Identifier
-
Mp00255:
Decorated permutations
—lower permutation⟶
Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000467: Graphs ⟶ ℤ
Values
=>
Cc0020;cc-rep
[+]=>[1]=>[1]=>([],1)=>0
[-]=>[1]=>[1]=>([],1)=>0
[-,+]=>[2,1]=>[2,1]=>([(0,1)],2)=>2
[-,+,+]=>[2,3,1]=>[3,1,2]=>([(0,2),(1,2)],3)=>10
[-,-,+]=>[3,1,2]=>[3,2,1]=>([(0,1),(0,2),(1,2)],3)=>6
[-,+,+,+]=>[2,3,4,1]=>[4,1,2,3]=>([(0,3),(1,3),(2,3)],4)=>24
[-,-,+,+]=>[3,4,1,2]=>[3,1,4,2]=>([(0,3),(1,2),(2,3)],4)=>30
[-,+,-,+]=>[2,4,1,3]=>[4,3,1,2]=>([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>16
[-,-,-,+]=>[4,1,2,3]=>[4,3,2,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>12
[-,3,2,+]=>[2,4,1,3]=>[4,3,1,2]=>([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>16
[4,-,+,1]=>[3,1,4,2]=>[4,2,1,3]=>([(0,3),(1,2),(1,3),(2,3)],4)=>20
[-,+,+,+,+]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>44
[-,-,+,+,+]=>[3,4,5,1,2]=>[5,2,4,1,3]=>([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)=>32
[-,+,-,+,+]=>[2,4,5,1,3]=>[4,1,2,5,3]=>([(0,4),(1,4),(2,3),(3,4)],5)=>56
[-,+,+,-,+]=>[2,3,5,1,4]=>[5,4,1,2,3]=>([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>32
[-,-,-,+,+]=>[4,5,1,2,3]=>[5,3,1,4,2]=>([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)=>32
[-,-,+,-,+]=>[3,5,1,2,4]=>[3,1,5,4,2]=>([(0,3),(1,2),(1,4),(2,4),(3,4)],5)=>52
[-,+,-,-,+]=>[2,5,1,3,4]=>[5,4,3,1,2]=>([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>24
[-,-,-,-,+]=>[5,1,2,3,4]=>[5,4,3,2,1]=>([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>20
[-,+,4,3,+]=>[2,3,5,1,4]=>[5,4,1,2,3]=>([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>32
[-,-,4,3,+]=>[3,5,1,2,4]=>[3,1,5,4,2]=>([(0,3),(1,2),(1,4),(2,4),(3,4)],5)=>52
[-,3,2,+,+]=>[2,4,5,1,3]=>[4,1,2,5,3]=>([(0,4),(1,4),(2,3),(3,4)],5)=>56
[-,3,2,-,+]=>[2,5,1,3,4]=>[5,4,3,1,2]=>([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>24
[-,3,4,2,+]=>[2,5,1,3,4]=>[5,4,3,1,2]=>([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>24
[-,4,2,3,+]=>[2,3,5,1,4]=>[5,4,1,2,3]=>([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>32
[-,4,+,2,+]=>[3,2,5,1,4]=>[2,5,4,1,3]=>([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)=>42
[-,4,-,2,+]=>[2,5,1,4,3]=>[4,5,3,1,2]=>([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)=>28
[-,5,-,2,4]=>[2,4,1,5,3]=>[5,3,1,2,4]=>([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>36
[-,5,-,+,2]=>[4,2,1,5,3]=>[2,5,3,1,4]=>([(0,4),(1,3),(2,3),(2,4),(3,4)],5)=>46
[2,4,+,1,+]=>[3,1,5,2,4]=>[5,4,2,1,3]=>([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>28
[2,5,-,+,1]=>[4,1,2,5,3]=>[5,3,2,1,4]=>([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>32
[4,-,+,1,+]=>[3,1,5,4,2]=>[4,5,2,1,3]=>([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)=>32
[5,-,+,1,4]=>[3,1,4,5,2]=>[5,2,1,3,4]=>([(0,4),(1,4),(2,3),(2,4),(3,4)],5)=>40
[5,-,+,+,1]=>[3,4,1,5,2]=>[3,1,5,2,4]=>([(0,4),(1,3),(2,3),(2,4)],5)=>70
[5,+,-,+,1]=>[2,4,1,5,3]=>[5,3,1,2,4]=>([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>36
[5,-,-,+,1]=>[4,1,5,2,3]=>[4,2,1,5,3]=>([(0,3),(1,2),(1,4),(2,4),(3,4)],5)=>52
[5,-,+,-,1]=>[3,1,5,2,4]=>[5,4,2,1,3]=>([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>28
[5,-,4,3,1]=>[3,1,5,2,4]=>[5,4,2,1,3]=>([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>28
[5,3,2,+,1]=>[2,4,1,5,3]=>[5,3,1,2,4]=>([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>36
[-,+,+,+,+,+]=>[2,3,4,5,6,1]=>[6,1,2,3,4,5]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>70
[-,-,+,+,+,+]=>[3,4,5,6,1,2]=>[5,1,3,6,2,4]=>([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>68
[-,+,-,+,+,+]=>[2,4,5,6,1,3]=>[6,3,5,1,2,4]=>([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>50
[-,+,+,-,+,+]=>[2,3,5,6,1,4]=>[5,1,2,3,6,4]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>88
[-,+,+,+,-,+]=>[2,3,4,6,1,5]=>[6,5,1,2,3,4]=>([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>54
[-,-,-,+,+,+]=>[4,5,6,1,2,3]=>[4,1,5,2,6,3]=>([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>84
[-,-,+,-,+,+]=>[3,5,6,1,2,4]=>[5,2,6,4,1,3]=>([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>50
[-,-,+,+,-,+]=>[3,4,6,1,2,5]=>[6,5,2,4,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>42
[-,+,-,-,+,+]=>[2,5,6,1,3,4]=>[6,4,1,2,5,3]=>([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>54
[-,+,-,+,-,+]=>[2,4,6,1,3,5]=>[4,1,2,6,5,3]=>([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>90
[-,+,+,-,-,+]=>[2,3,6,1,4,5]=>[6,5,4,1,2,3]=>([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>42
[-,-,-,-,+,+]=>[5,6,1,2,3,4]=>[5,3,1,6,4,2]=>([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>60
[-,-,-,+,-,+]=>[4,6,1,2,3,5]=>[6,5,3,1,4,2]=>([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>42
[-,-,+,-,-,+]=>[3,6,1,2,4,5]=>[3,1,6,5,4,2]=>([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>76
[-,+,-,-,-,+]=>[2,6,1,3,4,5]=>[6,5,4,3,1,2]=>([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>34
[-,-,-,-,-,+]=>[6,1,2,3,4,5]=>[6,5,4,3,2,1]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>30
[-,+,+,5,4,+]=>[2,3,4,6,1,5]=>[6,5,1,2,3,4]=>([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>54
[-,-,+,5,4,+]=>[3,4,6,1,2,5]=>[6,5,2,4,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>42
[-,+,-,5,4,+]=>[2,4,6,1,3,5]=>[4,1,2,6,5,3]=>([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>90
[-,-,-,5,4,+]=>[4,6,1,2,3,5]=>[6,5,3,1,4,2]=>([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>42
[-,+,4,3,+,+]=>[2,3,5,6,1,4]=>[5,1,2,3,6,4]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>88
[-,-,4,3,+,+]=>[3,5,6,1,2,4]=>[5,2,6,4,1,3]=>([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>50
[-,+,4,3,-,+]=>[2,3,6,1,4,5]=>[6,5,4,1,2,3]=>([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>42
[-,-,4,3,-,+]=>[3,6,1,2,4,5]=>[3,1,6,5,4,2]=>([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>76
[-,+,4,5,3,+]=>[2,3,6,1,4,5]=>[6,5,4,1,2,3]=>([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>42
[-,-,4,5,3,+]=>[3,6,1,2,4,5]=>[3,1,6,5,4,2]=>([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>76
[-,+,5,3,4,+]=>[2,3,4,6,1,5]=>[6,5,1,2,3,4]=>([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>54
[-,-,5,3,4,+]=>[3,4,6,1,2,5]=>[6,5,2,4,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>42
[-,+,5,+,3,+]=>[2,4,3,6,1,5]=>[3,6,5,1,2,4]=>([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>60
[-,-,5,+,3,+]=>[4,3,6,1,2,5]=>[4,1,6,5,2,3]=>([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>70
[-,+,5,-,3,+]=>[2,3,6,1,5,4]=>[5,6,4,1,2,3]=>([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>46
[-,-,5,-,3,+]=>[3,6,1,2,5,4]=>[3,1,5,6,4,2]=>([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)=>80
[-,+,6,-,3,5]=>[2,3,5,1,6,4]=>[6,4,1,2,3,5]=>([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>58
[-,-,6,-,3,5]=>[3,5,1,2,6,4]=>[3,1,6,4,2,5]=>([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)=>98
[-,+,6,-,+,3]=>[2,5,3,1,6,4]=>[3,6,4,1,2,5]=>([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>64
[-,-,6,-,+,3]=>[5,3,1,2,6,4]=>[6,4,2,3,1,5]=>([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>50
[-,3,2,+,+,+]=>[2,4,5,6,1,3]=>[6,3,5,1,2,4]=>([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>50
[-,3,2,-,+,+]=>[2,5,6,1,3,4]=>[6,4,1,2,5,3]=>([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>54
[-,3,2,+,-,+]=>[2,4,6,1,3,5]=>[4,1,2,6,5,3]=>([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>90
[-,3,2,-,-,+]=>[2,6,1,3,4,5]=>[6,5,4,3,1,2]=>([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>34
[-,3,2,5,4,+]=>[2,4,6,1,3,5]=>[4,1,2,6,5,3]=>([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>90
[-,3,4,2,+,+]=>[2,5,6,1,3,4]=>[6,4,1,2,5,3]=>([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>54
[-,3,4,2,-,+]=>[2,6,1,3,4,5]=>[6,5,4,3,1,2]=>([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>34
[-,3,4,5,2,+]=>[2,6,1,3,4,5]=>[6,5,4,3,1,2]=>([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>34
[-,3,5,2,4,+]=>[2,4,6,1,3,5]=>[4,1,2,6,5,3]=>([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>90
[-,3,5,+,2,+]=>[4,2,6,1,3,5]=>[2,4,1,6,5,3]=>([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)=>112
[-,3,5,-,2,+]=>[2,6,1,3,5,4]=>[5,6,4,3,1,2]=>([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>38
[-,3,6,-,2,5]=>[2,5,1,3,6,4]=>[6,4,3,1,2,5]=>([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>50
[-,3,6,-,+,2]=>[5,2,1,3,6,4]=>[2,6,4,3,1,5]=>([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>64
[-,4,2,3,+,+]=>[2,3,5,6,1,4]=>[5,1,2,3,6,4]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>88
[-,4,2,3,-,+]=>[2,3,6,1,4,5]=>[6,5,4,1,2,3]=>([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>42
[-,4,2,5,3,+]=>[2,3,6,1,4,5]=>[6,5,4,1,2,3]=>([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>42
[-,4,+,2,+,+]=>[3,2,5,6,1,4]=>[2,5,1,3,6,4]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>108
[-,4,-,2,+,+]=>[2,5,6,1,4,3]=>[5,4,1,2,6,3]=>([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>70
[-,4,+,2,-,+]=>[3,2,6,1,4,5]=>[2,6,5,4,1,3]=>([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>56
[-,4,-,2,-,+]=>[2,6,1,4,3,5]=>[4,6,5,3,1,2]=>([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>42
[-,4,+,5,2,+]=>[3,2,6,1,4,5]=>[2,6,5,4,1,3]=>([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>56
[-,4,-,5,2,+]=>[2,6,1,4,3,5]=>[4,6,5,3,1,2]=>([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>42
[-,4,5,2,3,+]=>[2,3,6,1,4,5]=>[6,5,4,1,2,3]=>([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>42
[-,4,5,3,2,+]=>[3,2,6,1,4,5]=>[2,6,5,4,1,3]=>([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>56
[-,5,2,3,4,+]=>[2,3,4,6,1,5]=>[6,5,1,2,3,4]=>([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>54
[-,5,2,+,3,+]=>[2,4,3,6,1,5]=>[3,6,5,1,2,4]=>([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>60
[-,5,2,-,3,+]=>[2,3,6,1,5,4]=>[5,6,4,1,2,3]=>([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>46
[-,5,+,2,4,+]=>[3,2,4,6,1,5]=>[2,6,5,1,3,4]=>([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>70
[-,5,-,2,4,+]=>[2,4,6,1,5,3]=>[4,1,2,5,6,3]=>([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>94
[-,5,+,+,2,+]=>[3,4,2,6,1,5]=>[6,5,1,3,2,4]=>([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>50
[-,5,-,+,2,+]=>[4,2,6,1,5,3]=>[2,4,1,5,6,3]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>116
[-,5,+,-,2,+]=>[3,2,6,1,5,4]=>[2,5,6,4,1,3]=>([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>60
[-,5,-,-,2,+]=>[2,6,1,5,3,4]=>[6,4,5,3,1,2]=>([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>38
[-,5,4,2,3,+]=>[2,3,6,1,5,4]=>[5,6,4,1,2,3]=>([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>46
[-,5,4,3,2,+]=>[3,2,6,1,5,4]=>[2,5,6,4,1,3]=>([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>60
[-,6,2,-,3,5]=>[2,3,5,1,6,4]=>[6,4,1,2,3,5]=>([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>58
[-,6,2,-,+,3]=>[2,5,3,1,6,4]=>[3,6,4,1,2,5]=>([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>64
[-,6,-,2,4,5]=>[2,4,5,1,6,3]=>[4,1,2,6,3,5]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>116
[-,6,-,2,+,4]=>[2,5,4,1,6,3]=>[6,3,4,1,2,5]=>([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>54
[-,6,-,2,-,4]=>[2,4,1,6,3,5]=>[6,5,3,1,2,4]=>([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[-,6,-,+,2,5]=>[4,2,5,1,6,3]=>[2,4,1,6,3,5]=>([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>140
[-,6,+,-,2,5]=>[3,2,5,1,6,4]=>[2,6,4,1,3,5]=>([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>74
[-,6,-,-,2,5]=>[2,5,1,6,3,4]=>[5,3,1,2,6,4]=>([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)=>80
[-,6,-,+,+,2]=>[4,5,2,1,6,3]=>[4,1,6,3,2,5]=>([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>88
[-,6,+,-,+,2]=>[3,5,2,1,6,4]=>[6,4,1,3,2,5]=>([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>54
[-,6,-,-,+,2]=>[5,2,1,6,3,4]=>[2,5,3,1,6,4]=>([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)=>98
[-,6,-,+,-,2]=>[4,2,1,6,3,5]=>[2,6,5,3,1,4]=>([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>60
[-,6,-,5,2,4]=>[2,4,1,6,3,5]=>[6,5,3,1,2,4]=>([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[-,6,-,5,4,2]=>[4,2,1,6,3,5]=>[2,6,5,3,1,4]=>([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>60
[-,6,4,2,3,5]=>[2,3,5,1,6,4]=>[6,4,1,2,3,5]=>([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>58
[-,6,4,2,+,3]=>[2,5,3,1,6,4]=>[3,6,4,1,2,5]=>([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>64
[-,6,4,3,2,5]=>[3,2,5,1,6,4]=>[2,6,4,1,3,5]=>([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>74
[-,6,4,3,+,2]=>[3,5,2,1,6,4]=>[6,4,1,3,2,5]=>([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>54
[2,3,5,+,1,+]=>[4,1,6,2,3,5]=>[4,2,1,6,5,3]=>([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)=>86
[2,3,6,-,+,1]=>[5,1,2,3,6,4]=>[6,4,3,2,1,5]=>([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[2,4,+,1,+,+]=>[3,1,5,6,2,4]=>[5,2,1,3,6,4]=>([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>84
[2,4,+,1,-,+]=>[3,1,6,2,4,5]=>[6,5,4,2,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>38
[2,4,+,5,1,+]=>[3,1,6,2,4,5]=>[6,5,4,2,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>38
[2,4,5,3,1,+]=>[3,1,6,2,4,5]=>[6,5,4,2,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>38
[2,5,+,1,4,+]=>[3,1,4,6,2,5]=>[6,5,2,1,3,4]=>([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>50
[2,5,+,+,1,+]=>[3,4,1,6,2,5]=>[3,1,6,5,2,4]=>([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)=>94
[2,5,-,+,1,+]=>[4,1,6,2,5,3]=>[4,2,1,5,6,3]=>([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>90
[2,5,+,-,1,+]=>[3,1,6,2,5,4]=>[5,6,4,2,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>42
[2,5,4,3,1,+]=>[3,1,6,2,5,4]=>[5,6,4,2,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>42
[2,6,-,+,1,5]=>[4,1,5,2,6,3]=>[4,2,1,6,3,5]=>([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)=>112
[2,6,+,-,1,5]=>[3,1,5,2,6,4]=>[6,4,2,1,3,5]=>([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>54
[2,6,-,+,+,1]=>[4,5,1,2,6,3]=>[6,3,1,4,2,5]=>([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>58
[2,6,+,-,+,1]=>[3,5,1,2,6,4]=>[3,1,6,4,2,5]=>([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)=>98
[2,6,-,-,+,1]=>[5,1,2,6,3,4]=>[5,3,2,1,6,4]=>([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>76
[2,6,-,+,-,1]=>[4,1,2,6,3,5]=>[6,5,3,2,1,4]=>([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>42
[2,6,-,5,4,1]=>[4,1,2,6,3,5]=>[6,5,3,2,1,4]=>([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>42
[2,6,4,3,1,5]=>[3,1,5,2,6,4]=>[6,4,2,1,3,5]=>([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>54
[2,6,4,3,+,1]=>[3,5,1,2,6,4]=>[3,1,6,4,2,5]=>([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)=>98
[3,+,5,+,1,+]=>[2,4,1,6,3,5]=>[6,5,3,1,2,4]=>([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[3,-,5,+,1,+]=>[4,1,6,3,2,5]=>[6,5,2,1,4,3]=>([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[3,+,6,-,+,1]=>[2,5,1,3,6,4]=>[6,4,3,1,2,5]=>([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>50
[3,-,6,-,+,1]=>[5,1,3,2,6,4]=>[3,6,4,2,1,5]=>([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>60
[3,5,2,+,1,+]=>[2,4,1,6,3,5]=>[6,5,3,1,2,4]=>([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[3,6,2,-,+,1]=>[2,5,1,3,6,4]=>[6,4,3,1,2,5]=>([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>50
[3,6,4,2,+,1]=>[2,5,1,3,6,4]=>[6,4,3,1,2,5]=>([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>50
[4,-,+,1,+,+]=>[3,1,5,6,4,2]=>[6,2,1,3,5,4]=>([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>62
[4,-,+,1,-,+]=>[3,1,6,4,2,5]=>[4,6,5,2,1,3]=>([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[4,-,+,5,1,+]=>[3,1,6,4,2,5]=>[4,6,5,2,1,3]=>([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[4,-,5,3,1,+]=>[3,1,6,4,2,5]=>[4,6,5,2,1,3]=>([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[4,6,-,2,+,1]=>[2,5,1,4,6,3]=>[4,6,3,1,2,5]=>([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>60
[5,-,+,1,4,+]=>[3,1,4,6,5,2]=>[5,6,2,1,3,4]=>([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>54
[5,-,+,+,1,+]=>[3,4,1,6,5,2]=>[3,1,5,6,2,4]=>([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)=>98
[5,+,-,+,1,+]=>[2,4,1,6,5,3]=>[5,6,3,1,2,4]=>([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>50
[5,-,-,+,1,+]=>[4,1,6,5,2,3]=>[5,2,1,4,6,3]=>([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>74
[5,-,+,-,1,+]=>[3,1,6,5,2,4]=>[6,4,5,2,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>42
[5,-,4,3,1,+]=>[3,1,6,5,2,4]=>[6,4,5,2,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>42
[5,-,6,-,3,1]=>[3,1,5,2,6,4]=>[6,4,2,1,3,5]=>([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>54
[5,3,2,+,1,+]=>[2,4,1,6,5,3]=>[5,6,3,1,2,4]=>([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>50
[5,6,-,2,4,1]=>[2,4,1,5,6,3]=>[6,3,1,2,4,5]=>([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>62
[5,6,-,+,1,2]=>[4,1,2,5,6,3]=>[6,3,2,1,4,5]=>([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>58
[5,6,-,+,2,1]=>[4,2,1,5,6,3]=>[2,6,3,1,4,5]=>([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>78
[6,-,+,1,4,5]=>[3,1,4,5,6,2]=>[6,2,1,3,4,5]=>([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>66
[6,-,+,1,+,4]=>[3,1,5,4,6,2]=>[4,6,2,1,3,5]=>([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>64
[6,-,+,1,-,4]=>[3,1,4,6,2,5]=>[6,5,2,1,3,4]=>([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>50
[6,-,+,+,1,5]=>[3,4,1,5,6,2]=>[3,1,6,2,4,5]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>116
[6,+,-,+,1,5]=>[2,4,1,5,6,3]=>[6,3,1,2,4,5]=>([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>62
[6,-,-,+,1,5]=>[4,1,5,6,2,3]=>[6,3,5,2,1,4]=>([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[6,-,+,-,1,5]=>[3,1,5,6,2,4]=>[5,2,1,3,6,4]=>([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>84
[6,-,+,+,+,1]=>[3,4,5,1,6,2]=>[6,2,4,1,3,5]=>([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>58
[6,+,-,+,+,1]=>[2,4,5,1,6,3]=>[4,1,2,6,3,5]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>116
[6,+,+,-,+,1]=>[2,3,5,1,6,4]=>[6,4,1,2,3,5]=>([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>58
[6,-,-,+,+,1]=>[4,5,1,6,2,3]=>[5,2,6,3,1,4]=>([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>54
[6,-,+,-,+,1]=>[3,5,1,6,2,4]=>[3,1,5,2,6,4]=>([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>140
[6,-,+,+,-,1]=>[3,4,1,6,2,5]=>[3,1,6,5,2,4]=>([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)=>94
[6,+,-,-,+,1]=>[2,5,1,6,3,4]=>[5,3,1,2,6,4]=>([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)=>80
[6,+,-,+,-,1]=>[2,4,1,6,3,5]=>[6,5,3,1,2,4]=>([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[6,-,-,-,+,1]=>[5,1,6,2,3,4]=>[6,4,2,1,5,3]=>([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>50
[6,-,-,+,-,1]=>[4,1,6,2,3,5]=>[4,2,1,6,5,3]=>([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)=>86
[6,-,+,-,-,1]=>[3,1,6,2,4,5]=>[6,5,4,2,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>38
[6,-,+,5,1,4]=>[3,1,4,6,2,5]=>[6,5,2,1,3,4]=>([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>50
[6,-,+,5,4,1]=>[3,4,1,6,2,5]=>[3,1,6,5,2,4]=>([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)=>94
[6,+,-,5,4,1]=>[2,4,1,6,3,5]=>[6,5,3,1,2,4]=>([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[6,-,-,5,4,1]=>[4,1,6,2,3,5]=>[4,2,1,6,5,3]=>([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)=>86
[6,-,4,3,1,5]=>[3,1,5,6,2,4]=>[5,2,1,3,6,4]=>([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>84
[6,+,4,3,+,1]=>[2,3,5,1,6,4]=>[6,4,1,2,3,5]=>([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>58
[6,-,4,3,+,1]=>[3,5,1,6,2,4]=>[3,1,5,2,6,4]=>([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>140
[6,-,4,3,-,1]=>[3,1,6,2,4,5]=>[6,5,4,2,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>38
[6,-,4,5,3,1]=>[3,1,6,2,4,5]=>[6,5,4,2,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>38
[6,-,5,3,1,4]=>[3,1,4,6,2,5]=>[6,5,2,1,3,4]=>([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>50
[6,-,5,3,4,1]=>[3,4,1,6,2,5]=>[3,1,6,5,2,4]=>([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)=>94
[6,-,5,+,1,3]=>[4,1,3,6,2,5]=>[3,6,5,2,1,4]=>([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>56
[6,-,5,+,3,1]=>[4,3,1,6,2,5]=>[6,5,2,3,1,4]=>([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[6,-,5,-,3,1]=>[3,1,6,2,5,4]=>[5,6,4,2,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>42
[6,3,2,+,1,5]=>[2,4,1,5,6,3]=>[6,3,1,2,4,5]=>([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>62
[6,3,2,+,+,1]=>[2,4,5,1,6,3]=>[4,1,2,6,3,5]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>116
[6,3,2,-,+,1]=>[2,5,1,6,3,4]=>[5,3,1,2,6,4]=>([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)=>80
[6,3,2,+,-,1]=>[2,4,1,6,3,5]=>[6,5,3,1,2,4]=>([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[6,3,2,5,4,1]=>[2,4,1,6,3,5]=>[6,5,3,1,2,4]=>([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[6,3,4,2,+,1]=>[2,5,1,6,3,4]=>[5,3,1,2,6,4]=>([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)=>80
[6,3,5,2,4,1]=>[2,4,1,6,3,5]=>[6,5,3,1,2,4]=>([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>46
[6,3,5,+,1,2]=>[4,1,2,6,3,5]=>[6,5,3,2,1,4]=>([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>42
[6,3,5,+,2,1]=>[4,2,1,6,3,5]=>[2,6,5,3,1,4]=>([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>60
[6,4,2,3,+,1]=>[2,3,5,1,6,4]=>[6,4,1,2,3,5]=>([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>58
[6,4,+,1,+,2]=>[3,1,5,2,6,4]=>[6,4,2,1,3,5]=>([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>54
[6,4,+,2,+,1]=>[3,2,5,1,6,4]=>[2,6,4,1,3,5]=>([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>74
[6,4,-,2,+,1]=>[2,5,1,6,4,3]=>[6,3,1,2,5,4]=>([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>58
[6,5,-,2,4,1]=>[2,4,1,6,5,3]=>[5,6,3,1,2,4]=>([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>50
[6,5,-,+,1,2]=>[4,1,2,6,5,3]=>[5,6,3,2,1,4]=>([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>46
[6,5,-,+,2,1]=>[4,2,1,6,5,3]=>[2,5,6,3,1,4]=>([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>64
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
Description
The hyper-Wiener index of a connected graph.
This is
$$ \sum_{\{u,v\}\subseteq V} d(u,v)+d(u,v)^2. $$
This is
$$ \sum_{\{u,v\}\subseteq V} d(u,v)+d(u,v)^2. $$
Map
lower permutation
Description
The lower bound in the Grassmann interval corresponding to the decorated permutation.
Let $I$ be the anti-exceedance set of a decorated permutation $w$. Let $v$ be the $k$-Grassmannian permutation determined by $v[k] = w^{-1}(I)$ and let $u$ be the permutation satisfying $u = wv$. Then $[u, v]$ is the Grassmann interval corresponding to $w$.
This map returns $u$.
Let $I$ be the anti-exceedance set of a decorated permutation $w$. Let $v$ be the $k$-Grassmannian permutation determined by $v[k] = w^{-1}(I)$ and let $u$ be the permutation satisfying $u = wv$. Then $[u, v]$ is the Grassmann interval corresponding to $w$.
This map returns $u$.
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!