Identifier
Values
[1] => [1,0] => [1] => [1] => 0
[2] => [1,0,1,0] => [2,1] => [2,1] => 0
[1,1] => [1,1,0,0] => [1,2] => [1,2] => 0
[3] => [1,0,1,0,1,0] => [2,3,1] => [3,1,2] => 0
[2,1] => [1,0,1,1,0,0] => [2,1,3] => [2,1,3] => 0
[1,1,1] => [1,1,0,1,0,0] => [3,1,2] => [1,3,2] => 0
[4] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => [4,1,2,3] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => [3,1,2,4] => 0
[2,2] => [1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [2,4,1,3] => [1,3,4,2] => 0
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [3,4,1,2] => [1,4,2,3] => 0
[5] => [1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [5,1,2,3,4] => 0
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [4,1,2,3,5] => 0
[3,2] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 0
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [1,3,4,5,2] => 0
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => [2,1,4,3] => 0
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [1,3,5,2,4] => 0
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [1,5,2,3,4] => 0
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 0
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => [5,1,2,3,4,6] => 0
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [3,1,2,4,5] => 0
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => [1,3,4,5,6,2] => 0
[3,3] => [1,1,1,0,1,0,0,0] => [4,1,2,3] => [1,2,4,3] => 0
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [3,2,1,5,4] => 1
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => [1,3,4,6,2,5] => 0
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [2,1,5,3,4] => 0
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => [1,3,6,2,4,5] => 0
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,1,2] => [1,6,2,3,4,5] => 0
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => 0
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,6,1,7] => [6,1,2,3,4,5,7] => 0
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => [4,1,2,3,5,6] => 0
[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,5,7,1,6] => [1,3,4,5,6,7,2] => 0
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [1,3,2,5,4] => 0
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => [4,2,3,1,6,5] => 0
[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,4,6,7,1,5] => [1,3,4,5,7,2,6] => 0
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [3,1,5,2,4] => 0
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => [3,2,1,6,4,5] => 1
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,5,6,7,1,4] => [1,3,4,7,2,5,6] => 0
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [2,3,1,5,4] => 0
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,5,6,2,3] => [2,1,6,3,4,5] => 0
[2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [2,4,5,6,7,1,3] => [1,3,7,2,4,5,6] => 0
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,7,1,2] => [1,7,2,3,4,5,6] => 0
[8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => 0
[7,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,6,7,1,8] => [7,1,2,3,4,5,6,8] => 0
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,5,1,6,7] => [5,1,2,3,4,6,7] => 0
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => [1,3,4,2,6,5] => 0
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [1,2,5,3,4] => 0
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => [4,2,1,6,3,5] => 1
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => [3,1,2,4,5,6] => 0
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [1,2,4,3,5] => 0
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [4,1,5,6,2,3] => [3,1,6,2,4,5] => 0
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [2,1,3,6,4,5] => [3,2,4,1,6,5] => 0
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [1,2,3,5,4] => 0
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,2,5,6,3,4] => [2,3,1,6,4,5] => 0
[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [1,4,5,6,7,2,3] => [2,1,7,3,4,5,6] => 0
[1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,7,8,1,2] => [1,8,2,3,4,5,6,7] => 0
[9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,7,8,9,1] => [9,1,2,3,4,5,6,7,8] => 0
[8,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,6,7,8,1,9] => [8,1,2,3,4,5,6,7,9] => 0
[7,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,5,6,1,7,8] => [6,1,2,3,4,5,7,8] => 0
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,4,7,1,5,6] => [1,3,4,5,2,7,6] => 0
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => [1,3,2,6,4,5] => 0
[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,4,1,5,6,7] => [4,1,2,3,5,6,7] => 0
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [4,5,1,6,2,3] => [4,1,6,2,3,5] => 0
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => [1,3,2,5,4,6] => 0
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,2,6,3,5] => [2,4,1,5,6,3] => 0
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [2,6,1,3,4,5] => [1,3,2,4,6,5] => 0
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [5,1,2,6,3,4] => [2,4,1,6,3,5] => 0
[2,2,1,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [1,4,5,6,7,8,2,3] => [2,1,8,3,4,5,6,7] => 0
[1,1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,7,8,9,1,2] => [1,9,2,3,4,5,6,7,8] => 0
[10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,7,8,9,10,1] => [10,1,2,3,4,5,6,7,8,9] => 0
[9,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,6,7,8,9,1,10] => [9,1,2,3,4,5,6,7,8,10] => 0
[8,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,5,6,7,1,8,9] => [7,1,2,3,4,5,6,8,9] => 0
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,6,7,1,4,5] => [1,3,4,2,7,5,6] => 0
[6,2,2] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1,6,7,8] => [5,1,2,3,4,6,7,8] => 0
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [4,5,6,1,2,3] => [1,2,6,3,4,5] => 0
[5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,6,1,4,5,7] => [1,3,4,2,6,5,7] => 0
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [4,5,1,2,3,6] => [1,2,5,3,4,6] => 0
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 0
[4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [2,3,7,1,4,5,6] => [1,3,4,2,5,7,6] => 0
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,2,3,6,4,5] => [2,3,4,1,6,5] => 0
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [4,6,1,2,3,5] => [1,2,3,5,6,4] => 0
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => [1,2,3,6,4,5] => 0
[2,2,1,1,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,4,5,6,7,8,9,2,3] => [2,1,9,3,4,5,6,7,8] => 0
[1,1,1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,7,8,9,10,1,2] => [1,10,2,3,4,5,6,7,8,9] => 0
[11] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,7,8,9,10,11,1] => [11,1,2,3,4,5,6,7,8,9,10] => 0
[10,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,6,7,8,9,10,1,11] => [10,1,2,3,4,5,6,7,8,9,11] => 0
[6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [2,5,6,7,1,3,4] => [1,3,2,7,4,5,6] => 0
[5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [2,5,6,1,3,4,7] => [1,3,2,6,4,5,7] => 0
[5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,1,4,5,6,7] => [3,1,2,4,5,6,7] => 0
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [4,1,2,3,5,6] => [1,2,4,3,5,6] => 0
[4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [2,5,7,1,3,4,6] => [1,3,2,4,6,7,5] => 0
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,6,2,3,4,5] => [2,1,3,4,6,5] => 0
[3,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [2,6,7,1,3,4,5] => [1,3,2,4,7,5,6] => 0
[1,1,1,1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,7,8,9,10,11,1,2] => [1,11,2,3,4,5,6,7,8,9,10] => 0
[6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 0
[5,5,2] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [4,5,6,1,2,3,7] => [1,2,6,3,4,5,7] => 0
[5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [2,5,1,3,4,6,7] => [1,3,2,5,4,6,7] => 0
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 0
>>> Load all 139 entries. <<<
[4,4,2,2] => [1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [4,5,7,1,2,3,6] => [1,2,3,5,6,7,4] => 0
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[3,3,2,2,2] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [4,6,7,1,2,3,5] => [1,2,3,5,7,4,6] => 0
[2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [5,6,7,1,2,3,4] => [1,2,3,7,4,5,6] => 0
[7,6] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [2,5,6,7,8,1,3,4] => [1,3,2,8,4,5,6,7] => 0
[6,6,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0] => [4,5,6,7,1,8,2,3] => [6,1,8,2,3,4,5,7] => 0
[5,5,3] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [4,5,1,2,3,6,7] => [1,2,5,3,4,6,7] => 0
[5,4,4] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,7,1,3,4,5,6] => [1,3,2,4,5,7,6] => 0
[4,4,2,2,1] => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0] => [4,5,7,1,2,8,3,6] => [2,6,1,4,5,8,3,7] => 0
[4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 0
[3,3,3,3,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,2,3,4,7,5,6] => [2,3,4,5,1,7,6] => 0
[3,3,3,2,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [1,6,7,2,3,4,5] => [2,1,3,4,7,5,6] => 0
[5,5,4] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [4,7,1,2,3,5,6] => [1,2,3,5,4,7,6] => 0
[5,5,3,1] => [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0] => [4,5,1,2,3,8,6,7] => [2,3,6,4,5,1,8,7] => 0
[5,5,2,2] => [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0] => [4,5,6,8,1,2,3,7] => [1,2,3,5,6,7,8,4] => 0
[4,4,3,3] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [4,1,2,3,5,6,7] => [1,2,4,3,5,6,7] => 0
[2,2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0] => [5,6,7,8,1,2,3,4] => [1,2,3,8,4,5,6,7] => 0
[5,5,5] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [6,7,1,2,3,4,5] => [1,2,3,4,7,5,6] => 0
[5,4,3,3] => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [2,5,1,3,4,6,7,8] => [1,3,2,5,4,6,7,8] => 0
[4,4,4,3] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [6,1,2,3,4,5,7] => [1,2,3,4,6,5,7] => 0
[3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 0
[6,5,5] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,7,8,1,3,4,5,6] => [1,3,2,4,5,8,6,7] => 0
[5,4,4,3] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [2,7,1,3,4,5,6,8] => [1,3,2,4,5,7,6,8] => 0
[4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[6,6,5] => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0] => [4,7,8,1,2,3,5,6] => [1,2,3,5,4,8,6,7] => 0
[5,5,5,2] => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0] => [6,7,1,8,2,3,4,5] => [4,1,2,3,8,5,6,7] => 0
[5,5,4,3] => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0] => [4,7,1,2,3,5,6,8] => [1,2,3,5,4,7,6,8] => 0
[5,4,4,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 0
[4,4,4,4,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [1,2,3,4,5,8,6,7] => [2,3,4,5,6,1,8,7] => 0
[] => [] => [] => [] => 0
[3,3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [7,8,1,2,3,4,5,6] => [1,2,3,4,5,8,6,7] => 0
[4,4,4,4,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[4,4,4,4,4,4] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => 0
[5,5,5,5] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => 0
[6,6,6,6,6] => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,10,9] => 0
[5,5,5,5,5,5] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 0
[5,5,5,5,5] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 0
[5,4,4,4,4] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => 0
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Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.