Identifier
Images
[1] => [1]
[1,1] => [1,1]
[2] => [2]
[1,1,1] => [1,1,1]
[1,2] => [1,2]
[2,1] => [2,1]
[3] => [3]
[1,1,1,1] => [1,1,1,1]
[1,1,2] => [1,1,2]
[1,2,1] => [2,1,1]
[1,3] => [1,3]
[2,1,1] => [1,2,1]
[2,2] => [2,2]
[3,1] => [3,1]
[4] => [4]
[1,1,1,1,1] => [1,1,1,1,1]
[1,1,1,2] => [1,1,1,2]
[1,1,2,1] => [2,1,1,1]
[1,1,3] => [1,1,3]
[1,2,1,1] => [1,2,1,1]
[1,2,2] => [1,2,2]
[1,3,1] => [3,1,1]
[1,4] => [1,4]
[2,1,1,1] => [1,1,2,1]
[2,1,2] => [2,1,2]
[2,2,1] => [2,2,1]
[2,3] => [2,3]
[3,1,1] => [1,3,1]
[3,2] => [3,2]
[4,1] => [4,1]
[5] => [5]
[1,1,1,1,1,1] => [1,1,1,1,1,1]
[1,1,1,1,2] => [1,1,1,1,2]
[1,1,1,2,1] => [2,1,1,1,1]
[1,1,1,3] => [1,1,1,3]
[1,1,2,1,1] => [1,2,1,1,1]
[1,1,2,2] => [1,1,2,2]
[1,1,3,1] => [3,1,1,1]
[1,1,4] => [1,1,4]
[1,2,1,1,1] => [1,1,2,1,1]
[1,2,1,2] => [2,1,1,2]
[1,2,2,1] => [2,2,1,1]
[1,2,3] => [1,2,3]
[1,3,1,1] => [1,3,1,1]
[1,3,2] => [3,1,2]
[1,4,1] => [4,1,1]
[1,5] => [1,5]
[2,1,1,1,1] => [1,1,1,2,1]
[2,1,1,2] => [1,2,1,2]
[2,1,2,1] => [1,2,2,1]
[2,1,3] => [2,1,3]
[2,2,1,1] => [2,1,2,1]
[2,2,2] => [2,2,2]
[2,3,1] => [2,3,1]
[2,4] => [2,4]
[3,1,1,1] => [1,1,3,1]
[3,1,2] => [1,3,2]
[3,2,1] => [3,2,1]
[3,3] => [3,3]
[4,1,1] => [1,4,1]
[4,2] => [4,2]
[5,1] => [5,1]
[6] => [6]
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
[1,1,1,1,1,2] => [1,1,1,1,1,2]
[1,1,1,1,2,1] => [2,1,1,1,1,1]
[1,1,1,1,3] => [1,1,1,1,3]
[1,1,1,2,1,1] => [1,2,1,1,1,1]
[1,1,1,2,2] => [1,1,1,2,2]
[1,1,1,3,1] => [3,1,1,1,1]
[1,1,1,4] => [1,1,1,4]
[1,1,2,1,1,1] => [1,1,2,1,1,1]
[1,1,2,1,2] => [2,1,1,1,2]
[1,1,2,2,1] => [2,2,1,1,1]
[1,1,2,3] => [1,1,2,3]
[1,1,3,1,1] => [1,3,1,1,1]
[1,1,3,2] => [3,1,1,2]
[1,1,4,1] => [4,1,1,1]
[1,1,5] => [1,1,5]
[1,2,1,1,1,1] => [1,1,1,2,1,1]
[1,2,1,1,2] => [1,2,1,1,2]
[1,2,1,2,1] => [1,2,2,1,1]
[1,2,1,3] => [2,1,1,3]
[1,2,2,1,1] => [2,1,2,1,1]
[1,2,2,2] => [1,2,2,2]
[1,2,3,1] => [2,3,1,1]
[1,2,4] => [1,2,4]
[1,3,1,1,1] => [1,1,3,1,1]
[1,3,1,2] => [1,3,1,2]
[1,3,2,1] => [3,2,1,1]
[1,3,3] => [1,3,3]
[1,4,1,1] => [1,4,1,1]
[1,4,2] => [4,1,2]
[1,5,1] => [5,1,1]
[1,6] => [1,6]
[2,1,1,1,1,1] => [1,1,1,1,2,1]
[2,1,1,1,2] => [1,1,2,1,2]
[2,1,1,2,1] => [1,1,2,2,1]
[2,1,1,3] => [1,2,1,3]
[2,1,2,1,1] => [2,1,1,2,1]
[2,1,2,2] => [2,1,2,2]
>>> Load all 1023 entries. <<<
[2,1,3,1] => [1,2,3,1]
[2,1,4] => [2,1,4]
[2,2,1,1,1] => [1,2,1,2,1]
[2,2,1,2] => [2,2,1,2]
[2,2,2,1] => [2,2,2,1]
[2,2,3] => [2,2,3]
[2,3,1,1] => [2,1,3,1]
[2,3,2] => [3,2,2]
[2,4,1] => [2,4,1]
[2,5] => [2,5]
[3,1,1,1,1] => [1,1,1,3,1]
[3,1,1,2] => [1,1,3,2]
[3,1,2,1] => [3,1,2,1]
[3,1,3] => [3,1,3]
[3,2,1,1] => [1,3,2,1]
[3,2,2] => [2,3,2]
[3,3,1] => [3,3,1]
[3,4] => [3,4]
[4,1,1,1] => [1,1,4,1]
[4,1,2] => [1,4,2]
[4,2,1] => [4,2,1]
[4,3] => [4,3]
[5,1,1] => [1,5,1]
[5,2] => [5,2]
[6,1] => [6,1]
[7] => [7]
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
[1,1,1,1,1,1,2] => [1,1,1,1,1,1,2]
[1,1,1,1,1,2,1] => [2,1,1,1,1,1,1]
[1,1,1,1,1,3] => [1,1,1,1,1,3]
[1,1,1,1,2,1,1] => [1,2,1,1,1,1,1]
[1,1,1,1,2,2] => [1,1,1,1,2,2]
[1,1,1,1,3,1] => [3,1,1,1,1,1]
[1,1,1,1,4] => [1,1,1,1,4]
[1,1,1,2,1,1,1] => [1,1,2,1,1,1,1]
[1,1,1,2,1,2] => [2,1,1,1,1,2]
[1,1,1,2,2,1] => [2,2,1,1,1,1]
[1,1,1,2,3] => [1,1,1,2,3]
[1,1,1,3,1,1] => [1,3,1,1,1,1]
[1,1,1,3,2] => [3,1,1,1,2]
[1,1,1,4,1] => [4,1,1,1,1]
[1,1,1,5] => [1,1,1,5]
[1,1,2,1,1,1,1] => [1,1,1,2,1,1,1]
[1,1,2,1,1,2] => [1,2,1,1,1,2]
[1,1,2,1,2,1] => [1,2,2,1,1,1]
[1,1,2,1,3] => [2,1,1,1,3]
[1,1,2,2,1,1] => [2,1,2,1,1,1]
[1,1,2,2,2] => [1,1,2,2,2]
[1,1,2,3,1] => [2,3,1,1,1]
[1,1,2,4] => [1,1,2,4]
[1,1,3,1,1,1] => [1,1,3,1,1,1]
[1,1,3,1,2] => [1,3,1,1,2]
[1,1,3,2,1] => [3,2,1,1,1]
[1,1,3,3] => [1,1,3,3]
[1,1,4,1,1] => [1,4,1,1,1]
[1,1,4,2] => [4,1,1,2]
[1,1,5,1] => [5,1,1,1]
[1,1,6] => [1,1,6]
[1,2,1,1,1,1,1] => [1,1,1,1,2,1,1]
[1,2,1,1,1,2] => [1,1,2,1,1,2]
[1,2,1,1,2,1] => [1,1,2,2,1,1]
[1,2,1,1,3] => [1,2,1,1,3]
[1,2,1,2,1,1] => [2,1,1,2,1,1]
[1,2,1,2,2] => [2,1,1,2,2]
[1,2,1,3,1] => [1,2,3,1,1]
[1,2,1,4] => [2,1,1,4]
[1,2,2,1,1,1] => [1,2,1,2,1,1]
[1,2,2,1,2] => [2,2,1,1,2]
[1,2,2,2,1] => [2,2,2,1,1]
[1,2,2,3] => [1,2,2,3]
[1,2,3,1,1] => [2,1,3,1,1]
[1,2,3,2] => [3,1,2,2]
[1,2,4,1] => [2,4,1,1]
[1,2,5] => [1,2,5]
[1,3,1,1,1,1] => [1,1,1,3,1,1]
[1,3,1,1,2] => [1,1,3,1,2]
[1,3,1,2,1] => [3,1,2,1,1]
[1,3,1,3] => [3,1,1,3]
[1,3,2,1,1] => [1,3,2,1,1]
[1,3,2,2] => [1,3,2,2]
[1,3,3,1] => [3,3,1,1]
[1,3,4] => [1,3,4]
[1,4,1,1,1] => [1,1,4,1,1]
[1,4,1,2] => [1,4,1,2]
[1,4,2,1] => [4,2,1,1]
[1,4,3] => [4,1,3]
[1,5,1,1] => [1,5,1,1]
[1,5,2] => [5,1,2]
[1,6,1] => [6,1,1]
[1,7] => [1,7]
[2,1,1,1,1,1,1] => [1,1,1,1,1,2,1]
[2,1,1,1,1,2] => [1,1,1,2,1,2]
[2,1,1,1,2,1] => [1,1,1,2,2,1]
[2,1,1,1,3] => [1,1,2,1,3]
[2,1,1,2,1,1] => [2,1,1,1,2,1]
[2,1,1,2,2] => [1,2,1,2,2]
[2,1,1,3,1] => [1,1,2,3,1]
[2,1,1,4] => [1,2,1,4]
[2,1,2,1,1,1] => [1,2,1,1,2,1]
[2,1,2,1,2] => [1,2,2,1,2]
[2,1,2,2,1] => [1,2,2,2,1]
[2,1,2,3] => [2,1,2,3]
[2,1,3,1,1] => [2,1,1,3,1]
[2,1,3,2] => [2,3,1,2]
[2,1,4,1] => [1,2,4,1]
[2,1,5] => [2,1,5]
[2,2,1,1,1,1] => [1,1,2,1,2,1]
[2,2,1,1,2] => [2,1,2,1,2]
[2,2,1,2,1] => [2,1,2,2,1]
[2,2,1,3] => [2,2,1,3]
[2,2,2,1,1] => [2,2,1,2,1]
[2,2,2,2] => [2,2,2,2]
[2,2,3,1] => [2,2,3,1]
[2,2,4] => [2,2,4]
[2,3,1,1,1] => [1,2,1,3,1]
[2,3,1,2] => [3,2,1,2]
[2,3,2,1] => [3,2,2,1]
[2,3,3] => [2,3,3]
[2,4,1,1] => [2,1,4,1]
[2,4,2] => [4,2,2]
[2,5,1] => [2,5,1]
[2,6] => [2,6]
[3,1,1,1,1,1] => [1,1,1,1,3,1]
[3,1,1,1,2] => [1,1,1,3,2]
[3,1,1,2,1] => [3,1,1,2,1]
[3,1,1,3] => [1,3,1,3]
[3,1,2,1,1] => [1,3,1,2,1]
[3,1,2,2] => [1,2,3,2]
[3,1,3,1] => [1,3,3,1]
[3,1,4] => [3,1,4]
[3,2,1,1,1] => [1,1,3,2,1]
[3,2,1,2] => [2,1,3,2]
[3,2,2,1] => [2,3,2,1]
[3,2,3] => [3,2,3]
[3,3,1,1] => [3,1,3,1]
[3,3,2] => [3,3,2]
[3,4,1] => [3,4,1]
[3,5] => [3,5]
[4,1,1,1,1] => [1,1,1,4,1]
[4,1,1,2] => [1,1,4,2]
[4,1,2,1] => [4,1,2,1]
[4,1,3] => [1,4,3]
[4,2,1,1] => [1,4,2,1]
[4,2,2] => [2,4,2]
[4,3,1] => [4,3,1]
[4,4] => [4,4]
[5,1,1,1] => [1,1,5,1]
[5,1,2] => [1,5,2]
[5,2,1] => [5,2,1]
[5,3] => [5,3]
[6,1,1] => [1,6,1]
[6,2] => [6,2]
[7,1] => [7,1]
[8] => [8]
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1]
[1,1,1,1,1,1,1,2] => [1,1,1,1,1,1,1,2]
[1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1]
[1,1,1,1,1,1,3] => [1,1,1,1,1,1,3]
[1,1,1,1,1,2,1,1] => [1,2,1,1,1,1,1,1]
[1,1,1,1,1,2,2] => [1,1,1,1,1,2,2]
[1,1,1,1,1,3,1] => [3,1,1,1,1,1,1]
[1,1,1,1,1,4] => [1,1,1,1,1,4]
[1,1,1,1,2,1,1,1] => [1,1,2,1,1,1,1,1]
[1,1,1,1,2,1,2] => [2,1,1,1,1,1,2]
[1,1,1,1,2,2,1] => [2,2,1,1,1,1,1]
[1,1,1,1,2,3] => [1,1,1,1,2,3]
[1,1,1,1,3,1,1] => [1,3,1,1,1,1,1]
[1,1,1,1,3,2] => [3,1,1,1,1,2]
[1,1,1,1,4,1] => [4,1,1,1,1,1]
[1,1,1,1,5] => [1,1,1,1,5]
[1,1,1,2,1,1,1,1] => [1,1,1,2,1,1,1,1]
[1,1,1,2,1,1,2] => [1,2,1,1,1,1,2]
[1,1,1,2,1,2,1] => [1,2,2,1,1,1,1]
[1,1,1,2,1,3] => [2,1,1,1,1,3]
[1,1,1,2,2,1,1] => [2,1,2,1,1,1,1]
[1,1,1,2,2,2] => [1,1,1,2,2,2]
[1,1,1,2,3,1] => [2,3,1,1,1,1]
[1,1,1,2,4] => [1,1,1,2,4]
[1,1,1,3,1,1,1] => [1,1,3,1,1,1,1]
[1,1,1,3,1,2] => [1,3,1,1,1,2]
[1,1,1,3,2,1] => [3,2,1,1,1,1]
[1,1,1,3,3] => [1,1,1,3,3]
[1,1,1,4,1,1] => [1,4,1,1,1,1]
[1,1,1,4,2] => [4,1,1,1,2]
[1,1,1,5,1] => [5,1,1,1,1]
[1,1,1,6] => [1,1,1,6]
[1,1,2,1,1,1,1,1] => [1,1,1,1,2,1,1,1]
[1,1,2,1,1,1,2] => [1,1,2,1,1,1,2]
[1,1,2,1,1,2,1] => [1,1,2,2,1,1,1]
[1,1,2,1,1,3] => [1,2,1,1,1,3]
[1,1,2,1,2,1,1] => [2,1,1,2,1,1,1]
[1,1,2,1,2,2] => [2,1,1,1,2,2]
[1,1,2,1,3,1] => [1,2,3,1,1,1]
[1,1,2,1,4] => [2,1,1,1,4]
[1,1,2,2,1,1,1] => [1,2,1,2,1,1,1]
[1,1,2,2,1,2] => [2,2,1,1,1,2]
[1,1,2,2,2,1] => [2,2,2,1,1,1]
[1,1,2,2,3] => [1,1,2,2,3]
[1,1,2,3,1,1] => [2,1,3,1,1,1]
[1,1,2,3,2] => [3,1,1,2,2]
[1,1,2,4,1] => [2,4,1,1,1]
[1,1,2,5] => [1,1,2,5]
[1,1,3,1,1,1,1] => [1,1,1,3,1,1,1]
[1,1,3,1,1,2] => [1,1,3,1,1,2]
[1,1,3,1,2,1] => [3,1,2,1,1,1]
[1,1,3,1,3] => [3,1,1,1,3]
[1,1,3,2,1,1] => [1,3,2,1,1,1]
[1,1,3,2,2] => [1,3,1,2,2]
[1,1,3,3,1] => [3,3,1,1,1]
[1,1,3,4] => [1,1,3,4]
[1,1,4,1,1,1] => [1,1,4,1,1,1]
[1,1,4,1,2] => [1,4,1,1,2]
[1,1,4,2,1] => [4,2,1,1,1]
[1,1,4,3] => [4,1,1,3]
[1,1,5,1,1] => [1,5,1,1,1]
[1,1,5,2] => [5,1,1,2]
[1,1,6,1] => [6,1,1,1]
[1,1,7] => [1,1,7]
[1,2,1,1,1,1,1,1] => [1,1,1,1,1,2,1,1]
[1,2,1,1,1,1,2] => [1,1,1,2,1,1,2]
[1,2,1,1,1,2,1] => [1,1,1,2,2,1,1]
[1,2,1,1,1,3] => [1,1,2,1,1,3]
[1,2,1,1,2,1,1] => [2,1,1,1,2,1,1]
[1,2,1,1,2,2] => [1,2,1,1,2,2]
[1,2,1,1,3,1] => [1,1,2,3,1,1]
[1,2,1,1,4] => [1,2,1,1,4]
[1,2,1,2,1,1,1] => [1,2,1,1,2,1,1]
[1,2,1,2,1,2] => [1,2,2,1,1,2]
[1,2,1,2,2,1] => [1,2,2,2,1,1]
[1,2,1,2,3] => [2,1,1,2,3]
[1,2,1,3,1,1] => [2,1,1,3,1,1]
[1,2,1,3,2] => [2,3,1,1,2]
[1,2,1,4,1] => [1,2,4,1,1]
[1,2,1,5] => [2,1,1,5]
[1,2,2,1,1,1,1] => [1,1,2,1,2,1,1]
[1,2,2,1,1,2] => [2,1,2,1,1,2]
[1,2,2,1,2,1] => [2,1,2,2,1,1]
[1,2,2,1,3] => [2,2,1,1,3]
[1,2,2,2,1,1] => [2,2,1,2,1,1]
[1,2,2,2,2] => [1,2,2,2,2]
[1,2,2,3,1] => [2,2,3,1,1]
[1,2,2,4] => [1,2,2,4]
[1,2,3,1,1,1] => [1,2,1,3,1,1]
[1,2,3,1,2] => [3,2,1,1,2]
[1,2,3,2,1] => [3,2,2,1,1]
[1,2,3,3] => [1,2,3,3]
[1,2,4,1,1] => [2,1,4,1,1]
[1,2,4,2] => [4,1,2,2]
[1,2,5,1] => [2,5,1,1]
[1,2,6] => [1,2,6]
[1,3,1,1,1,1,1] => [1,1,1,1,3,1,1]
[1,3,1,1,1,2] => [1,1,1,3,1,2]
[1,3,1,1,2,1] => [3,1,1,2,1,1]
[1,3,1,1,3] => [1,3,1,1,3]
[1,3,1,2,1,1] => [1,3,1,2,1,1]
[1,3,1,2,2] => [1,1,3,2,2]
[1,3,1,3,1] => [1,3,3,1,1]
[1,3,1,4] => [3,1,1,4]
[1,3,2,1,1,1] => [1,1,3,2,1,1]
[1,3,2,1,2] => [3,1,2,1,2]
[1,3,2,2,1] => [2,3,2,1,1]
[1,3,2,3] => [3,1,2,3]
[1,3,3,1,1] => [3,1,3,1,1]
[1,3,3,2] => [3,3,1,2]
[1,3,4,1] => [3,4,1,1]
[1,3,5] => [1,3,5]
[1,4,1,1,1,1] => [1,1,1,4,1,1]
[1,4,1,1,2] => [1,1,4,1,2]
[1,4,1,2,1] => [4,1,2,1,1]
[1,4,1,3] => [1,4,1,3]
[1,4,2,1,1] => [1,4,2,1,1]
[1,4,2,2] => [1,4,2,2]
[1,4,3,1] => [4,3,1,1]
[1,4,4] => [1,4,4]
[1,5,1,1,1] => [1,1,5,1,1]
[1,5,1,2] => [1,5,1,2]
[1,5,2,1] => [5,2,1,1]
[1,5,3] => [5,1,3]
[1,6,1,1] => [1,6,1,1]
[1,6,2] => [6,1,2]
[1,7,1] => [7,1,1]
[1,8] => [1,8]
[2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,2,1]
[2,1,1,1,1,1,2] => [1,1,1,1,2,1,2]
[2,1,1,1,1,2,1] => [1,1,1,1,2,2,1]
[2,1,1,1,1,3] => [1,1,1,2,1,3]
[2,1,1,1,2,1,1] => [2,1,1,1,1,2,1]
[2,1,1,1,2,2] => [1,1,2,1,2,2]
[2,1,1,1,3,1] => [1,1,1,2,3,1]
[2,1,1,1,4] => [1,1,2,1,4]
[2,1,1,2,1,1,1] => [1,2,1,1,1,2,1]
[2,1,1,2,1,2] => [1,1,2,2,1,2]
[2,1,1,2,2,1] => [1,1,2,2,2,1]
[2,1,1,2,3] => [1,2,1,2,3]
[2,1,1,3,1,1] => [2,1,1,1,3,1]
[2,1,1,3,2] => [1,2,3,1,2]
[2,1,1,4,1] => [1,1,2,4,1]
[2,1,1,5] => [1,2,1,5]
[2,1,2,1,1,1,1] => [1,1,2,1,1,2,1]
[2,1,2,1,1,2] => [2,1,1,2,1,2]
[2,1,2,1,2,1] => [2,1,1,2,2,1]
[2,1,2,1,3] => [1,2,2,1,3]
[2,1,2,2,1,1] => [2,2,1,1,2,1]
[2,1,2,2,2] => [2,1,2,2,2]
[2,1,2,3,1] => [1,2,2,3,1]
[2,1,2,4] => [2,1,2,4]
[2,1,3,1,1,1] => [1,2,1,1,3,1]
[2,1,3,1,2] => [2,1,3,1,2]
[2,1,3,2,1] => [3,1,2,2,1]
[2,1,3,3] => [2,1,3,3]
[2,1,4,1,1] => [2,1,1,4,1]
[2,1,4,2] => [2,4,1,2]
[2,1,5,1] => [1,2,5,1]
[2,1,6] => [2,1,6]
[2,2,1,1,1,1,1] => [1,1,1,2,1,2,1]
[2,2,1,1,1,2] => [1,2,1,2,1,2]
[2,2,1,1,2,1] => [1,2,1,2,2,1]
[2,2,1,1,3] => [2,1,2,1,3]
[2,2,1,2,1,1] => [1,2,2,1,2,1]
[2,2,1,2,2] => [2,2,1,2,2]
[2,2,1,3,1] => [2,1,2,3,1]
[2,2,1,4] => [2,2,1,4]
[2,2,2,1,1,1] => [2,1,2,1,2,1]
[2,2,2,1,2] => [2,2,2,1,2]
[2,2,2,2,1] => [2,2,2,2,1]
[2,2,2,3] => [2,2,2,3]
[2,2,3,1,1] => [2,2,1,3,1]
[2,2,3,2] => [3,2,2,2]
[2,2,4,1] => [2,2,4,1]
[2,2,5] => [2,2,5]
[2,3,1,1,1,1] => [1,1,2,1,3,1]
[2,3,1,1,2] => [1,3,2,1,2]
[2,3,1,2,1] => [2,3,1,2,1]
[2,3,1,3] => [2,3,1,3]
[2,3,2,1,1] => [3,2,1,2,1]
[2,3,2,2] => [2,3,2,2]
[2,3,3,1] => [2,3,3,1]
[2,3,4] => [2,3,4]
[2,4,1,1,1] => [1,2,1,4,1]
[2,4,1,2] => [4,2,1,2]
[2,4,2,1] => [4,2,2,1]
[2,4,3] => [4,2,3]
[2,5,1,1] => [2,1,5,1]
[2,5,2] => [5,2,2]
[2,6,1] => [2,6,1]
[2,7] => [2,7]
[3,1,1,1,1,1,1] => [1,1,1,1,1,3,1]
[3,1,1,1,1,2] => [1,1,1,1,3,2]
[3,1,1,1,2,1] => [3,1,1,1,2,1]
[3,1,1,1,3] => [1,1,3,1,3]
[3,1,1,2,1,1] => [1,3,1,1,2,1]
[3,1,1,2,2] => [1,1,2,3,2]
[3,1,1,3,1] => [1,1,3,3,1]
[3,1,1,4] => [1,3,1,4]
[3,1,2,1,1,1] => [1,1,3,1,2,1]
[3,1,2,1,2] => [2,1,1,3,2]
[3,1,2,2,1] => [1,3,2,2,1]
[3,1,2,3] => [1,3,2,3]
[3,1,3,1,1] => [3,1,1,3,1]
[3,1,3,2] => [1,3,3,2]
[3,1,4,1] => [1,3,4,1]
[3,1,5] => [3,1,5]
[3,2,1,1,1,1] => [1,1,1,3,2,1]
[3,2,1,1,2] => [1,2,1,3,2]
[3,2,1,2,1] => [1,2,3,2,1]
[3,2,1,3] => [3,2,1,3]
[3,2,2,1,1] => [2,1,3,2,1]
[3,2,2,2] => [2,2,3,2]
[3,2,3,1] => [3,2,3,1]
[3,2,4] => [3,2,4]
[3,3,1,1,1] => [1,3,1,3,1]
[3,3,1,2] => [3,1,3,2]
[3,3,2,1] => [3,3,2,1]
[3,3,3] => [3,3,3]
[3,4,1,1] => [3,1,4,1]
[3,4,2] => [3,4,2]
[3,5,1] => [3,5,1]
[3,6] => [3,6]
[4,1,1,1,1,1] => [1,1,1,1,4,1]
[4,1,1,1,2] => [1,1,1,4,2]
[4,1,1,2,1] => [4,1,1,2,1]
[4,1,1,3] => [1,1,4,3]
[4,1,2,1,1] => [1,4,1,2,1]
[4,1,2,2] => [1,2,4,2]
[4,1,3,1] => [4,1,3,1]
[4,1,4] => [4,1,4]
[4,2,1,1,1] => [1,1,4,2,1]
[4,2,1,2] => [2,1,4,2]
[4,2,2,1] => [2,4,2,1]
[4,2,3] => [2,4,3]
[4,3,1,1] => [1,4,3,1]
[4,3,2] => [4,3,2]
[4,4,1] => [4,4,1]
[4,5] => [4,5]
[5,1,1,1,1] => [1,1,1,5,1]
[5,1,1,2] => [1,1,5,2]
[5,1,2,1] => [5,1,2,1]
[5,1,3] => [1,5,3]
[5,2,1,1] => [1,5,2,1]
[5,2,2] => [2,5,2]
[5,3,1] => [5,3,1]
[5,4] => [5,4]
[6,1,1,1] => [1,1,6,1]
[6,1,2] => [1,6,2]
[6,2,1] => [6,2,1]
[6,3] => [6,3]
[7,1,1] => [1,7,1]
[7,2] => [7,2]
[8,1] => [8,1]
[9] => [9]
[1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1]
[1,1,1,1,1,1,1,1,2] => [1,1,1,1,1,1,1,1,2]
[1,1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1,1]
[1,1,1,1,1,1,1,3] => [1,1,1,1,1,1,1,3]
[1,1,1,1,1,1,2,1,1] => [1,2,1,1,1,1,1,1,1]
[1,1,1,1,1,1,2,2] => [1,1,1,1,1,1,2,2]
[1,1,1,1,1,1,3,1] => [3,1,1,1,1,1,1,1]
[1,1,1,1,1,1,4] => [1,1,1,1,1,1,4]
[1,1,1,1,1,2,1,1,1] => [1,1,2,1,1,1,1,1,1]
[1,1,1,1,1,2,1,2] => [2,1,1,1,1,1,1,2]
[1,1,1,1,1,2,2,1] => [2,2,1,1,1,1,1,1]
[1,1,1,1,1,2,3] => [1,1,1,1,1,2,3]
[1,1,1,1,1,3,1,1] => [1,3,1,1,1,1,1,1]
[1,1,1,1,1,3,2] => [3,1,1,1,1,1,2]
[1,1,1,1,1,4,1] => [4,1,1,1,1,1,1]
[1,1,1,1,1,5] => [1,1,1,1,1,5]
[1,1,1,1,2,1,1,1,1] => [1,1,1,2,1,1,1,1,1]
[1,1,1,1,2,1,1,2] => [1,2,1,1,1,1,1,2]
[1,1,1,1,2,1,2,1] => [1,2,2,1,1,1,1,1]
[1,1,1,1,2,1,3] => [2,1,1,1,1,1,3]
[1,1,1,1,2,2,1,1] => [2,1,2,1,1,1,1,1]
[1,1,1,1,2,2,2] => [1,1,1,1,2,2,2]
[1,1,1,1,2,3,1] => [2,3,1,1,1,1,1]
[1,1,1,1,2,4] => [1,1,1,1,2,4]
[1,1,1,1,3,1,1,1] => [1,1,3,1,1,1,1,1]
[1,1,1,1,3,1,2] => [1,3,1,1,1,1,2]
[1,1,1,1,3,2,1] => [3,2,1,1,1,1,1]
[1,1,1,1,3,3] => [1,1,1,1,3,3]
[1,1,1,1,4,1,1] => [1,4,1,1,1,1,1]
[1,1,1,1,4,2] => [4,1,1,1,1,2]
[1,1,1,1,5,1] => [5,1,1,1,1,1]
[1,1,1,1,6] => [1,1,1,1,6]
[1,1,1,2,1,1,1,1,1] => [1,1,1,1,2,1,1,1,1]
[1,1,1,2,1,1,1,2] => [1,1,2,1,1,1,1,2]
[1,1,1,2,1,1,2,1] => [1,1,2,2,1,1,1,1]
[1,1,1,2,1,1,3] => [1,2,1,1,1,1,3]
[1,1,1,2,1,2,1,1] => [2,1,1,2,1,1,1,1]
[1,1,1,2,1,2,2] => [2,1,1,1,1,2,2]
[1,1,1,2,1,3,1] => [1,2,3,1,1,1,1]
[1,1,1,2,1,4] => [2,1,1,1,1,4]
[1,1,1,2,2,1,1,1] => [1,2,1,2,1,1,1,1]
[1,1,1,2,2,1,2] => [2,2,1,1,1,1,2]
[1,1,1,2,2,2,1] => [2,2,2,1,1,1,1]
[1,1,1,2,2,3] => [1,1,1,2,2,3]
[1,1,1,2,3,1,1] => [2,1,3,1,1,1,1]
[1,1,1,2,3,2] => [3,1,1,1,2,2]
[1,1,1,2,4,1] => [2,4,1,1,1,1]
[1,1,1,2,5] => [1,1,1,2,5]
[1,1,1,3,1,1,1,1] => [1,1,1,3,1,1,1,1]
[1,1,1,3,1,1,2] => [1,1,3,1,1,1,2]
[1,1,1,3,1,2,1] => [3,1,2,1,1,1,1]
[1,1,1,3,1,3] => [3,1,1,1,1,3]
[1,1,1,3,2,1,1] => [1,3,2,1,1,1,1]
[1,1,1,3,2,2] => [1,3,1,1,2,2]
[1,1,1,3,3,1] => [3,3,1,1,1,1]
[1,1,1,3,4] => [1,1,1,3,4]
[1,1,1,4,1,1,1] => [1,1,4,1,1,1,1]
[1,1,1,4,1,2] => [1,4,1,1,1,2]
[1,1,1,4,2,1] => [4,2,1,1,1,1]
[1,1,1,4,3] => [4,1,1,1,3]
[1,1,1,5,1,1] => [1,5,1,1,1,1]
[1,1,1,5,2] => [5,1,1,1,2]
[1,1,1,6,1] => [6,1,1,1,1]
[1,1,1,7] => [1,1,1,7]
[1,1,2,1,1,1,1,1,1] => [1,1,1,1,1,2,1,1,1]
[1,1,2,1,1,1,1,2] => [1,1,1,2,1,1,1,2]
[1,1,2,1,1,1,2,1] => [1,1,1,2,2,1,1,1]
[1,1,2,1,1,1,3] => [1,1,2,1,1,1,3]
[1,1,2,1,1,2,1,1] => [2,1,1,1,2,1,1,1]
[1,1,2,1,1,2,2] => [1,2,1,1,1,2,2]
[1,1,2,1,1,3,1] => [1,1,2,3,1,1,1]
[1,1,2,1,1,4] => [1,2,1,1,1,4]
[1,1,2,1,2,1,1,1] => [1,2,1,1,2,1,1,1]
[1,1,2,1,2,1,2] => [1,2,2,1,1,1,2]
[1,1,2,1,2,2,1] => [1,2,2,2,1,1,1]
[1,1,2,1,2,3] => [2,1,1,1,2,3]
[1,1,2,1,3,1,1] => [2,1,1,3,1,1,1]
[1,1,2,1,3,2] => [2,3,1,1,1,2]
[1,1,2,1,4,1] => [1,2,4,1,1,1]
[1,1,2,1,5] => [2,1,1,1,5]
[1,1,2,2,1,1,1,1] => [1,1,2,1,2,1,1,1]
[1,1,2,2,1,1,2] => [2,1,2,1,1,1,2]
[1,1,2,2,1,2,1] => [2,1,2,2,1,1,1]
[1,1,2,2,1,3] => [2,2,1,1,1,3]
[1,1,2,2,2,1,1] => [2,2,1,2,1,1,1]
[1,1,2,2,2,2] => [1,1,2,2,2,2]
[1,1,2,2,3,1] => [2,2,3,1,1,1]
[1,1,2,2,4] => [1,1,2,2,4]
[1,1,2,3,1,1,1] => [1,2,1,3,1,1,1]
[1,1,2,3,1,2] => [3,2,1,1,1,2]
[1,1,2,3,2,1] => [3,2,2,1,1,1]
[1,1,2,3,3] => [1,1,2,3,3]
[1,1,2,4,1,1] => [2,1,4,1,1,1]
[1,1,2,4,2] => [4,1,1,2,2]
[1,1,2,5,1] => [2,5,1,1,1]
[1,1,2,6] => [1,1,2,6]
[1,1,3,1,1,1,1,1] => [1,1,1,1,3,1,1,1]
[1,1,3,1,1,1,2] => [1,1,1,3,1,1,2]
[1,1,3,1,1,2,1] => [3,1,1,2,1,1,1]
[1,1,3,1,1,3] => [1,3,1,1,1,3]
[1,1,3,1,2,1,1] => [1,3,1,2,1,1,1]
[1,1,3,1,2,2] => [1,1,3,1,2,2]
[1,1,3,1,3,1] => [1,3,3,1,1,1]
[1,1,3,1,4] => [3,1,1,1,4]
[1,1,3,2,1,1,1] => [1,1,3,2,1,1,1]
[1,1,3,2,1,2] => [3,1,2,1,1,2]
[1,1,3,2,2,1] => [2,3,2,1,1,1]
[1,1,3,2,3] => [3,1,1,2,3]
[1,1,3,3,1,1] => [3,1,3,1,1,1]
[1,1,3,3,2] => [3,3,1,1,2]
[1,1,3,4,1] => [3,4,1,1,1]
[1,1,3,5] => [1,1,3,5]
[1,1,4,1,1,1,1] => [1,1,1,4,1,1,1]
[1,1,4,1,1,2] => [1,1,4,1,1,2]
[1,1,4,1,2,1] => [4,1,2,1,1,1]
[1,1,4,1,3] => [1,4,1,1,3]
[1,1,4,2,1,1] => [1,4,2,1,1,1]
[1,1,4,2,2] => [1,4,1,2,2]
[1,1,4,3,1] => [4,3,1,1,1]
[1,1,4,4] => [1,1,4,4]
[1,1,5,1,1,1] => [1,1,5,1,1,1]
[1,1,5,1,2] => [1,5,1,1,2]
[1,1,5,2,1] => [5,2,1,1,1]
[1,1,5,3] => [5,1,1,3]
[1,1,6,1,1] => [1,6,1,1,1]
[1,1,6,2] => [6,1,1,2]
[1,1,7,1] => [7,1,1,1]
[1,1,8] => [1,1,8]
[1,2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,2,1,1]
[1,2,1,1,1,1,1,2] => [1,1,1,1,2,1,1,2]
[1,2,1,1,1,1,2,1] => [1,1,1,1,2,2,1,1]
[1,2,1,1,1,1,3] => [1,1,1,2,1,1,3]
[1,2,1,1,1,2,1,1] => [2,1,1,1,1,2,1,1]
[1,2,1,1,1,2,2] => [1,1,2,1,1,2,2]
[1,2,1,1,1,3,1] => [1,1,1,2,3,1,1]
[1,2,1,1,1,4] => [1,1,2,1,1,4]
[1,2,1,1,2,1,1,1] => [1,2,1,1,1,2,1,1]
[1,2,1,1,2,1,2] => [1,1,2,2,1,1,2]
[1,2,1,1,2,2,1] => [1,1,2,2,2,1,1]
[1,2,1,1,2,3] => [1,2,1,1,2,3]
[1,2,1,1,3,1,1] => [2,1,1,1,3,1,1]
[1,2,1,1,3,2] => [1,2,3,1,1,2]
[1,2,1,1,4,1] => [1,1,2,4,1,1]
[1,2,1,1,5] => [1,2,1,1,5]
[1,2,1,2,1,1,1,1] => [1,1,2,1,1,2,1,1]
[1,2,1,2,1,1,2] => [2,1,1,2,1,1,2]
[1,2,1,2,1,2,1] => [2,1,1,2,2,1,1]
[1,2,1,2,1,3] => [1,2,2,1,1,3]
[1,2,1,2,2,1,1] => [2,2,1,1,2,1,1]
[1,2,1,2,2,2] => [2,1,1,2,2,2]
[1,2,1,2,3,1] => [1,2,2,3,1,1]
[1,2,1,2,4] => [2,1,1,2,4]
[1,2,1,3,1,1,1] => [1,2,1,1,3,1,1]
[1,2,1,3,1,2] => [2,1,3,1,1,2]
[1,2,1,3,2,1] => [3,1,2,2,1,1]
[1,2,1,3,3] => [2,1,1,3,3]
[1,2,1,4,1,1] => [2,1,1,4,1,1]
[1,2,1,4,2] => [2,4,1,1,2]
[1,2,1,5,1] => [1,2,5,1,1]
[1,2,1,6] => [2,1,1,6]
[1,2,2,1,1,1,1,1] => [1,1,1,2,1,2,1,1]
[1,2,2,1,1,1,2] => [1,2,1,2,1,1,2]
[1,2,2,1,1,2,1] => [1,2,1,2,2,1,1]
[1,2,2,1,1,3] => [2,1,2,1,1,3]
[1,2,2,1,2,1,1] => [1,2,2,1,2,1,1]
[1,2,2,1,2,2] => [2,2,1,1,2,2]
[1,2,2,1,3,1] => [2,1,2,3,1,1]
[1,2,2,1,4] => [2,2,1,1,4]
[1,2,2,2,1,1,1] => [2,1,2,1,2,1,1]
[1,2,2,2,1,2] => [2,2,2,1,1,2]
[1,2,2,2,2,1] => [2,2,2,2,1,1]
[1,2,2,2,3] => [1,2,2,2,3]
[1,2,2,3,1,1] => [2,2,1,3,1,1]
[1,2,2,3,2] => [3,1,2,2,2]
[1,2,2,4,1] => [2,2,4,1,1]
[1,2,2,5] => [1,2,2,5]
[1,2,3,1,1,1,1] => [1,1,2,1,3,1,1]
[1,2,3,1,1,2] => [1,3,2,1,1,2]
[1,2,3,1,2,1] => [2,3,1,2,1,1]
[1,2,3,1,3] => [2,3,1,1,3]
[1,2,3,2,1,1] => [3,2,1,2,1,1]
[1,2,3,2,2] => [1,3,2,2,2]
[1,2,3,3,1] => [2,3,3,1,1]
[1,2,3,4] => [1,2,3,4]
[1,2,4,1,1,1] => [1,2,1,4,1,1]
[1,2,4,1,2] => [4,2,1,1,2]
[1,2,4,2,1] => [4,2,2,1,1]
[1,2,4,3] => [4,1,2,3]
[1,2,5,1,1] => [2,1,5,1,1]
[1,2,5,2] => [5,1,2,2]
[1,2,6,1] => [2,6,1,1]
[1,2,7] => [1,2,7]
[1,3,1,1,1,1,1,1] => [1,1,1,1,1,3,1,1]
[1,3,1,1,1,1,2] => [1,1,1,1,3,1,2]
[1,3,1,1,1,2,1] => [3,1,1,1,2,1,1]
[1,3,1,1,1,3] => [1,1,3,1,1,3]
[1,3,1,1,2,1,1] => [1,3,1,1,2,1,1]
[1,3,1,1,2,2] => [1,1,1,3,2,2]
[1,3,1,1,3,1] => [1,1,3,3,1,1]
[1,3,1,1,4] => [1,3,1,1,4]
[1,3,1,2,1,1,1] => [1,1,3,1,2,1,1]
[1,3,1,2,1,2] => [3,1,1,2,1,2]
[1,3,1,2,2,1] => [1,3,2,2,1,1]
[1,3,1,2,3] => [1,3,1,2,3]
[1,3,1,3,1,1] => [3,1,1,3,1,1]
[1,3,1,3,2] => [1,3,3,1,2]
[1,3,1,4,1] => [1,3,4,1,1]
[1,3,1,5] => [3,1,1,5]
[1,3,2,1,1,1,1] => [1,1,1,3,2,1,1]
[1,3,2,1,1,2] => [1,3,1,2,1,2]
[1,3,2,1,2,1] => [1,2,3,2,1,1]
[1,3,2,1,3] => [3,2,1,1,3]
[1,3,2,2,1,1] => [2,1,3,2,1,1]
[1,3,2,2,2] => [1,2,3,2,2]
[1,3,2,3,1] => [3,2,3,1,1]
[1,3,2,4] => [3,1,2,4]
[1,3,3,1,1,1] => [1,3,1,3,1,1]
[1,3,3,1,2] => [3,1,3,1,2]
[1,3,3,2,1] => [3,3,2,1,1]
[1,3,3,3] => [1,3,3,3]
[1,3,4,1,1] => [3,1,4,1,1]
[1,3,4,2] => [3,4,1,2]
[1,3,5,1] => [3,5,1,1]
[1,3,6] => [1,3,6]
[1,4,1,1,1,1,1] => [1,1,1,1,4,1,1]
[1,4,1,1,1,2] => [1,1,1,4,1,2]
[1,4,1,1,2,1] => [4,1,1,2,1,1]
[1,4,1,1,3] => [1,1,4,1,3]
[1,4,1,2,1,1] => [1,4,1,2,1,1]
[1,4,1,2,2] => [1,1,4,2,2]
[1,4,1,3,1] => [4,1,3,1,1]
[1,4,1,4] => [4,1,1,4]
[1,4,2,1,1,1] => [1,1,4,2,1,1]
[1,4,2,1,2] => [4,1,2,1,2]
[1,4,2,2,1] => [2,4,2,1,1]
[1,4,2,3] => [1,4,2,3]
[1,4,3,1,1] => [1,4,3,1,1]
[1,4,3,2] => [4,3,1,2]
[1,4,4,1] => [4,4,1,1]
[1,4,5] => [1,4,5]
[1,5,1,1,1,1] => [1,1,1,5,1,1]
[1,5,1,1,2] => [1,1,5,1,2]
[1,5,1,2,1] => [5,1,2,1,1]
[1,5,1,3] => [1,5,1,3]
[1,5,2,1,1] => [1,5,2,1,1]
[1,5,2,2] => [1,5,2,2]
[1,5,3,1] => [5,3,1,1]
[1,5,4] => [5,1,4]
[1,6,1,1,1] => [1,1,6,1,1]
[1,6,1,2] => [1,6,1,2]
[1,6,2,1] => [6,2,1,1]
[1,6,3] => [6,1,3]
[1,7,1,1] => [1,7,1,1]
[1,7,2] => [7,1,2]
[1,8,1] => [8,1,1]
[1,9] => [1,9]
[2,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2,1]
[2,1,1,1,1,1,1,2] => [1,1,1,1,1,2,1,2]
[2,1,1,1,1,1,2,1] => [1,1,1,1,1,2,2,1]
[2,1,1,1,1,1,3] => [1,1,1,1,2,1,3]
[2,1,1,1,1,2,1,1] => [2,1,1,1,1,1,2,1]
[2,1,1,1,1,2,2] => [1,1,1,2,1,2,2]
[2,1,1,1,1,3,1] => [1,1,1,1,2,3,1]
[2,1,1,1,1,4] => [1,1,1,2,1,4]
[2,1,1,1,2,1,1,1] => [1,2,1,1,1,1,2,1]
[2,1,1,1,2,1,2] => [1,1,1,2,2,1,2]
[2,1,1,1,2,2,1] => [1,1,1,2,2,2,1]
[2,1,1,1,2,3] => [1,1,2,1,2,3]
[2,1,1,1,3,1,1] => [2,1,1,1,1,3,1]
[2,1,1,1,3,2] => [1,1,2,3,1,2]
[2,1,1,1,4,1] => [1,1,1,2,4,1]
[2,1,1,1,5] => [1,1,2,1,5]
[2,1,1,2,1,1,1,1] => [1,1,2,1,1,1,2,1]
[2,1,1,2,1,1,2] => [2,1,1,1,2,1,2]
[2,1,1,2,1,2,1] => [2,1,1,1,2,2,1]
[2,1,1,2,1,3] => [1,1,2,2,1,3]
[2,1,1,2,2,1,1] => [2,2,1,1,1,2,1]
[2,1,1,2,2,2] => [1,2,1,2,2,2]
[2,1,1,2,3,1] => [1,1,2,2,3,1]
[2,1,1,2,4] => [1,2,1,2,4]
[2,1,1,3,1,1,1] => [1,2,1,1,1,3,1]
[2,1,1,3,1,2] => [2,1,1,3,1,2]
[2,1,1,3,2,1] => [3,1,1,2,2,1]
[2,1,1,3,3] => [1,2,1,3,3]
[2,1,1,4,1,1] => [2,1,1,1,4,1]
[2,1,1,4,2] => [1,2,4,1,2]
[2,1,1,5,1] => [1,1,2,5,1]
[2,1,1,6] => [1,2,1,6]
[2,1,2,1,1,1,1,1] => [1,1,1,2,1,1,2,1]
[2,1,2,1,1,1,2] => [1,2,1,1,2,1,2]
[2,1,2,1,1,2,1] => [1,2,1,1,2,2,1]
[2,1,2,1,1,3] => [2,1,1,2,1,3]
[2,1,2,1,2,1,1] => [1,2,2,1,1,2,1]
[2,1,2,1,2,2] => [1,2,2,1,2,2]
[2,1,2,1,3,1] => [2,1,1,2,3,1]
[2,1,2,1,4] => [1,2,2,1,4]
[2,1,2,2,1,1,1] => [2,1,2,1,1,2,1]
[2,1,2,2,1,2] => [1,2,2,2,1,2]
[2,1,2,2,2,1] => [1,2,2,2,2,1]
[2,1,2,2,3] => [2,1,2,2,3]
[2,1,2,3,1,1] => [2,2,1,1,3,1]
[2,1,2,3,2] => [2,3,1,2,2]
[2,1,2,4,1] => [1,2,2,4,1]
[2,1,2,5] => [2,1,2,5]
[2,1,3,1,1,1,1] => [1,1,2,1,1,3,1]
[2,1,3,1,1,2] => [1,2,1,3,1,2]
[2,1,3,1,2,1] => [2,3,1,1,2,1]
[2,1,3,1,3] => [1,2,3,1,3]
[2,1,3,2,1,1] => [3,2,1,1,2,1]
[2,1,3,2,2] => [3,2,1,2,2]
[2,1,3,3,1] => [1,2,3,3,1]
[2,1,3,4] => [2,1,3,4]
[2,1,4,1,1,1] => [1,2,1,1,4,1]
[2,1,4,1,2] => [2,1,4,1,2]
[2,1,4,2,1] => [4,1,2,2,1]
[2,1,4,3] => [2,4,1,3]
[2,1,5,1,1] => [2,1,1,5,1]
[2,1,5,2] => [2,5,1,2]
[2,1,6,1] => [1,2,6,1]
[2,1,7] => [2,1,7]
[2,2,1,1,1,1,1,1] => [1,1,1,1,2,1,2,1]
[2,2,1,1,1,1,2] => [1,1,2,1,2,1,2]
[2,2,1,1,1,2,1] => [1,1,2,1,2,2,1]
[2,2,1,1,1,3] => [1,2,1,2,1,3]
[2,2,1,1,2,1,1] => [1,1,2,2,1,2,1]
[2,2,1,1,2,2] => [2,1,2,1,2,2]
[2,2,1,1,3,1] => [1,2,1,2,3,1]
[2,2,1,1,4] => [2,1,2,1,4]
[2,2,1,2,1,1,1] => [2,1,1,2,1,2,1]
[2,2,1,2,1,2] => [2,1,2,2,1,2]
[2,2,1,2,2,1] => [2,1,2,2,2,1]
[2,2,1,2,3] => [2,2,1,2,3]
[2,2,1,3,1,1] => [1,2,2,1,3,1]
[2,2,1,3,2] => [2,2,3,1,2]
[2,2,1,4,1] => [2,1,2,4,1]
[2,2,1,5] => [2,2,1,5]
[2,2,2,1,1,1,1] => [1,2,1,2,1,2,1]
[2,2,2,1,1,2] => [2,2,1,2,1,2]
[2,2,2,1,2,1] => [2,2,1,2,2,1]
[2,2,2,1,3] => [2,2,2,1,3]
[2,2,2,2,1,1] => [2,2,2,1,2,1]
[2,2,2,2,2] => [2,2,2,2,2]
[2,2,2,3,1] => [2,2,2,3,1]
[2,2,2,4] => [2,2,2,4]
[2,2,3,1,1,1] => [2,1,2,1,3,1]
[2,2,3,1,2] => [3,2,2,1,2]
[2,2,3,2,1] => [3,2,2,2,1]
[2,2,3,3] => [2,2,3,3]
[2,2,4,1,1] => [2,2,1,4,1]
[2,2,4,2] => [4,2,2,2]
[2,2,5,1] => [2,2,5,1]
[2,2,6] => [2,2,6]
[2,3,1,1,1,1,1] => [1,1,1,2,1,3,1]
[2,3,1,1,1,2] => [1,1,3,2,1,2]
[2,3,1,1,2,1] => [1,2,3,1,2,1]
[2,3,1,1,3] => [2,1,3,1,3]
[2,3,1,2,1,1] => [2,1,3,1,2,1]
[2,3,1,2,2] => [2,1,3,2,2]
[2,3,1,3,1] => [2,1,3,3,1]
[2,3,1,4] => [2,3,1,4]
[2,3,2,1,1,1] => [1,3,2,1,2,1]
[2,3,2,1,2] => [2,3,2,1,2]
[2,3,2,2,1] => [2,3,2,2,1]
[2,3,2,3] => [3,2,2,3]
[2,3,3,1,1] => [2,3,1,3,1]
[2,3,3,2] => [3,3,2,2]
[2,3,4,1] => [2,3,4,1]
[2,3,5] => [2,3,5]
[2,4,1,1,1,1] => [1,1,2,1,4,1]
[2,4,1,1,2] => [1,4,2,1,2]
[2,4,1,2,1] => [2,4,1,2,1]
[2,4,1,3] => [4,2,1,3]
[2,4,2,1,1] => [4,2,1,2,1]
[2,4,2,2] => [2,4,2,2]
[2,4,3,1] => [4,2,3,1]
[2,4,4] => [2,4,4]
[2,5,1,1,1] => [1,2,1,5,1]
[2,5,1,2] => [5,2,1,2]
[2,5,2,1] => [5,2,2,1]
[2,5,3] => [5,2,3]
[2,6,1,1] => [2,1,6,1]
[2,6,2] => [6,2,2]
[2,7,1] => [2,7,1]
[2,8] => [2,8]
[3,1,1,1,1,1,1,1] => [1,1,1,1,1,1,3,1]
[3,1,1,1,1,1,2] => [1,1,1,1,1,3,2]
[3,1,1,1,1,2,1] => [3,1,1,1,1,2,1]
[3,1,1,1,1,3] => [1,1,1,3,1,3]
[3,1,1,1,2,1,1] => [1,3,1,1,1,2,1]
[3,1,1,1,2,2] => [1,1,1,2,3,2]
[3,1,1,1,3,1] => [1,1,1,3,3,1]
[3,1,1,1,4] => [1,1,3,1,4]
[3,1,1,2,1,1,1] => [1,1,3,1,1,2,1]
[3,1,1,2,1,2] => [2,1,1,1,3,2]
[3,1,1,2,2,1] => [1,3,1,2,2,1]
[3,1,1,2,3] => [1,1,3,2,3]
[3,1,1,3,1,1] => [3,1,1,1,3,1]
[3,1,1,3,2] => [1,1,3,3,2]
[3,1,1,4,1] => [1,1,3,4,1]
[3,1,1,5] => [1,3,1,5]
[3,1,2,1,1,1,1] => [1,1,1,3,1,2,1]
[3,1,2,1,1,2] => [1,2,1,1,3,2]
[3,1,2,1,2,1] => [1,1,3,2,2,1]
[3,1,2,1,3] => [3,1,2,1,3]
[3,1,2,2,1,1] => [3,1,2,1,2,1]
[3,1,2,2,2] => [1,2,2,3,2]
[3,1,2,3,1] => [3,1,2,3,1]
[3,1,2,4] => [1,3,2,4]
[3,1,3,1,1,1] => [1,3,1,1,3,1]
[3,1,3,1,2] => [3,1,1,3,2]
[3,1,3,2,1] => [3,3,1,2,1]
[3,1,3,3] => [3,1,3,3]
[3,1,4,1,1] => [3,1,1,4,1]
[3,1,4,2] => [1,3,4,2]
[3,1,5,1] => [1,3,5,1]
[3,1,6] => [3,1,6]
[3,2,1,1,1,1,1] => [1,1,1,1,3,2,1]
[3,2,1,1,1,2] => [1,1,2,1,3,2]
[3,2,1,1,2,1] => [1,1,2,3,2,1]
[3,2,1,1,3] => [1,3,2,1,3]
[3,2,1,2,1,1] => [2,1,1,3,2,1]
[3,2,1,2,2] => [2,1,2,3,2]
[3,2,1,3,1] => [1,3,2,3,1]
[3,2,1,4] => [3,2,1,4]
[3,2,2,1,1,1] => [1,2,1,3,2,1]
[3,2,2,1,2] => [2,2,1,3,2]
[3,2,2,2,1] => [2,2,3,2,1]
[3,2,2,3] => [2,3,2,3]
[3,2,3,1,1] => [3,2,1,3,1]
[3,2,3,2] => [2,3,3,2]
[3,2,4,1] => [3,2,4,1]
[3,2,5] => [3,2,5]
[3,3,1,1,1,1] => [1,1,3,1,3,1]
[3,3,1,1,2] => [1,3,1,3,2]
[3,3,1,2,1] => [1,3,3,2,1]
[3,3,1,3] => [3,3,1,3]
[3,3,2,1,1] => [3,1,3,2,1]
[3,3,2,2] => [3,2,3,2]
[3,3,3,1] => [3,3,3,1]
[3,3,4] => [3,3,4]
[3,4,1,1,1] => [1,3,1,4,1]
[3,4,1,2] => [3,1,4,2]
[3,4,2,1] => [3,4,2,1]
[3,4,3] => [4,3,3]
[3,5,1,1] => [3,1,5,1]
[3,5,2] => [3,5,2]
[3,6,1] => [3,6,1]
[3,7] => [3,7]
[4,1,1,1,1,1,1] => [1,1,1,1,1,4,1]
[4,1,1,1,1,2] => [1,1,1,1,4,2]
[4,1,1,1,2,1] => [4,1,1,1,2,1]
[4,1,1,1,3] => [1,1,1,4,3]
[4,1,1,2,1,1] => [1,4,1,1,2,1]
[4,1,1,2,2] => [1,1,2,4,2]
[4,1,1,3,1] => [4,1,1,3,1]
[4,1,1,4] => [1,4,1,4]
[4,1,2,1,1,1] => [1,1,4,1,2,1]
[4,1,2,1,2] => [2,1,1,4,2]
[4,1,2,2,1] => [1,4,2,2,1]
[4,1,2,3] => [1,2,4,3]
[4,1,3,1,1] => [1,4,1,3,1]
[4,1,3,2] => [4,1,3,2]
[4,1,4,1] => [1,4,4,1]
[4,1,5] => [4,1,5]
[4,2,1,1,1,1] => [1,1,1,4,2,1]
[4,2,1,1,2] => [1,2,1,4,2]
[4,2,1,2,1] => [1,2,4,2,1]
[4,2,1,3] => [2,1,4,3]
[4,2,2,1,1] => [2,1,4,2,1]
[4,2,2,2] => [2,2,4,2]
[4,2,3,1] => [2,4,3,1]
[4,2,4] => [4,2,4]
[4,3,1,1,1] => [1,1,4,3,1]
[4,3,1,2] => [1,4,3,2]
[4,3,2,1] => [4,3,2,1]
[4,3,3] => [3,4,3]
[4,4,1,1] => [4,1,4,1]
[4,4,2] => [4,4,2]
[4,5,1] => [4,5,1]
[4,6] => [4,6]
[5,1,1,1,1,1] => [1,1,1,1,5,1]
[5,1,1,1,2] => [1,1,1,5,2]
[5,1,1,2,1] => [5,1,1,2,1]
[5,1,1,3] => [1,1,5,3]
[5,1,2,1,1] => [1,5,1,2,1]
[5,1,2,2] => [1,2,5,2]
[5,1,3,1] => [5,1,3,1]
[5,1,4] => [1,5,4]
[5,2,1,1,1] => [1,1,5,2,1]
[5,2,1,2] => [2,1,5,2]
[5,2,2,1] => [2,5,2,1]
[5,2,3] => [2,5,3]
[5,3,1,1] => [1,5,3,1]
[5,3,2] => [5,3,2]
[5,4,1] => [5,4,1]
[5,5] => [5,5]
[6,1,1,1,1] => [1,1,1,6,1]
[6,1,1,2] => [1,1,6,2]
[6,1,2,1] => [6,1,2,1]
[6,1,3] => [1,6,3]
[6,2,1,1] => [1,6,2,1]
[6,2,2] => [2,6,2]
[6,3,1] => [6,3,1]
[6,4] => [6,4]
[7,1,1,1] => [1,1,7,1]
[7,1,2] => [1,7,2]
[7,2,1] => [7,2,1]
[7,3] => [7,3]
[8,1,1] => [1,8,1]
[8,2] => [8,2]
[9,1] => [9,1]
[10] => [10]
Identities
click to show experimental identities
(only identities of compositions of up to three maps are shown)
Inverses
Mp00314Foata bijection ∘ Mp00315inverse Foata bijection = id : Integer compositions ⟶ Integer compositions
Mp00315inverse Foata bijection ∘ Mp00314Foata bijection = id : Integer compositions ⟶ Integer compositions
Absorbing relations
Mp00040to partition ∘ Mp00315inverse Foata bijection = Mp00040to partition : Integer compositions ⟶ Integer partitions
Mp00315inverse Foata bijection ∘ Mp00071descent composition ∘ Mp00081reading word permutation = Mp00071descent composition ∘ Mp00081reading word permutation : Standard tableaux ⟶ Integer compositions
Mp00315inverse Foata bijection ∘ Mp00071descent composition ∘ Mp00075reading word permutation = Mp00071descent composition ∘ Mp00075reading word permutation : Semistandard tableaux ⟶ Integer compositions
Mp00315inverse Foata bijection ∘ Mp00207horizontal strip sizes ∘ Mp00045reading tableau = Mp00207horizontal strip sizes ∘ Mp00045reading tableau : Integer partitions ⟶ Integer compositions
Mp00315inverse Foata bijection ∘ Mp00100touch composition ∘ Mp00230parallelogram polyomino = Mp00100touch composition ∘ Mp00230parallelogram polyomino : Integer partitions ⟶ Integer compositions
Mp00037to partition of connected components ∘ Mp00184to threshold graph ∘ Mp00315inverse Foata bijection = Mp00037to partition of connected components ∘ Mp00184to threshold graph : Integer compositions ⟶ Integer partitions
Mp00154core ∘ Mp00184to threshold graph ∘ Mp00315inverse Foata bijection = Mp00154core ∘ Mp00184to threshold graph : Integer compositions ⟶ Graphs
Mp00324chromatic difference sequence ∘ Mp00184to threshold graph ∘ Mp00315inverse Foata bijection = Mp00324chromatic difference sequence ∘ Mp00184to threshold graph : Integer compositions ⟶ Integer compositions
Mp00315inverse Foata bijection ∘ Mp00324chromatic difference sequence ∘ Mp00011to graph = Mp00324chromatic difference sequence ∘ Mp00011to graph : Binary trees ⟶ Integer compositions
Mp00315inverse Foata bijection ∘ Mp00324chromatic difference sequence ∘ Mp00074to graph = Mp00324chromatic difference sequence ∘ Mp00074to graph : Posets ⟶ Integer compositions
Mp00315inverse Foata bijection ∘ Mp00133delta morphism ∘ Mp00324chromatic difference sequence = Mp00133delta morphism ∘ Mp00324chromatic difference sequence : Graphs ⟶ Integer compositions
Mp00315inverse Foata bijection ∘ Mp00324chromatic difference sequence ∘ Mp00046to graph = Mp00324chromatic difference sequence ∘ Mp00046to graph : Ordered trees ⟶ Integer compositions
Mp00315inverse Foata bijection ∘ Mp00319to composition ∘ Mp00052to non-decreasing parking function = Mp00319to composition ∘ Mp00052to non-decreasing parking function : Parking functions ⟶ Integer compositions
Mp00147square ∘ Mp00184to threshold graph ∘ Mp00315inverse Foata bijection = Mp00147square ∘ Mp00184to threshold graph : Integer compositions ⟶ Graphs
Mp00250clique graph ∘ Mp00184to threshold graph ∘ Mp00315inverse Foata bijection = Mp00250clique graph ∘ Mp00184to threshold graph : Integer compositions ⟶ Graphs
Mp00315inverse Foata bijection ∘ Mp00315inverse Foata bijection ∘ Mp00038reverse ∘ Mp00324chromatic difference sequence = Mp00038reverse ∘ Mp00324chromatic difference sequence : Graphs ⟶ Integer compositions
Mp00315inverse Foata bijection ∘ Mp00315inverse Foata bijection ∘ Mp00133delta morphism ∘ Mp00057to touch composition = Mp00133delta morphism ∘ Mp00057to touch composition : Parking functions ⟶ Integer compositions
Mp00315inverse Foata bijection ∘ Mp00315inverse Foata bijection ∘ Mp00133delta morphism ∘ Mp00054to inverse des composition = Mp00133delta morphism ∘ Mp00054to inverse des composition : Parking functions ⟶ Integer compositions
Mp00315inverse Foata bijection ∘ Mp00071descent composition ∘ Mp00256upper permutation = Mp00071descent composition ∘ Mp00256upper permutation : Decorated permutations ⟶ Integer compositions
Mp00315inverse Foata bijection ∘ Mp00315inverse Foata bijection ∘ Mp00133delta morphism ∘ Mp00181row lengths = Mp00133delta morphism ∘ Mp00181row lengths : Skew partitions ⟶ Integer compositions
Other relations
Mp00315inverse Foata bijection ∘ Mp00100touch composition ∘ Mp00127left-to-right-maxima to Dyck path = Mp00315inverse Foata bijection ∘ Mp00178to composition ∘ Mp00114connectivity set : Permutations ⟶ Integer compositions
Mp00314Foata bijection ∘ Mp00038reverse ∘ Mp00324chromatic difference sequence = Mp00315inverse Foata bijection ∘ Mp00038reverse ∘ Mp00324chromatic difference sequence : Graphs ⟶ Integer compositions
Mp00314Foata bijection ∘ Mp00133delta morphism ∘ Mp00057to touch composition = Mp00315inverse Foata bijection ∘ Mp00133delta morphism ∘ Mp00057to touch composition : Parking functions ⟶ Integer compositions
Mp00314Foata bijection ∘ Mp00133delta morphism ∘ Mp00054to inverse des composition = Mp00315inverse Foata bijection ∘ Mp00133delta morphism ∘ Mp00054to inverse des composition : Parking functions ⟶ Integer compositions
Mp00314Foata bijection ∘ Mp00133delta morphism ∘ Mp00181row lengths = Mp00315inverse Foata bijection ∘ Mp00133delta morphism ∘ Mp00181row lengths : Skew partitions ⟶ Integer compositions
Mp00314Foata bijection ∘ Mp00071descent composition ∘ Mp00223runsort = Mp00315inverse Foata bijection ∘ Mp00071descent composition ∘ Mp00223runsort : Permutations ⟶ Integer compositions
Mp00314Foata bijection ∘ Mp00294peak composition ∘ Mp00045reading tableau = Mp00315inverse Foata bijection ∘ Mp00294peak composition ∘ Mp00045reading tableau : Integer partitions ⟶ Integer compositions
Mp00314Foata bijection ∘ Mp00295valley composition ∘ Mp00042initial tableau = Mp00315inverse Foata bijection ∘ Mp00295valley composition ∘ Mp00042initial tableau : Integer partitions ⟶ Integer compositions
Mp00314Foata bijection ∘ Mp00248DEX composition ∘ Mp00129to 321-avoiding permutation (Billey-Jockusch-Stanley) = Mp00315inverse Foata bijection ∘ Mp00248DEX composition ∘ Mp00129to 321-avoiding permutation (Billey-Jockusch-Stanley) : Dyck paths ⟶ Integer compositions
Mp00314Foata bijection ∘ Mp00324chromatic difference sequence ∘ Mp00184to threshold graph = Mp00315inverse Foata bijection ∘ Mp00324chromatic difference sequence ∘ Mp00184to threshold graph : Integer compositions ⟶ Integer compositions
Mp00315inverse Foata bijection ∘ Mp00207horizontal strip sizes ∘ Mp00070Robinson-Schensted recording tableau ∘ Mp00081reading word permutation = Mp00071descent composition ∘ Mp00081reading word permutation : Standard tableaux ⟶ Integer compositions
Mp00314Foata bijection ∘ Mp00248DEX composition ∘ Mp00075reading word permutation = Mp00315inverse Foata bijection ∘ Mp00248DEX composition ∘ Mp00075reading word permutation : Semistandard tableaux ⟶ Integer compositions
Mp00314Foata bijection ∘ Mp00173rotate front to back ∘ Mp00324chromatic difference sequence = Mp00315inverse Foata bijection ∘ Mp00173rotate front to back ∘ Mp00324chromatic difference sequence : Graphs ⟶ Integer compositions
Mp00314Foata bijection ∘ Mp00071descent composition ∘ Mp00245standardize = Mp00315inverse Foata bijection ∘ Mp00071descent composition ∘ Mp00245standardize : Signed permutations ⟶ Integer compositions
Mp00314Foata bijection ∘ Mp00133delta morphism ∘ Mp00287to composition = Mp00315inverse Foata bijection ∘ Mp00133delta morphism ∘ Mp00287to composition : Ordered set partitions ⟶ Integer compositions
Description
The inverse of Foata's bijection.
See Mp00314Foata bijection.
See Mp00314Foata bijection.
Properties
bijective, graded
Sage code
def mapping(self):
"""
sage: all(mapping(foata(pi)) == pi for n in range(1, 9) for pi in Compositions(n))
True
"""
self = Words()(self)
s = self.standard_permutation()
ordered_alphabet = sorted(self.letters(),
key=self.parent().sortkey_letters)
eval_dict = self.evaluation_dict()
weight = [eval_dict[a] for a in ordered_alphabet]
return Composition((s.foata_bijection_inverse()).destandardize(weight, ordered_alphabet=ordered_alphabet))
Weight
21
Created
Sep 12, 2023 at 11:42 by Martin Rubey
Updated
Sep 12, 2023 at 11:42 by Martin Rubey
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