Identifier
-
Mp00255:
Decorated permutations
—lower permutation⟶
Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤ
Values
=>
Cc0005;cc-rep
[-,+]=>[2,1]=>[1,2]=>[1,0,1,0]=>2
[-,+,+]=>[2,3,1]=>[2,1,3]=>[1,1,0,0,1,0]=>2
[-,-,+]=>[3,1,2]=>[1,3,2]=>[1,0,1,1,0,0]=>2
[-,+,-]=>[2,1,3]=>[2,3,1]=>[1,1,0,1,0,0]=>2
[-,3,2]=>[2,1,3]=>[2,3,1]=>[1,1,0,1,0,0]=>2
[3,+,1]=>[2,1,3]=>[2,3,1]=>[1,1,0,1,0,0]=>2
[-,+,+,+]=>[2,3,4,1]=>[3,2,1,4]=>[1,1,1,0,0,0,1,0]=>2
[-,-,+,+]=>[3,4,1,2]=>[2,1,4,3]=>[1,1,0,0,1,1,0,0]=>2
[-,+,-,+]=>[2,4,1,3]=>[3,1,4,2]=>[1,1,1,0,0,1,0,0]=>2
[-,+,+,-]=>[2,3,1,4]=>[3,2,4,1]=>[1,1,1,0,0,1,0,0]=>2
[-,-,-,+]=>[4,1,2,3]=>[1,4,3,2]=>[1,0,1,1,1,0,0,0]=>2
[-,-,+,-]=>[3,1,2,4]=>[2,4,3,1]=>[1,1,0,1,1,0,0,0]=>2
[-,+,-,-]=>[2,1,3,4]=>[3,4,2,1]=>[1,1,1,0,1,0,0,0]=>2
[-,+,4,3]=>[2,3,1,4]=>[3,2,4,1]=>[1,1,1,0,0,1,0,0]=>2
[-,-,4,3]=>[3,1,2,4]=>[2,4,3,1]=>[1,1,0,1,1,0,0,0]=>2
[-,3,2,+]=>[2,4,1,3]=>[3,1,4,2]=>[1,1,1,0,0,1,0,0]=>2
[-,3,2,-]=>[2,1,3,4]=>[3,4,2,1]=>[1,1,1,0,1,0,0,0]=>2
[-,3,4,2]=>[2,1,3,4]=>[3,4,2,1]=>[1,1,1,0,1,0,0,0]=>2
[-,4,2,3]=>[2,3,1,4]=>[3,2,4,1]=>[1,1,1,0,0,1,0,0]=>2
[-,4,+,2]=>[3,2,1,4]=>[2,3,4,1]=>[1,1,0,1,0,1,0,0]=>3
[-,4,-,2]=>[2,1,4,3]=>[3,4,1,2]=>[1,1,1,0,1,0,0,0]=>2
[2,4,+,1]=>[3,1,2,4]=>[2,4,3,1]=>[1,1,0,1,1,0,0,0]=>2
[3,+,1,+]=>[2,1,4,3]=>[3,4,1,2]=>[1,1,1,0,1,0,0,0]=>2
[3,+,1,-]=>[2,1,3,4]=>[3,4,2,1]=>[1,1,1,0,1,0,0,0]=>2
[3,+,4,1]=>[2,1,3,4]=>[3,4,2,1]=>[1,1,1,0,1,0,0,0]=>2
[3,4,2,1]=>[2,1,3,4]=>[3,4,2,1]=>[1,1,1,0,1,0,0,0]=>2
[4,+,1,3]=>[2,1,3,4]=>[3,4,2,1]=>[1,1,1,0,1,0,0,0]=>2
[4,+,+,1]=>[2,3,1,4]=>[3,2,4,1]=>[1,1,1,0,0,1,0,0]=>2
[4,-,+,1]=>[3,1,4,2]=>[2,4,1,3]=>[1,1,0,1,1,0,0,0]=>2
[4,+,-,1]=>[2,1,4,3]=>[3,4,1,2]=>[1,1,1,0,1,0,0,0]=>2
[4,3,2,1]=>[2,1,4,3]=>[3,4,1,2]=>[1,1,1,0,1,0,0,0]=>2
[-,+,+,+,+]=>[2,3,4,5,1]=>[4,3,2,1,5]=>[1,1,1,1,0,0,0,0,1,0]=>2
[-,-,+,+,+]=>[3,4,5,1,2]=>[3,2,1,5,4]=>[1,1,1,0,0,0,1,1,0,0]=>2
[-,+,-,+,+]=>[2,4,5,1,3]=>[4,2,1,5,3]=>[1,1,1,1,0,0,0,1,0,0]=>2
[-,+,+,-,+]=>[2,3,5,1,4]=>[4,3,1,5,2]=>[1,1,1,1,0,0,0,1,0,0]=>2
[-,+,+,+,-]=>[2,3,4,1,5]=>[4,3,2,5,1]=>[1,1,1,1,0,0,0,1,0,0]=>2
[-,-,-,+,+]=>[4,5,1,2,3]=>[2,1,5,4,3]=>[1,1,0,0,1,1,1,0,0,0]=>2
[-,-,+,-,+]=>[3,5,1,2,4]=>[3,1,5,4,2]=>[1,1,1,0,0,1,1,0,0,0]=>2
[-,-,+,+,-]=>[3,4,1,2,5]=>[3,2,5,4,1]=>[1,1,1,0,0,1,1,0,0,0]=>2
[-,+,-,-,+]=>[2,5,1,3,4]=>[4,1,5,3,2]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,+,-,+,-]=>[2,4,1,3,5]=>[4,2,5,3,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,+,+,-,-]=>[2,3,1,4,5]=>[4,3,5,2,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,-,-,-,+]=>[5,1,2,3,4]=>[1,5,4,3,2]=>[1,0,1,1,1,1,0,0,0,0]=>2
[-,-,-,+,-]=>[4,1,2,3,5]=>[2,5,4,3,1]=>[1,1,0,1,1,1,0,0,0,0]=>2
[-,-,+,-,-]=>[3,1,2,4,5]=>[3,5,4,2,1]=>[1,1,1,0,1,1,0,0,0,0]=>2
[-,+,-,-,-]=>[2,1,3,4,5]=>[4,5,3,2,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[-,+,+,5,4]=>[2,3,4,1,5]=>[4,3,2,5,1]=>[1,1,1,1,0,0,0,1,0,0]=>2
[-,-,+,5,4]=>[3,4,1,2,5]=>[3,2,5,4,1]=>[1,1,1,0,0,1,1,0,0,0]=>2
[-,+,-,5,4]=>[2,4,1,3,5]=>[4,2,5,3,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,-,-,5,4]=>[4,1,2,3,5]=>[2,5,4,3,1]=>[1,1,0,1,1,1,0,0,0,0]=>2
[-,+,4,3,+]=>[2,3,5,1,4]=>[4,3,1,5,2]=>[1,1,1,1,0,0,0,1,0,0]=>2
[-,-,4,3,+]=>[3,5,1,2,4]=>[3,1,5,4,2]=>[1,1,1,0,0,1,1,0,0,0]=>2
[-,+,4,3,-]=>[2,3,1,4,5]=>[4,3,5,2,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,-,4,3,-]=>[3,1,2,4,5]=>[3,5,4,2,1]=>[1,1,1,0,1,1,0,0,0,0]=>2
[-,+,4,5,3]=>[2,3,1,4,5]=>[4,3,5,2,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,-,4,5,3]=>[3,1,2,4,5]=>[3,5,4,2,1]=>[1,1,1,0,1,1,0,0,0,0]=>2
[-,+,5,3,4]=>[2,3,4,1,5]=>[4,3,2,5,1]=>[1,1,1,1,0,0,0,1,0,0]=>2
[-,-,5,3,4]=>[3,4,1,2,5]=>[3,2,5,4,1]=>[1,1,1,0,0,1,1,0,0,0]=>2
[-,+,5,+,3]=>[2,4,3,1,5]=>[4,2,3,5,1]=>[1,1,1,1,0,0,0,1,0,0]=>2
[-,-,5,+,3]=>[4,3,1,2,5]=>[2,3,5,4,1]=>[1,1,0,1,0,1,1,0,0,0]=>3
[-,+,5,-,3]=>[2,3,1,5,4]=>[4,3,5,1,2]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,-,5,-,3]=>[3,1,2,5,4]=>[3,5,4,1,2]=>[1,1,1,0,1,1,0,0,0,0]=>2
[-,3,2,+,+]=>[2,4,5,1,3]=>[4,2,1,5,3]=>[1,1,1,1,0,0,0,1,0,0]=>2
[-,3,2,-,+]=>[2,5,1,3,4]=>[4,1,5,3,2]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,3,2,+,-]=>[2,4,1,3,5]=>[4,2,5,3,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,3,2,-,-]=>[2,1,3,4,5]=>[4,5,3,2,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[-,3,2,5,4]=>[2,4,1,3,5]=>[4,2,5,3,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,3,4,2,+]=>[2,5,1,3,4]=>[4,1,5,3,2]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,3,4,2,-]=>[2,1,3,4,5]=>[4,5,3,2,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[-,3,4,5,2]=>[2,1,3,4,5]=>[4,5,3,2,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[-,3,5,2,4]=>[2,4,1,3,5]=>[4,2,5,3,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,3,5,+,2]=>[4,2,1,3,5]=>[2,4,5,3,1]=>[1,1,0,1,1,0,1,0,0,0]=>3
[-,3,5,-,2]=>[2,1,3,5,4]=>[4,5,3,1,2]=>[1,1,1,1,0,1,0,0,0,0]=>2
[-,4,2,3,+]=>[2,3,5,1,4]=>[4,3,1,5,2]=>[1,1,1,1,0,0,0,1,0,0]=>2
[-,4,2,3,-]=>[2,3,1,4,5]=>[4,3,5,2,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,4,2,5,3]=>[2,3,1,4,5]=>[4,3,5,2,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,4,+,2,+]=>[3,2,5,1,4]=>[3,4,1,5,2]=>[1,1,1,0,1,0,0,1,0,0]=>3
[-,4,-,2,+]=>[2,5,1,4,3]=>[4,1,5,2,3]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,4,+,2,-]=>[3,2,1,4,5]=>[3,4,5,2,1]=>[1,1,1,0,1,0,1,0,0,0]=>3
[-,4,-,2,-]=>[2,1,4,3,5]=>[4,5,2,3,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[-,4,+,5,2]=>[3,2,1,4,5]=>[3,4,5,2,1]=>[1,1,1,0,1,0,1,0,0,0]=>3
[-,4,-,5,2]=>[2,1,4,3,5]=>[4,5,2,3,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[-,4,5,2,3]=>[2,3,1,4,5]=>[4,3,5,2,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,4,5,3,2]=>[3,2,1,4,5]=>[3,4,5,2,1]=>[1,1,1,0,1,0,1,0,0,0]=>3
[-,5,2,3,4]=>[2,3,4,1,5]=>[4,3,2,5,1]=>[1,1,1,1,0,0,0,1,0,0]=>2
[-,5,2,+,3]=>[2,4,3,1,5]=>[4,2,3,5,1]=>[1,1,1,1,0,0,0,1,0,0]=>2
[-,5,2,-,3]=>[2,3,1,5,4]=>[4,3,5,1,2]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,5,+,2,4]=>[3,2,4,1,5]=>[3,4,2,5,1]=>[1,1,1,0,1,0,0,1,0,0]=>3
[-,5,-,2,4]=>[2,4,1,5,3]=>[4,2,5,1,3]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,5,+,+,2]=>[3,4,2,1,5]=>[3,2,4,5,1]=>[1,1,1,0,0,1,0,1,0,0]=>3
[-,5,-,+,2]=>[4,2,1,5,3]=>[2,4,5,1,3]=>[1,1,0,1,1,0,1,0,0,0]=>3
[-,5,+,-,2]=>[3,2,1,5,4]=>[3,4,5,1,2]=>[1,1,1,0,1,0,1,0,0,0]=>3
[-,5,-,-,2]=>[2,1,5,3,4]=>[4,5,1,3,2]=>[1,1,1,1,0,1,0,0,0,0]=>2
[-,5,4,2,3]=>[2,3,1,5,4]=>[4,3,5,1,2]=>[1,1,1,1,0,0,1,0,0,0]=>2
[-,5,4,3,2]=>[3,2,1,5,4]=>[3,4,5,1,2]=>[1,1,1,0,1,0,1,0,0,0]=>3
[2,3,5,+,1]=>[4,1,2,3,5]=>[2,5,4,3,1]=>[1,1,0,1,1,1,0,0,0,0]=>2
[2,4,+,1,+]=>[3,1,5,2,4]=>[3,5,1,4,2]=>[1,1,1,0,1,1,0,0,0,0]=>2
[2,4,+,1,-]=>[3,1,2,4,5]=>[3,5,4,2,1]=>[1,1,1,0,1,1,0,0,0,0]=>2
[2,4,+,5,1]=>[3,1,2,4,5]=>[3,5,4,2,1]=>[1,1,1,0,1,1,0,0,0,0]=>2
[2,4,5,3,1]=>[3,1,2,4,5]=>[3,5,4,2,1]=>[1,1,1,0,1,1,0,0,0,0]=>2
[2,5,+,1,4]=>[3,1,4,2,5]=>[3,5,2,4,1]=>[1,1,1,0,1,1,0,0,0,0]=>2
[2,5,+,+,1]=>[3,4,1,2,5]=>[3,2,5,4,1]=>[1,1,1,0,0,1,1,0,0,0]=>2
[2,5,-,+,1]=>[4,1,2,5,3]=>[2,5,4,1,3]=>[1,1,0,1,1,1,0,0,0,0]=>2
[2,5,+,-,1]=>[3,1,2,5,4]=>[3,5,4,1,2]=>[1,1,1,0,1,1,0,0,0,0]=>2
[2,5,4,3,1]=>[3,1,2,5,4]=>[3,5,4,1,2]=>[1,1,1,0,1,1,0,0,0,0]=>2
[3,+,1,+,+]=>[2,1,4,5,3]=>[4,5,2,1,3]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,+,1,-,+]=>[2,1,5,3,4]=>[4,5,1,3,2]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,+,1,+,-]=>[2,1,4,3,5]=>[4,5,2,3,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,+,1,-,-]=>[2,1,3,4,5]=>[4,5,3,2,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,+,1,5,4]=>[2,1,4,3,5]=>[4,5,2,3,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,+,4,1,+]=>[2,1,5,3,4]=>[4,5,1,3,2]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,+,4,1,-]=>[2,1,3,4,5]=>[4,5,3,2,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,+,4,5,1]=>[2,1,3,4,5]=>[4,5,3,2,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,+,5,1,4]=>[2,1,4,3,5]=>[4,5,2,3,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,+,5,+,1]=>[2,4,1,3,5]=>[4,2,5,3,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[3,-,5,+,1]=>[4,1,3,2,5]=>[2,5,3,4,1]=>[1,1,0,1,1,1,0,0,0,0]=>2
[3,+,5,-,1]=>[2,1,3,5,4]=>[4,5,3,1,2]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,4,2,1,+]=>[2,1,5,3,4]=>[4,5,1,3,2]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,4,2,1,-]=>[2,1,3,4,5]=>[4,5,3,2,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,4,2,5,1]=>[2,1,3,4,5]=>[4,5,3,2,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,4,5,2,1]=>[2,1,3,4,5]=>[4,5,3,2,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,5,2,1,4]=>[2,1,4,3,5]=>[4,5,2,3,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,5,2,+,1]=>[2,4,1,3,5]=>[4,2,5,3,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[3,5,2,-,1]=>[2,1,3,5,4]=>[4,5,3,1,2]=>[1,1,1,1,0,1,0,0,0,0]=>2
[3,5,4,2,1]=>[2,1,3,5,4]=>[4,5,3,1,2]=>[1,1,1,1,0,1,0,0,0,0]=>2
[4,+,1,3,+]=>[2,1,3,5,4]=>[4,5,3,1,2]=>[1,1,1,1,0,1,0,0,0,0]=>2
[4,+,1,3,-]=>[2,1,3,4,5]=>[4,5,3,2,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[4,+,1,5,3]=>[2,1,3,4,5]=>[4,5,3,2,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[4,+,+,1,+]=>[2,3,1,5,4]=>[4,3,5,1,2]=>[1,1,1,1,0,0,1,0,0,0]=>2
[4,-,+,1,+]=>[3,1,5,4,2]=>[3,5,1,2,4]=>[1,1,1,0,1,1,0,0,0,0]=>2
[4,+,-,1,+]=>[2,1,5,4,3]=>[4,5,1,2,3]=>[1,1,1,1,0,1,0,0,0,0]=>2
[4,+,+,1,-]=>[2,3,1,4,5]=>[4,3,5,2,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[4,-,+,1,-]=>[3,1,4,2,5]=>[3,5,2,4,1]=>[1,1,1,0,1,1,0,0,0,0]=>2
[4,+,-,1,-]=>[2,1,4,3,5]=>[4,5,2,3,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[4,+,+,5,1]=>[2,3,1,4,5]=>[4,3,5,2,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[4,-,+,5,1]=>[3,1,4,2,5]=>[3,5,2,4,1]=>[1,1,1,0,1,1,0,0,0,0]=>2
[4,+,-,5,1]=>[2,1,4,3,5]=>[4,5,2,3,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[4,+,5,1,3]=>[2,1,3,4,5]=>[4,5,3,2,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[4,+,5,3,1]=>[2,3,1,4,5]=>[4,3,5,2,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[4,-,5,3,1]=>[3,1,4,2,5]=>[3,5,2,4,1]=>[1,1,1,0,1,1,0,0,0,0]=>2
[4,3,2,1,+]=>[2,1,5,4,3]=>[4,5,1,2,3]=>[1,1,1,1,0,1,0,0,0,0]=>2
[4,3,2,1,-]=>[2,1,4,3,5]=>[4,5,2,3,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[4,3,2,5,1]=>[2,1,4,3,5]=>[4,5,2,3,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[4,3,5,2,1]=>[2,1,4,3,5]=>[4,5,2,3,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[4,5,2,1,3]=>[2,1,3,4,5]=>[4,5,3,2,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[4,5,2,3,1]=>[2,3,1,4,5]=>[4,3,5,2,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[4,5,+,1,2]=>[3,1,2,4,5]=>[3,5,4,2,1]=>[1,1,1,0,1,1,0,0,0,0]=>2
[4,5,+,2,1]=>[3,2,1,4,5]=>[3,4,5,2,1]=>[1,1,1,0,1,0,1,0,0,0]=>3
[4,5,-,2,1]=>[2,1,4,5,3]=>[4,5,2,1,3]=>[1,1,1,1,0,1,0,0,0,0]=>2
[5,+,1,3,4]=>[2,1,3,4,5]=>[4,5,3,2,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[5,+,1,+,3]=>[2,1,4,3,5]=>[4,5,2,3,1]=>[1,1,1,1,0,1,0,0,0,0]=>2
[5,+,1,-,3]=>[2,1,3,5,4]=>[4,5,3,1,2]=>[1,1,1,1,0,1,0,0,0,0]=>2
[5,+,+,1,4]=>[2,3,1,4,5]=>[4,3,5,2,1]=>[1,1,1,1,0,0,1,0,0,0]=>2
[5,-,+,1,4]=>[3,1,4,5,2]=>[3,5,2,1,4]=>[1,1,1,0,1,1,0,0,0,0]=>2
[5,+,-,1,4]=>[2,1,4,5,3]=>[4,5,2,1,3]=>[1,1,1,1,0,1,0,0,0,0]=>2
[5,+,+,+,1]=>[2,3,4,1,5]=>[4,3,2,5,1]=>[1,1,1,1,0,0,0,1,0,0]=>2
[5,-,+,+,1]=>[3,4,1,5,2]=>[3,2,5,1,4]=>[1,1,1,0,0,1,1,0,0,0]=>2
[5,+,-,+,1]=>[2,4,1,5,3]=>[4,2,5,1,3]=>[1,1,1,1,0,0,1,0,0,0]=>2
[5,+,+,-,1]=>[2,3,1,5,4]=>[4,3,5,1,2]=>[1,1,1,1,0,0,1,0,0,0]=>2
[5,-,-,+,1]=>[4,1,5,2,3]=>[2,5,1,4,3]=>[1,1,0,1,1,1,0,0,0,0]=>2
[5,-,+,-,1]=>[3,1,5,2,4]=>[3,5,1,4,2]=>[1,1,1,0,1,1,0,0,0,0]=>2
[5,+,-,-,1]=>[2,1,5,3,4]=>[4,5,1,3,2]=>[1,1,1,1,0,1,0,0,0,0]=>2
[5,+,4,1,3]=>[2,1,3,5,4]=>[4,5,3,1,2]=>[1,1,1,1,0,1,0,0,0,0]=>2
[5,+,4,3,1]=>[2,3,1,5,4]=>[4,3,5,1,2]=>[1,1,1,1,0,0,1,0,0,0]=>2
[5,-,4,3,1]=>[3,1,5,2,4]=>[3,5,1,4,2]=>[1,1,1,0,1,1,0,0,0,0]=>2
[5,3,2,1,4]=>[2,1,4,5,3]=>[4,5,2,1,3]=>[1,1,1,1,0,1,0,0,0,0]=>2
[5,3,2,+,1]=>[2,4,1,5,3]=>[4,2,5,1,3]=>[1,1,1,1,0,0,1,0,0,0]=>2
[5,3,2,-,1]=>[2,1,5,3,4]=>[4,5,1,3,2]=>[1,1,1,1,0,1,0,0,0,0]=>2
[5,3,4,2,1]=>[2,1,5,3,4]=>[4,5,1,3,2]=>[1,1,1,1,0,1,0,0,0,0]=>2
[5,4,2,1,3]=>[2,1,3,5,4]=>[4,5,3,1,2]=>[1,1,1,1,0,1,0,0,0,0]=>2
[5,4,2,3,1]=>[2,3,1,5,4]=>[4,3,5,1,2]=>[1,1,1,1,0,0,1,0,0,0]=>2
[5,4,+,1,2]=>[3,1,2,5,4]=>[3,5,4,1,2]=>[1,1,1,0,1,1,0,0,0,0]=>2
[5,4,+,2,1]=>[3,2,1,5,4]=>[3,4,5,1,2]=>[1,1,1,0,1,0,1,0,0,0]=>3
[5,4,-,2,1]=>[2,1,5,4,3]=>[4,5,1,2,3]=>[1,1,1,1,0,1,0,0,0,0]=>2
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Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
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 $u$.
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