,6,3,5,4,2]=>1
[1,6,4,2,3,5]=>20
[1,6,4,2,5,3]=>6
[1,6,4,3,2,5]=>4
[1,6,4,3,5,2]=>1
[1,6,4,5,2,3]=>20
[1,6,4,5,3,2]=>4
[1,6,5,2,3,4]=>4
[1,6,5,2,4,3]=>20
[1,6,5,3,2,4]=>20
[1,6,5,3,4,2]=>4
[1,6,5,4,2,3]=>6
[1,6,5,4,3,2]=>1
[2,1,3,4,5,6]=>0
[2,1,3,4,6,5]=>1
[2,1,3,5,4,6]=>1
[2,1,3,5,6,4]=>4
[2,1,3,6,4,5]=>4
[2,1,3,6,5,4]=>1
[2,1,4,3,5,6]=>1
[2,1,4,3,6,5]=>3
[2,1,4,5,3,6]=>4
[2,1,4,5,6,3]=>12
[2,1,4,6,3,5]=>12
[2,1,4,6,5,3]=>4
[2,1,5,3,4,6]=>4
[2,1,5,3,6,4]=>12
[2,1,5,4,3,6]=>1
[2,1,5,4,6,3]=>4
[2,1,5,6,3,4]=>3
[2,1,5,6,4,3]=>12
[2,1,6,3,4,5]=>12
[2,1,6,3,5,4]=>4
[2,1,6,4,3,5]=>4
[2,1,6,4,5,3]=>1
[2,1,6,5,3,4]=>12
[2,1,6,5,4,3]=>3
[2,3,1,4,5,6]=>1
[2,3,1,4,6,5]=>4
[2,3,1,5,4,6]=>4
[2,3,1,5,6,4]=>11
[2,3,1,6,4,5]=>11
[2,3,1,6,5,4]=>4
[2,3,4,1,5,6]=>6
[2,3,4,1,6,5]=>12
[2,3,4,5,1,6]=>20
[2,3,4,5,6,1]=>50
[2,3,4,6,1,5]=>50
[2,3,4,6,5,1]=>20
[2,3,5,1,4,6]=>20
[2,3,5,1,6,4]=>50
[2,3,5,4,1,6]=>6
[2,3,5,4,6,1]=>20
[2,3,5,6,1,4]=>12
[2,3,5,6,4,1]=>50
[2,3,6,1,4,5]=>50
[2,3,6,1,5,4]=>20
[2,3,6,4,1,5]=>20
[2,3,6,4,5,1]=>6
[2,3,6,5,1,4]=>50
[2,3,6,5,4,1]=>12
[2,4,1,3,5,6]=>6
[2,4,1,3,6,5]=>12
[2,4,1,5,3,6]=>20
[2,4,1,5,6,3]=>50
[2,4,1,6,3,5]=>50
[2,4,1,6,5,3]=>20
[2,4,3,1,5,6]=>1
[2,4,3,1,6,5]=>4
[2,4,3,5,1,6]=>6
[2,4,3,5,6,1]=>20
[2,4,3,6,1,5]=>20
[2,4,3,6,5,1]=>6
[2,4,5,1,3,6]=>4
[2,4,5,1,6,3]=>11
[2,4,5,3,1,6]=>20
[2,4,5,3,6,1]=>50
[2,4,5,6,1,3]=>50
[2,4,5,6,3,1]=>12
[2,4,6,1,3,5]=>11
[2,4,6,1,5,3]=>4
[2,4,6,3,1,5]=>50
[2,4,6,3,5,1]=>20
[2,4,6,5,1,3]=>12
[2,4,6,5,3,1]=>50
[2,5,1,3,4,6]=>20
[2,5,1,3,6,4]=>50
[2,5,1,4,3,6]=>6
[2,5,1,4,6,3]=>20
[2,5,1,6,3,4]=>12
[2,5,1,6,4,3]=>50
[2,5,3,1,4,6]=>6
[2,5,3,1,6,4]=>20
[2,5,3,4,1,6]=>1
[2,5,3,4,6,1]=>6
[2,5,3,6,1,4]=>4
[2,5,3,6,4,1]=>20
[2,5,4,1,3,6]=>20
[2,5,4,1,6,3]=>50
[2,5,4,3,1,6]=>4
[2,5,4,3,6,1]=>12
[2,5,4,6,1,3]=>11
[2,5,4,6,3,1]=>50
[2,5,6,1,3,4]=>50
[2,5,6,1,4,3]=>12
[2,5,6,3,1,4]=>11
[2,5,6,3,4,1]=>50
[2,5,6,4,1,3]=>4
[2,5,6,4,3,1]=>20
[2,6,1,3,4,5]=>50
[2,6,1,3,5,4]=>20
[2,6,1,4,3,5]=>20
[2,6,1,4,5,3]=>6
[2,6,1,5,3,4]=>50
[2,6,1,5,4,3]=>12
[2,6,3,1,4,5]=>20
[2,6,3,1,5,4]=>6
[2,6,3,4,1,5]=>6
[2,6,3,4,5,1]=>1
[2,6,3,5,1,4]=>20
[2,6,3,5,4,1]=>4
[2,6,4,1,3,5]=>50
[2,6,4,1,5,3]=>20
[2,6,4,3,1,5]=>12
[2,6,4,3,5,1]=>4
[2,6,4,5,1,3]=>50
[2,6,4,5,3,1]=>11
[2,6,5,1,3,4]=>12
[2,6,5,1,4,3]=>50
[2,6,5,3,1,4]=>50
[2,6,5,3,4,1]=>11
[2,6,5,4,1,3]=>20
[2,6,5,4,3,1]=>4
[3,1,2,4,5,6]=>1
[3,1,2,4,6,5]=>4
[3,1,2,5,4,6]=>4
[3,1,2,5,6,4]=>11
[3,1,2,6,4,5]=>11
[3,1,2,6,5,4]=>4
[3,1,4,2,5,6]=>6
[3,1,4,2,6,5]=>12
[3,1,4,5,2,6]=>20
[3,1,4,5,6,2]=>50
[3,1,4,6,2,5]=>50
[3,1,4,6,5,2]=>20
[3,1,5,2,4,6]=>20
[3,1,5,2,6,4]=>50
[3,1,5,4,2,6]=>6
[3,1,5,4,6,2]=>20
[3,1,5,6,2,4]=>12
[3,1,5,6,4,2]=>50
[3,1,6,2,4,5]=>50
[3,1,6,2,5,4]=>20
[3,1,6,4,2,5]=>20
[3,1,6,4,5,2]=>6
[3,1,6,5,2,4]=>50
[3,1,6,5,4,2]=>12
[3,2,1,4,5,6]=>0
[3,2,1,4,6,5]=>1
[3,2,1,5,4,6]=>1
[3,2,1,5,6,4]=>4
[3,2,1,6,4,5]=>4
[3,2,1,6,5,4]=>1
[3,2,4,1,5,6]=>1
[3,2,4,1,6,5]=>4
[3,2,4,5,1,6]=>6
[3,2,4,5,6,1]=>20
[3,2,4,6,1,5]=>20
[3,2,4,6,5,1]=>6
[3,2,5,1,4,6]=>6
[3,2,5,1,6,4]=>20
[3,2,5,4,1,6]=>1
[3,2,5,4,6,1]=>6
[3,2,5,6,1,4]=>4
[3,2,5,6,4,1]=>20
[3,2,6,1,4,5]=>20
[3,2,6,1,5,4]=>6
[3,2,6,4,1,5]=>6
[3,2,6,4,5,1]=>1
[3,2,6,5,1,4]=>20
[3,2,6,5,4,1]=>4
[3,4,1,2,5,6]=>1
[3,4,1,2,6,5]=>3
[3,4,1,5,2,6]=>4
[3,4,1,5,6,2]=>12
[3,4,1,6,2,5]=>12
[3,4,1,6,5,2]=>4
[3,4,2,1,5,6]=>6
[3,4,2,1,6,5]=>12
[3,4,2,5,1,6]=>20
[3,4,2,5,6,1]=>50
[3,4,2,6,1,5]=>50
[3,4,2,6,5,1]=>20
[3,4,5,1,2,6]=>20
[3,4,5,1,6,2]=>50
[3,4,5,2,1,6]=>4
[3,4,5,2,6,1]=>12
[3,4,5,6,1,2]=>11
[3,4,5,6,2,1]=>50
[3,4,6,1,2,5]=>50
[3,4,6,1,5,2]=>20
[3,4,6,2,1,5]=>12
[3,4,6,2,5,1]=>4
[3,4,6,5,1,2]=>50
[3,4,6,5,2,1]=>11
[3,5,1,2,4,6]=>4
[3,5,1,2,6,4]=>12
[3,5,1,4,2,6]=>1
[3,5,1,4,6,2]=>4
[3,5,1,6,2,4]=>3
[3,5,1,6,4,2]=>12
[3,5,2,1,4,6]=>20
[3,5,2,1,6,4]=>50
[3,5,2,4,1,6]=>6
[3,5,2,4,6,1]=>20
[3,5,2,6,1,4]=>12
[3,5,2,6,4,1]=>50
[3,5,4,1,2,6]=>4
[3,5,4,1,6,2]=>11
[3,5,4,2,1,6]=>20
[3,5,4,2,6,1]=>50
[3,5,4,6,1,2]=>50
[3,5,4,6,2,1]=>12
[3,5,6,1,2,4]=>12
[3,5,6,1,4,2]=>50
[3,5,6,2,1,4]=>50
[3,5,6,2,4,1]=>11
[3,5,6,4,1,2]=>20
[3,5,6,4,2,1]=>4
[3,6,1,2,4,5]=>12
[3,6,1,2,5,4]=>4
[3,6,1,4,2,5]=>4
[3,6,1,4,5,2]=>1
[3,6,1,5,2,4]=>12
[3,6,1,5,4,2]=>3
[3,6,2,1,4,5]=>50
[3,6,2,1,5,4]=>20
[3,6,2,4,1,5]=>20
[3,6,2,4,5,1]=>6
[3,6,2,5,1,4]=>50
[3,6,2,5,4,1]=>12
[3,6,4,1,2,5]=>11
[3,6,4,1,5,2]=>4
[3,6,4,2,1,5]=>50
[3,6,4,2,5,1]=>20
[3,6,4,5,1,2]=>12
[3,6,4,5,2,1]=>50
[3,6,5,1,2,4]=>50
[3,6,5,1,4,2]=>12
[3,6,5,2,1,4]=>11
[3,6,5,2,4,1]=>50
[3,6,5,4,1,2]=>4
[3,6,5,4,2,1]=>20
[4,1,2,3,5,6]=>6
[4,1,2,3,6,5]=>12
[4,1,2,5,3,6]=>20
[4,1,2,5,6,3]=>50
[4,1,2,6,3,5]=>50
[4,1,2,6,5,3]=>20
[4,1,3,2,5,6]=>1
[4,1,3,2,6,5]=>4
[4,1,3,5,2,6]=>6
[4,1,3,5,6,2]=>20
[4,1,3,6,2,5]=>20
[4,1,3,6,5,2]=>6
[4,1,5,2,3,6]=>4
[4,1,5,2,6,3]=>11
[4,1,5,3,2,6]=>20
[4,1,5,3,6,2]=>50
[4,1,5,6,2,3]=>50
[4,1,5,6,3,2]=>12
[4,1,6,2,3,5]=>11
[4,1,6,2,5,3]=>4
[4,1,6,3,2,5]=>50
[4,1,6,3,5,2]=>20
[4,1,6,5,2,3]=>12
[4,1,6,5,3,2]=>50
[4,2,1,3,5,6]=>1
[4,2,1,3,6,5]=>4
[4,2,1,5,3,6]=>6
[4,2,1,5,6,3]=>20
[4,2,1,6,3,5]=>20
[4,2,1,6,5,3]=>6
[4,2,3,1,5,6]=>0
[4,2,3,1,6,5]=>1
[4,2,3,5,1,6]=>1
[4,2,3,5,6,1]=>6
[4,2,3,6,1,5]=>6
[4,2,3,6,5,1]=>1
[4,2,5,1,3,6]=>1
[4,2,5,1,6,3]=>4
[4,2,5,3,1,6]=>6
[4,2,5,3,6,1]=>20
[4,2,5,6,1,3]=>20
[4,2,5,6,3,1]=>4
[4,2,6,1,3,5]=>4
[4,2,6,1,5,3]=>1
[4,2,6,3,1,5]=>20
[4,2,6,3,5,1]=>6
[4,2,6,5,1,3]=>4
[4,2,6,5,3,1]=>20
[4,3,1,2,5,6]=>6
[4,3,1,2,6,5]=>12
[4,3,1,5,2,6]=>20
[4,3,1,5,6,2]=>50
[4,3,1,6,2,5]=>50
[4,3,1,6,5,2]=>20
[4,3,2,1,5,6]=>1
[4,3,2,1,6,5]=>3
[4,3,2,5,1,6]=>4
[4,3,2,5,6,1]=>12
[4,3,2,6,1,5]=>12
[4,3,2,6,5,1]=>4
[4,3,5,1,2,6]=>4
[4,3,5,1,6,2]=>12
[4,3,5,2,1,6]=>20
[4,3,5,2,6,1]=>50
[4,3,5,6,1,2]=>50
[4,3,5,6,2,1]=>11
[4,3,6,1,2,5]=>12
[4,3,6,1,5,2]=>4
[4,3,6,2,1,5]=>50
[4,3,6,2,5,1]=>20
[4,3,6,5,1,2]=>11
[4,3,6,5,2,1]=>50
[4,5,1,2,3,6]=>20
[4,5,1,2,6,3]=>50
[4,5,1,3,2,6]=>4
[4,5,1,3,6,2]=>11
[4,5,1,6,2,3]=>12
[4,5,1,6,3,2]=>50
[4,5,2,1,3,6]=>4
[4,5,2,1,6,3]=>12
[4,5,2,3,1,6]=>20
[4,5,2,3,6,1]=>50
[4,5,2,6,1,3]=>50
[4,5,2,6,3,1]=>11
[4,5,3,1,2,6]=>1
[4,5,3,1,6,2]=>4
[4,5,3,2,1,6]=>6
[4,5,3,2,6,1]=>20
[4,5,3,6,1,2]=>20
[4,5,3,6,2,1]=>4
[4,5,6,1,2,3]=>3
[4,5,6,1,3,2]=>12
[4,5,6,2,1,3]=>12
[4,5,6,2,3,1]=>50
[4,5,6,3,1,2]=>50
[4,5,6,3,2,1]=>12
[4,6,1,2,3,5]=>50
[4,6,1,2,5,3]=>20
[4,6,1,3,2,5]=>11
[4,6,1,3,5,2]=>4
[4,6,1,5,2,3]=>50
[4,6,1,5,3,2]=>12
[4,6,2,1,3,5]=>12
[4,6,2,1,5,3]=>4
[4,6,2,3,1,5]=>50
[4,6,2,3,5,1]=>20
[4,6,2,5,1,3]=>11
[4,6,2,5,3,1]=>50
[4,6,3,1,2,5]=>4
[4,6,3,1,5,2]=>1
[4,6,3,2,1,5]=>20
[4,6,3,2,5,1]=>6
[4,6,3,5,1,2]=>4
[4,6,3,5,2,1]=>20
[4,6,5,1,2,3]=>12
[4,6,5,1,3,2]=>3
[4,6,5,2,1,3]=>50
[4,6,5,2,3,1]=>12
[4,6,5,3,1,2]=>12
[4,6,5,3,2,1]=>50
[5,1,2,3,4,6]=>20
[5,1,2,3,6,4]=>50
[5,1,2,4,3,6]=>6
[5,1,2,4,6,3]=>20
[5,1,2,6,3,4]=>12
[5,1,2,6,4,3]=>50
[5,1,3,2,4,6]=>6
[5,1,3,2,6,4]=>20
[5,1,3,4,2,6]=>1
[5,1,3,4,6,2]=>6
[5,1,3,6,2,4]=>4
[5,1,3,6,4,2]=>20
[5,1,4,2,3,6]=>20
[5,1,4,2,6,3]=>50
[5,1,4,3,2,6]=>4
[5,1,4,3,6,2]=>12
[5,1,4,6,2,3]=>11
[5,1,4,6,3,2]=>50
[5,1,6,2,3,4]=>50
[5,1,6,2,4,3]=>12
[5,1,6,3,2,4]=>11
[5,1,6,3,4,2]=>50
[5,1,6,4,2,3]=>4
[5,1,6,4,3,2]=>20
[5,2,1,3,4,6]=>6
[5,2,1,3,6,4]=>20
[5,2,1,4,3,6]=>1
[5,2,1,4,6,3]=>6
[5,2,1,6,3,4]=>4
[5,2,1,6,4,3]=>20
[5,2,3,1,4,6]=>1
[5,2,3,1,6,4]=>6
[5,2,3,4,1,6]=>0
[5,2,3,4,6,1]=>1
[5,2,3,6,1,4]=>1
[5,2,3,6,4,1]=>6
[5,2,4,1,3,6]=>6
[5,2,4,1,6,3]=>20
[5,2,4,3,1,6]=>1
[5,2,4,3,6,1]=>4
[5,2,4,6,1,3]=>4
[5,2,4,6,3,1]=>20
[5,2,6,1,3,4]=>20
[5,2,6,1,4,3]=>4
[5,2,6,3,1,4]=>4
[5,2,6,3,4,1]=>20
[5,2,6,4,1,3]=>1
[5,2,6,4,3,1]=>6
[5,3,1,2,4,6]=>20
[5,3,1,2,6,4]=>50
[5,3,1,4,2,6]=>6
[5,3,1,4,6,2]=>20
[5,3,1,6,2,4]=>12
[5,3,1,6,4,2]=>50
[5,3,2,1,4,6]=>4
[5,3,2,1,6,4]=>12
[5,3,2,4,1,6]=>1
[5,3,2,4,6,1]=>4
[5,3,2,6,1,4]=>3
[5,3,2,6,4,1]=>12
[5,3,4,1,2,6]=>20
[5,3,4,1,6,2]=>50
[5,3,4,2,1,6]=>4
[5,3,4,2,6,1]=>11
[5,3,4,6,1,2]=>12
[5,3,4,6,2,1]=>50
[5,3,6,1,2,4]=>50
[5,3,6,1,4,2]=>11
[5,3,6,2,1,4]=>12
[5,3,6,2,4,1]=>50
[5,3,6,4,1,2]=>4
[5,3,6,4,2,1]=>20
[5,4,1,2,3,6]=>4
[5,4,1,2,6,3]=>12
[5,4,1,3,2,6]=>20
[5,4,1,3,6,2]=>50
[5,4,1,6,2,3]=>50
[5,4,1,6,3,2]=>11
[5,4,2,1,3,6]=>20
[5,4,2,1,6,3]=>50
[5,4,2,3,1,6]=>4
[5,4,2,3,6,1]=>11
[5,4,2,6,1,3]=>12
[5,4,2,6,3,1]=>50
[5,4,3,1,2,6]=>6
[5,4,3,1,6,2]=>20
[5,4,3,2,1,6]=>1
[5,4,3,2,6,1]=>4
[5,4,3,6,1,2]=>4
[5,4,3,6,2,1]=>20
[5,4,6,1,2,3]=>12
[5,4,6,1,3,2]=>50
[5,4,6,2,1,3]=>3
[5,4,6,2,3,1]=>12
[5,4,6,3,1,2]=>12
[5,4,6,3,2,1]=>50
[5,6,1,2,3,4]=>11
[5,6,1,2,4,3]=>50
[5,6,1,3,2,4]=>50
[5,6,1,3,4,2]=>12
[5,6,1,4,2,3]=>20
[5,6,1,4,3,2]=>4
[5,6,2,1,3,4]=>50
[5,6,2,1,4,3]=>11
[5,6,2,3,1,4]=>12
[5,6,2,3,4,1]=>50
[5,6,2,4,1,3]=>4
[5,6,2,4,3,1]=>20
[5,6,3,1,2,4]=>20
[5,6,3,1,4,2]=>4
[5,6,3,2,1,4]=>4
[5,6,3,2,4,1]=>20
[5,6,3,4,1,2]=>1
[5,6,3,4,2,1]=>6
[5,6,4,1,2,3]=>50
[5,6,4,1,3,2]=>12
[5,6,4,2,1,3]=>12
[5,6,4,2,3,1]=>50
[5,6,4,3,1,2]=>3
[5,6,4,3,2,1]=>12
[6,1,2,3,4,5]=>50
[6,1,2,3,5,4]=>20
[6,1,2,4,3,5]=>20
[6,1,2,4,5,3]=>6
[6,1,2,5,3,4]=>50
[6,1,2,5,4,3]=>12
[6,1,3,2,4,5]=>20
[6,1,3,2,5,4]=>6
[6,1,3,4,2,5]=>6
[6,1,3,4,5,2]=>1
[6,1,3,5,2,4]=>20
[6,1,3,5,4,2]=>4
[6,1,4,2,3,5]=>50
[6,1,4,2,5,3]=>20
[6,1,4,3,2,5]=>12
[6,1,4,3,5,2]=>4
[6,1,4,5,2,3]=>50
[6,1,4,5,3,2]=>11
[6,1,5,2,3,4]=>12
[6,1,5,2,4,3]=>50
[6,1,5,3,2,4]=>50
[6,1,5,3,4,2]=>11
[6,1,5,4,2,3]=>20
[6,1,5,4,3,2]=>4
[6,2,1,3,4,5]=>20
[6,2,1,3,5,4]=>6
[6,2,1,4,3,5]=>6
[6,2,1,4,5,3]=>1
[6,2,1,5,3,4]=>20
[6,2,1,5,4,3]=>4
[6,2,3,1,4,5]=>6
[6,2,3,1,5,4]=>1
[6,2,3,4,1,5]=>1
[6,2,3,4,5,1]=>0
[6,2,3,5,1,4]=>6
[6,2,3,5,4,1]=>1
[6,2,4,1,3,5]=>20
[6,2,4,1,5,3]=>6
[6,2,4,3,1,5]=>4
[6,2,4,3,5,1]=>1
[6,2,4,5,1,3]=>20
[6,2,4,5,3,1]=>4
[6,2,5,1,3,4]=>4
[6,2,5,1,4,3]=>20
[6,2,5,3,1,4]=>20
[6,2,5,3,4,1]=>4
[6,2,5,4,1,3]=>6
[6,2,5,4,3,1]=>1
[6,3,1,2,4,5]=>50
[6,3,1,2,5,4]=>20
[6,3,1,4,2,5]=>20
[6,3,1,4,5,2]=>6
[6,3,1,5,2,4]=>50
[6,3,1,5,4,2]=>12
[6,3,2,1,4,5]=>12
[6,3,2,1,5,4]=>4
[6,3,2,4,1,5]=>4
[6,3,2,4,5,1]=>1
[6,3,2,5,1,4]=>12
[6,3,2,5,4,1]=>3
[6,3,4,1,2,5]=>50
[6,3,4,1,5,2]=>20
[6,3,4,2,1,5]=>11
[6,3,4,2,5,1]=>4
[6,3,4,5,1,2]=>50
[6,3,4,5,2,1]=>12
[6,3,5,1,2,4]=>11
[6,3,5,1,4,2]=>50
[6,3,5,2,1,4]=>50
[6,3,5,2,4,1]=>12
[6,3,5,4,1,2]=>20
[6,3,5,4,2,1]=>4
[6,4,1,2,3,5]=>12
[6,4,1,2,5,3]=>4
[6,4,1,3,2,5]=>50
[6,4,1,3,5,2]=>20
[6,4,1,5,2,3]=>11
[6,4,1,5,3,2]=>50
[6,4,2,1,3,5]=>50
[6,4,2,1,5,3]=>20
[6,4,2,3,1,5]=>11
[6,4,2,3,5,1]=>4
[6,4,2,5,1,3]=>50
[6,4,2,5,3,1]=>12
[6,4,3,1,2,5]=>20
[6,4,3,1,5,2]=>6
[6,4,3,2,1,5]=>4
[6,4,3,2,5,1]=>1
[6,4,3,5,1,2]=>20
[6,4,3,5,2,1]=>4
[6,4,5,1,2,3]=>50
[6,4,5,1,3,2]=>12
[6,4,5,2,1,3]=>12
[6,4,5,2,3,1]=>3
[6,4,5,3,1,2]=>50
[6,4,5,3,2,1]=>12
[6,5,1,2,3,4]=>50
[6,5,1,2,4,3]=>11
[6,5,1,3,2,4]=>12
[6,5,1,3,4,2]=>50
[6,5,1,4,2,3]=>4
[6,5,1,4,3,2]=>20
[6,5,2,1,3,4]=>11
[6,5,2,1,4,3]=>50
[6,5,2,3,1,4]=>50
[6,5,2,3,4,1]=>12
[6,5,2,4,1,3]=>20
[6,5,2,4,3,1]=>4
[6,5,3,1,2,4]=>4
[6,5,3,1,4,2]=>20
[6,5,3,2,1,4]=>20
[6,5,3,2,4,1]=>4
[6,5,3,4,1,2]=>6
[6,5,3,4,2,1]=>1
[6,5,4,1,2,3]=>12
[6,5,4,1,3,2]=>50
[6,5,4,2,1,3]=>50
[6,5,4,2,3,1]=>12
[6,5,4,3,1,2]=>12
[6,5,4,3,2,1]=>3
                    

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Description

The factorization defect of a permutation.
The factorization poset of a permutation $\sigma$ is the principal order ideal generated by $\sigma$ in the absolute order. In particular, the maximal chains in the factorization poset of $\sigma$ are in bijection with the reduced factorizations of $\sigma$ into transpositions. The factorization defect of $\sigma$ is the number of rank-2 elements in its factorization poset.
This is the statistic "d" defined in [1, Section 6.1], where it was called the minimal size of a feedback arc set in the cycle graph. The fact that this equals the number of rank-2 elements in the factorization poset follows from [1, Proposition 6.3] together with the shellability of the factorization poset.

References

[1] Mühle, H., Ripoll, V. Connectivity Properties of Factorization Posets in Generated Groups DOI:10.1007/s11083-019-09496-1

Code

def narayana_number(n,k):
    return 1/n*binomial(n,k)*binomial(n,k-1)

def statistic(pi):
    c = tuple(pi.cycle_type())
    return sum([narayana_number(c[T[0]],2)*narayana_number(c[T[1]],2) for T in Subsets(range(len(c)),2)]) + sum([narayana_number(t,3) for t in c])

Created

Jul 07, 2021 at 10:23 by Henri Mühle

Updated

Nov 18, 2024 at 19:29 by Nupur Jain