Identifier
Values
[1,0] => [2,1] => [1,2] => [1,2] => 1
[1,0,1,0] => [3,1,2] => [1,3,2] => [1,3,2] => 2
[1,1,0,0] => [2,3,1] => [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0] => [4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 2
[1,0,1,1,0,0] => [3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 3
[1,1,0,0,1,0] => [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,1,0,1,0,0] => [4,3,1,2] => [1,4,2,3] => [1,4,2,3] => 2
[1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,5,4,3,2] => [1,4,3,5,2] => 2
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [1,4,5,3,2] => [1,5,3,4,2] => 3
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [1,3,5,4,2] => [1,4,5,3,2] => 3
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,5,3,4,2] => [1,3,4,5,2] => 2
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [1,3,4,5,2] => [1,5,4,3,2] => 4
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 3
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,5,4,2,3] => [1,4,2,5,3] => 2
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [1,5,3,2,4] => [1,3,5,2,4] => 2
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [1,4,5,2,3] => [1,5,2,4,3] => 3
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [1,2,5,3,4] => [1,2,5,3,4] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [1,5,2,3,4] => [1,5,2,3,4] => 2
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => 2
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,4,6,3,5,2] => 3
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,5,3,6,4,2] => 3
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,5,4,3,6,2] => 2
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,6,3,5,4,2] => 4
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,5,4,6,3,2] => 3
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,6,4,5,3,2] => 4
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,5,3,4,6,2] => 2
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,4,2,6,3,5] => 2
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,6,3,4,5,2] => 3
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,5,6,4,3,2] => 4
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,4,5,6,3,2] => 3
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,3,4,5,6,2] => 2
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,6,5,4,3,2] => 5
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => 2
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,6,4,5,3] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,5,6,4,3] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,4,5,6,3] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,6,5,4,3] => 4
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,4,5,2,6,3] => 2
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,4,6,2,5,3] => 3
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,5,3,6,2,4] => 2
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,3,6,5,2,4] => 3
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,6,3,5,2,4] => 3
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,5,2,6,4,3] => 3
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,5,4,2,6,3] => 2
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,3,6,2,4,5] => 2
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,6,2,5,4,3] => 4
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => 2
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 3
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [1,2,6,5,3,4] => [1,2,5,3,6,4] => 2
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [1,2,6,4,3,5] => [1,2,4,6,3,5] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [1,2,5,6,3,4] => [1,2,6,3,5,4] => 3
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,6,5,2,3,4] => [1,5,2,6,3,4] => 2
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [1,6,4,2,3,5] => [1,4,2,6,3,5] => 2
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [1,6,3,4,2,5] => [1,3,4,6,2,5] => 2
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [1,5,6,2,3,4] => [1,6,2,5,3,4] => 3
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 2
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => 2
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => 2
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [1,7,4,5,6,3,2] => [1,6,5,4,3,7,2] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [1,4,5,6,7,3,2] => [1,7,3,6,5,4,2] => 5
[1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [1,3,7,6,5,4,2] => [1,5,6,4,7,3,2] => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => [1,3,7,5,6,4,2] => [1,6,5,4,7,3,2] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => [1,3,5,6,7,4,2] => [1,7,4,6,5,3,2] => 5
[1,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,7,2,5,6] => [1,3,4,7,6,5,2] => [1,6,5,7,4,3,2] => 4
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => [1,3,6,7,4,5,2] => [1,7,4,5,6,3,2] => 4
[1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => [1,7,5,2,3,4,6] => [1,5,2,7,3,4,6] => 2
[1,0,1,1,1,1,0,0,0,0,1,0] => [3,1,4,5,7,2,6] => [1,3,4,5,7,6,2] => [1,6,7,5,4,3,2] => 5
[1,0,1,1,1,1,0,0,0,1,0,0] => [3,1,4,7,6,2,5] => [1,3,4,7,5,6,2] => [1,5,6,7,4,3,2] => 4
[1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [1,3,4,5,6,7,2] => [1,7,6,5,4,3,2] => 6
[1,1,0,1,1,0,0,1,0,1,0,0] => [7,3,1,6,2,4,5] => [1,7,5,2,3,4,6] => [1,5,2,7,3,4,6] => 2
[1,1,0,1,1,1,0,1,0,0,0,0] => [7,4,1,5,6,2,3] => [1,7,3,2,4,5,6] => [1,3,7,2,4,5,6] => 2
[1,1,1,1,0,0,0,1,0,0,1,0] => [2,3,7,5,1,4,6] => [1,2,3,7,6,4,5] => [1,2,3,6,4,7,5] => 2
[1,1,1,1,0,1,0,0,0,0,1,0] => [7,3,4,5,1,2,6] => [1,7,6,2,3,4,5] => [1,6,2,7,3,4,5] => 2
[1,1,1,1,0,1,0,0,0,1,0,0] => [7,3,4,6,1,2,5] => [1,7,5,2,3,4,6] => [1,5,2,7,3,4,6] => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => 2
[1,1,1,1,1,0,1,0,0,0,0,0] => [7,3,4,5,6,1,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => 2
[1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [3,1,5,2,6,7,8,4] => [1,3,5,6,7,8,4,2] => [1,8,4,7,6,5,3,2] => 6
[1,0,1,1,0,1,1,1,0,1,0,0,0,0] => [8,1,5,2,6,7,3,4] => [1,8,4,2,3,5,6,7] => [1,4,2,8,3,5,6,7] => 2
[1,0,1,1,1,0,1,0,0,1,0,0,1,0] => [8,1,4,6,2,3,5,7] => [1,8,7,5,2,3,4,6] => [1,5,2,7,3,8,4,6] => 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [8,1,6,5,2,7,3,4] => [1,8,4,5,2,3,6,7] => [1,5,4,2,8,3,6,7] => 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [3,1,4,5,8,2,6,7] => [1,3,4,5,8,7,6,2] => [1,7,6,8,5,4,3,2] => 5
[1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [3,1,4,5,7,2,8,6] => [1,3,4,5,7,8,6,2] => [1,8,6,7,5,4,3,2] => 6
[1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [8,1,4,5,7,2,3,6] => [1,8,6,2,3,4,5,7] => [1,6,2,8,3,4,5,7] => 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [3,1,4,5,6,8,2,7] => [1,3,4,5,6,8,7,2] => [1,7,8,6,5,4,3,2] => 6
[1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [3,1,4,5,8,7,2,6] => [1,3,4,5,8,6,7,2] => [1,6,7,8,5,4,3,2] => 5
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [3,1,4,5,6,7,8,2] => [1,3,4,5,6,7,8,2] => [1,8,7,6,5,4,3,2] => 7
[1,1,0,0,1,1,1,1,0,0,0,1,0,0] => [2,4,1,5,8,7,3,6] => [1,2,4,5,8,6,7,3] => [1,2,6,7,8,5,4,3] => 4
[1,1,0,1,1,0,0,1,0,1,0,0,1,0] => [8,3,1,6,2,4,5,7] => [1,8,7,5,2,3,4,6] => [1,5,2,7,3,8,4,6] => 2
[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [8,3,1,5,7,2,4,6] => [1,8,6,2,3,4,5,7] => [1,6,2,8,3,4,5,7] => 2
[1,1,1,0,0,0,1,1,0,1,0,1,0,0] => [2,3,8,1,7,4,5,6] => [1,2,3,8,6,4,5,7] => [1,2,3,6,4,8,5,7] => 2
[1,1,1,0,0,0,1,1,1,0,1,0,0,0] => [2,3,8,1,6,7,4,5] => [1,2,3,8,5,6,7,4] => [1,2,3,5,6,7,8,4] => 2
[1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [8,3,4,1,7,2,5,6] => [1,8,6,2,3,4,5,7] => [1,6,2,8,3,4,5,7] => 2
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [8,3,5,1,6,7,2,4] => [1,8,4,2,3,5,6,7] => [1,4,2,8,3,5,6,7] => 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0] => [8,3,4,5,1,2,6,7] => [1,8,7,6,2,3,4,5] => [1,6,2,7,3,8,4,5] => 2
>>> Load all 132 entries. <<<
[1,1,1,1,0,1,0,0,0,0,1,1,0,0] => [7,3,4,5,1,2,8,6] => [1,7,8,6,2,3,4,5] => [1,6,2,8,3,7,4,5] => 3
[1,1,1,1,0,1,0,0,0,1,0,0,1,0] => [8,3,4,6,1,2,5,7] => [1,8,7,5,2,3,4,6] => [1,5,2,7,3,8,4,6] => 2
[1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [6,3,4,5,1,8,2,7] => [1,6,8,7,2,3,4,5] => [1,7,2,8,3,6,4,5] => 3
[1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [8,3,4,5,1,7,2,6] => [1,8,6,7,2,3,4,5] => [1,7,2,6,3,8,4,5] => 2
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [8,3,6,5,1,7,2,4] => [1,8,4,5,2,3,6,7] => [1,5,4,2,8,3,6,7] => 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [6,3,4,5,1,7,8,2] => [1,6,7,8,2,3,4,5] => [1,8,2,7,3,6,4,5] => 4
[1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [2,3,8,5,6,1,4,7] => [1,2,3,8,7,4,5,6] => [1,2,3,7,4,8,5,6] => 2
[1,1,1,1,1,0,0,0,1,0,0,1,0,0] => [2,3,8,5,7,1,4,6] => [1,2,3,8,6,4,5,7] => [1,2,3,6,4,8,5,7] => 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [2,3,7,5,6,1,8,4] => [1,2,3,7,8,4,5,6] => [1,2,3,8,4,7,5,6] => 3
[1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [8,3,4,5,6,1,2,7] => [1,8,7,2,3,4,5,6] => [1,7,2,8,3,4,5,6] => 2
[1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [8,3,4,5,7,1,2,6] => [1,8,6,2,3,4,5,7] => [1,6,2,8,3,4,5,7] => 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [7,3,4,5,6,1,8,2] => [1,7,8,2,3,4,5,6] => [1,8,2,7,3,4,5,6] => 3
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,3,4,5,6,8,1,7] => [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => 2
[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [2,3,4,5,8,7,1,6] => [1,2,3,4,5,8,6,7] => [1,2,3,4,5,8,6,7] => 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [2,3,8,5,6,7,1,4] => [1,2,3,8,4,5,6,7] => [1,2,3,8,4,5,6,7] => 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [8,3,4,5,6,7,1,2] => [1,8,2,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [3,1,4,5,6,7,9,2,8] => [1,3,4,5,6,7,9,8,2] => [1,8,9,7,6,5,4,3,2] => 7
[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] => [1,3,4,5,6,7,8,9,2] => [1,9,8,7,6,5,4,3,2] => 8
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,9,1,8] => [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,9,8] => 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [9,3,4,5,6,7,8,1,2] => [1,9,2,3,4,5,6,7,8] => [1,9,2,3,4,5,6,7,8] => 2
[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] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,8,10,1,9] => [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,10,9] => 2
[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] => [1,3,4,5,6,7,8,9,10,2] => [1,10,9,8,7,6,5,4,3,2] => 9
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,8,9,11,1,10] => [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,11,10] => 2
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [3,1,4,5,6,7,8,9,10,11,2] => [1,3,4,5,6,7,8,9,10,11,2] => [1,11,10,9,8,7,6,5,4,3,2] => 10
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [3,1,4,5,6,7,8,10,2,9] => [1,3,4,5,6,7,8,10,9,2] => [1,9,10,8,7,6,5,4,3,2] => 8
[] => [1] => [1] => [1] => 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [10,3,4,5,6,7,8,9,1,2] => [1,10,2,3,4,5,6,7,8,9] => [1,10,2,3,4,5,6,7,8,9] => 2
[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] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [11,3,4,5,6,7,8,9,10,1,2] => [1,11,2,3,4,5,6,7,8,9,10] => [1,11,2,3,4,5,6,7,8,9,10] => 2
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Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.