Identifier
-
Mp00255:
Decorated permutations
—lower permutation⟶
Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤ
Values
=>
Cc0020;cc-rep
[+]=>[1]=>[1]=>([],1)=>1
[-]=>[1]=>[1]=>([],1)=>1
[-,+]=>[2,1]=>[2,1]=>([(0,1)],2)=>1
[-,+,+]=>[2,3,1]=>[3,1,2]=>([(0,2),(1,2)],3)=>1
[-,-,+]=>[3,1,2]=>[3,2,1]=>([(0,1),(0,2),(1,2)],3)=>2
[-,+,+,+]=>[2,3,4,1]=>[4,1,2,3]=>([(0,3),(1,3),(2,3)],4)=>2
[-,-,+,+]=>[3,4,1,2]=>[3,1,4,2]=>([(0,3),(1,2),(2,3)],4)=>1
[-,+,-,+]=>[2,4,1,3]=>[4,3,1,2]=>([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>2
[-,-,-,+]=>[4,1,2,3]=>[4,3,2,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>3
[-,3,2,+]=>[2,4,1,3]=>[4,3,1,2]=>([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>2
[4,-,+,1]=>[3,1,4,2]=>[4,2,1,3]=>([(0,3),(1,2),(1,3),(2,3)],4)=>1
[-,+,+,+,+]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>3
[-,-,+,+,+]=>[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)=>1
[-,+,-,+,+]=>[2,4,5,1,3]=>[4,1,2,5,3]=>([(0,4),(1,4),(2,3),(3,4)],5)=>1
[-,+,+,-,+]=>[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)=>2
[-,-,-,+,+]=>[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)=>1
[-,-,+,-,+]=>[3,5,1,2,4]=>[3,1,5,4,2]=>([(0,3),(1,2),(1,4),(2,4),(3,4)],5)=>1
[-,+,-,-,+]=>[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)=>3
[-,-,-,-,+]=>[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)=>4
[-,+,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)=>2
[-,-,4,3,+]=>[3,5,1,2,4]=>[3,1,5,4,2]=>([(0,3),(1,2),(1,4),(2,4),(3,4)],5)=>1
[-,3,2,+,+]=>[2,4,5,1,3]=>[4,1,2,5,3]=>([(0,4),(1,4),(2,3),(3,4)],5)=>1
[-,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)=>3
[-,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)=>3
[-,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)=>2
[-,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)=>1
[-,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)=>2
[-,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)=>1
[-,5,-,+,2]=>[4,2,1,5,3]=>[2,5,3,1,4]=>([(0,4),(1,3),(2,3),(2,4),(3,4)],5)=>1
[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)=>2
[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)=>2
[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)=>1
[5,-,+,1,4]=>[3,1,4,5,2]=>[5,2,1,3,4]=>([(0,4),(1,4),(2,3),(2,4),(3,4)],5)=>2
[5,-,+,+,1]=>[3,4,1,5,2]=>[3,1,5,2,4]=>([(0,4),(1,3),(2,3),(2,4)],5)=>1
[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)=>1
[5,-,-,+,1]=>[4,1,5,2,3]=>[4,2,1,5,3]=>([(0,3),(1,2),(1,4),(2,4),(3,4)],5)=>1
[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)=>2
[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)=>2
[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)=>1
[-,+,+,+,+,+]=>[2,3,4,5,6,1]=>[6,1,2,3,4,5]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>4
[-,-,+,+,+,+]=>[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)=>1
[-,+,-,+,+,+]=>[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)=>1
[-,+,+,-,+,+]=>[2,3,5,6,1,4]=>[5,1,2,3,6,4]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2
[-,+,+,+,-,+]=>[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)=>3
[-,-,-,+,+,+]=>[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)=>1
[-,-,+,-,+,+]=>[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)=>1
[-,-,+,+,-,+]=>[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)=>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)=>1
[-,+,-,+,-,+]=>[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)=>1
[-,+,+,-,-,+]=>[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)=>3
[-,-,-,-,+,+]=>[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)=>1
[-,-,-,+,-,+]=>[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)=>2
[-,-,+,-,-,+]=>[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)=>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)=>4
[-,-,-,-,-,+]=>[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)=>5
[-,+,+,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)=>3
[-,-,+,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)=>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)=>1
[-,-,-,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)=>2
[-,+,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)=>2
[-,-,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)=>1
[-,+,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)=>3
[-,-,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)=>2
[-,+,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)=>3
[-,-,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)=>2
[-,+,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)=>3
[-,-,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)=>2
[-,+,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)=>2
[-,-,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)=>3
[-,+,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)=>2
[-,-,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)=>1
[-,+,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)=>2
[-,-,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)=>1
[-,+,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)=>1
[-,-,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)=>2
[-,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)=>1
[-,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)=>1
[-,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)=>1
[-,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)=>4
[-,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)=>1
[-,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)=>1
[-,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)=>4
[-,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)=>4
[-,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)=>1
[-,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)=>1
[-,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)=>3
[-,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)=>2
[-,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)=>1
[-,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)=>2
[-,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)=>3
[-,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)=>3
[-,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)=>1
[-,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)=>2
[-,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)=>2
[-,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)=>2
[-,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)=>2
[-,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)=>2
[-,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)=>3
[-,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)=>2
[-,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)=>3
[-,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)=>2
[-,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)=>2
[-,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)=>2
[-,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)=>2
[-,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)=>2
[-,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)=>1
[-,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)=>1
[-,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)=>3
[-,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)=>2
[-,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)=>1
[-,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)=>2
[-,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)=>1
[-,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)=>1
[-,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)=>2
[-,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)=>2
[-,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)=>1
[-,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)=>1
[-,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)=>1
[-,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)=>2
[-,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)=>1
[-,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)=>1
[-,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)=>1
[-,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)=>2
[-,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)=>1
[-,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)=>2
[-,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)=>1
[-,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)=>1
[-,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)=>1
[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)=>2
[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)=>3
[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)=>1
[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)=>3
[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)=>3
[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)=>3
[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)=>2
[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)=>1
[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)=>1
[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)=>2
[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)=>2
[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)=>1
[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)=>1
[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)=>1
[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)=>1
[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)=>2
[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)=>2
[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)=>2
[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)=>1
[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)=>1
[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)=>2
[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)=>2
[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)=>2
[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)=>1
[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)=>2
[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)=>2
[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)=>2
[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)=>2
[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)=>2
[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)=>2
[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)=>2
[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)=>1
[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)=>2
[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)=>1
[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)=>1
[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)=>1
[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)=>2
[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)=>2
[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)=>1
[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)=>1
[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)=>2
[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)=>2
[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)=>1
[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)=>3
[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)=>1
[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)=>2
[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)=>1
[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)=>2
[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)=>1
[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)=>1
[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)=>1
[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)=>1
[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)=>2
[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)=>1
[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)=>1
[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)=>1
[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)=>1
[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)=>2
[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)=>1
[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)=>2
[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)=>3
[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)=>2
[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)=>1
[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)=>2
[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)=>2
[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)=>1
[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)=>2
[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)=>1
[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)=>3
[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)=>3
[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)=>2
[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)=>1
[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)=>2
[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)=>2
[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)=>2
[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)=>2
[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)=>1
[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)=>1
[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)=>2
[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)=>2
[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)=>1
[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)=>2
[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)=>2
[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)=>1
[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)=>2
[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)=>1
[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)=>1
[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)=>1
[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)=>1
[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)=>2
[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)=>1
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Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
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.
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