Identifier
Values
[1] => [1,0] => [1] => [1] => 1
[1,1] => [1,0,1,0] => [2,1] => [1,2] => 1
[2] => [1,1,0,0] => [1,2] => [1,2] => 1
[1,1,1] => [1,0,1,0,1,0] => [2,1,3] => [1,2,3] => 1
[1,2] => [1,0,1,1,0,0] => [2,3,1] => [1,2,3] => 1
[2,1] => [1,1,0,0,1,0] => [3,1,2] => [1,3,2] => 2
[3] => [1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [2,1,4,3] => [1,2,3,4] => 1
[1,1,2] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => [1,2,4,3] => 2
[1,2,1] => [1,0,1,1,0,0,1,0] => [2,1,3,4] => [1,2,3,4] => 1
[1,3] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => 1
[2,1,1] => [1,1,0,0,1,0,1,0] => [3,1,4,2] => [1,3,4,2] => 2
[2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [1,3,2,4] => 2
[3,1] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [1,4,3,2] => 2
[4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => [1,2,3,4,5] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [1,2,4,3,5] => 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => [1,2,3,4,5] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [1,2,4,3,5] => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => [1,2,3,5,4] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [1,2,5,4,3] => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [1,2,3,4,5] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => [1,3,4,2,5] => 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => [1,3,2,4,5] => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [1,3,4,5,2] => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [1,3,5,2,4] => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => [1,4,2,3,5] => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [1,4,2,5,3] => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [1,5,4,3,2] => 2
[5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => [1,2,4,3,5,6] => 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => [1,2,3,4,6,5] => 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => [1,2,4,3,6,5] => 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => [1,2,4,3,5,6] => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => [1,2,3,4,5,6] => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => [1,2,4,6,3,5] => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => [1,2,3,5,6,4] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [1,2,5,6,4,3] => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => [1,2,3,5,4,6] => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => [1,2,5,3,6,4] => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => [1,2,3,6,5,4] => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => [1,2,6,5,4,3] => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,5,6] => [1,2,3,4,5,6] => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [3,1,4,2,6,5] => [1,3,4,2,5,6] => 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [3,4,1,2,6,5] => [1,3,2,4,5,6] => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [3,1,4,6,2,5] => [1,3,4,6,5,2] => 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [3,4,6,1,2,5] => [1,3,6,5,2,4] => 3
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [3,1,4,2,5,6] => [1,3,4,2,5,6] => 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [3,4,1,2,5,6] => [1,3,2,4,5,6] => 2
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [3,1,4,5,6,2] => [1,3,4,5,6,2] => 2
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [1,3,5,2,4,6] => 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [4,1,5,2,6,3] => [1,4,2,3,5,6] => 2
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [4,5,1,2,6,3] => [1,4,2,5,6,3] => 2
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [4,1,5,6,2,3] => [1,4,6,3,5,2] => 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => [1,4,2,5,3,6] => 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [5,1,6,2,3,4] => [1,5,3,6,4,2] => 2
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => [1,5,3,2,6,4] => 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => 2
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5,7] => [1,2,3,4,5,6,7] => 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5,7] => [1,2,4,3,5,6,7] => 2
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5,7] => [1,2,3,4,6,5,7] => 2
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5,7] => [1,2,4,3,6,5,7] => 2
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,6,7,5] => [1,2,3,4,5,6,7] => 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,6,7,5] => [1,2,4,3,5,6,7] => 2
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,6,7,3,5] => [1,2,3,4,6,5,7] => 2
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,6,7,1,3,5] => [1,2,4,7,5,3,6] => 2
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,3,7,5,6] => [1,2,3,4,5,7,6] => 2
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [2,4,1,3,7,5,6] => [1,2,4,3,5,7,6] => 2
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,7,3,5,6] => [1,2,3,4,7,6,5] => 2
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [2,4,7,1,3,5,6] => [1,2,4,3,7,6,5] => 3
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,4,3,5,6,7] => [1,2,3,4,5,6,7] => 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [2,4,1,3,5,6,7] => [1,2,4,3,5,6,7] => 2
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,4,5,6,7,3] => [1,2,3,4,5,6,7] => 1
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [2,4,5,6,7,1,3] => [1,2,4,6,3,5,7] => 2
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,5,3,6,4,7] => [1,2,3,5,6,4,7] => 2
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [2,5,1,3,6,4,7] => [1,2,5,6,4,3,7] => 2
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,5,6,3,4,7] => [1,2,3,5,4,6,7] => 2
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,6,1,3,4,7] => [1,2,5,3,6,4,7] => 2
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,5,3,6,7,4] => [1,2,3,5,6,7,4] => 2
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [2,5,1,3,6,7,4] => [1,2,5,6,7,4,3] => 2
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,6,7,3,4] => [1,2,3,5,7,4,6] => 2
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [2,5,6,7,1,3,4] => [1,2,5,3,6,4,7] => 2
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [2,1,6,3,7,4,5] => [1,2,3,6,4,5,7] => 2
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [2,6,1,3,7,4,5] => [1,2,6,4,3,5,7] => 2
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [2,1,6,7,3,4,5] => [1,2,3,6,4,7,5] => 2
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [2,6,7,1,3,4,5] => [1,2,6,4,3,7,5] => 2
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [2,1,7,3,4,5,6] => [1,2,3,7,6,5,4] => 2
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [2,7,1,3,4,5,6] => [1,2,7,6,5,4,3] => 2
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [2,1,3,4,5,6,7] => [1,2,3,4,5,6,7] => 1
[1,6] => [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] => 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [3,1,4,2,6,5,7] => [1,3,4,2,5,6,7] => 2
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [3,4,1,2,6,5,7] => [1,3,2,4,5,6,7] => 2
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [3,1,4,6,2,5,7] => [1,3,4,6,5,2,7] => 2
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [3,4,6,1,2,5,7] => [1,3,6,5,2,4,7] => 3
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [3,1,4,2,6,7,5] => [1,3,4,2,5,6,7] => 2
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [3,4,1,2,6,7,5] => [1,3,2,4,5,6,7] => 2
>>> Load all 168 entries. <<<
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,7,2,5] => [1,3,4,6,2,5,7] => 2
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [3,4,6,7,1,2,5] => [1,3,6,2,4,7,5] => 2
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,7,5,6] => [1,3,4,2,5,7,6] => 2
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,7,5,6] => [1,3,2,4,5,7,6] => 2
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [3,1,4,7,2,5,6] => [1,3,4,7,6,5,2] => 2
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [3,4,7,1,2,5,6] => [1,3,7,6,5,2,4] => 3
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,2,5,6,7] => [1,3,4,2,5,6,7] => 2
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [3,4,1,2,5,6,7] => [1,3,2,4,5,6,7] => 2
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [3,1,4,5,6,7,2] => [1,3,4,5,6,7,2] => 2
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [3,4,5,6,7,1,2] => [1,3,5,7,2,4,6] => 2
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [4,1,5,2,6,3,7] => [1,4,2,3,5,6,7] => 2
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [4,5,1,2,6,3,7] => [1,4,2,5,6,3,7] => 2
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [4,1,5,6,2,3,7] => [1,4,6,3,5,2,7] => 2
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [4,5,6,1,2,3,7] => [1,4,2,5,3,6,7] => 2
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [4,1,5,2,6,7,3] => [1,4,2,3,5,6,7] => 2
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [4,5,1,2,6,7,3] => [1,4,2,5,6,7,3] => 2
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [4,1,5,6,7,2,3] => [1,4,6,2,3,5,7] => 2
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => 2
[7] => [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] => 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] => [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => 1
[1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5,8,7] => [1,2,4,3,6,5,7,8] => 2
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,6,8,5,7] => [1,2,3,4,5,6,8,7] => 2
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,3,6,5,7,8] => [1,2,3,4,5,6,7,8] => 1
[1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [2,4,6,1,3,5,7,8] => [1,2,4,3,6,5,7,8] => 2
[1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,4,3,6,7,8,5] => [1,2,3,4,5,6,7,8] => 1
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,7,8,5,6] => [1,2,3,4,5,7,6,8] => 2
[1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [2,1,4,3,5,6,7,8] => [1,2,3,4,5,6,7,8] => 1
[1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [2,1,4,5,6,7,8,3] => [1,2,3,4,5,6,7,8] => 1
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [2,1,5,6,3,4,8,7] => [1,2,3,5,4,6,7,8] => 2
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4,7,8] => [1,2,3,5,4,6,7,8] => 2
[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [2,1,5,3,6,7,8,4] => [1,2,3,5,6,7,8,4] => 2
[1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [2,1,6,3,7,4,8,5] => [1,2,3,6,4,5,7,8] => 2
[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [2,1,6,7,8,3,4,5] => [1,2,3,6,4,7,5,8] => 2
[1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,1,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 1
[1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => 1
[2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [3,4,1,2,6,5,8,7] => [1,3,2,4,5,6,7,8] => 2
[2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,6,5,7,8] => [1,3,2,4,5,6,7,8] => 2
[2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0] => [3,4,1,2,6,7,8,5] => [1,3,2,4,5,6,7,8] => 2
[2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [3,4,1,2,7,8,5,6] => [1,3,2,4,5,7,6,8] => 2
[2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => [3,4,1,2,5,6,7,8] => [1,3,2,4,5,6,7,8] => 2
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [4,1,5,2,6,3,8,7] => [1,4,2,3,5,6,7,8] => 2
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,1,2,3,8,7] => [1,4,2,5,3,6,7,8] => 2
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0] => [4,1,5,2,6,3,7,8] => [1,4,2,3,5,6,7,8] => 2
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3,7,8] => [1,4,2,5,3,6,7,8] => 2
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,6,7,8,3] => [1,4,2,3,5,6,7,8] => 2
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => 2
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [8,1,2,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => 2
[8] => [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] => 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] => [2,1,4,3,6,5,8,7,9] => [1,2,3,4,5,6,7,8,9] => 1
[1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5,9,7,8] => [1,2,3,4,5,6,7,9,8] => 2
[1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [2,1,4,3,6,5,7,8,9] => [1,2,3,4,5,6,7,8,9] => 1
[1,1,5,1,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [2,1,4,3,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 1
[1,1,6,1] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,1,4,5,6,7,8,9,3] => [1,2,3,4,5,6,7,8,9] => 1
[1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [2,1,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 1
[1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => [1,2,3,4,5,6,7,8,9] => 1
[8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [9,1,2,3,4,5,6,7,8] => [1,9,8,7,6,5,4,3,2] => 2
[9] => [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] => 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] => [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => 1
[1,1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,3,6,5,8,10,7,9] => [1,2,3,4,5,6,7,8,10,9] => 2
[1,1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5,8,7,9,10] => [1,2,3,4,5,6,7,8,9,10] => 1
[1,1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [2,1,4,3,6,5,8,9,10,7] => [1,2,3,4,5,6,7,8,9,10] => 1
[1,1,7,1] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [2,1,4,5,6,7,8,9,10,3] => [1,2,3,4,5,6,7,8,9,10] => 1
[1,8,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [2,1,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 1
[1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => [1,2,3,4,5,6,7,8,9,10] => 1
[9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => [1,10,9,8,7,6,5,4,3,2] => 2
[10] => [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] => 1
[10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => [1,11,10,9,8,7,6,5,4,3,2] => 2
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Description
The number of parts of the shifted shape of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of parts of the shifted shape.
Map
bounce path
Description
The bounce path determined by an integer composition.
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
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.