Identifier
-
Mp00256:
Decorated permutations
—upper permutation⟶
Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤ
Values
=>
Cc0020;cc-rep
[+]=>[1]=>[1]=>([],1)=>0
[-]=>[1]=>[1]=>([],1)=>0
[+,+]=>[1,2]=>[1,2]=>([],2)=>0
[-,+]=>[2,1]=>[2,1]=>([(0,1)],2)=>1
[+,-]=>[1,2]=>[1,2]=>([],2)=>0
[-,-]=>[1,2]=>[1,2]=>([],2)=>0
[2,1]=>[2,1]=>[2,1]=>([(0,1)],2)=>1
[+,+,+]=>[1,2,3]=>[1,2,3]=>([],3)=>0
[+,-,+]=>[1,3,2]=>[1,3,2]=>([(1,2)],3)=>1
[+,+,-]=>[1,2,3]=>[1,2,3]=>([],3)=>0
[-,-,+]=>[3,1,2]=>[3,2,1]=>([(0,1),(0,2),(1,2)],3)=>2
[-,+,-]=>[2,1,3]=>[2,1,3]=>([(1,2)],3)=>1
[+,-,-]=>[1,2,3]=>[1,2,3]=>([],3)=>0
[-,-,-]=>[1,2,3]=>[1,2,3]=>([],3)=>0
[+,3,2]=>[1,3,2]=>[1,3,2]=>([(1,2)],3)=>1
[-,3,2]=>[3,1,2]=>[3,2,1]=>([(0,1),(0,2),(1,2)],3)=>2
[2,1,-]=>[2,1,3]=>[2,1,3]=>([(1,2)],3)=>1
[2,3,1]=>[3,1,2]=>[3,2,1]=>([(0,1),(0,2),(1,2)],3)=>2
[3,-,1]=>[3,1,2]=>[3,2,1]=>([(0,1),(0,2),(1,2)],3)=>2
[+,+,+,+]=>[1,2,3,4]=>[1,2,3,4]=>([],4)=>0
[+,+,-,+]=>[1,2,4,3]=>[1,2,4,3]=>([(2,3)],4)=>1
[+,+,+,-]=>[1,2,3,4]=>[1,2,3,4]=>([],4)=>0
[+,-,-,+]=>[1,4,2,3]=>[1,4,3,2]=>([(1,2),(1,3),(2,3)],4)=>2
[+,-,+,-]=>[1,3,2,4]=>[1,3,2,4]=>([(2,3)],4)=>1
[+,+,-,-]=>[1,2,3,4]=>[1,2,3,4]=>([],4)=>0
[-,-,-,+]=>[4,1,2,3]=>[4,3,2,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>3
[-,-,+,-]=>[3,1,2,4]=>[3,2,1,4]=>([(1,2),(1,3),(2,3)],4)=>2
[-,+,-,-]=>[2,1,3,4]=>[2,1,3,4]=>([(2,3)],4)=>1
[+,-,-,-]=>[1,2,3,4]=>[1,2,3,4]=>([],4)=>0
[-,-,-,-]=>[1,2,3,4]=>[1,2,3,4]=>([],4)=>0
[+,+,4,3]=>[1,2,4,3]=>[1,2,4,3]=>([(2,3)],4)=>1
[+,-,4,3]=>[1,4,2,3]=>[1,4,3,2]=>([(1,2),(1,3),(2,3)],4)=>2
[-,-,4,3]=>[4,1,2,3]=>[4,3,2,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>3
[+,3,2,-]=>[1,3,2,4]=>[1,3,2,4]=>([(2,3)],4)=>1
[-,3,2,-]=>[3,1,2,4]=>[3,2,1,4]=>([(1,2),(1,3),(2,3)],4)=>2
[+,3,4,2]=>[1,4,2,3]=>[1,4,3,2]=>([(1,2),(1,3),(2,3)],4)=>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
[+,4,-,2]=>[1,4,2,3]=>[1,4,3,2]=>([(1,2),(1,3),(2,3)],4)=>2
[-,4,-,2]=>[4,1,2,3]=>[4,3,2,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>3
[2,1,-,-]=>[2,1,3,4]=>[2,1,3,4]=>([(2,3)],4)=>1
[2,3,1,-]=>[3,1,2,4]=>[3,2,1,4]=>([(1,2),(1,3),(2,3)],4)=>2
[2,3,4,1]=>[4,1,2,3]=>[4,3,2,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>3
[2,4,-,1]=>[4,1,2,3]=>[4,3,2,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>3
[3,-,1,-]=>[3,1,2,4]=>[3,2,1,4]=>([(1,2),(1,3),(2,3)],4)=>2
[3,-,4,1]=>[4,1,2,3]=>[4,3,2,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>3
[4,-,-,1]=>[4,1,2,3]=>[4,3,2,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>3
[+,+,+,+,+]=>[1,2,3,4,5]=>[1,2,3,4,5]=>([],5)=>0
[-,+,+,+,+]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[+,+,+,-,+]=>[1,2,3,5,4]=>[1,2,3,5,4]=>([(3,4)],5)=>1
[+,+,+,+,-]=>[1,2,3,4,5]=>[1,2,3,4,5]=>([],5)=>0
[-,+,+,-,+]=>[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)=>3
[+,+,-,-,+]=>[1,2,5,3,4]=>[1,2,5,4,3]=>([(2,3),(2,4),(3,4)],5)=>2
[+,+,-,+,-]=>[1,2,4,3,5]=>[1,2,4,3,5]=>([(3,4)],5)=>1
[+,+,+,-,-]=>[1,2,3,4,5]=>[1,2,3,4,5]=>([],5)=>0
[+,-,-,-,+]=>[1,5,2,3,4]=>[1,5,4,3,2]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
[+,-,-,+,-]=>[1,4,2,3,5]=>[1,4,3,2,5]=>([(2,3),(2,4),(3,4)],5)=>2
[+,-,+,-,-]=>[1,3,2,4,5]=>[1,3,2,4,5]=>([(3,4)],5)=>1
[+,+,-,-,-]=>[1,2,3,4,5]=>[1,2,3,4,5]=>([],5)=>0
[-,-,-,-,+]=>[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
[-,-,-,+,-]=>[4,1,2,3,5]=>[4,3,2,1,5]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
[-,-,+,-,-]=>[3,1,2,4,5]=>[3,2,1,4,5]=>([(2,3),(2,4),(3,4)],5)=>2
[-,+,-,-,-]=>[2,1,3,4,5]=>[2,1,3,4,5]=>([(3,4)],5)=>1
[+,-,-,-,-]=>[1,2,3,4,5]=>[1,2,3,4,5]=>([],5)=>0
[-,-,-,-,-]=>[1,2,3,4,5]=>[1,2,3,4,5]=>([],5)=>0
[+,+,+,5,4]=>[1,2,3,5,4]=>[1,2,3,5,4]=>([(3,4)],5)=>1
[-,+,+,5,4]=>[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)=>3
[+,+,-,5,4]=>[1,2,5,3,4]=>[1,2,5,4,3]=>([(2,3),(2,4),(3,4)],5)=>2
[+,-,-,5,4]=>[1,5,2,3,4]=>[1,5,4,3,2]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
[-,-,-,5,4]=>[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,-]=>[1,2,4,3,5]=>[1,2,4,3,5]=>([(3,4)],5)=>1
[+,-,4,3,-]=>[1,4,2,3,5]=>[1,4,3,2,5]=>([(2,3),(2,4),(3,4)],5)=>2
[-,-,4,3,-]=>[4,1,2,3,5]=>[4,3,2,1,5]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
[+,+,4,5,3]=>[1,2,5,3,4]=>[1,2,5,4,3]=>([(2,3),(2,4),(3,4)],5)=>2
[+,-,4,5,3]=>[1,5,2,3,4]=>[1,5,4,3,2]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>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
[+,+,5,-,3]=>[1,2,5,3,4]=>[1,2,5,4,3]=>([(2,3),(2,4),(3,4)],5)=>2
[+,-,5,-,3]=>[1,5,2,3,4]=>[1,5,4,3,2]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
[-,-,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
[+,3,2,-,-]=>[1,3,2,4,5]=>[1,3,2,4,5]=>([(3,4)],5)=>1
[-,3,2,-,-]=>[3,1,2,4,5]=>[3,2,1,4,5]=>([(2,3),(2,4),(3,4)],5)=>2
[+,3,4,2,-]=>[1,4,2,3,5]=>[1,4,3,2,5]=>([(2,3),(2,4),(3,4)],5)=>2
[-,3,4,2,-]=>[4,1,2,3,5]=>[4,3,2,1,5]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
[+,3,4,5,2]=>[1,5,2,3,4]=>[1,5,4,3,2]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
[-,3,4,5,2]=>[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
[+,3,5,-,2]=>[1,5,2,3,4]=>[1,5,4,3,2]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
[-,3,5,-,2]=>[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,-,2,-]=>[1,4,2,3,5]=>[1,4,3,2,5]=>([(2,3),(2,4),(3,4)],5)=>2
[-,4,-,2,-]=>[4,1,2,3,5]=>[4,3,2,1,5]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
[+,4,-,5,2]=>[1,5,2,3,4]=>[1,5,4,3,2]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
[-,4,-,5,2]=>[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
[+,5,-,-,2]=>[1,5,2,3,4]=>[1,5,4,3,2]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
[-,5,-,-,2]=>[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
[2,1,+,+,+]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[2,1,+,-,+]=>[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)=>3
[2,1,-,-,-]=>[2,1,3,4,5]=>[2,1,3,4,5]=>([(3,4)],5)=>1
[2,1,+,5,4]=>[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)=>3
[2,3,1,-,-]=>[3,1,2,4,5]=>[3,2,1,4,5]=>([(2,3),(2,4),(3,4)],5)=>2
[2,3,4,1,-]=>[4,1,2,3,5]=>[4,3,2,1,5]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
[2,3,4,5,1]=>[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
[2,3,5,-,1]=>[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
[2,4,-,1,-]=>[4,1,2,3,5]=>[4,3,2,1,5]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
[2,4,-,5,1]=>[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
[2,5,-,-,1]=>[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
[3,1,2,+,+]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[3,1,2,-,+]=>[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)=>3
[3,1,2,5,4]=>[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)=>3
[3,+,1,+,+]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[3,+,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)=>3
[3,-,1,-,-]=>[3,1,2,4,5]=>[3,2,1,4,5]=>([(2,3),(2,4),(3,4)],5)=>2
[3,+,1,5,4]=>[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)=>3
[3,-,4,1,-]=>[4,1,2,3,5]=>[4,3,2,1,5]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
[3,-,4,5,1]=>[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
[3,-,5,-,1]=>[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
[4,1,2,3,+]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[4,1,2,5,3]=>[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)=>3
[4,1,+,2,+]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[4,1,+,5,2]=>[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)=>3
[4,+,1,3,+]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[4,+,1,5,3]=>[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)=>3
[4,+,+,1,+]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[4,-,-,1,-]=>[4,1,2,3,5]=>[4,3,2,1,5]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>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)=>3
[4,-,-,5,1]=>[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
[5,1,2,3,4]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[5,1,2,+,3]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[5,1,2,-,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)=>3
[5,1,+,2,4]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[5,1,+,+,2]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[5,1,+,-,2]=>[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)=>3
[5,+,1,3,4]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[5,+,1,+,3]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[5,+,1,-,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)=>3
[5,+,+,1,4]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[5,+,+,+,1]=>[2,3,4,5,1]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2
[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)=>3
[5,-,-,-,1]=>[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
[+,+,+,+,+,+]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>([],6)=>0
[+,-,+,+,+,+]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,+,+,+,-,+]=>[1,2,3,4,6,5]=>[1,2,3,4,6,5]=>([(4,5)],6)=>1
[+,+,+,+,+,-]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>([],6)=>0
[-,+,+,+,+,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,-,+,+,-,+]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,+,+,-,-,+]=>[1,2,3,6,4,5]=>[1,2,3,6,5,4]=>([(3,4),(3,5),(4,5)],6)=>2
[+,+,+,-,+,-]=>[1,2,3,5,4,6]=>[1,2,3,5,4,6]=>([(4,5)],6)=>1
[+,+,+,+,-,-]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>([],6)=>0
[-,+,+,-,+,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,+,-,-,-,+]=>[1,2,6,3,4,5]=>[1,2,6,5,4,3]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,+,-,-,+,-]=>[1,2,5,3,4,6]=>[1,2,5,4,3,6]=>([(3,4),(3,5),(4,5)],6)=>2
[+,+,-,+,-,-]=>[1,2,4,3,5,6]=>[1,2,4,3,5,6]=>([(4,5)],6)=>1
[+,+,+,-,-,-]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>([],6)=>0
[+,-,-,-,-,+]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[+,-,-,-,+,-]=>[1,5,2,3,4,6]=>[1,5,4,3,2,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,-,-,+,-,-]=>[1,4,2,3,5,6]=>[1,4,3,2,5,6]=>([(3,4),(3,5),(4,5)],6)=>2
[+,-,+,-,-,-]=>[1,3,2,4,5,6]=>[1,3,2,4,5,6]=>([(4,5)],6)=>1
[+,+,-,-,-,-]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>([],6)=>0
[-,-,-,-,-,+]=>[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
[-,-,-,-,+,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[-,-,-,+,-,-]=>[4,1,2,3,5,6]=>[4,3,2,1,5,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[-,-,+,-,-,-]=>[3,1,2,4,5,6]=>[3,2,1,4,5,6]=>([(3,4),(3,5),(4,5)],6)=>2
[-,+,-,-,-,-]=>[2,1,3,4,5,6]=>[2,1,3,4,5,6]=>([(4,5)],6)=>1
[+,-,-,-,-,-]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>([],6)=>0
[-,-,-,-,-,-]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>([],6)=>0
[+,+,+,+,6,5]=>[1,2,3,4,6,5]=>[1,2,3,4,6,5]=>([(4,5)],6)=>1
[+,-,+,+,6,5]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,+,+,-,6,5]=>[1,2,3,6,4,5]=>[1,2,3,6,5,4]=>([(3,4),(3,5),(4,5)],6)=>2
[+,+,-,-,6,5]=>[1,2,6,3,4,5]=>[1,2,6,5,4,3]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,-,-,-,6,5]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[-,-,-,-,6,5]=>[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,-]=>[1,2,3,5,4,6]=>[1,2,3,5,4,6]=>([(4,5)],6)=>1
[-,+,+,5,4,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,+,-,5,4,-]=>[1,2,5,3,4,6]=>[1,2,5,4,3,6]=>([(3,4),(3,5),(4,5)],6)=>2
[+,-,-,5,4,-]=>[1,5,2,3,4,6]=>[1,5,4,3,2,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[-,-,-,5,4,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[+,+,+,5,6,4]=>[1,2,3,6,4,5]=>[1,2,3,6,5,4]=>([(3,4),(3,5),(4,5)],6)=>2
[+,+,-,5,6,4]=>[1,2,6,3,4,5]=>[1,2,6,5,4,3]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,-,-,5,6,4]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>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
[+,+,+,6,-,4]=>[1,2,3,6,4,5]=>[1,2,3,6,5,4]=>([(3,4),(3,5),(4,5)],6)=>2
[+,+,-,6,-,4]=>[1,2,6,3,4,5]=>[1,2,6,5,4,3]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,-,-,6,-,4]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[-,-,-,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
[+,+,4,3,-,-]=>[1,2,4,3,5,6]=>[1,2,4,3,5,6]=>([(4,5)],6)=>1
[+,-,4,3,-,-]=>[1,4,2,3,5,6]=>[1,4,3,2,5,6]=>([(3,4),(3,5),(4,5)],6)=>2
[-,-,4,3,-,-]=>[4,1,2,3,5,6]=>[4,3,2,1,5,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,+,4,5,3,-]=>[1,2,5,3,4,6]=>[1,2,5,4,3,6]=>([(3,4),(3,5),(4,5)],6)=>2
[+,-,4,5,3,-]=>[1,5,2,3,4,6]=>[1,5,4,3,2,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[-,-,4,5,3,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[+,+,4,5,6,3]=>[1,2,6,3,4,5]=>[1,2,6,5,4,3]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,-,4,5,6,3]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[-,-,4,5,6,3]=>[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
[+,+,4,6,-,3]=>[1,2,6,3,4,5]=>[1,2,6,5,4,3]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,-,4,6,-,3]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[-,-,4,6,-,3]=>[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,-,3,-]=>[1,2,5,3,4,6]=>[1,2,5,4,3,6]=>([(3,4),(3,5),(4,5)],6)=>2
[+,-,5,-,3,-]=>[1,5,2,3,4,6]=>[1,5,4,3,2,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[-,-,5,-,3,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[+,+,5,-,6,3]=>[1,2,6,3,4,5]=>[1,2,6,5,4,3]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,-,5,-,6,3]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[-,-,5,-,6,3]=>[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
[+,+,6,-,-,3]=>[1,2,6,3,4,5]=>[1,2,6,5,4,3]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,-,6,-,-,3]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[-,-,6,-,-,3]=>[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
[+,3,2,+,+,+]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,3,2,+,-,+]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,3,2,-,-,-]=>[1,3,2,4,5,6]=>[1,3,2,4,5,6]=>([(4,5)],6)=>1
[-,3,2,-,-,-]=>[3,1,2,4,5,6]=>[3,2,1,4,5,6]=>([(3,4),(3,5),(4,5)],6)=>2
[+,3,2,+,6,5]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,3,4,2,-,-]=>[1,4,2,3,5,6]=>[1,4,3,2,5,6]=>([(3,4),(3,5),(4,5)],6)=>2
[-,3,4,2,-,-]=>[4,1,2,3,5,6]=>[4,3,2,1,5,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,3,4,5,2,-]=>[1,5,2,3,4,6]=>[1,5,4,3,2,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[-,3,4,5,2,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[+,3,4,5,6,2]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[-,3,4,5,6,2]=>[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
[+,3,4,6,-,2]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[-,3,4,6,-,2]=>[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
[+,3,5,-,2,-]=>[1,5,2,3,4,6]=>[1,5,4,3,2,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[-,3,5,-,2,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[+,3,5,-,6,2]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[-,3,5,-,6,2]=>[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
[+,3,6,-,-,2]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[-,3,6,-,-,2]=>[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
[+,4,2,3,+,+]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,4,2,3,-,+]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,4,2,3,6,5]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,4,+,2,+,+]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,4,+,2,-,+]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,4,-,2,-,-]=>[1,4,2,3,5,6]=>[1,4,3,2,5,6]=>([(3,4),(3,5),(4,5)],6)=>2
[-,4,-,2,-,-]=>[4,1,2,3,5,6]=>[4,3,2,1,5,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,4,+,2,6,5]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,4,-,5,2,-]=>[1,5,2,3,4,6]=>[1,5,4,3,2,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[-,4,-,5,2,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[+,4,-,5,6,2]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[-,4,-,5,6,2]=>[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
[+,4,-,6,-,2]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[-,4,-,6,-,2]=>[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
[+,5,2,3,4,+]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,5,2,3,6,4]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,5,2,+,3,+]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,5,2,+,6,3]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,5,+,2,4,+]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,5,+,2,6,4]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,5,+,+,2,+]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,5,-,-,2,-]=>[1,5,2,3,4,6]=>[1,5,4,3,2,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[-,5,-,-,2,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[+,5,+,+,6,2]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,5,-,-,6,2]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[-,5,-,-,6,2]=>[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
[+,6,2,3,4,5]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,6,2,3,+,4]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,6,2,3,-,4]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,6,2,+,3,5]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,6,2,+,+,3]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,6,2,+,-,3]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,6,+,2,4,5]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,6,+,2,+,4]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,6,+,2,-,4]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,6,+,+,2,5]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,6,+,+,+,2]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[+,6,+,+,-,2]=>[1,3,4,6,2,5]=>[1,6,5,2,3,4]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[+,6,-,-,-,2]=>[1,6,2,3,4,5]=>[1,6,5,4,3,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[-,6,-,-,-,2]=>[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
[2,1,+,+,+,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[2,1,+,-,+,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[2,1,-,-,-,-]=>[2,1,3,4,5,6]=>[2,1,3,4,5,6]=>([(4,5)],6)=>1
[2,1,+,5,4,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[2,3,1,-,-,-]=>[3,1,2,4,5,6]=>[3,2,1,4,5,6]=>([(3,4),(3,5),(4,5)],6)=>2
[2,3,4,1,-,-]=>[4,1,2,3,5,6]=>[4,3,2,1,5,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[2,3,4,5,1,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[2,3,4,5,6,1]=>[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
[2,3,4,6,-,1]=>[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
[2,3,5,-,1,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[2,3,5,-,6,1]=>[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
[2,3,6,-,-,1]=>[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
[2,4,-,1,-,-]=>[4,1,2,3,5,6]=>[4,3,2,1,5,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[2,4,-,5,1,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[2,4,-,5,6,1]=>[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
[2,4,-,6,-,1]=>[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
[2,5,-,-,1,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[2,5,-,-,6,1]=>[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
[2,6,-,-,-,1]=>[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
[3,1,2,+,+,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[3,1,2,-,+,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[3,1,2,5,4,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[3,+,1,+,+,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[3,+,1,-,+,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[3,-,1,-,-,-]=>[3,1,2,4,5,6]=>[3,2,1,4,5,6]=>([(3,4),(3,5),(4,5)],6)=>2
[3,+,1,5,4,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[3,-,4,1,-,-]=>[4,1,2,3,5,6]=>[4,3,2,1,5,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[3,-,4,5,1,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[3,-,4,5,6,1]=>[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
[3,-,4,6,-,1]=>[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
[3,-,5,-,1,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[3,-,5,-,6,1]=>[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
[3,-,6,-,-,1]=>[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
[4,1,2,3,+,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[4,1,2,5,3,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[4,1,+,2,+,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[4,1,+,5,2,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[4,+,1,3,+,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[4,+,1,5,3,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[4,+,+,1,+,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[4,-,-,1,-,-]=>[4,1,2,3,5,6]=>[4,3,2,1,5,6]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[4,+,+,5,1,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[4,-,-,5,1,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[4,-,-,5,6,1]=>[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
[4,-,-,6,-,1]=>[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
[5,1,2,3,4,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[5,1,2,+,3,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[5,1,2,-,3,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[5,1,+,2,4,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[5,1,+,+,2,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[5,1,+,-,2,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[5,+,1,3,4,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[5,+,1,+,3,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[5,+,1,-,3,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[5,+,+,1,4,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[5,+,+,+,1,-]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6]=>([(1,5),(2,5),(3,5),(4,5)],6)=>2
[5,+,+,-,1,-]=>[2,3,5,1,4,6]=>[5,4,1,2,3,6]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
[5,-,-,-,1,-]=>[5,1,2,3,4,6]=>[5,4,3,2,1,6]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
[5,-,-,-,6,1]=>[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
[6,-,-,-,-,1]=>[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
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 largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
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.
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
upper permutation
Description
The upper 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 $v$.
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 $v$.
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!