Identifier
Values
[1,0] => [1,1,0,0] => [1,0,1,0] => [1,2] => 0
[1,0,1,0] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => [1,2,3] => 0
[1,1,0,0] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => [2,3,1] => 2
[1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 2
[1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 2
[1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 3
[1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => [3,4,2,1] => 3
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 2
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 3
[1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 3
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 2
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 4
[1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 3
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 4
[1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => 4
[1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => 3
[1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 4
[1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => 4
[1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 2
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 2
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 3
[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 2
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => 4
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 3
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 4
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => 4
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => 4
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => 2
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => 4
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,3,5,1,6] => 4
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,3,5,6,1] => 5
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [2,5,4,6,3,1] => 5
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => 3
[1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,4,6,1] => 5
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => 4
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => 5
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,5,6,4,1] => 5
[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => [2,4,5,3,1,6] => 4
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [2,4,5,3,6,1] => 5
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [2,4,5,6,3,1] => 5
[1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [2,5,6,4,3,1] => 5
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => [3,4,2,1,5,6] => 3
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,4,2,6,1] => 5
[1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [1,1,1,0,1,0,0,1,0,0,1,0] => [3,4,2,5,1,6] => 4
[1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,2,5,6,1] => 5
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [3,5,4,6,2,1] => 5
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => [3,4,5,2,1,6] => 4
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,2,6,1] => 5
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,2,1] => 5
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,4,2,1] => 5
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => [4,5,3,2,1,6] => 4
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,3,2,6,1] => 5
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [4,5,3,6,2,1] => 5
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [4,5,6,3,2,1] => 5
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [5,6,4,3,2,1] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,4,6,7,5] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,3,5,6,4,7] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,3,5,6,7,4] => 3
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,3,6,7,5,4] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [1,2,4,5,3,6,7] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0] => [1,2,4,6,5,7,3] => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [1,2,4,5,6,3,7] => 3
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => 4
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0] => [1,2,4,6,7,5,3] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0] => [1,2,5,6,4,3,7] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,2,5,6,4,7,3] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,2,5,6,7,4,3] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,2,6,7,5,4,3] => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [1,3,4,2,5,6,7] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [1,3,4,2,6,7,5] => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [1,0,1,1,0,1,1,0,0,1,0,0,1,0] => [1,3,5,4,6,2,7] => 4
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,1,0,1,0,0] => [1,3,5,4,6,7,2] => 5
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0] => [1,3,6,5,7,4,2] => 5
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0,1,0,1,0] => [1,3,4,5,2,6,7] => 3
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,1,1,0,0,1,0,0] => [1,3,4,6,5,7,2] => 5
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [1,3,4,5,6,2,7] => 4
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,7,2] => 5
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,1,0,1,0,0,0] => [1,3,4,6,7,5,2] => 5
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0,1,0] => [1,3,5,6,4,2,7] => 4
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0] => [1,3,5,6,4,7,2] => 5
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0] => [1,3,5,6,7,4,2] => 5
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [1,0,1,1,0,1,1,1,0,1,0,0,0,0] => [1,3,6,7,5,4,2] => 5
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0] => [1,4,5,3,2,6,7] => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [1,4,6,5,3,7,2] => 5
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [1,0,1,1,1,0,1,0,0,1,0,0,1,0] => [1,4,5,3,6,2,7] => 4
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,5,3,6,7,2] => 5
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0,1,0] => [1,4,5,6,3,2,7] => 4
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [1,4,5,6,3,7,2] => 5
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,4,5,6,7,3,2] => 5
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Description
The number of Bruhat lower covers of a permutation.
This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$.
This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.
Map
peaks-to-valleys
Description
Return the path that has a valley wherever the original path has a peak of height at least one.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
to 312-avoiding permutation
Description
Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).