Identifier
-
Mp00255:
Decorated permutations
—lower permutation⟶
Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤ
Values
=>
Cc0020;cc-rep
[-,+]=>[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)=>3
[-,+,+,+]=>[2,3,4,1]=>[4,1,2,3]=>([(0,3),(1,3),(2,3)],4)=>1
[-,-,+,+]=>[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)=>4
[-,-,-,+]=>[4,1,2,3]=>[4,3,2,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>6
[-,3,2,+]=>[2,4,1,3]=>[4,3,1,2]=>([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>4
[4,-,+,1]=>[3,1,4,2]=>[4,2,1,3]=>([(0,3),(1,2),(1,3),(2,3)],4)=>2
[-,+,+,+,+]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1
[-,-,+,+,+]=>[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)=>5
[-,+,-,+,+]=>[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)=>5
[-,-,-,+,+]=>[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)=>5
[-,-,+,-,+]=>[3,5,1,2,4]=>[3,1,5,4,2]=>([(0,3),(1,2),(1,4),(2,4),(3,4)],5)=>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)=>8
[-,-,-,-,+]=>[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)=>10
[-,+,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)=>5
[-,-,4,3,+]=>[3,5,1,2,4]=>[3,1,5,4,2]=>([(0,3),(1,2),(1,4),(2,4),(3,4)],5)=>2
[-,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)=>8
[-,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)=>8
[-,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)=>5
[-,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)=>3
[-,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)=>6
[-,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)=>3
[-,5,-,+,2]=>[4,2,1,5,3]=>[2,5,3,1,4]=>([(0,4),(1,3),(2,3),(2,4),(3,4)],5)=>2
[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)=>6
[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)=>4
[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)=>4
[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)=>3
[5,-,-,+,1]=>[4,1,5,2,3]=>[4,2,1,5,3]=>([(0,3),(1,2),(1,4),(2,4),(3,4)],5)=>2
[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)=>6
[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)=>6
[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)=>3
[-,+,+,+,+,+]=>[2,3,4,5,6,1]=>[6,1,2,3,4,5]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1
[-,-,+,+,+,+]=>[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)=>3
[-,+,-,+,+,+]=>[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)=>7
[-,+,+,-,+,+]=>[2,3,5,6,1,4]=>[5,1,2,3,6,4]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>1
[-,+,+,+,-,+]=>[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)=>6
[-,-,-,+,+,+]=>[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)=>2
[-,-,+,-,+,+]=>[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)=>6
[-,-,+,+,-,+]=>[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)=>10
[-,+,-,-,+,+]=>[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)=>6
[-,+,-,+,-,+]=>[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)=>2
[-,+,+,-,-,+]=>[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)=>10
[-,-,-,-,+,+]=>[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)=>5
[-,-,-,+,-,+]=>[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)=>10
[-,-,+,-,-,+]=>[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)=>4
[-,+,-,-,-,+]=>[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)=>13
[-,-,-,-,-,+]=>[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)=>15
[-,+,+,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)=>6
[-,-,+,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)=>10
[-,+,-,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)=>2
[-,-,-,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)=>10
[-,+,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)=>1
[-,-,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)=>6
[-,+,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)=>10
[-,-,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)=>4
[-,+,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)=>10
[-,-,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)=>4
[-,+,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)=>6
[-,-,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)=>10
[-,+,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)=>5
[-,-,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)=>4
[-,+,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)=>8
[-,-,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)=>3
[-,+,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)=>4
[-,-,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)=>2
[-,+,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)=>4
[-,-,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)=>6
[-,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)=>7
[-,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)=>6
[-,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)=>2
[-,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)=>13
[-,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)=>2
[-,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)=>6
[-,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)=>13
[-,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)=>13
[-,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)=>2
[-,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)=>2
[-,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)=>11
[-,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)=>6
[-,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)=>4
[-,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)=>1
[-,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)=>10
[-,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)=>10
[-,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)=>4
[-,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)=>6
[-,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)=>9
[-,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)=>6
[-,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)=>9
[-,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)=>10
[-,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)=>6
[-,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)=>6
[-,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)=>5
[-,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)=>8
[-,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)=>4
[-,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)=>1
[-,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)=>7
[-,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)=>5
[-,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)=>11
[-,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)=>8
[-,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)=>5
[-,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)=>4
[-,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)=>4
[-,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)=>5
[-,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)=>8
[-,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)=>3
[-,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)=>3
[-,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)=>3
[-,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)=>5
[-,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)=>2
[-,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)=>5
[-,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)=>8
[-,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)=>5
[-,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)=>4
[-,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)=>4
[-,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)=>3
[-,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)=>5
[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)=>3
[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)=>7
[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)=>2
[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)=>11
[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)=>11
[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)=>11
[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)=>7
[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)=>3
[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)=>2
[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)=>9
[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)=>9
[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)=>2
[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)=>5
[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)=>4
[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)=>2
[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)=>4
[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)=>9
[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)=>9
[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)=>5
[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)=>2
[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)=>8
[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)=>8
[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)=>6
[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)=>5
[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)=>8
[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)=>6
[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)=>6
[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)=>3
[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)=>7
[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)=>7
[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)=>7
[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)=>5
[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)=>5
[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)=>2
[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)=>6
[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)=>3
[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)=>9
[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)=>9
[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)=>5
[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)=>6
[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)=>3
[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)=>4
[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)=>2
[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)=>2
[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)=>4
[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)=>7
[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)=>3
[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)=>8
[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)=>2
[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)=>4
[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)=>4
[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)=>5
[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)=>3
[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)=>3
[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)=>8
[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)=>7
[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)=>3
[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)=>11
[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)=>7
[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)=>3
[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)=>8
[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)=>3
[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)=>2
[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)=>4
[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)=>11
[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)=>11
[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)=>7
[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)=>3
[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)=>6
[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)=>8
[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)=>9
[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)=>3
[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)=>3
[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)=>8
[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)=>8
[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)=>3
[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)=>8
[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)=>9
[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)=>5
[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)=>4
[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)=>5
[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)=>3
[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)=>4
[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)=>6
[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)=>7
[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)=>4
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Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
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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