Identifier
Values
[1,2] => [1,0,1,0] => [1,2] => [2,1] => 0
[2,1] => [1,1,0,0] => [2,1] => [1,2] => 0
[1,2,3] => [1,0,1,0,1,0] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,0,1,1,0,0] => [1,3,2] => [3,1,2] => 0
[2,1,3] => [1,1,0,0,1,0] => [2,1,3] => [2,3,1] => 0
[2,3,1] => [1,1,0,1,0,0] => [2,3,1] => [2,1,3] => 0
[3,1,2] => [1,1,1,0,0,0] => [3,2,1] => [1,2,3] => 0
[3,2,1] => [1,1,1,0,0,0] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => [4,3,1,2] => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => [4,2,3,1] => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => [4,2,1,3] => 0
[1,4,2,3] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => [4,1,2,3] => 0
[1,4,3,2] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => [4,1,2,3] => 0
[2,1,3,4] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => [3,4,2,1] => 0
[2,1,4,3] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => [3,4,1,2] => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,2,4,1] => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => [3,2,1,4] => 0
[2,4,1,3] => [1,1,0,1,1,0,0,0] => [2,4,3,1] => [3,1,2,4] => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0] => [2,4,3,1] => [3,1,2,4] => 0
[3,1,2,4] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => [2,3,4,1] => 0
[3,1,4,2] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => [2,3,1,4] => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => [2,3,4,1] => 0
[3,2,4,1] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => [2,3,1,4] => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0] => [4,2,3,1] => [1,3,2,4] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0] => [4,2,3,1] => [1,3,2,4] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => [1,2,3,4] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => [1,2,3,4] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => [1,2,3,4] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => [1,2,3,4] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [5,4,2,3,1] => 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [5,4,1,2,3] => 0
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [5,4,1,2,3] => 0
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [5,3,4,2,1] => 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [5,3,4,1,2] => 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [5,3,2,4,1] => 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [5,3,2,1,4] => 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [5,3,1,2,4] => 0
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [5,3,1,2,4] => 0
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [5,2,3,4,1] => 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [5,2,3,1,4] => 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [5,2,3,4,1] => 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [5,2,3,1,4] => 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => [5,1,3,2,4] => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => [5,1,3,2,4] => 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [4,5,3,2,1] => 0
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [4,5,3,1,2] => 0
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [4,5,2,3,1] => 0
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [4,5,2,1,3] => 0
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [4,3,5,2,1] => 0
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [4,3,5,1,2] => 0
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [4,3,2,5,1] => 0
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [4,3,2,1,5] => 0
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [4,3,1,2,5] => 0
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [4,3,1,2,5] => 0
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [4,2,3,5,1] => 0
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [4,2,3,1,5] => 0
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [4,2,3,5,1] => 0
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [4,2,3,1,5] => 0
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => [4,1,3,2,5] => 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => [4,1,3,2,5] => 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [4,1,2,3,5] => 0
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [4,1,2,3,5] => 0
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [4,1,2,3,5] => 0
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [4,1,2,3,5] => 0
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [4,1,2,3,5] => 0
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [4,1,2,3,5] => 0
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [3,4,5,2,1] => 0
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [3,4,5,1,2] => 0
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [3,4,2,5,1] => 0
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [3,4,2,1,5] => 0
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [3,4,1,2,5] => 0
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [3,4,1,2,5] => 0
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [3,4,5,2,1] => 0
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [3,4,5,1,2] => 0
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [3,4,2,5,1] => 0
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [3,4,2,1,5] => 0
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [3,4,1,2,5] => 0
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [3,4,1,2,5] => 0
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => [2,4,3,5,1] => 1
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => [2,4,3,1,5] => 1
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => [2,4,3,5,1] => 1
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => [2,4,3,1,5] => 1
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => [1,4,3,2,5] => 2
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => [1,4,3,2,5] => 2
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [1,4,2,3,5] => 1
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [1,4,2,3,5] => 1
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [1,4,2,3,5] => 1
>>> Load all 152 entries. <<<
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [1,4,2,3,5] => 1
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [1,4,2,3,5] => 1
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [1,4,2,3,5] => 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [2,3,4,5,1] => 0
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [2,3,4,1,5] => 0
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [2,3,4,5,1] => 0
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [2,3,4,1,5] => 0
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [1,3,4,2,5] => 1
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [1,3,4,2,5] => 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [2,3,4,5,1] => 0
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [2,3,4,1,5] => 0
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [2,3,4,5,1] => 0
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [2,3,4,1,5] => 0
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [1,3,4,2,5] => 1
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [1,3,4,2,5] => 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [2,3,4,5,1] => 0
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [2,3,4,1,5] => 0
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [2,3,4,5,1] => 0
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [2,3,4,1,5] => 0
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [1,3,4,2,5] => 1
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [1,3,4,2,5] => 1
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [1,3,2,4,5] => 1
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [1,3,2,4,5] => 1
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [1,3,2,4,5] => 1
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [1,3,2,4,5] => 1
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [1,3,2,4,5] => 1
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [1,3,2,4,5] => 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => 0
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Description
The Castelnuovo-Mumford regularity of a permutation.
The Castelnuovo-Mumford regularity of a permutation $\sigma$ is the Castelnuovo-Mumford regularity of the matrix Schubert variety $X_\sigma$.
Equivalently, it is the difference between the degrees of the Grothendieck polynomial and the Schubert polynomial for $\sigma$. It can be computed by subtracting the Coxeter length St000018The number of inversions of a permutation. from the Rajchgot index St001759The Rajchgot index of a permutation..
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.