Identifier
- St001818: Decorated permutations ⟶ ℤ
Values
=>
[+]=>0
[-]=>0
[+,+]=>0
[-,+]=>1
[+,-]=>1
[-,-]=>0
[2,1]=>0
[+,+,+]=>0
[-,+,+]=>2
[+,-,+]=>2
[+,+,-]=>2
[-,-,+]=>2
[-,+,-]=>2
[+,-,-]=>2
[-,-,-]=>0
[+,3,2]=>1
[-,3,2]=>1
[2,1,+]=>1
[2,1,-]=>1
[2,3,1]=>0
[3,1,2]=>0
[3,+,1]=>1
[3,-,1]=>1
[+,+,+,+]=>0
[-,+,+,+]=>3
[+,-,+,+]=>3
[+,+,-,+]=>3
[+,+,+,-]=>3
[-,-,+,+]=>4
[-,+,-,+]=>4
[-,+,+,-]=>4
[+,-,-,+]=>4
[+,-,+,-]=>4
[+,+,-,-]=>4
[-,-,-,+]=>3
[-,-,+,-]=>3
[-,+,-,-]=>3
[+,-,-,-]=>3
[-,-,-,-]=>0
[+,+,4,3]=>2
[-,+,4,3]=>3
[+,-,4,3]=>3
[-,-,4,3]=>2
[+,3,2,+]=>2
[-,3,2,+]=>3
[+,3,2,-]=>3
[-,3,2,-]=>2
[+,3,4,2]=>2
[-,3,4,2]=>1
[+,4,2,3]=>1
[-,4,2,3]=>2
[+,4,+,2]=>2
[-,4,+,2]=>3
[+,4,-,2]=>3
[-,4,-,2]=>2
[2,1,+,+]=>2
[2,1,-,+]=>3
[2,1,+,-]=>3
[2,1,-,-]=>2
[2,1,4,3]=>2
[2,3,1,+]=>2
[2,3,1,-]=>1
[2,3,4,1]=>0
[2,4,1,3]=>1
[2,4,+,1]=>2
[2,4,-,1]=>1
[3,1,2,+]=>1
[3,1,2,-]=>2
[3,1,4,2]=>1
[3,+,1,+]=>2
[3,-,1,+]=>3
[3,+,1,-]=>3
[3,-,1,-]=>2
[3,+,4,1]=>2
[3,-,4,1]=>1
[3,4,1,2]=>0
[3,4,2,1]=>1
[4,1,2,3]=>0
[4,1,+,2]=>1
[4,1,-,2]=>2
[4,+,1,3]=>1
[4,-,1,3]=>2
[4,+,+,1]=>2
[4,-,+,1]=>3
[4,+,-,1]=>3
[4,-,-,1]=>2
[4,3,1,2]=>1
[4,3,2,1]=>2
[+,+,+,+,+]=>0
[-,+,+,+,+]=>4
[+,-,+,+,+]=>4
[+,+,-,+,+]=>4
[+,+,+,-,+]=>4
[+,+,+,+,-]=>4
[-,-,+,+,+]=>6
[-,+,-,+,+]=>6
[-,+,+,-,+]=>6
[-,+,+,+,-]=>6
[+,-,-,+,+]=>6
[+,-,+,-,+]=>6
[+,-,+,+,-]=>6
[+,+,-,-,+]=>6
[+,+,-,+,-]=>6
[+,+,+,-,-]=>6
[-,-,-,+,+]=>6
[-,-,+,-,+]=>6
[-,-,+,+,-]=>6
[-,+,-,-,+]=>6
[-,+,-,+,-]=>6
[-,+,+,-,-]=>6
[+,-,-,-,+]=>6
[+,-,-,+,-]=>6
[+,-,+,-,-]=>6
[+,+,-,-,-]=>6
[-,-,-,-,+]=>4
[-,-,-,+,-]=>4
[-,-,+,-,-]=>4
[-,+,-,-,-]=>4
[+,-,-,-,-]=>4
[-,-,-,-,-]=>0
[+,+,+,5,4]=>3
[-,+,+,5,4]=>5
[+,-,+,5,4]=>5
[+,+,-,5,4]=>5
[-,-,+,5,4]=>5
[-,+,-,5,4]=>5
[+,-,-,5,4]=>5
[-,-,-,5,4]=>3
[+,+,4,3,+]=>3
[-,+,4,3,+]=>5
[+,-,4,3,+]=>5
[+,+,4,3,-]=>5
[-,-,4,3,+]=>5
[-,+,4,3,-]=>5
[+,-,4,3,-]=>5
[-,-,4,3,-]=>3
[+,+,4,5,3]=>4
[-,+,4,5,3]=>4
[+,-,4,5,3]=>4
[-,-,4,5,3]=>2
[+,+,5,3,4]=>2
[-,+,5,3,4]=>4
[+,-,5,3,4]=>4
[-,-,5,3,4]=>4
[+,+,5,+,3]=>3
[-,+,5,+,3]=>5
[+,-,5,+,3]=>5
[+,+,5,-,3]=>5
[-,-,5,+,3]=>5
[-,+,5,-,3]=>5
[+,-,5,-,3]=>5
[-,-,5,-,3]=>3
[+,3,2,+,+]=>3
[-,3,2,+,+]=>5
[+,3,2,-,+]=>5
[+,3,2,+,-]=>5
[-,3,2,-,+]=>5
[-,3,2,+,-]=>5
[+,3,2,-,-]=>5
[-,3,2,-,-]=>3
[+,3,2,5,4]=>4
[-,3,2,5,4]=>4
[+,3,4,2,+]=>4
[-,3,4,2,+]=>4
[+,3,4,2,-]=>4
[-,3,4,2,-]=>2
[+,3,4,5,2]=>3
[-,3,4,5,2]=>1
[+,3,5,2,4]=>3
[-,3,5,2,4]=>3
[+,3,5,+,2]=>4
[-,3,5,+,2]=>4
[+,3,5,-,2]=>4
[-,3,5,-,2]=>2
[+,4,2,3,+]=>2
[-,4,2,3,+]=>4
[+,4,2,3,-]=>4
[-,4,2,3,-]=>4
[+,4,2,5,3]=>3
[-,4,2,5,3]=>3
[+,4,+,2,+]=>3
[-,4,+,2,+]=>5
[+,4,-,2,+]=>5
[+,4,+,2,-]=>5
[-,4,-,2,+]=>5
[-,4,+,2,-]=>5
[+,4,-,2,-]=>5
[-,4,-,2,-]=>3
[+,4,+,5,2]=>4
[-,4,+,5,2]=>4
[+,4,-,5,2]=>4
[-,4,-,5,2]=>2
[+,4,5,2,3]=>2
[-,4,5,2,3]=>2
[+,4,5,3,2]=>3
[-,4,5,3,2]=>3
[+,5,2,3,4]=>1
[-,5,2,3,4]=>3
[+,5,2,+,3]=>2
[-,5,2,+,3]=>4
[+,5,2,-,3]=>4
[-,5,2,-,3]=>4
[+,5,+,2,4]=>2
[-,5,+,2,4]=>4
[+,5,-,2,4]=>4
[-,5,-,2,4]=>4
[+,5,+,+,2]=>3
[-,5,+,+,2]=>5
[+,5,-,+,2]=>5
[+,5,+,-,2]=>5
[-,5,-,+,2]=>5
[-,5,+,-,2]=>5
[+,5,-,-,2]=>5
[-,5,-,-,2]=>3
[+,5,4,2,3]=>3
[-,5,4,2,3]=>3
[+,5,4,3,2]=>4
[-,5,4,3,2]=>4
[2,1,+,+,+]=>3
[2,1,-,+,+]=>5
[2,1,+,-,+]=>5
[2,1,+,+,-]=>5
[2,1,-,-,+]=>5
[2,1,-,+,-]=>5
[2,1,+,-,-]=>5
[2,1,-,-,-]=>3
[2,1,+,5,4]=>4
[2,1,-,5,4]=>4
[2,1,4,3,+]=>4
[2,1,4,3,-]=>4
[2,1,4,5,3]=>3
[2,1,5,3,4]=>3
[2,1,5,+,3]=>4
[2,1,5,-,3]=>4
[2,3,1,+,+]=>4
[2,3,1,-,+]=>4
[2,3,1,+,-]=>4
[2,3,1,-,-]=>2
[2,3,1,5,4]=>3
[2,3,4,1,+]=>3
[2,3,4,1,-]=>1
[2,3,4,5,1]=>0
[2,3,5,1,4]=>2
[2,3,5,+,1]=>3
[2,3,5,-,1]=>1
[2,4,1,3,+]=>3
[2,4,1,3,-]=>3
[2,4,1,5,3]=>2
[2,4,+,1,+]=>4
[2,4,-,1,+]=>4
[2,4,+,1,-]=>4
[2,4,-,1,-]=>2
[2,4,+,5,1]=>3
[2,4,-,5,1]=>1
[2,4,5,1,3]=>1
[2,4,5,3,1]=>2
[2,5,1,3,4]=>2
[2,5,1,+,3]=>3
[2,5,1,-,3]=>3
[2,5,+,1,4]=>3
[2,5,-,1,4]=>3
[2,5,+,+,1]=>4
[2,5,-,+,1]=>4
[2,5,+,-,1]=>4
[2,5,-,-,1]=>2
[2,5,4,1,3]=>2
[2,5,4,3,1]=>3
[3,1,2,+,+]=>2
[3,1,2,-,+]=>4
[3,1,2,+,-]=>4
[3,1,2,-,-]=>4
[3,1,2,5,4]=>3
[3,1,4,2,+]=>3
[3,1,4,2,-]=>3
[3,1,4,5,2]=>2
[3,1,5,2,4]=>2
[3,1,5,+,2]=>3
[3,1,5,-,2]=>3
[3,+,1,+,+]=>3
[3,-,1,+,+]=>5
[3,+,1,-,+]=>5
[3,+,1,+,-]=>5
[3,-,1,-,+]=>5
[3,-,1,+,-]=>5
[3,+,1,-,-]=>5
[3,-,1,-,-]=>3
[3,+,1,5,4]=>4
[3,-,1,5,4]=>4
[3,+,4,1,+]=>4
[3,-,4,1,+]=>4
[3,+,4,1,-]=>4
[3,-,4,1,-]=>2
[3,+,4,5,1]=>3
[3,-,4,5,1]=>1
[3,+,5,1,4]=>3
[3,-,5,1,4]=>3
[3,+,5,+,1]=>4
[3,-,5,+,1]=>4
[3,+,5,-,1]=>4
[3,-,5,-,1]=>2
[3,4,1,2,+]=>2
[3,4,1,2,-]=>2
[3,4,1,5,2]=>1
[3,4,2,1,+]=>3
[3,4,2,1,-]=>3
[3,4,2,5,1]=>2
[3,4,5,1,2]=>0
[3,4,5,2,1]=>1
[3,5,1,2,4]=>1
[3,5,1,+,2]=>2
[3,5,1,-,2]=>2
[3,5,2,1,4]=>2
[3,5,2,+,1]=>3
[3,5,2,-,1]=>3
[3,5,4,1,2]=>1
[3,5,4,2,1]=>2
[4,1,2,3,+]=>1
[4,1,2,3,-]=>3
[4,1,2,5,3]=>2
[4,1,+,2,+]=>2
[4,1,-,2,+]=>4
[4,1,+,2,-]=>4
[4,1,-,2,-]=>4
[4,1,+,5,2]=>3
[4,1,-,5,2]=>3
[4,1,5,2,3]=>1
[4,1,5,3,2]=>2
[4,+,1,3,+]=>2
[4,-,1,3,+]=>4
[4,+,1,3,-]=>4
[4,-,1,3,-]=>4
[4,+,1,5,3]=>3
[4,-,1,5,3]=>3
[4,+,+,1,+]=>3
[4,-,+,1,+]=>5
[4,+,-,1,+]=>5
[4,+,+,1,-]=>5
[4,-,-,1,+]=>5
[4,-,+,1,-]=>5
[4,+,-,1,-]=>5
[4,-,-,1,-]=>3
[4,+,+,5,1]=>4
[4,-,+,5,1]=>4
[4,+,-,5,1]=>4
[4,-,-,5,1]=>2
[4,+,5,1,3]=>2
[4,-,5,1,3]=>2
[4,+,5,3,1]=>3
[4,-,5,3,1]=>3
[4,3,1,2,+]=>3
[4,3,1,2,-]=>3
[4,3,1,5,2]=>2
[4,3,2,1,+]=>4
[4,3,2,1,-]=>4
[4,3,2,5,1]=>3
[4,3,5,1,2]=>1
[4,3,5,2,1]=>2
[4,5,1,2,3]=>0
[4,5,1,3,2]=>1
[4,5,2,1,3]=>1
[4,5,2,3,1]=>2
[4,5,+,1,2]=>2
[4,5,-,1,2]=>2
[4,5,+,2,1]=>3
[4,5,-,2,1]=>3
[5,1,2,3,4]=>0
[5,1,2,+,3]=>1
[5,1,2,-,3]=>3
[5,1,+,2,4]=>1
[5,1,-,2,4]=>3
[5,1,+,+,2]=>2
[5,1,-,+,2]=>4
[5,1,+,-,2]=>4
[5,1,-,-,2]=>4
[5,1,4,2,3]=>2
[5,1,4,3,2]=>3
[5,+,1,3,4]=>1
[5,-,1,3,4]=>3
[5,+,1,+,3]=>2
[5,-,1,+,3]=>4
[5,+,1,-,3]=>4
[5,-,1,-,3]=>4
[5,+,+,1,4]=>2
[5,-,+,1,4]=>4
[5,+,-,1,4]=>4
[5,-,-,1,4]=>4
[5,+,+,+,1]=>3
[5,-,+,+,1]=>5
[5,+,-,+,1]=>5
[5,+,+,-,1]=>5
[5,-,-,+,1]=>5
[5,-,+,-,1]=>5
[5,+,-,-,1]=>5
[5,-,-,-,1]=>3
[5,+,4,1,3]=>3
[5,-,4,1,3]=>3
[5,+,4,3,1]=>4
[5,-,4,3,1]=>4
[5,3,1,2,4]=>2
[5,3,1,+,2]=>3
[5,3,1,-,2]=>3
[5,3,2,1,4]=>3
[5,3,2,+,1]=>4
[5,3,2,-,1]=>4
[5,3,4,1,2]=>2
[5,3,4,2,1]=>3
[5,4,1,2,3]=>1
[5,4,1,3,2]=>2
[5,4,2,1,3]=>2
[5,4,2,3,1]=>3
[5,4,+,1,2]=>3
[5,4,-,1,2]=>3
[5,4,+,2,1]=>4
[5,4,-,2,1]=>4
[+,+,+,+,+,+]=>0
[-,+,+,+,+,+]=>5
[+,-,+,+,+,+]=>5
[+,+,-,+,+,+]=>5
[+,+,+,-,+,+]=>5
[+,+,+,+,-,+]=>5
[+,+,+,+,+,-]=>5
[-,-,+,+,+,+]=>8
[-,+,-,+,+,+]=>8
[-,+,+,-,+,+]=>8
[-,+,+,+,-,+]=>8
[-,+,+,+,+,-]=>8
[+,-,-,+,+,+]=>8
[+,-,+,-,+,+]=>8
[+,-,+,+,-,+]=>8
[+,-,+,+,+,-]=>8
[+,+,-,-,+,+]=>8
[+,+,-,+,-,+]=>8
[+,+,-,+,+,-]=>8
[+,+,+,-,-,+]=>8
[+,+,+,-,+,-]=>8
[+,+,+,+,-,-]=>8
[-,-,-,+,+,+]=>9
[-,-,+,-,+,+]=>9
[-,-,+,+,-,+]=>9
[-,-,+,+,+,-]=>9
[-,+,-,-,+,+]=>9
[-,+,-,+,-,+]=>9
[-,+,-,+,+,-]=>9
[-,+,+,-,-,+]=>9
[-,+,+,-,+,-]=>9
[-,+,+,+,-,-]=>9
[+,-,-,-,+,+]=>9
[+,-,-,+,-,+]=>9
[+,-,-,+,+,-]=>9
[+,-,+,-,-,+]=>9
[+,-,+,-,+,-]=>9
[+,-,+,+,-,-]=>9
[+,+,-,-,-,+]=>9
[+,+,-,-,+,-]=>9
[+,+,-,+,-,-]=>9
[+,+,+,-,-,-]=>9
[-,-,-,-,+,+]=>8
[-,-,-,+,-,+]=>8
[-,-,-,+,+,-]=>8
[-,-,+,-,-,+]=>8
[-,-,+,-,+,-]=>8
[-,-,+,+,-,-]=>8
[-,+,-,-,-,+]=>8
[-,+,-,-,+,-]=>8
[-,+,-,+,-,-]=>8
[-,+,+,-,-,-]=>8
[+,-,-,-,-,+]=>8
[+,-,-,-,+,-]=>8
[+,-,-,+,-,-]=>8
[+,-,+,-,-,-]=>8
[+,+,-,-,-,-]=>8
[-,-,-,-,-,+]=>5
[-,-,-,-,+,-]=>5
[-,-,-,+,-,-]=>5
[-,-,+,-,-,-]=>5
[-,+,-,-,-,-]=>5
[+,-,-,-,-,-]=>5
[-,-,-,-,-,-]=>0
[+,+,+,+,6,5]=>4
[-,+,+,+,6,5]=>7
[+,-,+,+,6,5]=>7
[+,+,-,+,6,5]=>7
[+,+,+,-,6,5]=>7
[-,-,+,+,6,5]=>8
[-,+,-,+,6,5]=>8
[-,+,+,-,6,5]=>8
[+,-,-,+,6,5]=>8
[+,-,+,-,6,5]=>8
[+,+,-,-,6,5]=>8
[-,-,-,+,6,5]=>7
[-,-,+,-,6,5]=>7
[-,+,-,-,6,5]=>7
[+,-,-,-,6,5]=>7
[-,-,-,-,6,5]=>4
[+,+,+,5,4,+]=>4
[-,+,+,5,4,+]=>7
[+,-,+,5,4,+]=>7
[+,+,-,5,4,+]=>7
[+,+,+,5,4,-]=>7
[-,-,+,5,4,+]=>8
[-,+,-,5,4,+]=>8
[-,+,+,5,4,-]=>8
[+,-,-,5,4,+]=>8
[+,-,+,5,4,-]=>8
[+,+,-,5,4,-]=>8
[-,-,-,5,4,+]=>7
[-,-,+,5,4,-]=>7
[-,+,-,5,4,-]=>7
[+,-,-,5,4,-]=>7
[-,-,-,5,4,-]=>4
[+,+,+,5,6,4]=>6
[-,+,+,5,6,4]=>7
[+,-,+,5,6,4]=>7
[+,+,-,5,6,4]=>7
[-,-,+,5,6,4]=>6
[-,+,-,5,6,4]=>6
[+,-,-,5,6,4]=>6
[-,-,-,5,6,4]=>3
[+,+,+,6,4,5]=>3
[-,+,+,6,4,5]=>6
[+,-,+,6,4,5]=>6
[+,+,-,6,4,5]=>6
[-,-,+,6,4,5]=>7
[-,+,-,6,4,5]=>7
[+,-,-,6,4,5]=>7
[-,-,-,6,4,5]=>6
[+,+,+,6,+,4]=>4
[-,+,+,6,+,4]=>7
[+,-,+,6,+,4]=>7
[+,+,-,6,+,4]=>7
[+,+,+,6,-,4]=>7
[-,-,+,6,+,4]=>8
[-,+,-,6,+,4]=>8
[-,+,+,6,-,4]=>8
[+,-,-,6,+,4]=>8
[+,-,+,6,-,4]=>8
[+,+,-,6,-,4]=>8
[-,-,-,6,+,4]=>7
[-,-,+,6,-,4]=>7
[-,+,-,6,-,4]=>7
[+,-,-,6,-,4]=>7
[-,-,-,6,-,4]=>4
[+,+,4,3,+,+]=>4
[-,+,4,3,+,+]=>7
[+,-,4,3,+,+]=>7
[+,+,4,3,-,+]=>7
[+,+,4,3,+,-]=>7
[-,-,4,3,+,+]=>8
[-,+,4,3,-,+]=>8
[-,+,4,3,+,-]=>8
[+,-,4,3,-,+]=>8
[+,-,4,3,+,-]=>8
[+,+,4,3,-,-]=>8
[-,-,4,3,-,+]=>7
[-,-,4,3,+,-]=>7
[-,+,4,3,-,-]=>7
[+,-,4,3,-,-]=>7
[-,-,4,3,-,-]=>4
[+,+,4,3,6,5]=>6
[-,+,4,3,6,5]=>7
[+,-,4,3,6,5]=>7
[-,-,4,3,6,5]=>6
[+,+,4,5,3,+]=>6
[-,+,4,5,3,+]=>7
[+,-,4,5,3,+]=>7
[+,+,4,5,3,-]=>7
[-,-,4,5,3,+]=>6
[-,+,4,5,3,-]=>6
[+,-,4,5,3,-]=>6
[-,-,4,5,3,-]=>3
[+,+,4,5,6,3]=>6
[-,+,4,5,6,3]=>5
[+,-,4,5,6,3]=>5
[-,-,4,5,6,3]=>2
[+,+,4,6,3,5]=>5
[-,+,4,6,3,5]=>6
[+,-,4,6,3,5]=>6
[-,-,4,6,3,5]=>5
[+,+,4,6,+,3]=>6
[-,+,4,6,+,3]=>7
[+,-,4,6,+,3]=>7
[+,+,4,6,-,3]=>7
[-,-,4,6,+,3]=>6
[-,+,4,6,-,3]=>6
[+,-,4,6,-,3]=>6
[-,-,4,6,-,3]=>3
[+,+,5,3,4,+]=>3
[-,+,5,3,4,+]=>6
[+,-,5,3,4,+]=>6
[+,+,5,3,4,-]=>6
[-,-,5,3,4,+]=>7
[-,+,5,3,4,-]=>7
[+,-,5,3,4,-]=>7
[-,-,5,3,4,-]=>6
[+,+,5,3,6,4]=>5
[-,+,5,3,6,4]=>6
[+,-,5,3,6,4]=>6
[-,-,5,3,6,4]=>5
[+,+,5,+,3,+]=>4
[-,+,5,+,3,+]=>7
[+,-,5,+,3,+]=>7
[+,+,5,-,3,+]=>7
[+,+,5,+,3,-]=>7
[-,-,5,+,3,+]=>8
[-,+,5,-,3,+]=>8
[-,+,5,+,3,-]=>8
[+,-,5,-,3,+]=>8
[+,-,5,+,3,-]=>8
[+,+,5,-,3,-]=>8
[-,-,5,-,3,+]=>7
[-,-,5,+,3,-]=>7
[-,+,5,-,3,-]=>7
[+,-,5,-,3,-]=>7
[-,-,5,-,3,-]=>4
[+,+,5,+,6,3]=>6
[-,+,5,+,6,3]=>7
[+,-,5,+,6,3]=>7
[+,+,5,-,6,3]=>7
[-,-,5,+,6,3]=>6
[-,+,5,-,6,3]=>6
[+,-,5,-,6,3]=>6
[-,-,5,-,6,3]=>3
[+,+,5,6,3,4]=>4
[-,+,5,6,3,4]=>5
[+,-,5,6,3,4]=>5
[-,-,5,6,3,4]=>4
[+,+,5,6,4,3]=>5
[-,+,5,6,4,3]=>6
[+,-,5,6,4,3]=>6
[-,-,5,6,4,3]=>5
[+,+,6,3,4,5]=>2
[-,+,6,3,4,5]=>5
[+,-,6,3,4,5]=>5
[-,-,6,3,4,5]=>6
[+,+,6,3,+,4]=>3
[-,+,6,3,+,4]=>6
[+,-,6,3,+,4]=>6
[+,+,6,3,-,4]=>6
[-,-,6,3,+,4]=>7
[-,+,6,3,-,4]=>7
[+,-,6,3,-,4]=>7
[-,-,6,3,-,4]=>6
[+,+,6,+,3,5]=>3
[-,+,6,+,3,5]=>6
[+,-,6,+,3,5]=>6
[+,+,6,-,3,5]=>6
[-,-,6,+,3,5]=>7
[-,+,6,-,3,5]=>7
[+,-,6,-,3,5]=>7
[-,-,6,-,3,5]=>6
[+,+,6,+,+,3]=>4
[-,+,6,+,+,3]=>7
[+,-,6,+,+,3]=>7
[+,+,6,-,+,3]=>7
[+,+,6,+,-,3]=>7
[-,-,6,+,+,3]=>8
[-,+,6,-,+,3]=>8
[-,+,6,+,-,3]=>8
[+,-,6,-,+,3]=>8
[+,-,6,+,-,3]=>8
[+,+,6,-,-,3]=>8
[-,-,6,-,+,3]=>7
[-,-,6,+,-,3]=>7
[-,+,6,-,-,3]=>7
[+,-,6,-,-,3]=>7
[-,-,6,-,-,3]=>4
[+,+,6,5,3,4]=>5
[-,+,6,5,3,4]=>6
[+,-,6,5,3,4]=>6
[-,-,6,5,3,4]=>5
[+,+,6,5,4,3]=>6
[-,+,6,5,4,3]=>7
[+,-,6,5,4,3]=>7
[-,-,6,5,4,3]=>6
[+,3,2,+,+,+]=>4
[-,3,2,+,+,+]=>7
[+,3,2,-,+,+]=>7
[+,3,2,+,-,+]=>7
[+,3,2,+,+,-]=>7
[-,3,2,-,+,+]=>8
[-,3,2,+,-,+]=>8
[-,3,2,+,+,-]=>8
[+,3,2,-,-,+]=>8
[+,3,2,-,+,-]=>8
[+,3,2,+,-,-]=>8
[-,3,2,-,-,+]=>7
[-,3,2,-,+,-]=>7
[-,3,2,+,-,-]=>7
[+,3,2,-,-,-]=>7
[-,3,2,-,-,-]=>4
[+,3,2,+,6,5]=>6
[-,3,2,+,6,5]=>7
[+,3,2,-,6,5]=>7
[-,3,2,-,6,5]=>6
[+,3,2,5,4,+]=>6
[-,3,2,5,4,+]=>7
[+,3,2,5,4,-]=>7
[-,3,2,5,4,-]=>6
[+,3,2,5,6,4]=>6
[-,3,2,5,6,4]=>5
[+,3,2,6,4,5]=>5
[-,3,2,6,4,5]=>6
[+,3,2,6,+,4]=>6
[-,3,2,6,+,4]=>7
[+,3,2,6,-,4]=>7
[-,3,2,6,-,4]=>6
[+,3,4,2,+,+]=>6
[-,3,4,2,+,+]=>7
[+,3,4,2,-,+]=>7
[+,3,4,2,+,-]=>7
[-,3,4,2,-,+]=>6
[-,3,4,2,+,-]=>6
[+,3,4,2,-,-]=>6
[-,3,4,2,-,-]=>3
[+,3,4,2,6,5]=>6
[-,3,4,2,6,5]=>5
[+,3,4,5,2,+]=>6
[-,3,4,5,2,+]=>5
[+,3,4,5,2,-]=>5
[-,3,4,5,2,-]=>2
[+,3,4,5,6,2]=>4
[-,3,4,5,6,2]=>1
[+,3,4,6,2,5]=>5
[-,3,4,6,2,5]=>4
[+,3,4,6,+,2]=>6
[-,3,4,6,+,2]=>5
[+,3,4,6,-,2]=>5
[-,3,4,6,-,2]=>2
[+,3,5,2,4,+]=>5
[-,3,5,2,4,+]=>6
[+,3,5,2,4,-]=>6
[-,3,5,2,4,-]=>5
[+,3,5,2,6,4]=>5
[-,3,5,2,6,4]=>4
[+,3,5,+,2,+]=>6
[-,3,5,+,2,+]=>7
[+,3,5,-,2,+]=>7
[+,3,5,+,2,-]=>7
[-,3,5,-,2,+]=>6
[-,3,5,+,2,-]=>6
[+,3,5,-,2,-]=>6
[-,3,5,-,2,-]=>3
[+,3,5,+,6,2]=>6
[-,3,5,+,6,2]=>5
[+,3,5,-,6,2]=>5
[-,3,5,-,6,2]=>2
[+,3,5,6,2,4]=>4
[-,3,5,6,2,4]=>3
[+,3,5,6,4,2]=>5
[-,3,5,6,4,2]=>4
[+,3,6,2,4,5]=>4
[-,3,6,2,4,5]=>5
[+,3,6,2,+,4]=>5
[-,3,6,2,+,4]=>6
[+,3,6,2,-,4]=>6
[-,3,6,2,-,4]=>5
[+,3,6,+,2,5]=>5
[-,3,6,+,2,5]=>6
[+,3,6,-,2,5]=>6
[-,3,6,-,2,5]=>5
[+,3,6,+,+,2]=>6
[-,3,6,+,+,2]=>7
[+,3,6,-,+,2]=>7
[+,3,6,+,-,2]=>7
[-,3,6,-,+,2]=>6
[-,3,6,+,-,2]=>6
[+,3,6,-,-,2]=>6
[-,3,6,-,-,2]=>3
[+,3,6,5,2,4]=>5
[-,3,6,5,2,4]=>4
[+,3,6,5,4,2]=>6
[-,3,6,5,4,2]=>5
[+,4,2,3,+,+]=>3
[-,4,2,3,+,+]=>6
[+,4,2,3,-,+]=>6
[+,4,2,3,+,-]=>6
[-,4,2,3,-,+]=>7
[-,4,2,3,+,-]=>7
[+,4,2,3,-,-]=>7
[-,4,2,3,-,-]=>6
[+,4,2,3,6,5]=>5
[-,4,2,3,6,5]=>6
[+,4,2,5,3,+]=>5
[-,4,2,5,3,+]=>6
[+,4,2,5,3,-]=>6
[-,4,2,5,3,-]=>5
[+,4,2,5,6,3]=>5
[-,4,2,5,6,3]=>4
[+,4,2,6,3,5]=>4
[-,4,2,6,3,5]=>5
[+,4,2,6,+,3]=>5
[-,4,2,6,+,3]=>6
[+,4,2,6,-,3]=>6
[-,4,2,6,-,3]=>5
[+,4,+,2,+,+]=>4
[-,4,+,2,+,+]=>7
[+,4,-,2,+,+]=>7
[+,4,+,2,-,+]=>7
[+,4,+,2,+,-]=>7
[-,4,-,2,+,+]=>8
[-,4,+,2,-,+]=>8
[-,4,+,2,+,-]=>8
[+,4,-,2,-,+]=>8
[+,4,-,2,+,-]=>8
[+,4,+,2,-,-]=>8
[-,4,-,2,-,+]=>7
[-,4,-,2,+,-]=>7
[-,4,+,2,-,-]=>7
[+,4,-,2,-,-]=>7
[-,4,-,2,-,-]=>4
[+,4,+,2,6,5]=>6
[-,4,+,2,6,5]=>7
[+,4,-,2,6,5]=>7
[-,4,-,2,6,5]=>6
[+,4,+,5,2,+]=>6
[-,4,+,5,2,+]=>7
[+,4,-,5,2,+]=>7
[+,4,+,5,2,-]=>7
[-,4,-,5,2,+]=>6
[-,4,+,5,2,-]=>6
[+,4,-,5,2,-]=>6
[-,4,-,5,2,-]=>3
[+,4,+,5,6,2]=>6
[-,4,+,5,6,2]=>5
[+,4,-,5,6,2]=>5
[-,4,-,5,6,2]=>2
[+,4,+,6,2,5]=>5
[-,4,+,6,2,5]=>6
[+,4,-,6,2,5]=>6
[-,4,-,6,2,5]=>5
[+,4,+,6,+,2]=>6
[-,4,+,6,+,2]=>7
[+,4,-,6,+,2]=>7
[+,4,+,6,-,2]=>7
[-,4,-,6,+,2]=>6
[-,4,+,6,-,2]=>6
[+,4,-,6,-,2]=>6
[-,4,-,6,-,2]=>3
[+,4,5,2,3,+]=>4
[-,4,5,2,3,+]=>5
[+,4,5,2,3,-]=>5
[-,4,5,2,3,-]=>4
[+,4,5,2,6,3]=>4
[-,4,5,2,6,3]=>3
[+,4,5,3,2,+]=>5
[-,4,5,3,2,+]=>6
[+,4,5,3,2,-]=>6
[-,4,5,3,2,-]=>5
[+,4,5,3,6,2]=>5
[-,4,5,3,6,2]=>4
[+,4,5,6,2,3]=>3
[-,4,5,6,2,3]=>2
[+,4,5,6,3,2]=>4
[-,4,5,6,3,2]=>3
[+,4,6,2,3,5]=>3
[-,4,6,2,3,5]=>4
[+,4,6,2,+,3]=>4
[-,4,6,2,+,3]=>5
[+,4,6,2,-,3]=>5
[-,4,6,2,-,3]=>4
[+,4,6,3,2,5]=>4
[-,4,6,3,2,5]=>5
[+,4,6,3,+,2]=>5
[-,4,6,3,+,2]=>6
[+,4,6,3,-,2]=>6
[-,4,6,3,-,2]=>5
[+,4,6,5,2,3]=>4
[-,4,6,5,2,3]=>3
[+,4,6,5,3,2]=>5
[-,4,6,5,3,2]=>4
[+,5,2,3,4,+]=>2
[-,5,2,3,4,+]=>5
[+,5,2,3,4,-]=>5
[-,5,2,3,4,-]=>6
[+,5,2,3,6,4]=>4
[-,5,2,3,6,4]=>5
[+,5,2,+,3,+]=>3
[-,5,2,+,3,+]=>6
[+,5,2,-,3,+]=>6
[+,5,2,+,3,-]=>6
[-,5,2,-,3,+]=>7
[-,5,2,+,3,-]=>7
[+,5,2,-,3,-]=>7
[-,5,2,-,3,-]=>6
[+,5,2,+,6,3]=>5
[-,5,2,+,6,3]=>6
[+,5,2,-,6,3]=>6
[-,5,2,-,6,3]=>5
[+,5,2,6,3,4]=>3
[-,5,2,6,3,4]=>4
[+,5,2,6,4,3]=>4
[-,5,2,6,4,3]=>5
[+,5,+,2,4,+]=>3
[-,5,+,2,4,+]=>6
[+,5,-,2,4,+]=>6
[+,5,+,2,4,-]=>6
[-,5,-,2,4,+]=>7
[-,5,+,2,4,-]=>7
[+,5,-,2,4,-]=>7
[-,5,-,2,4,-]=>6
[+,5,+,2,6,4]=>5
[-,5,+,2,6,4]=>6
[+,5,-,2,6,4]=>6
[-,5,-,2,6,4]=>5
[+,5,+,+,2,+]=>4
[-,5,+,+,2,+]=>7
[+,5,-,+,2,+]=>7
[+,5,+,-,2,+]=>7
[+,5,+,+,2,-]=>7
[-,5,-,+,2,+]=>8
[-,5,+,-,2,+]=>8
[-,5,+,+,2,-]=>8
[+,5,-,-,2,+]=>8
[+,5,-,+,2,-]=>8
[+,5,+,-,2,-]=>8
[-,5,-,-,2,+]=>7
[-,5,-,+,2,-]=>7
[-,5,+,-,2,-]=>7
[+,5,-,-,2,-]=>7
[-,5,-,-,2,-]=>4
[+,5,+,+,6,2]=>6
[-,5,+,+,6,2]=>7
[+,5,-,+,6,2]=>7
[+,5,+,-,6,2]=>7
[-,5,-,+,6,2]=>6
[-,5,+,-,6,2]=>6
[+,5,-,-,6,2]=>6
[-,5,-,-,6,2]=>3
[+,5,+,6,2,4]=>4
[-,5,+,6,2,4]=>5
[+,5,-,6,2,4]=>5
[-,5,-,6,2,4]=>4
[+,5,+,6,4,2]=>5
[-,5,+,6,4,2]=>6
[+,5,-,6,4,2]=>6
[-,5,-,6,4,2]=>5
[+,5,4,2,3,+]=>5
[-,5,4,2,3,+]=>6
[+,5,4,2,3,-]=>6
[-,5,4,2,3,-]=>5
[+,5,4,2,6,3]=>5
[-,5,4,2,6,3]=>4
[+,5,4,3,2,+]=>6
[-,5,4,3,2,+]=>7
[+,5,4,3,2,-]=>7
[-,5,4,3,2,-]=>6
[+,5,4,3,6,2]=>6
[-,5,4,3,6,2]=>5
[+,5,4,6,2,3]=>4
[-,5,4,6,2,3]=>3
[+,5,4,6,3,2]=>5
[-,5,4,6,3,2]=>4
[+,5,6,2,3,4]=>2
[-,5,6,2,3,4]=>3
[+,5,6,2,4,3]=>3
[-,5,6,2,4,3]=>4
[+,5,6,3,2,4]=>3
[-,5,6,3,2,4]=>4
[+,5,6,3,4,2]=>4
[-,5,6,3,4,2]=>5
[+,5,6,+,2,3]=>4
[-,5,6,+,2,3]=>5
[+,5,6,-,2,3]=>5
[-,5,6,-,2,3]=>4
[+,5,6,+,3,2]=>5
[-,5,6,+,3,2]=>6
[+,5,6,-,3,2]=>6
[-,5,6,-,3,2]=>5
[+,6,2,3,4,5]=>1
[-,6,2,3,4,5]=>4
[+,6,2,3,+,4]=>2
[-,6,2,3,+,4]=>5
[+,6,2,3,-,4]=>5
[-,6,2,3,-,4]=>6
[+,6,2,+,3,5]=>2
[-,6,2,+,3,5]=>5
[+,6,2,-,3,5]=>5
[-,6,2,-,3,5]=>6
[+,6,2,+,+,3]=>3
[-,6,2,+,+,3]=>6
[+,6,2,-,+,3]=>6
[+,6,2,+,-,3]=>6
[-,6,2,-,+,3]=>7
[-,6,2,+,-,3]=>7
[+,6,2,-,-,3]=>7
[-,6,2,-,-,3]=>6
[+,6,2,5,3,4]=>4
[-,6,2,5,3,4]=>5
[+,6,2,5,4,3]=>5
[-,6,2,5,4,3]=>6
[+,6,+,2,4,5]=>2
[-,6,+,2,4,5]=>5
[+,6,-,2,4,5]=>5
[-,6,-,2,4,5]=>6
[+,6,+,2,+,4]=>3
[-,6,+,2,+,4]=>6
[+,6,-,2,+,4]=>6
[+,6,+,2,-,4]=>6
[-,6,-,2,+,4]=>7
[-,6,+,2,-,4]=>7
[+,6,-,2,-,4]=>7
[-,6,-,2,-,4]=>6
[+,6,+,+,2,5]=>3
[-,6,+,+,2,5]=>6
[+,6,-,+,2,5]=>6
[+,6,+,-,2,5]=>6
[-,6,-,+,2,5]=>7
[-,6,+,-,2,5]=>7
[+,6,-,-,2,5]=>7
[-,6,-,-,2,5]=>6
[+,6,+,+,+,2]=>4
[-,6,+,+,+,2]=>7
[+,6,-,+,+,2]=>7
[+,6,+,-,+,2]=>7
[+,6,+,+,-,2]=>7
[-,6,-,+,+,2]=>8
[-,6,+,-,+,2]=>8
[-,6,+,+,-,2]=>8
[+,6,-,-,+,2]=>8
[+,6,-,+,-,2]=>8
[+,6,+,-,-,2]=>8
[-,6,-,-,+,2]=>7
[-,6,-,+,-,2]=>7
[-,6,+,-,-,2]=>7
[+,6,-,-,-,2]=>7
[-,6,-,-,-,2]=>4
[+,6,+,5,2,4]=>5
[-,6,+,5,2,4]=>6
[+,6,-,5,2,4]=>6
[-,6,-,5,2,4]=>5
[+,6,+,5,4,2]=>6
[-,6,+,5,4,2]=>7
[+,6,-,5,4,2]=>7
[-,6,-,5,4,2]=>6
[+,6,4,2,3,5]=>4
[-,6,4,2,3,5]=>5
[+,6,4,2,+,3]=>5
[-,6,4,2,+,3]=>6
[+,6,4,2,-,3]=>6
[-,6,4,2,-,3]=>5
[+,6,4,3,2,5]=>5
[-,6,4,3,2,5]=>6
[+,6,4,3,+,2]=>6
[-,6,4,3,+,2]=>7
[+,6,4,3,-,2]=>7
[-,6,4,3,-,2]=>6
[+,6,4,5,2,3]=>5
[-,6,4,5,2,3]=>4
[+,6,4,5,3,2]=>6
[-,6,4,5,3,2]=>5
[+,6,5,2,3,4]=>3
[-,6,5,2,3,4]=>4
[+,6,5,2,4,3]=>4
[-,6,5,2,4,3]=>5
[+,6,5,3,2,4]=>4
[-,6,5,3,2,4]=>5
[+,6,5,3,4,2]=>5
[-,6,5,3,4,2]=>6
[+,6,5,+,2,3]=>5
[-,6,5,+,2,3]=>6
[+,6,5,-,2,3]=>6
[-,6,5,-,2,3]=>5
[+,6,5,+,3,2]=>6
[-,6,5,+,3,2]=>7
[+,6,5,-,3,2]=>7
[-,6,5,-,3,2]=>6
[2,1,+,+,+,+]=>4
[2,1,-,+,+,+]=>7
[2,1,+,-,+,+]=>7
[2,1,+,+,-,+]=>7
[2,1,+,+,+,-]=>7
[2,1,-,-,+,+]=>8
[2,1,-,+,-,+]=>8
[2,1,-,+,+,-]=>8
[2,1,+,-,-,+]=>8
[2,1,+,-,+,-]=>8
[2,1,+,+,-,-]=>8
[2,1,-,-,-,+]=>7
[2,1,-,-,+,-]=>7
[2,1,-,+,-,-]=>7
[2,1,+,-,-,-]=>7
[2,1,-,-,-,-]=>4
[2,1,+,+,6,5]=>6
[2,1,-,+,6,5]=>7
[2,1,+,-,6,5]=>7
[2,1,-,-,6,5]=>6
[2,1,+,5,4,+]=>6
[2,1,-,5,4,+]=>7
[2,1,+,5,4,-]=>7
[2,1,-,5,4,-]=>6
[2,1,+,5,6,4]=>6
[2,1,-,5,6,4]=>5
[2,1,+,6,4,5]=>5
[2,1,-,6,4,5]=>6
[2,1,+,6,+,4]=>6
[2,1,-,6,+,4]=>7
[2,1,+,6,-,4]=>7
[2,1,-,6,-,4]=>6
[2,1,4,3,+,+]=>6
[2,1,4,3,-,+]=>7
[2,1,4,3,+,-]=>7
[2,1,4,3,-,-]=>6
[2,1,4,3,6,5]=>6
[2,1,4,5,3,+]=>6
[2,1,4,5,3,-]=>5
[2,1,4,5,6,3]=>4
[2,1,4,6,3,5]=>5
[2,1,4,6,+,3]=>6
[2,1,4,6,-,3]=>5
[2,1,5,3,4,+]=>5
[2,1,5,3,4,-]=>6
[2,1,5,3,6,4]=>5
[2,1,5,+,3,+]=>6
[2,1,5,-,3,+]=>7
[2,1,5,+,3,-]=>7
[2,1,5,-,3,-]=>6
[2,1,5,+,6,3]=>6
[2,1,5,-,6,3]=>5
[2,1,5,6,3,4]=>4
[2,1,5,6,4,3]=>5
[2,1,6,3,4,5]=>4
[2,1,6,3,+,4]=>5
[2,1,6,3,-,4]=>6
[2,1,6,+,3,5]=>5
[2,1,6,-,3,5]=>6
[2,1,6,+,+,3]=>6
[2,1,6,-,+,3]=>7
[2,1,6,+,-,3]=>7
[2,1,6,-,-,3]=>6
[2,1,6,5,3,4]=>5
[2,1,6,5,4,3]=>6
[2,3,1,+,+,+]=>6
[2,3,1,-,+,+]=>7
[2,3,1,+,-,+]=>7
[2,3,1,+,+,-]=>7
[2,3,1,-,-,+]=>6
[2,3,1,-,+,-]=>6
[2,3,1,+,-,-]=>6
[2,3,1,-,-,-]=>3
[2,3,1,+,6,5]=>6
[2,3,1,-,6,5]=>5
[2,3,1,5,4,+]=>6
[2,3,1,5,4,-]=>5
[2,3,1,5,6,4]=>4
[2,3,1,6,4,5]=>5
[2,3,1,6,+,4]=>6
[2,3,1,6,-,4]=>5
[2,3,4,1,+,+]=>6
[2,3,4,1,-,+]=>5
[2,3,4,1,+,-]=>5
[2,3,4,1,-,-]=>2
[2,3,4,1,6,5]=>4
[2,3,4,5,1,+]=>4
[2,3,4,5,1,-]=>1
[2,3,4,5,6,1]=>0
[2,3,4,6,1,5]=>3
[2,3,4,6,+,1]=>4
[2,3,4,6,-,1]=>1
[2,3,5,1,4,+]=>5
[2,3,5,1,4,-]=>4
[2,3,5,1,6,4]=>3
[2,3,5,+,1,+]=>6
[2,3,5,-,1,+]=>5
[2,3,5,+,1,-]=>5
[2,3,5,-,1,-]=>2
[2,3,5,+,6,1]=>4
[2,3,5,-,6,1]=>1
[2,3,5,6,1,4]=>2
[2,3,5,6,4,1]=>3
[2,3,6,1,4,5]=>4
[2,3,6,1,+,4]=>5
[2,3,6,1,-,4]=>4
[2,3,6,+,1,5]=>5
[2,3,6,-,1,5]=>4
[2,3,6,+,+,1]=>6
[2,3,6,-,+,1]=>5
[2,3,6,+,-,1]=>5
[2,3,6,-,-,1]=>2
[2,3,6,5,1,4]=>3
[2,3,6,5,4,1]=>4
[2,4,1,3,+,+]=>5
[2,4,1,3,-,+]=>6
[2,4,1,3,+,-]=>6
[2,4,1,3,-,-]=>5
[2,4,1,3,6,5]=>5
[2,4,1,5,3,+]=>5
[2,4,1,5,3,-]=>4
[2,4,1,5,6,3]=>3
[2,4,1,6,3,5]=>4
[2,4,1,6,+,3]=>5
[2,4,1,6,-,3]=>4
[2,4,+,1,+,+]=>6
[2,4,-,1,+,+]=>7
[2,4,+,1,-,+]=>7
[2,4,+,1,+,-]=>7
[2,4,-,1,-,+]=>6
[2,4,-,1,+,-]=>6
[2,4,+,1,-,-]=>6
[2,4,-,1,-,-]=>3
[2,4,+,1,6,5]=>6
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Description
The number of alignments of a decorated permutation.
An alignment in a decorated permutation $w$ is a pair of directed edges $(i\mapsto w(i), j\mapsto w(j))$ which can be drawn as distinct noncrossing arcs oriented in the same direction in the circular chord diagram.
According to [1], the number of alignments of a decorated permutation on $n$ letters with $k$ anit-exceedances equals and associated Grassmann interval $[u, v]$ equals $k(n - k) - [\ell(v) − \ell(u)]$.
An alignment in a decorated permutation $w$ is a pair of directed edges $(i\mapsto w(i), j\mapsto w(j))$ which can be drawn as distinct noncrossing arcs oriented in the same direction in the circular chord diagram.
According to [1], the number of alignments of a decorated permutation on $n$ letters with $k$ anit-exceedances equals and associated Grassmann interval $[u, v]$ equals $k(n - k) - [\ell(v) − \ell(u)]$.
References
[1] A. Postnikov, Total positivity, Grassmannians, and networks. 27 Sep 2006. Postnikov, A. Total positivity, Grassmannians, and networks arXiv:math/0609764
[2] Billey, S. C., Weaver, J. E. Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs arXiv:2207.06508
[2] Billey, S. C., Weaver, J. E. Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs arXiv:2207.06508
Code
# please simplify def cyclic_interval(a, b, n): if a <= b: return list(range(a, b+1)) return list(range(a, n+1)) + list(range(1, b+1)) def alignments(x): pi = x.to_signed_permutation().permutation() n = len(pi) al = [] for i, e in enumerate(pi, 1): for j, f in enumerate(pi, 1): if i == j: continue a, b = sorted([i, e]) c, d = sorted([j, f]) if a <= c <= b <= d or c <= a <= d <= b: continue if (e not in cyclic_interval(i, f-1, n) or f not in cyclic_interval(e+1, j, n)): continue if j == f and x[j-1] < 0: continue if i == e and x[i-1] > 0: continue al.append((i, e, j, f)) return al def statistic(x): return len(alignments(x))
Created
Jul 15, 2022 at 18:19 by Martin Rubey
Updated
Jul 15, 2022 at 18:19 by Martin Rubey
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