Identifier
Values
[1] => [1,0,1,0] => [2,1] => [1,2] => 0
[2] => [1,1,0,0,1,0] => [3,1,2] => [1,3,2] => 0
[1,1] => [1,0,1,1,0,0] => [2,3,1] => [3,1,2] => 0
[3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [1,3,4,2] => 0
[2,1] => [1,0,1,0,1,0] => [2,1,3] => [3,2,1] => 1
[1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [3,4,1,2] => 0
[4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [1,3,4,5,2] => 0
[3,1] => [1,1,0,1,0,0,1,0] => [3,1,2,4] => [4,2,3,1] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,1,3,2] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [2,3,1,4] => [3,4,2,1] => 2
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [3,4,5,1,2] => 0
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => [1,3,4,5,6,2] => 0
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => [5,2,3,4,1] => 0
[3,2] => [1,1,0,0,1,0,1,0] => [3,1,4,2] => [4,2,1,3] => 1
[3,1,1] => [1,0,1,1,0,0,1,0] => [2,1,3,4] => [3,2,4,1] => 1
[2,2,1] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => [3,1,4,2] => 0
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => [3,4,5,2,1] => 3
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [3,4,5,6,1,2] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1,2,3,4,5,6] => [1,3,4,5,6,7,2] => 0
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [5,1,2,3,4,6] => [6,2,3,4,5,1] => 0
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => [5,2,3,1,4] => 0
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => [4,2,3,5,1] => 0
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [5,1,3,4,2] => 0
[3,2,1] => [1,0,1,0,1,0,1,0] => [2,1,4,3] => [3,2,1,4] => 2
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => [3,4,2,5,1] => 2
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [4,5,1,3,2] => 0
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => [3,4,1,5,2] => 0
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [2,3,4,5,1,6] => [3,4,5,6,2,1] => 4
[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [5,1,2,3,6,4] => [6,2,3,4,1,5] => 0
[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [4,1,2,3,5,6] => [5,2,3,4,6,1] => 0
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => [5,2,1,4,3] => 1
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => [4,2,3,1,5] => 0
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [3,2,4,5,1] => 1
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => [4,1,3,5,2] => 0
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => [4,5,2,1,3] => 2
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => [3,4,2,1,5] => 3
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [2,3,4,1,5,6] => [3,4,5,2,6,1] => 3
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [3,5,1,4,2] => 0
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => [3,4,5,1,6,2] => 0
[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [5,1,2,6,3,4] => [6,2,3,1,5,4] => 0
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [4,1,2,3,6,5] => [5,2,3,4,1,6] => 0
[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => [3,1,2,4,5,6] => [4,2,3,5,6,1] => 0
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => [6,1,3,4,5,2] => 0
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => [4,2,1,5,3] => 2
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [4,2,5,1,3] => 1
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => [3,2,4,1,5] => 1
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [2,3,1,4,5,6] => [3,4,2,5,6,1] => 2
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => [4,5,2,3,1] => 0
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [3,1,4,5,2] => 0
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => [3,5,2,1,4] => 3
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => [3,4,5,2,1,6] => 4
[2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [4,5,6,1,3,2] => 0
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [2,3,5,6,1,4] => [3,4,6,1,5,2] => 0
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => [5,1,6,2,3,4] => [6,2,1,4,5,3] => 1
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [4,1,2,6,3,5] => [5,2,3,1,6,4] => 0
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => [4,1,2,5,6,3] => [5,2,3,6,1,4] => 0
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [3,1,2,4,6,5] => [4,2,3,5,1,6] => 0
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,5,6] => [3,2,4,5,6,1] => 1
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [4,6,1,2,3,5] => [5,1,3,4,6,2] => 0
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => [4,2,5,3,1] => 1
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => [3,2,1,5,4] => 3
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => [3,2,5,1,4] => 2
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => [3,4,2,5,1,6] => 2
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => [5,6,1,3,4,2] => 0
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [3,5,2,4,1] => 1
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => [3,4,1,5,6,2] => 0
[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => [3,4,5,1,6,2] => [4,5,6,2,1,3] => 3
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,1,6,4] => [3,4,6,2,1,5] => 4
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => [3,5,6,1,4,2] => 0
[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => [4,1,6,2,3,5] => [5,2,1,4,6,3] => 2
[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [4,1,2,5,3,6] => [5,2,3,6,4,1] => 1
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [3,1,2,6,4,5] => [4,2,3,1,6,5] => 0
[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => [3,1,2,5,6,4] => [4,2,3,6,1,5] => 0
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => [3,2,4,5,1,6] => 1
[4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => [4,5,1,2,3,6] => [5,6,2,3,4,1] => 0
[4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => [3,6,1,2,4,5] => [4,1,3,5,6,2] => 0
[4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => [4,5,1,6,2,3] => [5,6,2,1,4,3] => 2
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => [3,2,5,4,1] => 2
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => [3,4,2,1,6,5] => 4
[4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => [3,4,1,5,6,2] => [4,5,2,6,1,3] => 2
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [2,3,1,5,6,4] => [3,4,2,6,1,5] => 3
[3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => [3,5,6,1,2,4] => [4,6,1,3,5,2] => 0
[3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => [3,4,5,1,2,6] => [4,5,6,2,3,1] => 0
[3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [3,4,6,2,5,1] => 2
[3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => [3,5,6,2,1,4] => 4
[5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => [4,1,5,2,3,6] => [5,2,6,3,4,1] => 0
[5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0] => [3,1,6,2,4,5] => [4,2,1,5,6,3] => 3
[5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0] => [4,1,5,6,2,3] => [5,2,6,1,4,3] => 1
[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4,6] => [4,2,3,6,5,1] => 1
[5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,3,6,4,5] => [3,2,4,1,6,5] => 1
[5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0] => [3,1,4,5,6,2] => [4,2,5,6,1,3] => 1
[5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,0] => [2,1,3,5,6,4] => [3,2,4,6,1,5] => 1
[4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => [4,5,1,2,6,3] => [5,6,2,3,1,4] => 0
[4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4,6] => [4,6,2,3,5,1] => 0
[4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => [3,1,4,5,6,2] => 0
[4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => [3,5,1,6,2,4] => [4,6,2,1,5,3] => 3
[4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0] => [3,4,1,5,2,6] => [4,5,2,6,3,1] => 1
[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => [3,4,2,6,5,1] => 3
[4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => [3,5,2,6,1,4] => 3
[3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0] => [3,4,6,1,2,5] => [4,5,1,3,6,2] => 0
[3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => [3,6,1,4,5,2] => 0
>>> Load all 132 entries. <<<
[3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [3,5,6,2,4,1] => 1
[5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0] => [4,1,5,2,6,3] => [5,2,6,3,1,4] => 1
[5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4,6] => [4,2,6,3,5,1] => 0
[5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => [3,2,1,5,6,4] => 4
[5,3,3,1] => [1,1,0,1,0,0,1,1,0,0,1,0] => [3,1,5,6,2,4] => [4,2,6,1,5,3] => 2
[5,3,2,2] => [1,1,0,0,1,1,0,1,0,0,1,0] => [3,1,4,5,2,6] => [4,2,5,6,3,1] => 2
[5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,6] => [3,2,4,6,5,1] => 2
[5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => [3,2,5,6,1,4] => 2
[4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,0] => [3,5,1,2,6,4] => [4,6,2,3,1,5] => 0
[4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [3,4,1,2,5,6] => [4,5,2,3,6,1] => 0
[4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,0,0] => [2,5,1,3,4,6] => [3,6,2,4,5,1] => 1
[4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,0] => [3,4,1,6,2,5] => [4,5,2,1,6,3] => 3
[4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => [3,6,2,1,5,4] => 4
[4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => [3,5,2,6,4,1] => 2
[3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => [3,5,1,4,6,2] => 0
[5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => [3,1,5,2,6,4] => [4,2,6,3,1,5] => 1
[5,4,2,2] => [1,1,0,0,1,1,0,0,1,0,1,0] => [3,1,4,2,5,6] => [4,2,5,3,6,1] => 1
[5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0] => [2,1,5,3,4,6] => [3,2,6,4,5,1] => 1
[5,3,3,2] => [1,1,0,0,1,0,1,1,0,0,1,0] => [3,1,4,6,2,5] => [4,2,5,1,6,3] => 1
[5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => [3,2,6,1,5,4] => 3
[5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [3,2,5,6,4,1] => 3
[4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [3,4,1,2,6,5] => [4,5,2,3,1,6] => 0
[4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [3,6,2,4,1,5] => 1
[4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => [3,5,2,4,6,1] => 1
[4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => [3,5,2,1,6,4] => 4
[5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0] => [3,1,4,2,6,5] => [4,2,5,3,1,6] => 2
[5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => [3,2,6,4,1,5] => 2
[5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [3,2,5,4,6,1] => 2
[5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => [3,2,5,1,6,4] => 2
[4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => [3,5,2,4,1,6] => 1
[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => [3,2,5,4,1,6] => 3
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Description
The number of simsun double descents of a permutation.
The restriction of a permutation $\pi$ to $[k] = \{1,\ldots,k\}$ is given in one-line notation by the subword of $\pi$ of letters in $[k]$.
A simsun double descent of a permutation $\pi$ is a double descent of any restriction of $\pi$ to $[1,\ldots,k]$ for some $k$. (Note here that the same double descent can appear in multiple restrictions!)
Map
Lehmer code rotation
Description
Sends a permutation $\pi$ to the unique permutation $\tau$ (of the same length) such that every entry in the Lehmer code of $\tau$ is cyclically one larger than the Lehmer code of $\pi$.
Map
to 321-avoiding permutation
Description
Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.