Identifier
Values
[1] => [1] => [1,0] => [1] => 1
[1,1] => [1,1] => [1,0,1,0] => [2,1] => 1
[2] => [2] => [1,1,0,0] => [1,2] => 1
[1,1,1] => [1,1,1] => [1,0,1,0,1,0] => [3,2,1] => 1
[1,2] => [2,1] => [1,1,0,0,1,0] => [3,1,2] => 2
[2,1] => [1,2] => [1,0,1,1,0,0] => [2,3,1] => 1
[3] => [3] => [1,1,1,0,0,0] => [1,2,3] => 1
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 1
[1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 2
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 2
[1,3] => [3,1] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 2
[2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 1
[2,2] => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 2
[3,1] => [1,3] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[4] => [4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 1
[1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 1
[1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 2
[1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 2
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 2
[1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 2
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 2
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 2
[1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 2
[2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 1
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 2
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 2
[2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 2
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 1
[3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 2
[4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[5] => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => 1
[1,1,1,1,2] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [6,5,4,3,1,2] => 2
[1,1,1,2,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => 2
[1,1,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [6,5,4,1,2,3] => 2
[1,1,2,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => 2
[1,1,2,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [6,5,3,4,1,2] => 2
[1,1,3,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => 2
[1,1,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => 2
[1,2,1,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => 2
[1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [6,4,5,3,1,2] => 2
[1,2,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,5,1,2,3] => 3
[1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => 2
[1,3,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => 2
[1,4,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => 2
[1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 2
[2,1,1,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => 1
[2,1,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [5,6,4,3,1,2] => 2
[2,1,2,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => 2
[2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => 2
[2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => 2
[2,3,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => 2
[2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 2
[3,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => 2
[3,2,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => 2
[3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => 2
[4,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => 1
[4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => 2
[5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
[6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [7,6,5,4,3,2,1] => 1
[1,1,1,1,1,2] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [7,6,5,4,3,1,2] => 2
[1,1,1,1,3] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [7,6,5,4,1,2,3] => 2
[1,1,1,2,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [7,6,5,3,4,1,2] => 2
[1,1,1,4] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [7,6,5,1,2,3,4] => 2
[1,1,2,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [7,6,4,5,1,2,3] => 3
[1,1,5] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [7,6,1,2,3,4,5] => 2
[1,2,2,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [7,5,6,3,4,1,2] => 2
[1,2,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [7,5,6,1,2,3,4] => 3
[1,3,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,1,2,3] => 3
[1,6] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1,2,3,4,5,6] => 2
[2,2,3] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => 3
[2,5] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [6,7,1,2,3,4,5] => 2
[3,4] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [5,6,7,1,2,3,4] => 2
[5,2] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [3,4,5,6,7,1,2] => 2
[6,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 1
[7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,7,6,5,4,3,2,1] => 1
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,7,6,5,4,3,1,2] => 2
[1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [8,7,6,5,4,1,2,3] => 2
[1,1,1,1,2,2] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => [8,7,6,5,3,4,1,2] => 2
[1,1,1,1,4] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0] => [8,7,6,5,1,2,3,4] => 2
[1,1,1,2,1,2] => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0] => [8,7,6,4,5,3,1,2] => 2
[1,1,1,2,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [8,7,6,4,5,2,3,1] => 2
[1,1,1,2,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0] => [8,7,6,4,5,1,2,3] => 3
[1,1,1,5] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0] => [8,7,6,1,2,3,4,5] => 2
[1,1,2,1,1,2] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0] => [8,7,5,6,4,3,1,2] => 2
[1,1,2,1,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [8,7,5,6,4,2,3,1] => 2
[1,1,2,2,1,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [8,7,5,6,3,4,2,1] => 2
[1,1,2,2,2] => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0] => [8,7,5,6,3,4,1,2] => 2
[1,1,2,4] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0] => [8,7,5,6,1,2,3,4] => 3
[1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0] => [8,7,4,5,6,1,2,3] => 3
[1,1,6] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => [8,7,1,2,3,4,5,6] => 2
[1,2,1,1,1,2] => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => [8,6,7,5,4,3,1,2] => 2
[1,2,1,1,2,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [8,6,7,5,4,2,3,1] => 2
[1,2,1,2,1,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [8,6,7,5,3,4,2,1] => 2
[1,2,2,1,1,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [8,6,7,4,5,3,2,1] => 2
[1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0] => [8,6,7,4,5,1,2,3] => 3
>>> Load all 185 entries. <<<
[1,2,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0] => [8,6,7,1,2,3,4,5] => 3
[1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0] => [8,5,6,7,4,1,2,3] => 3
[1,3,3,1] => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [8,5,6,7,2,3,4,1] => 3
[1,3,4] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0] => [8,5,6,7,1,2,3,4] => 3
[1,7] => [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] => 2
[2,1,1,1,1,2] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,8,6,5,4,3,1,2] => 2
[2,1,1,1,2,1] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [7,8,6,5,4,2,3,1] => 2
[2,1,1,2,1,1] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [7,8,6,5,3,4,2,1] => 2
[2,1,2,1,1,1] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [7,8,6,4,5,3,2,1] => 2
[2,2,1,1,1,1] => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [7,8,5,6,4,3,2,1] => 2
[2,2,2,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [7,8,5,6,3,4,1,2] => 2
[2,2,4] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0] => [7,8,5,6,1,2,3,4] => 3
[2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0] => [7,8,4,5,6,1,2,3] => 3
[2,4,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => [7,8,3,4,5,6,1,2] => 2
[2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [7,8,1,2,3,4,5,6] => 2
[3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0] => [6,7,8,5,4,1,2,3] => 2
[3,1,3,1] => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [6,7,8,5,2,3,4,1] => 2
[3,3,1,1] => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [6,7,8,3,4,5,2,1] => 2
[3,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0] => [6,7,8,1,2,3,4,5] => 2
[4,2,2] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => [5,6,7,8,3,4,1,2] => 2
[4,4] => [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] => 2
[6,2] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [3,4,5,6,7,8,1,2] => 2
[7,1] => [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
[8] => [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
[1,1,1,1,1,1,1,1,1] => [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] => [9,8,7,6,5,4,3,2,1] => 1
[1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,8,7,6,5,4,3,1,2] => 2
[1,1,1,1,1,1,3] => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,8,7,6,5,4,1,2,3] => 2
[1,1,1,1,1,2,2] => [2,2,1,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [9,8,7,6,5,3,4,1,2] => 2
[1,1,1,1,1,4] => [4,1,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0] => [9,8,7,6,5,1,2,3,4] => 2
[1,1,1,1,2,3] => [3,2,1,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => [9,8,7,6,4,5,1,2,3] => 3
[1,1,1,1,5] => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0] => [9,8,7,6,1,2,3,4,5] => 2
[1,1,1,2,2,2] => [2,2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [9,8,7,5,6,3,4,1,2] => 2
[1,1,1,2,4] => [4,2,1,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0] => [9,8,7,5,6,1,2,3,4] => 3
[1,1,1,3,3] => [3,3,1,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0] => [9,8,7,4,5,6,1,2,3] => 3
[1,1,1,6] => [6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0] => [9,8,7,1,2,3,4,5,6] => 2
[1,1,2,5] => [5,2,1,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0] => [9,8,6,7,1,2,3,4,5] => 3
[1,1,3,4] => [4,3,1,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0] => [9,8,5,6,7,1,2,3,4] => 3
[1,1,7] => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0] => [9,8,1,2,3,4,5,6,7] => 2
[1,2,2,2,2] => [2,2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [9,7,8,5,6,3,4,1,2] => 2
[1,2,2,4] => [4,2,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0] => [9,7,8,5,6,1,2,3,4] => 3
[1,2,6] => [6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0] => [9,7,8,1,2,3,4,5,6] => 3
[1,3,5] => [5,3,1] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0] => [9,6,7,8,1,2,3,4,5] => 3
[1,4,4] => [4,4,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0] => [9,5,6,7,8,1,2,3,4] => 3
[1,8] => [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] => 2
[2,2,2,3] => [3,2,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [8,9,6,7,4,5,1,2,3] => 3
[2,3,4] => [4,3,2] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0] => [8,9,5,6,7,1,2,3,4] => 3
[2,7] => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0] => [8,9,1,2,3,4,5,6,7] => 2
[3,3,3] => [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0] => [7,8,9,4,5,6,1,2,3] => 3
[3,6] => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0] => [7,8,9,1,2,3,4,5,6] => 2
[4,5] => [5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0] => [6,7,8,9,1,2,3,4,5] => 2
[8,1] => [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
[9] => [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
[1,1,1,1,1,1,1,1,1,1] => [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] => [10,9,8,7,6,5,4,3,2,1] => 1
[1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,9,8,7,6,5,4,3,1,2] => 2
[1,1,1,1,1,1,1,3] => [3,1,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,9,8,7,6,5,4,1,2,3] => 2
[1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,9,8,7,6,5,3,4,1,2] => 2
[1,1,1,1,1,1,4] => [4,1,1,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,9,8,7,6,5,1,2,3,4] => 2
[1,1,1,1,1,2,3] => [3,2,1,1,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [10,9,8,7,6,4,5,1,2,3] => 3
[1,1,1,1,1,5] => [5,1,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0] => [10,9,8,7,6,1,2,3,4,5] => 2
[1,1,1,1,2,2,2] => [2,2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => [10,9,8,7,5,6,3,4,1,2] => 2
[1,1,1,1,6] => [6,1,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0] => [10,9,8,7,1,2,3,4,5,6] => 2
[1,1,1,7] => [7,1,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0] => [10,9,8,1,2,3,4,5,6,7] => 2
[1,1,2,2,2,2] => [2,2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0] => [10,9,7,8,5,6,3,4,1,2] => 2
[1,1,4,4] => [4,4,1,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0] => [10,9,5,6,7,8,1,2,3,4] => 3
[1,1,8] => [8,1,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0] => [10,9,1,2,3,4,5,6,7,8] => 2
[1,2,2,2,3] => [3,2,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [10,8,9,6,7,4,5,1,2,3] => 3
[1,2,2,5] => [5,2,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0] => [10,8,9,6,7,1,2,3,4,5] => 3
[1,2,7] => [7,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0] => [10,8,9,1,2,3,4,5,6,7] => 3
[1,3,6] => [6,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0] => [10,7,8,9,1,2,3,4,5,6] => 3
[1,4,5] => [5,4,1] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0] => [10,6,7,8,9,1,2,3,4,5] => 3
[1,9] => [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] => 2
[2,2,2,2,2] => [2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [9,10,7,8,5,6,3,4,1,2] => 2
[2,2,2,4] => [4,2,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [9,10,7,8,5,6,1,2,3,4] => 3
[2,2,3,3] => [3,3,2,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [9,10,7,8,4,5,6,1,2,3] => 3
[2,4,4] => [4,4,2] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => [9,10,5,6,7,8,1,2,3,4] => 3
[2,8] => [8,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0] => [9,10,1,2,3,4,5,6,7,8] => 2
[3,3,4] => [4,3,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0] => [8,9,10,5,6,7,1,2,3,4] => 3
[3,7] => [7,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0] => [8,9,10,1,2,3,4,5,6,7] => 2
[4,6] => [6,4] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 2
[5,5] => [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0] => [6,7,8,9,10,1,2,3,4,5] => 2
[9,1] => [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
[10] => [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
[1,10] => [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] => 2
[10,1] => [1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
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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
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
Map
reverse
Description
Return the reversal of a composition.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.