Identifier
Values
[1,1,0,0,1,0] => [[2,2],[1]] => [1] => 0
[1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => [1] => 0
[1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => [1,1] => 0
[1,1,0,0,1,1,0,0] => [[3,2],[1]] => [1] => 0
[1,1,0,1,0,0,1,0] => [[3,3],[2]] => [2] => 1
[1,1,0,1,1,0,0,0] => [[3,3],[1]] => [1] => 0
[1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => [1] => 0
[1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => [1] => 0
[1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => [1,1] => 0
[1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]] => [1] => 0
[1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => [2] => 1
[1,0,1,1,0,1,1,0,0,0] => [[3,3,1],[1]] => [1] => 0
[1,0,1,1,1,0,0,0,1,0] => [[2,2,2,1],[1]] => [1] => 0
[1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => [1,1,1] => 0
[1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => [1,1] => 0
[1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => [2,1] => 1
[1,1,0,0,1,1,0,1,0,0] => [[4,2],[1]] => [1] => 0
[1,1,0,0,1,1,1,0,0,0] => [[3,3,2],[1,1]] => [1,1] => 0
[1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => [2,2] => 1
[1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]] => [2] => 1
[1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => [3] => 2
[1,1,0,1,0,1,1,0,0,0] => [[4,4],[2]] => [2] => 1
[1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [2,1] => 1
[1,1,0,1,1,0,0,1,0,0] => [[4,3],[1]] => [1] => 0
[1,1,0,1,1,0,1,0,0,0] => [[3,3,3],[1,1]] => [1,1] => 0
[1,1,0,1,1,1,0,0,0,0] => [[4,4],[1]] => [1] => 0
[1,1,1,0,0,0,1,0,1,0] => [[2,2,2,2],[1,1]] => [1,1] => 0
[1,1,1,0,0,0,1,1,0,0] => [[3,2,2],[1]] => [1] => 0
[1,1,1,0,0,1,0,0,1,0] => [[3,3,2],[2]] => [2] => 1
[1,1,1,0,0,1,1,0,0,0] => [[3,3,2],[1]] => [1] => 0
[1,1,1,0,1,0,0,0,1,0] => [[2,2,2,2],[1]] => [1] => 0
[1,1,1,1,0,0,0,0,1,0] => [[3,3,3],[2]] => [2] => 1
[1,1,1,1,0,0,1,0,0,0] => [[3,3,3],[1]] => [1] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1,1],[1]] => [1] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1,1],[1,1]] => [1,1] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [[3,2,1,1],[1]] => [1] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [[3,3,1,1],[2]] => [2] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [[3,3,1,1],[1]] => [1] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [[2,2,2,1,1],[1]] => [1] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [[3,2,2,1],[1,1]] => [1,1] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [[3,3,2,1],[2,1]] => [2,1] => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [[4,2,1],[1]] => [1] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [[3,3,2,1],[1,1]] => [1,1] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [[3,3,3,1],[2,2]] => [2,2] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [[4,3,1],[2]] => [2] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [[4,4,1],[3]] => [3] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [[4,4,1],[2]] => [2] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [[3,3,3,1],[2,1]] => [2,1] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [[4,3,1],[1]] => [1] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [[3,3,3,1],[1,1]] => [1,1] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [[4,4,1],[1]] => [1] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [[2,2,2,2,1],[1,1]] => [1,1] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [[3,2,2,1],[1]] => [1] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [[3,3,2,1],[2]] => [2] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [[3,3,2,1],[1]] => [1] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [[2,2,2,2,1],[1]] => [1] => 0
[1,0,1,1,1,1,0,0,0,0,1,0] => [[3,3,3,1],[2]] => [2] => 1
[1,0,1,1,1,1,0,0,1,0,0,0] => [[3,3,3,1],[1]] => [1] => 0
[1,1,0,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 0
[1,1,0,0,1,0,1,0,1,1,0,0] => [[3,2,2,2],[1,1,1]] => [1,1,1] => 0
[1,1,0,0,1,0,1,1,0,0,1,0] => [[3,3,2,2],[2,1,1]] => [2,1,1] => 1
[1,1,0,0,1,0,1,1,0,1,0,0] => [[4,2,2],[1,1]] => [1,1] => 0
[1,1,0,0,1,0,1,1,1,0,0,0] => [[3,3,2,2],[1,1,1]] => [1,1,1] => 0
[1,1,0,0,1,1,0,0,1,0,1,0] => [[3,3,3,2],[2,2,1]] => [2,2,1] => 1
[1,1,0,0,1,1,0,0,1,1,0,0] => [[4,3,2],[2,1]] => [2,1] => 1
[1,1,0,0,1,1,0,1,0,0,1,0] => [[4,4,2],[3,1]] => [3,1] => 2
[1,1,0,0,1,1,0,1,0,1,0,0] => [[5,2],[1]] => [1] => 0
[1,1,0,0,1,1,0,1,1,0,0,0] => [[4,4,2],[2,1]] => [2,1] => 1
[1,1,0,0,1,1,1,0,0,0,1,0] => [[3,3,3,2],[2,1,1]] => [2,1,1] => 1
[1,1,0,0,1,1,1,0,0,1,0,0] => [[4,3,2],[1,1]] => [1,1] => 0
[1,1,0,0,1,1,1,0,1,0,0,0] => [[3,3,3,2],[1,1,1]] => [1,1,1] => 0
[1,1,0,0,1,1,1,1,0,0,0,0] => [[4,4,2],[1,1]] => [1,1] => 0
[1,1,0,1,0,0,1,0,1,0,1,0] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 1
[1,1,0,1,0,0,1,0,1,1,0,0] => [[4,3,3],[2,2]] => [2,2] => 1
[1,1,0,1,0,0,1,1,0,0,1,0] => [[4,4,3],[3,2]] => [3,2] => 2
[1,1,0,1,0,0,1,1,0,1,0,0] => [[5,3],[2]] => [2] => 1
[1,1,0,1,0,0,1,1,1,0,0,0] => [[4,4,3],[2,2]] => [2,2] => 1
[1,1,0,1,0,1,0,0,1,0,1,0] => [[4,4,4],[3,3]] => [3,3] => 3
[1,1,0,1,0,1,0,0,1,1,0,0] => [[5,4],[3]] => [3] => 2
[1,1,0,1,0,1,0,1,0,0,1,0] => [[5,5],[4]] => [4] => 3
[1,1,0,1,0,1,0,1,1,0,0,0] => [[5,5],[3]] => [3] => 2
[1,1,0,1,0,1,1,0,0,0,1,0] => [[4,4,4],[3,2]] => [3,2] => 2
[1,1,0,1,0,1,1,0,0,1,0,0] => [[5,4],[2]] => [2] => 1
[1,1,0,1,0,1,1,0,1,0,0,0] => [[4,4,4],[2,2]] => [2,2] => 1
[1,1,0,1,0,1,1,1,0,0,0,0] => [[5,5],[2]] => [2] => 1
[1,1,0,1,1,0,0,0,1,0,1,0] => [[3,3,3,3],[2,2,1]] => [2,2,1] => 1
[1,1,0,1,1,0,0,0,1,1,0,0] => [[4,3,3],[2,1]] => [2,1] => 1
[1,1,0,1,1,0,0,1,0,0,1,0] => [[4,4,3],[3,1]] => [3,1] => 2
[1,1,0,1,1,0,0,1,0,1,0,0] => [[5,3],[1]] => [1] => 0
[1,1,0,1,1,0,0,1,1,0,0,0] => [[4,4,3],[2,1]] => [2,1] => 1
[1,1,0,1,1,0,1,0,0,0,1,0] => [[3,3,3,3],[2,1,1]] => [2,1,1] => 1
[1,1,0,1,1,0,1,0,0,1,0,0] => [[4,3,3],[1,1]] => [1,1] => 0
[1,1,0,1,1,0,1,0,1,0,0,0] => [[3,3,3,3],[1,1,1]] => [1,1,1] => 0
[1,1,0,1,1,0,1,1,0,0,0,0] => [[4,4,3],[1,1]] => [1,1] => 0
[1,1,0,1,1,1,0,0,0,0,1,0] => [[4,4,4],[3,1]] => [3,1] => 2
[1,1,0,1,1,1,0,0,0,1,0,0] => [[5,4],[1]] => [1] => 0
[1,1,0,1,1,1,0,0,1,0,0,0] => [[4,4,4],[2,1]] => [2,1] => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => [[5,5],[1]] => [1] => 0
[1,1,0,1,1,1,1,0,0,0,0,0] => [[4,4,4],[1,1]] => [1,1] => 0
[1,1,1,0,0,0,1,0,1,0,1,0] => [[2,2,2,2,2],[1,1,1]] => [1,1,1] => 0
>>> Load all 469 entries. <<<
[1,1,1,0,0,0,1,0,1,1,0,0] => [[3,2,2,2],[1,1]] => [1,1] => 0
[1,1,1,0,0,0,1,1,0,0,1,0] => [[3,3,2,2],[2,1]] => [2,1] => 1
[1,1,1,0,0,0,1,1,0,1,0,0] => [[4,2,2],[1]] => [1] => 0
[1,1,1,0,0,0,1,1,1,0,0,0] => [[3,3,2,2],[1,1]] => [1,1] => 0
[1,1,1,0,0,1,0,0,1,0,1,0] => [[3,3,3,2],[2,2]] => [2,2] => 1
[1,1,1,0,0,1,0,0,1,1,0,0] => [[4,3,2],[2]] => [2] => 1
[1,1,1,0,0,1,0,1,0,0,1,0] => [[4,4,2],[3]] => [3] => 2
[1,1,1,0,0,1,0,1,1,0,0,0] => [[4,4,2],[2]] => [2] => 1
[1,1,1,0,0,1,1,0,0,0,1,0] => [[3,3,3,2],[2,1]] => [2,1] => 1
[1,1,1,0,0,1,1,0,0,1,0,0] => [[4,3,2],[1]] => [1] => 0
[1,1,1,0,0,1,1,0,1,0,0,0] => [[3,3,3,2],[1,1]] => [1,1] => 0
[1,1,1,0,0,1,1,1,0,0,0,0] => [[4,4,2],[1]] => [1] => 0
[1,1,1,0,1,0,0,0,1,0,1,0] => [[2,2,2,2,2],[1,1]] => [1,1] => 0
[1,1,1,0,1,0,0,0,1,1,0,0] => [[3,2,2,2],[1]] => [1] => 0
[1,1,1,0,1,0,0,1,0,0,1,0] => [[3,3,2,2],[2]] => [2] => 1
[1,1,1,0,1,0,0,1,1,0,0,0] => [[3,3,2,2],[1]] => [1] => 0
[1,1,1,0,1,0,1,0,0,0,1,0] => [[2,2,2,2,2],[1]] => [1] => 0
[1,1,1,0,1,1,0,0,0,0,1,0] => [[3,3,3,2],[2]] => [2] => 1
[1,1,1,0,1,1,0,0,1,0,0,0] => [[3,3,3,2],[1]] => [1] => 0
[1,1,1,1,0,0,0,0,1,0,1,0] => [[3,3,3,3],[2,2]] => [2,2] => 1
[1,1,1,1,0,0,0,0,1,1,0,0] => [[4,3,3],[2]] => [2] => 1
[1,1,1,1,0,0,0,1,0,0,1,0] => [[4,4,3],[3]] => [3] => 2
[1,1,1,1,0,0,0,1,1,0,0,0] => [[4,4,3],[2]] => [2] => 1
[1,1,1,1,0,0,1,0,0,0,1,0] => [[3,3,3,3],[2,1]] => [2,1] => 1
[1,1,1,1,0,0,1,0,0,1,0,0] => [[4,3,3],[1]] => [1] => 0
[1,1,1,1,0,0,1,0,1,0,0,0] => [[3,3,3,3],[1,1]] => [1,1] => 0
[1,1,1,1,0,0,1,1,0,0,0,0] => [[4,4,3],[1]] => [1] => 0
[1,1,1,1,0,1,0,0,0,0,1,0] => [[4,4,4],[3]] => [3] => 2
[1,1,1,1,0,1,0,0,1,0,0,0] => [[4,4,4],[2]] => [2] => 1
[1,1,1,1,0,1,1,0,0,0,0,0] => [[4,4,4],[1]] => [1] => 0
[1,1,1,1,1,0,0,0,0,0,1,0] => [[3,3,3,3],[2]] => [2] => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [[3,3,3,3],[1]] => [1] => 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1,1,1],[1]] => [1] => 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1,1,1],[1,1]] => [1,1] => 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [[3,2,1,1,1],[1]] => [1] => 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [[3,3,1,1,1],[2]] => [2] => 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [[3,2,2,1,1],[1,1]] => [1,1] => 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[3,3,2,1,1],[2,1]] => [2,1] => 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0] => [[4,2,1,1],[1]] => [1] => 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [[3,3,3,1,1],[2,2]] => [2,2] => 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [[4,3,1,1],[2]] => [2] => 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [[4,4,1,1],[3]] => [3] => 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0] => [[4,2,2,1],[1,1]] => [1,1] => 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[4,3,2,1],[2,1]] => [2,1] => 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [[4,4,2,1],[3,1]] => [3,1] => 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0] => [[5,2,1],[1]] => [1] => 0
[1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0] => [[4,3,3,1],[2,2]] => [2,2] => 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0] => [[4,4,3,1],[3,2]] => [3,2] => 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [[5,3,1],[2]] => [2] => 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0] => [[4,4,4,1],[3,3]] => [3,3] => 3
[1,0,1,1,0,1,0,1,0,0,1,1,0,0] => [[5,4,1],[3]] => [3] => 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [[5,5,1],[4]] => [4] => 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [[4,4,4,1],[1,1]] => [1,1] => 0
[1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [[4,4,3,1],[1]] => [1] => 0
[1,0,1,1,1,1,0,1,0,0,1,0,0,0] => [[4,4,4,1],[2]] => [2] => 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [[4,4,4,1],[1]] => [1] => 0
[1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [[3,3,3,3,1],[1]] => [1] => 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => 0
[1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 0
[1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [[4,2,2,2],[1,1,1]] => [1,1,1] => 0
[1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[4,3,2,2],[2,1,1]] => [2,1,1] => 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [[4,4,2,2],[3,1,1]] => [3,1,1] => 2
[1,1,0,0,1,0,1,1,0,1,0,1,0,0] => [[5,2,2],[1,1]] => [1,1] => 0
[1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[4,3,3,2],[2,2,1]] => [2,2,1] => 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[4,4,3,2],[3,2,1]] => [3,2,1] => 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0] => [[5,3,2],[2,1]] => [2,1] => 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0] => [[4,4,4,2],[3,3,1]] => [3,3,1] => 3
[1,1,0,0,1,1,0,1,0,0,1,1,0,0] => [[5,4,2],[3,1]] => [3,1] => 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0] => [[5,5,2],[4,1]] => [4,1] => 3
[1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [[6,2],[1]] => [1] => 0
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [[4,4,4,2],[1,1,1]] => [1,1,1] => 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [[4,3,3,3],[2,2,2]] => [2,2,2] => 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0] => [[4,4,3,3],[3,2,2]] => [3,2,2] => 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [[5,3,3],[2,2]] => [2,2] => 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0] => [[4,4,4,3],[3,3,2]] => [3,3,2] => 3
[1,1,0,1,0,0,1,1,0,0,1,1,0,0] => [[5,4,3],[3,2]] => [3,2] => 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0] => [[5,5,3],[4,2]] => [4,2] => 3
[1,1,0,1,0,0,1,1,0,1,0,1,0,0] => [[6,3],[2]] => [2] => 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [[4,4,4,4],[3,3,3]] => [3,3,3] => 3
[1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [[5,4,4],[3,3]] => [3,3] => 3
[1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [[5,5,4],[4,3]] => [4,3] => 4
[1,1,0,1,0,1,0,0,1,1,0,1,0,0] => [[6,4],[3]] => [3] => 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0] => [[5,5,5],[4,4]] => [4,4] => 5
[1,1,0,1,0,1,0,1,0,0,1,1,0,0] => [[6,5],[4]] => [4] => 3
[1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [[6,6],[5]] => [5] => 4
[1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [[6,6],[4]] => [4] => 3
[1,1,0,1,0,1,0,1,1,0,0,0,1,0] => [[5,5,5],[4,3]] => [4,3] => 4
[1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [[6,5],[3]] => [3] => 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [[5,5,5],[3,3]] => [3,3] => 3
[1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [[6,6],[3]] => [3] => 2
[1,1,0,1,0,1,1,0,0,0,1,0,1,0] => [[4,4,4,4],[3,3,2]] => [3,3,2] => 3
[1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [[6,4],[2]] => [2] => 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [[5,5,4],[3,2]] => [3,2] => 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0] => [[4,4,4,4],[3,2,2]] => [3,2,2] => 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [[5,4,4],[2,2]] => [2,2] => 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [[4,4,4,4],[2,2,2]] => [2,2,2] => 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [[5,5,4],[2,2]] => [2,2] => 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0] => [[5,5,5],[4,2]] => [4,2] => 3
[1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [[6,5],[2]] => [2] => 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [[5,5,5],[3,2]] => [3,2] => 2
[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [[6,6],[2]] => [2] => 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [[5,5,5],[2,2]] => [2,2] => 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0] => [[3,3,3,3,3],[2,2,2,1]] => [2,2,2,1] => 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [[6,3],[1]] => [1] => 0
[1,1,0,1,1,0,0,1,0,1,1,0,0,0] => [[5,5,3],[3,1]] => [3,1] => 2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [[5,4,3],[2,1]] => [2,1] => 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [[4,4,4,3],[2,2,1]] => [2,2,1] => 1
[1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [[5,5,3],[2,1]] => [2,1] => 1
[1,1,0,1,1,0,1,0,0,0,1,0,1,0] => [[3,3,3,3,3],[2,2,1,1]] => [2,2,1,1] => 1
[1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [[5,3,3],[1,1]] => [1,1] => 0
[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [[4,4,3,3],[2,1,1]] => [2,1,1] => 1
[1,1,0,1,1,0,1,0,1,0,0,0,1,0] => [[3,3,3,3,3],[2,1,1,1]] => [2,1,1,1] => 1
[1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [[4,3,3,3],[1,1,1]] => [1,1,1] => 0
[1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [[3,3,3,3,3],[1,1,1,1]] => [1,1,1,1] => 0
[1,1,0,1,1,0,1,0,1,1,0,0,0,0] => [[4,4,3,3],[1,1,1]] => [1,1,1] => 0
[1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [[5,4,3],[1,1]] => [1,1] => 0
[1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [[4,4,4,3],[2,1,1]] => [2,1,1] => 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [[5,5,3],[1,1]] => [1,1] => 0
[1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [[4,4,4,3],[1,1,1]] => [1,1,1] => 0
[1,1,0,1,1,1,0,0,0,0,1,0,1,0] => [[4,4,4,4],[3,3,1]] => [3,3,1] => 3
[1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [[6,4],[1]] => [1] => 0
[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [[5,5,4],[3,1]] => [3,1] => 2
[1,1,0,1,1,1,0,0,1,0,0,0,1,0] => [[4,4,4,4],[3,2,1]] => [3,2,1] => 2
[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [[5,4,4],[2,1]] => [2,1] => 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0] => [[4,4,4,4],[2,2,1]] => [2,2,1] => 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [[5,5,4],[2,1]] => [2,1] => 1
[1,1,0,1,1,1,0,1,0,0,0,0,1,0] => [[5,5,5],[4,1]] => [4,1] => 3
[1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [[6,5],[1]] => [1] => 0
[1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [[5,5,5],[3,1]] => [3,1] => 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0] => [[6,6],[1]] => [1] => 0
[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [[5,5,5],[2,1]] => [2,1] => 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [[4,4,4,4],[3,1,1]] => [3,1,1] => 2
[1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [[5,4,4],[1,1]] => [1,1] => 0
[1,1,0,1,1,1,1,0,0,0,1,0,0,0] => [[4,4,4,4],[2,1,1]] => [2,1,1] => 1
[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [[5,5,4],[1,1]] => [1,1] => 0
[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [[4,4,4,4],[1,1,1]] => [1,1,1] => 0
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [[5,5,5],[1,1]] => [1,1] => 0
[1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 0
[1,1,1,0,0,1,0,1,0,1,1,0,0,0] => [[5,5,2],[3]] => [3] => 2
[1,1,1,0,0,1,0,1,1,0,0,1,0,0] => [[5,4,2],[2]] => [2] => 1
[1,1,1,0,0,1,0,1,1,0,1,0,0,0] => [[4,4,4,2],[2,2]] => [2,2] => 1
[1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [[5,5,2],[2]] => [2] => 1
[1,1,1,0,0,1,1,0,0,1,0,1,0,0] => [[5,3,2],[1]] => [1] => 0
[1,1,1,0,0,1,1,0,0,1,1,0,0,0] => [[4,4,3,2],[2,1]] => [2,1] => 1
[1,1,1,0,0,1,1,0,1,0,0,1,0,0] => [[4,3,3,2],[1,1]] => [1,1] => 0
[1,1,1,0,0,1,1,0,1,0,1,0,0,0] => [[3,3,3,3,2],[1,1,1]] => [1,1,1] => 0
[1,1,1,0,0,1,1,0,1,1,0,0,0,0] => [[4,4,3,2],[1,1]] => [1,1] => 0
[1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [[5,4,2],[1]] => [1] => 0
[1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [[4,4,4,2],[2,1]] => [2,1] => 1
[1,1,1,0,0,1,1,1,0,1,0,0,0,0] => [[5,5,2],[1]] => [1] => 0
[1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [[4,4,4,2],[1,1]] => [1,1] => 0
[1,1,1,0,1,0,0,0,1,0,1,0,1,0] => [[2,2,2,2,2,2],[1,1,1]] => [1,1,1] => 0
[1,1,1,0,1,0,0,1,0,1,1,0,0,0] => [[4,4,2,2],[2]] => [2] => 1
[1,1,1,0,1,0,0,1,1,0,0,1,0,0] => [[4,3,2,2],[1]] => [1] => 0
[1,1,1,0,1,0,0,1,1,0,1,0,0,0] => [[3,3,3,2,2],[1,1]] => [1,1] => 0
[1,1,1,0,1,0,0,1,1,1,0,0,0,0] => [[4,4,2,2],[1]] => [1] => 0
[1,1,1,0,1,0,1,0,0,0,1,0,1,0] => [[2,2,2,2,2,2],[1,1]] => [1,1] => 0
[1,1,1,0,1,0,1,0,0,1,1,0,0,0] => [[3,3,2,2,2],[1]] => [1] => 0
[1,1,1,0,1,0,1,1,0,0,1,0,0,0] => [[3,3,3,2,2],[1]] => [1] => 0
[1,1,1,0,1,1,0,0,0,1,1,0,0,0] => [[4,4,3,2],[2]] => [2] => 1
[1,1,1,0,1,1,0,0,1,0,0,1,0,0] => [[4,3,3,2],[1]] => [1] => 0
[1,1,1,0,1,1,0,0,1,0,1,0,0,0] => [[3,3,3,3,2],[1,1]] => [1,1] => 0
[1,1,1,0,1,1,0,0,1,1,0,0,0,0] => [[4,4,3,2],[1]] => [1] => 0
[1,1,1,0,1,1,0,1,0,0,0,0,1,0] => [[4,4,4,2],[3]] => [3] => 2
[1,1,1,0,1,1,0,1,0,0,1,0,0,0] => [[4,4,4,2],[2]] => [2] => 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [[4,4,4,2],[1]] => [1] => 0
[1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [[3,3,3,3,2],[2]] => [2] => 1
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [[3,3,3,3,2],[1]] => [1] => 0
[1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [[3,3,3,3,3],[2,2,2]] => [2,2,2] => 1
[1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [[5,5,3],[3]] => [3] => 2
[1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [[5,4,3],[2]] => [2] => 1
[1,1,1,1,0,0,0,1,1,0,1,0,0,0] => [[4,4,4,3],[2,2]] => [2,2] => 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [[5,5,3],[2]] => [2] => 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0] => [[3,3,3,3,3],[2,2,1]] => [2,2,1] => 1
[1,1,1,1,0,0,1,0,0,1,0,1,0,0] => [[5,3,3],[1]] => [1] => 0
[1,1,1,1,0,0,1,0,0,1,1,0,0,0] => [[4,4,3,3],[2,1]] => [2,1] => 1
[1,1,1,1,0,0,1,0,1,0,0,0,1,0] => [[3,3,3,3,3],[2,1,1]] => [2,1,1] => 1
[1,1,1,1,0,0,1,0,1,0,0,1,0,0] => [[4,3,3,3],[1,1]] => [1,1] => 0
[1,1,1,1,0,0,1,0,1,0,1,0,0,0] => [[3,3,3,3,3],[1,1,1]] => [1,1,1] => 0
[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [[4,4,3,3],[1,1]] => [1,1] => 0
[1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [[4,4,4,3],[3,1]] => [3,1] => 2
[1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [[5,4,3],[1]] => [1] => 0
[1,1,1,1,0,0,1,1,0,0,1,0,0,0] => [[4,4,4,3],[2,1]] => [2,1] => 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [[5,5,3],[1]] => [1] => 0
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [[4,4,4,3],[1,1]] => [1,1] => 0
[1,1,1,1,0,1,0,0,0,0,1,0,1,0] => [[4,4,4,4],[3,3]] => [3,3] => 3
[1,1,1,1,0,1,0,0,0,1,1,0,0,0] => [[5,5,4],[3]] => [3] => 2
[1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [[4,4,4,4],[3,2]] => [3,2] => 2
[1,1,1,1,0,1,0,0,1,0,0,1,0,0] => [[5,4,4],[2]] => [2] => 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0] => [[4,4,4,4],[2,2]] => [2,2] => 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [[5,5,4],[2]] => [2] => 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [[5,5,5],[4]] => [4] => 3
[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [[5,5,5],[3]] => [3] => 2
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [[5,5,5],[2]] => [2] => 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [[4,4,4,4],[3,1]] => [3,1] => 2
[1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [[5,4,4],[1]] => [1] => 0
[1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [[4,4,4,4],[2,1]] => [2,1] => 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [[5,5,4],[1]] => [1] => 0
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [[4,4,4,4],[1,1]] => [1,1] => 0
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [[5,5,5],[1]] => [1] => 0
[1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [[3,3,3,3,3],[2,2]] => [2,2] => 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [[4,3,3,3],[2]] => [2] => 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,0] => [[4,4,3,3],[3]] => [3] => 2
[1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [[4,4,3,3],[2]] => [2] => 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [[3,3,3,3,3],[2,1]] => [2,1] => 1
[1,1,1,1,1,0,0,0,1,0,0,1,0,0] => [[4,3,3,3],[1]] => [1] => 0
[1,1,1,1,1,0,0,0,1,0,1,0,0,0] => [[3,3,3,3,3],[1,1]] => [1,1] => 0
[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [[4,4,3,3],[1]] => [1] => 0
[1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [[4,4,4,3],[3]] => [3] => 2
[1,1,1,1,1,0,0,1,0,0,1,0,0,0] => [[4,4,4,3],[2]] => [2] => 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [[4,4,4,3],[1]] => [1] => 0
[1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [[3,3,3,3,3],[2]] => [2] => 1
[1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [[3,3,3,3,3],[1]] => [1] => 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [[4,4,4,4],[3]] => [3] => 2
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [[4,4,4,4],[2]] => [2] => 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [[4,4,4,4],[1]] => [1] => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1,1,1,1],[1]] => [1] => 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1,1,1,1],[1,1]] => [1,1] => 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [[3,2,1,1,1,1],[1]] => [1] => 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2,1,1,1],[1,1,1]] => [1,1,1] => 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [[3,2,2,1,1,1],[1,1]] => [1,1] => 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[3,3,2,1,1,1],[2,1]] => [2,1] => 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [[4,3,1,1,1],[2]] => [2] => 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,1,1],[1,1,1,1]] => [1,1,1,1] => 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [[3,2,2,2,1,1],[1,1,1]] => [1,1,1] => 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [[3,3,2,2,1,1],[2,1,1]] => [2,1,1] => 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[3,3,3,2,1,1],[2,2,1]] => [2,2,1] => 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[4,3,2,1,1],[2,1]] => [2,1] => 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [[4,4,2,1,1],[3,1]] => [3,1] => 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0] => [[4,3,3,1,1],[2,2]] => [2,2] => 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0] => [[4,4,3,1,1],[3,2]] => [3,2] => 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [[5,3,1,1],[2]] => [2] => 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,2,1],[1,1,1,1,1]] => [1,1,1,1,1] => 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[3,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[3,3,2,2,2,1],[2,1,1,1]] => [2,1,1,1] => 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[3,3,3,2,2,1],[2,2,1,1]] => [2,2,1,1] => 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[4,3,2,2,1],[2,1,1]] => [2,1,1] => 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [[4,4,2,2,1],[3,1,1]] => [3,1,1] => 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[3,3,3,3,2,1],[2,2,2,1]] => [2,2,2,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[4,3,3,2,1],[2,2,1]] => [2,2,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[4,4,3,2,1],[3,2,1]] => [3,2,1] => 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0] => [[5,3,2,1],[2,1]] => [2,1] => 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0] => [[4,4,4,2,1],[3,3,1]] => [3,3,1] => 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0] => [[5,4,2,1],[3,1]] => [3,1] => 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [[4,3,3,3,1],[2,2,2]] => [2,2,2] => 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0] => [[4,4,3,3,1],[3,2,2]] => [3,2,2] => 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [[5,3,3,1],[2,2]] => [2,2] => 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0] => [[4,4,4,3,1],[3,3,2]] => [3,3,2] => 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0] => [[5,4,3,1],[3,2]] => [3,2] => 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0] => [[5,5,3,1],[4,2]] => [4,2] => 3
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [[4,4,4,4,1],[3,3,3]] => [3,3,3] => 3
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [[5,4,4,1],[3,3]] => [3,3] => 3
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [[5,5,4,1],[4,3]] => [4,3] => 4
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0] => [[6,4,1],[3]] => [3] => 2
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [[5,5,5,1],[1]] => [1] => 0
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [[4,4,4,4,1],[1]] => [1] => 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]] => [1,1,1,1,1,1] => 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[3,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => 0
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[3,3,2,2,2,2],[2,1,1,1,1]] => [2,1,1,1,1] => 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0] => [[3,3,3,2,2,2],[2,2,1,1,1]] => [2,2,1,1,1] => 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0] => [[4,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0] => [[4,4,2,2,2],[3,1,1,1]] => [3,1,1,1] => 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0] => [[3,3,3,3,2,2],[2,2,2,1,1]] => [2,2,2,1,1] => 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0] => [[4,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[4,4,3,2,2],[3,2,1,1]] => [3,2,1,1] => 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0] => [[5,3,2,2],[2,1,1]] => [2,1,1] => 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0] => [[4,4,4,2,2],[3,3,1,1]] => [3,3,1,1] => 3
[1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0] => [[5,4,2,2],[3,1,1]] => [3,1,1] => 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[3,3,3,3,3,2],[2,2,2,2,1]] => [2,2,2,2,1] => 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0] => [[4,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0] => [[4,4,3,3,2],[3,2,2,1]] => [3,2,2,1] => 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0] => [[5,3,3,2],[2,2,1]] => [2,2,1] => 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[4,4,4,3,2],[3,3,2,1]] => [3,3,2,1] => 3
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[5,4,3,2],[3,2,1]] => [3,2,1] => 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0] => [[5,5,3,2],[4,2,1]] => [4,2,1] => 3
[1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0] => [[4,4,4,4,2],[3,3,3,1]] => [3,3,3,1] => 3
[1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0] => [[5,4,4,2],[3,3,1]] => [3,3,1] => 3
[1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0] => [[5,5,4,2],[4,3,1]] => [4,3,1] => 4
[1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0] => [[6,4,2],[3,1]] => [3,1] => 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0] => [[5,5,5,2],[4,4,1]] => [4,4,1] => 5
[1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0] => [[6,5,2],[4,1]] => [4,1] => 3
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[3,3,3,3,3,3],[2,2,2,2,2]] => [2,2,2,2,2] => 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[4,4,3,3,3],[3,2,2,2]] => [3,2,2,2] => 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0] => [[5,3,3,3],[2,2,2]] => [2,2,2] => 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[4,4,4,3,3],[3,3,2,2]] => [3,3,2,2] => 3
[1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[5,4,3,3],[3,2,2]] => [3,2,2] => 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0] => [[5,5,3,3],[4,2,2]] => [4,2,2] => 3
[1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[5,4,4,3],[3,3,2]] => [3,3,2] => 3
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[5,5,4,3],[4,3,2]] => [4,3,2] => 4
[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0] => [[6,4,3],[3,2]] => [3,2] => 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0] => [[5,5,5,3],[4,4,2]] => [4,4,2] => 5
[1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0] => [[6,5,3],[4,2]] => [4,2] => 3
[1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0] => [[6,6,3],[5,2]] => [5,2] => 4
[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0] => [[5,4,4,4],[3,3,3]] => [3,3,3] => 3
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0] => [[5,5,4,4],[4,3,3]] => [4,3,3] => 4
[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0] => [[6,4,4],[3,3]] => [3,3] => 3
[1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0] => [[6,5,4],[4,3]] => [4,3] => 4
[1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0] => [[6,6,4],[5,3]] => [5,3] => 5
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0] => [[7,4],[3]] => [3] => 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0] => [[6,6,6],[5,5]] => [5,5] => 7
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [[7,7],[6]] => [6] => 5
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [[7,7],[5]] => [5] => 4
[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0] => [[6,6,6],[5,4]] => [5,4] => 6
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0] => [[6,6,6],[4,4]] => [4,4] => 5
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0] => [[7,7],[4]] => [4] => 3
[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0] => [[5,5,5,5],[4,3,3]] => [4,3,3] => 4
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0] => [[5,5,5,5],[3,3,3]] => [3,3,3] => 3
[1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0] => [[6,6,6],[5,3]] => [5,3] => 5
[1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0] => [[4,4,4,4,4],[3,3,2,2]] => [3,3,2,2] => 3
[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0] => [[4,4,4,4,4],[3,2,2,2]] => [3,2,2,2] => 2
[1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0] => [[5,5,5,5],[4,4,2]] => [4,4,2] => 5
[1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0] => [[5,5,5,5],[4,3,2]] => [4,3,2] => 4
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0] => [[5,5,5,5],[3,3,2]] => [3,3,2] => 3
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0] => [[6,6,6],[4,2]] => [4,2] => 3
[1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0] => [[5,5,5,5],[4,2,2]] => [4,2,2] => 3
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0] => [[5,5,5,5],[3,2,2]] => [3,2,2] => 2
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0] => [[3,3,3,3,3,3],[2,2,2,2,1]] => [2,2,2,2,1] => 1
[1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0] => [[3,3,3,3,3,3],[2,2,2,1,1]] => [2,2,2,1,1] => 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0] => [[4,4,4,4,4],[3,3,3,1]] => [3,3,3,1] => 3
[1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0] => [[4,4,4,4,4],[3,3,2,1]] => [3,3,2,1] => 3
[1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0] => [[4,4,4,4,4],[3,2,2,1]] => [3,2,2,1] => 2
[1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0] => [[5,5,5,5],[4,4,1]] => [4,4,1] => 5
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0] => [[5,5,5,5],[4,3,1]] => [4,3,1] => 4
[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0] => [[5,5,5,5],[3,3,1]] => [3,3,1] => 3
[1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0] => [[5,5,5,5],[4,2,1]] => [4,2,1] => 3
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0] => [[5,5,5,5],[3,2,1]] => [3,2,1] => 2
[1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0] => [[4,4,4,4,4],[3,3,1,1]] => [3,3,1,1] => 3
[1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0] => [[4,4,4,4,4],[3,2,1,1]] => [3,2,1,1] => 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => 0
[1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 0
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0] => [[3,3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => 1
[1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0] => [[3,3,3,3,3,3],[2,2,1,1]] => [2,2,1,1] => 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0] => [[4,4,4,4,4],[3,3,3]] => [3,3,3] => 3
[1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0] => [[4,4,4,4,4],[3,3,2]] => [3,3,2] => 3
[1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0] => [[4,4,4,4,4],[3,2,2]] => [3,2,2] => 2
[1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0] => [[4,4,4,4,4],[3,3,1]] => [3,3,1] => 3
[1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0] => [[4,4,4,4,4],[3,2,1]] => [3,2,1] => 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0] => [[5,5,5,5],[4,1]] => [4,1] => 3
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0] => [[3,3,3,3,3,3],[2]] => [2] => 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0] => [[4,4,4,4,3],[3]] => [3] => 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => [[5,5,5,4],[4]] => [4] => 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0] => [[4,4,4,4,4],[3,1]] => [3,1] => 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => [[5,5,5,5],[4]] => [4] => 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [[4,4,4,4,4],[3]] => [3] => 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [[5,5,5,5,5],[4]] => [4] => 3
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [[5,5,5,5,5,5],[4]] => [4] => 3
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0] => [[4,4,4,4,4,4],[3]] => [3] => 2
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [[6,6,6,6,6,6],[5]] => [5] => 4
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0] => [[6,6,6,6,6],[5]] => [5] => 4
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0] => [[5,5,5,5,4],[4]] => [4] => 3
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0] => [[5,5,5,5,5],[4,1]] => [4,1] => 3
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [[5,5,5,5,1],[1]] => [1] => 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [[8,8],[6]] => [6] => 5
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]] => [1,1,1,1,1,1,1] => 0
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1]] => [1,1,1,1,1,1] => 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]] => [1,1,1,1,1,1,1,1] => 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [[8,8],[7]] => [7] => 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [[9,9],[8]] => [8] => 7
search for individual values
searching the database for the individual values of this statistic
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Map
inner shape
Description
The inner shape of a skew partition.
Map
skew partition
Description
The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of $\gamma(D)$.