Identifier
Values
[1] => [1,0,1,0] => [2,1] => [1,2] => 0
[2] => [1,1,0,0,1,0] => [3,1,2] => [3,1,2] => 1
[1,1] => [1,0,1,1,0,0] => [2,3,1] => [1,2,3] => 0
[3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [3,4,1,2] => 0
[2,1] => [1,0,1,0,1,0] => [3,2,1] => [2,1,3] => 1
[1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => 0
[4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [3,4,5,1,2] => 1
[3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => [2,4,1,3] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,1,2,3] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => [2,1,3,4] => 1
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => [3,4,5,6,1,2] => 0
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [2,4,5,1,3] => 1
[3,2] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => [4,2,1,3] => 1
[3,1,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => [2,3,1,4] => 0
[2,2,1] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => [3,1,2,4] => 1
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [2,1,3,4,5] => 1
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [6,2,1,3,4,5] => [2,4,5,6,1,3] => 0
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [4,2,5,1,3] => 2
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [2,3,5,1,4] => 1
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [4,5,1,2,3] => 0
[3,2,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => [3,2,1,4] => 1
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [2,3,1,4,5] => 0
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,1,2,3,4] => 1
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [3,1,2,4,5] => 1
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,4,5,6,1] => [2,1,3,4,5,6] => 1
[1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 0
[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [6,3,1,2,4,5] => [4,2,5,6,1,3] => 1
[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [6,2,3,1,4,5] => [2,3,5,6,1,4] => 0
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [4,5,2,1,3] => 1
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [3,2,5,1,4] => 2
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [2,3,4,1,5] => 1
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [3,5,1,2,4] => 0
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [5,2,1,3,4] => 1
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [3,2,1,4,5] => 1
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [4,2,3,5,6,1] => [2,3,1,4,5,6] => 0
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [4,1,2,3,5] => 1
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [3,1,2,4,5,6] => 1
[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,5,2,6,1,3] => 0
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [6,3,2,1,4,5] => [3,2,5,6,1,4] => 1
[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => [6,2,3,4,1,5] => [2,3,4,6,1,5] => 0
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => [4,5,6,1,2,3] => 1
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [3,5,2,1,4] => 1
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [5,2,3,1,4] => 2
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [3,2,4,1,5] => 2
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [5,2,3,4,6,1] => [2,3,4,1,5,6] => 1
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [5,3,1,2,4] => 1
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [3,4,1,2,5] => 0
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [4,2,1,3,5] => 1
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [3,2,1,4,5,6] => 1
[2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [6,1,2,3,4,5] => 1
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [4,1,2,3,5,6] => 1
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => [4,5,6,2,1,3] => 1
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [6,4,2,1,3,5] => [3,5,2,6,1,4] => 0
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => [6,3,4,1,2,5] => [5,2,3,6,1,4] => 1
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [6,3,2,4,1,5] => [3,2,4,6,1,5] => 1
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [2,3,4,5,1,6] => 0
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [5,6,2,1,3,4] => [3,5,6,1,2,4] => 1
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [5,3,2,1,4] => 2
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [3,4,2,1,5] => 1
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [4,2,3,1,5] => 2
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [3,2,4,1,5,6] => 2
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => [5,6,1,2,3,4] => 0
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [4,3,1,2,5] => 1
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [3,4,1,2,5,6] => 0
[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => [4,3,5,6,1,2] => [6,2,1,3,4,5] => 1
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [4,2,1,3,5,6] => 1
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [5,1,2,3,4,6] => 1
[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => [6,5,2,1,3,4] => [3,5,6,2,1,4] => 1
[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [6,4,3,1,2,5] => [5,3,2,6,1,4] => 1
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [6,4,2,3,1,5] => [3,4,2,6,1,5] => 0
[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => [6,3,4,2,1,5] => [4,2,3,6,1,5] => 1
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [3,2,4,5,1,6] => 1
[4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => [5,6,3,1,2,4] => [5,3,6,1,2,4] => 2
[4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => [5,6,2,3,1,4] => [3,4,6,1,2,5] => 1
[4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => [5,4,6,1,2,3] => [5,6,2,1,3,4] => 1
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [4,3,2,1,5] => 2
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [3,4,2,1,5,6] => 1
[4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => [5,3,4,6,1,2] => [6,2,3,1,4,5] => 2
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [4,2,3,1,5,6] => 2
[3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => [4,5,6,2,1,3] => [4,6,1,2,3,5] => 0
[3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => [4,5,3,6,1,2] => [6,3,1,2,4,5] => 1
[3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [4,3,1,2,5,6] => 1
[3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [5,2,1,3,4,6] => 1
[5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => [6,5,3,1,2,4] => [5,3,6,2,1,4] => 2
[5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0] => [6,5,2,3,1,4] => [3,4,6,2,1,5] => 1
[5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,5,1,2,3] => [5,6,2,3,1,4] => 0
[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [6,4,3,2,1,5] => [4,3,2,6,1,5] => 1
[5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [3,4,2,5,1,6] => 0
[5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => [6,2,3,4,1,5] => 1
[5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [4,2,3,5,1,6] => 1
[4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => [5,6,3,1,2,4] => 1
[4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => [5,6,3,2,1,4] => [4,3,6,1,2,5] => 2
[4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [3,4,5,1,2,6] => 1
[4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => [5,4,6,2,1,3] => [4,6,2,1,3,5] => 1
[4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0] => [5,4,3,6,1,2] => [6,3,2,1,4,5] => 2
[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [4,3,2,1,5,6] => 2
[4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [5,2,3,1,4,6] => 2
[3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => [6,4,1,2,3,5] => 1
[3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [4,5,1,2,3,6] => 0
>>> Load all 132 entries. <<<
[3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [5,3,1,2,4,6] => 1
[5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0] => [6,5,4,1,2,3] => [5,6,3,2,1,4] => 1
[5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => [6,5,3,2,1,4] => [4,3,6,2,1,5] => 2
[5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [3,4,5,2,1,6] => 1
[5,3,3,1] => [1,1,0,1,0,0,1,1,0,0,1,0] => [6,4,5,2,1,3] => [4,6,2,3,1,5] => 0
[5,3,2,2] => [1,1,0,0,1,1,0,1,0,0,1,0] => [6,4,3,5,1,2] => [6,3,2,4,1,5] => 1
[5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [4,3,2,5,1,6] => 1
[5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [5,2,3,4,1,6] => 1
[4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,0] => [5,6,4,2,1,3] => [4,6,3,1,2,5] => 1
[4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => [6,3,4,1,2,5] => 2
[4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [4,3,5,1,2,6] => 2
[4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,0] => [5,4,6,3,1,2] => [6,4,2,1,3,5] => 2
[4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [4,5,2,1,3,6] => 1
[4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [5,3,2,1,4,6] => 2
[3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [5,4,1,2,3,6] => 1
[5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => [6,5,4,2,1,3] => [4,6,3,2,1,5] => 1
[5,4,2,2] => [1,1,0,0,1,1,0,0,1,0,1,0] => [6,5,3,4,1,2] => [6,3,4,2,1,5] => 2
[5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [4,3,5,2,1,6] => 2
[5,3,3,2] => [1,1,0,0,1,0,1,1,0,0,1,0] => [6,4,5,3,1,2] => [6,4,2,3,1,5] => 1
[5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [4,5,2,3,1,6] => 0
[5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [5,3,2,4,1,6] => 1
[4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [5,6,4,3,1,2] => [6,4,3,1,2,5] => 2
[4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [4,5,3,1,2,6] => 1
[4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [5,3,4,1,2,6] => 2
[4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [5,4,2,1,3,6] => 2
[5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0] => [6,5,4,3,1,2] => [6,4,3,2,1,5] => 2
[5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [4,5,3,2,1,6] => 1
[5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [5,3,4,2,1,6] => 2
[5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [5,4,2,3,1,6] => 1
[4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [5,4,3,1,2,6] => 2
[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [5,4,3,2,1,6] => 2
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Description
The number of odd descents of a permutation.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
Inverse Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $c\pi^{-1}$ where $c = (1,\ldots,n)$ is the long cycle.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].