Identifier
Values
[1,0] => [2,1] => [2,1] => [2,1] => 1
[1,0,1,0] => [3,1,2] => [3,2,1] => [3,2,1] => 1
[1,1,0,0] => [2,3,1] => [3,1,2] => [2,3,1] => 2
[1,0,1,0,1,0] => [4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,1,0,0] => [3,1,4,2] => [4,2,1,3] => [3,2,4,1] => 2
[1,1,0,0,1,0] => [2,4,1,3] => [4,3,1,2] => [3,4,2,1] => 2
[1,1,0,1,0,0] => [4,3,1,2] => [4,2,3,1] => [4,2,3,1] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,3,2,1,4] => [4,3,2,5,1] => 2
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [5,4,2,1,3] => [4,3,5,2,1] => 2
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [5,3,4,2,1] => [5,4,2,3,1] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,2,1,3,4] => [3,2,4,5,1] => 3
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [5,4,3,1,2] => [4,5,3,2,1] => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [5,3,1,2,4] => [3,4,2,5,1] => 3
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [5,4,2,3,1] => [5,3,4,2,1] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,2,5,3,1] => [5,2,4,1,3] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [5,2,3,1,4] => [4,2,3,5,1] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,4,1,2,3] => [3,4,5,2,1] => 3
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [5,3,4,1,2] => [4,5,2,3,1] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [5,2,3,4,1] => [5,2,3,4,1] => 1
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [6,4,3,2,1,5] => [5,4,3,2,6,1] => 2
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [6,5,3,2,1,4] => [5,4,3,6,2,1] => 2
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [6,4,5,3,2,1] => [6,5,4,2,3,1] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [6,3,2,1,4,5] => [4,3,2,5,6,1] => 3
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [6,5,4,2,1,3] => [5,4,6,3,2,1] => 2
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [6,4,2,1,3,5] => [4,3,5,2,6,1] => 3
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [6,5,3,4,2,1] => [6,5,3,4,2,1] => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [5,3,6,4,2,1] => [6,5,2,4,1,3] => 1
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [6,3,4,2,1,5] => [5,4,2,3,6,1] => 2
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [6,5,2,1,3,4] => [4,3,5,6,2,1] => 3
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [6,4,5,2,1,3] => [5,4,6,2,3,1] => 2
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [6,3,4,5,2,1] => [6,5,2,3,4,1] => 1
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [6,2,1,3,4,5] => [3,2,4,5,6,1] => 4
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [6,5,4,3,1,2] => [5,6,4,3,2,1] => 2
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [6,4,3,1,2,5] => [4,5,3,2,6,1] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [6,5,3,1,2,4] => [4,5,3,6,2,1] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [6,4,5,3,1,2] => [5,6,4,2,3,1] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [6,3,1,2,4,5] => [3,4,2,5,6,1] => 4
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [6,5,4,2,3,1] => [6,4,5,3,2,1] => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [6,4,2,3,1,5] => [5,3,4,2,6,1] => 2
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,2,6,5,3,1] => [6,2,5,1,4,3] => 1
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => [3,6,2,5,1,4] => 2
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,2,6,3,1,5] => [5,2,4,1,6,3] => 2
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [6,5,2,3,1,4] => [5,3,4,6,2,1] => 2
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [6,4,5,2,3,1] => [6,4,5,2,3,1] => 1
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [5,2,4,6,3,1] => [6,2,5,3,1,4] => 1
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [6,2,3,1,4,5] => [4,2,3,5,6,1] => 3
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [6,5,4,1,2,3] => [4,5,6,3,2,1] => 3
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [6,4,1,2,3,5] => [3,4,5,2,6,1] => 4
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [6,5,3,4,1,2] => [5,6,3,4,2,1] => 2
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,3,6,4,1,2] => [5,6,2,4,1,3] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [6,3,4,1,2,5] => [4,5,2,3,6,1] => 3
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [6,5,2,3,4,1] => [6,3,4,5,2,1] => 1
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [5,2,3,6,4,1] => [6,2,3,5,1,4] => 1
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [5,2,6,3,4,1] => [6,2,4,5,1,3] => 1
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [6,2,3,4,1,5] => [5,2,3,4,6,1] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [6,5,1,2,3,4] => [3,4,5,6,2,1] => 4
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [6,4,5,1,2,3] => [4,5,6,2,3,1] => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [6,3,4,5,1,2] => [5,6,2,3,4,1] => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [6,2,3,4,5,1] => [6,2,3,4,5,1] => 1
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [7,3,2,1,4,5,6] => [4,3,2,5,6,7,1] => 4
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => [7,4,2,1,3,5,6] => [4,3,5,2,6,7,1] => 4
[1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [7,2,1,3,4,5,6] => [3,2,4,5,6,7,1] => 5
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => [7,3,1,2,4,5,6] => [3,4,2,5,6,7,1] => 5
[1,1,0,1,1,1,1,0,0,0,0,0] => [4,3,1,5,6,7,2] => [7,2,3,1,4,5,6] => [4,2,3,5,6,7,1] => 4
[1,1,1,0,0,0,1,1,1,0,0,0] => [2,3,5,1,6,7,4] => [7,4,1,2,3,5,6] => [3,4,5,2,6,7,1] => 5
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,5,4,1,6,7,3] => [7,3,4,1,2,5,6] => [4,5,2,3,6,7,1] => 4
[1,1,1,0,1,1,1,0,0,0,0,0] => [5,3,4,1,6,7,2] => [7,2,3,4,1,5,6] => [5,2,3,4,6,7,1] => 3
[1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => 5
[1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,4,5,1,7,2] => [7,2,3,4,5,1,6] => [6,2,3,4,5,7,1] => 2
[1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,1,2,3,4,5,8,6] => [8,6,5,4,3,2,1,7] => [7,6,5,4,3,2,8,1] => 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [5,1,2,3,8,4,6,7] => [8,7,6,4,3,2,1,5] => [7,6,5,4,8,3,2,1] => 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [5,1,2,3,7,4,8,6] => [8,6,4,3,2,1,5,7] => [6,5,4,3,7,2,8,1] => 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0] => [8,1,2,5,3,7,4,6] => [8,6,7,4,5,3,2,1] => [8,7,6,4,5,2,3,1] => 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [4,1,2,5,6,7,8,3] => [8,3,2,1,4,5,6,7] => [4,3,2,5,6,7,8,1] => 5
[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [3,1,8,2,4,5,6,7] => [8,7,6,5,4,2,1,3] => [7,6,8,5,4,3,2,1] => 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [3,1,7,2,4,5,8,6] => [8,6,5,4,2,1,3,7] => [6,5,7,4,3,2,8,1] => 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [3,1,5,2,8,4,6,7] => [8,7,6,4,2,1,3,5] => [6,5,7,4,8,3,2,1] => 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [3,1,5,2,7,4,8,6] => [8,6,4,2,1,3,5,7] => [5,4,6,3,7,2,8,1] => 4
[1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [8,1,4,2,3,7,5,6] => [8,6,7,5,3,4,2,1] => [8,7,5,6,4,2,3,1] => 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0] => [8,1,4,2,6,3,5,7] => [8,7,5,6,3,4,2,1] => [8,7,5,6,3,4,2,1] => 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [3,1,4,5,6,7,8,2] => [8,2,1,3,4,5,6,7] => [3,2,4,5,6,7,8,1] => 6
[1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [2,8,1,3,4,7,5,6] => [8,6,7,5,4,3,1,2] => [7,8,6,5,4,2,3,1] => 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [2,8,1,3,6,4,5,7] => [8,7,5,6,4,3,1,2] => [7,8,6,5,3,4,2,1] => 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0] => [2,8,1,5,3,4,6,7] => [8,7,6,4,5,3,1,2] => [7,8,6,4,5,3,2,1] => 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,4,1,5,6,7,8,3] => [8,3,1,2,4,5,6,7] => [3,4,2,5,6,7,8,1] => 6
[1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [8,3,1,2,4,7,5,6] => [8,6,7,5,4,2,3,1] => [8,6,7,5,4,2,3,1] => 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0] => [8,3,1,2,6,4,5,7] => [8,7,5,6,4,2,3,1] => [8,6,7,5,3,4,2,1] => 1
[1,1,0,1,1,0,0,1,0,0,1,0,1,0] => [8,3,1,5,2,4,6,7] => [8,7,6,4,5,2,3,1] => [8,6,7,4,5,3,2,1] => 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [7,4,1,5,6,2,8,3] => [6,2,4,5,8,3,1,7] => [7,2,6,3,4,1,8,5] => 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [4,3,1,5,6,7,8,2] => [8,2,3,1,4,5,6,7] => [4,2,3,5,6,7,8,1] => 5
[1,1,1,0,0,0,1,1,1,0,1,0,0,0] => [2,3,8,1,6,7,4,5] => [8,5,6,7,4,1,2,3] => [6,7,8,5,2,3,4,1] => 3
[1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [2,3,5,1,6,7,8,4] => [8,4,1,2,3,5,6,7] => [3,4,5,2,6,7,8,1] => 6
[1,1,1,0,0,1,0,0,1,0,1,0,1,0] => [2,8,4,1,3,5,6,7] => [8,7,6,5,3,4,1,2] => [7,8,5,6,4,3,2,1] => 2
>>> Load all 116 entries. <<<
[1,1,1,0,0,1,1,0,1,0,0,0,1,0] => [2,8,5,1,6,3,4,7] => [6,3,5,8,7,4,1,2] => [7,8,2,6,3,1,5,4] => 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [8,3,4,1,6,7,2,5] => [8,5,6,7,2,3,4,1] => [8,5,6,7,2,3,4,1] => 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [5,3,4,1,6,7,8,2] => [8,2,3,4,1,5,6,7] => [5,2,3,4,6,7,8,1] => 4
[1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [2,3,8,5,6,1,4,7] => [8,7,4,5,6,1,2,3] => [6,7,8,3,4,5,2,1] => 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => [2,3,4,5,6,7,8,1] => 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [3,1,4,5,6,7,8,9,2] => [9,2,1,3,4,5,6,7,8] => [3,2,4,5,6,7,8,9,1] => 7
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,4,1,5,6,7,8,9,3] => [9,3,1,2,4,5,6,7,8] => [3,4,2,5,6,7,8,9,1] => 7
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0] => [4,3,1,5,6,7,8,9,2] => [9,2,3,1,4,5,6,7,8] => [4,2,3,5,6,7,8,9,1] => 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => [9,1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,9,1] => 8
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [3,1,4,5,6,7,8,9,10,2] => [10,2,1,3,4,5,6,7,8,9] => [3,2,4,5,6,7,8,9,10,1] => 8
[] => [1] => [1] => [1] => 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => [10,1,2,3,4,5,6,7,8,9] => [2,3,4,5,6,7,8,9,10,1] => 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => [11,1,2,3,4,5,6,7,8,9,10] => [2,3,4,5,6,7,8,9,10,11,1] => 10
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Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
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.
Map
inverse
Description
Sends a permutation to its inverse.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.