- FindStat

Identifier

Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000006: Dyck paths ⟶ ℤ

Values

[1,1,0,0,1,0] => [[2,2],[1]] => [1] => [1,0] => 0

[1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => [1] => [1,0] => 0

[1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,0,0,1,1,0,0] => [[3,2],[1]] => [1] => [1,0] => 0

[1,1,0,1,0,0,1,0] => [[3,3],[2]] => [2] => [1,0,1,0] => 1

[1,1,0,1,1,0,0,0] => [[3,3],[1]] => [1] => [1,0] => 0

[1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => [1] => [1,0] => 0

[1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => [1] => [1,0] => 0

[1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]] => [1] => [1,0] => 0

[1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => [2] => [1,0,1,0] => 1

[1,0,1,1,0,1,1,0,0,0] => [[3,3,1],[1]] => [1] => [1,0] => 0

[1,0,1,1,1,0,0,0,1,0] => [[2,2,2,1],[1]] => [1] => [1,0] => 0

[1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1

[1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,0,0,1,1,0,1,0,0] => [[4,2],[1]] => [1] => [1,0] => 0

[1,1,0,0,1,1,1,0,0,0] => [[3,3,2],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]] => [2] => [1,0,1,0] => 1

[1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,0,1,0,1,1,0,0,0] => [[4,4],[2]] => [2] => [1,0,1,0] => 1

[1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,0,1,1,0,0,1,0,0] => [[4,3],[1]] => [1] => [1,0] => 0

[1,1,0,1,1,0,1,0,0,0] => [[3,3,3],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,0,1,1,1,0,0,0,0] => [[4,4],[1]] => [1] => [1,0] => 0

[1,1,1,0,0,0,1,0,1,0] => [[2,2,2,2],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,1,0,0,0,1,1,0,0] => [[3,2,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,0,1,0,0,1,0] => [[3,3,2],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,0,0,1,1,0,0,0] => [[3,3,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,1,0,0,0,1,0] => [[2,2,2,2],[1]] => [1] => [1,0] => 0

[1,1,1,1,0,0,0,0,1,0] => [[3,3,3],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,1,0,0,1,0,0,0] => [[3,3,3],[1]] => [1] => [1,0] => 0

[1,0,1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1,1],[1]] => [1] => [1,0] => 0

[1,0,1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1,1],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,0,1,0,1,1,0,0,1,1,0,0] => [[3,2,1,1],[1]] => [1] => [1,0] => 0

[1,0,1,0,1,1,0,1,0,0,1,0] => [[3,3,1,1],[2]] => [2] => [1,0,1,0] => 1

[1,0,1,0,1,1,0,1,1,0,0,0] => [[3,3,1,1],[1]] => [1] => [1,0] => 0

[1,0,1,0,1,1,1,0,0,0,1,0] => [[2,2,2,1,1],[1]] => [1] => [1,0] => 0

[1,0,1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1

[1,0,1,1,0,0,1,0,1,1,0,0] => [[3,2,2,1],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,0,1,1,0,0,1,1,0,0,1,0] => [[3,3,2,1],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,0,1,1,0,0,1,1,0,1,0,0] => [[4,2,1],[1]] => [1] => [1,0] => 0

[1,0,1,1,0,0,1,1,1,0,0,0] => [[3,3,2,1],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,0,1,1,0,1,0,0,1,0,1,0] => [[3,3,3,1],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,0,1,1,0,1,0,0,1,1,0,0] => [[4,3,1],[2]] => [2] => [1,0,1,0] => 1

[1,0,1,1,0,1,0,1,0,0,1,0] => [[4,4,1],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,0,1,1,0,1,0,1,1,0,0,0] => [[4,4,1],[2]] => [2] => [1,0,1,0] => 1

[1,0,1,1,0,1,1,0,0,0,1,0] => [[3,3,3,1],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,0,1,1,0,1,1,0,0,1,0,0] => [[4,3,1],[1]] => [1] => [1,0] => 0

[1,0,1,1,0,1,1,0,1,0,0,0] => [[3,3,3,1],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,0,1,1,0,1,1,1,0,0,0,0] => [[4,4,1],[1]] => [1] => [1,0] => 0

[1,0,1,1,1,0,0,0,1,0,1,0] => [[2,2,2,2,1],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,0,1,1,1,0,0,0,1,1,0,0] => [[3,2,2,1],[1]] => [1] => [1,0] => 0

[1,0,1,1,1,0,0,1,0,0,1,0] => [[3,3,2,1],[2]] => [2] => [1,0,1,0] => 1

[1,0,1,1,1,0,0,1,1,0,0,0] => [[3,3,2,1],[1]] => [1] => [1,0] => 0

[1,0,1,1,1,0,1,0,0,0,1,0] => [[2,2,2,2,1],[1]] => [1] => [1,0] => 0

[1,0,1,1,1,1,0,0,0,0,1,0] => [[3,3,3,1],[2]] => [2] => [1,0,1,0] => 1

[1,0,1,1,1,1,0,0,1,0,0,0] => [[3,3,3,1],[1]] => [1] => [1,0] => 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] => [1,1,0,1,0,1,0,0] => 3

[1,1,0,0,1,0,1,0,1,1,0,0] => [[3,2,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1

[1,1,0,0,1,0,1,1,0,0,1,0] => [[3,3,2,2],[2,1,1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2

[1,1,0,0,1,0,1,1,0,1,0,0] => [[4,2,2],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,0,0,1,0,1,1,1,0,0,0] => [[3,3,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1

[1,1,0,0,1,1,0,0,1,0,1,0] => [[3,3,3,2],[2,2,1]] => [2,2,1] => [1,1,1,0,0,1,0,0] => 2

[1,1,0,0,1,1,0,0,1,1,0,0] => [[4,3,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,0,0,1,1,0,1,0,0,1,0] => [[4,4,2],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 3

[1,1,0,0,1,1,0,1,0,1,0,0] => [[5,2],[1]] => [1] => [1,0] => 0

[1,1,0,0,1,1,0,1,1,0,0,0] => [[4,4,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,0,0,1,1,1,0,0,0,1,0] => [[3,3,3,2],[2,1,1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2

[1,1,0,0,1,1,1,0,0,1,0,0] => [[4,3,2],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,0,0,1,1,1,0,1,0,0,0] => [[3,3,3,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1

[1,1,0,0,1,1,1,1,0,0,0,0] => [[4,4,2],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,0,1,0,0,1,0,1,0,1,0] => [[3,3,3,3],[2,2,2]] => [2,2,2] => [1,1,1,1,0,0,0,0] => 0

[1,1,0,1,0,0,1,0,1,1,0,0] => [[4,3,3],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,0,1,0,0,1,1,0,0,1,0] => [[4,4,3],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1

[1,1,0,1,0,0,1,1,0,1,0,0] => [[5,3],[2]] => [2] => [1,0,1,0] => 1

[1,1,0,1,0,0,1,1,1,0,0,0] => [[4,4,3],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,0,1,0,1,0,0,1,0,1,0] => [[4,4,4],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 1

[1,1,0,1,0,1,0,0,1,1,0,0] => [[5,4],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,0,1,0,1,0,1,0,0,1,0] => [[5,5],[4]] => [4] => [1,0,1,0,1,0,1,0] => 6

[1,1,0,1,0,1,0,1,1,0,0,0] => [[5,5],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,0,1,0,1,1,0,0,0,1,0] => [[4,4,4],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1

[1,1,0,1,0,1,1,0,0,1,0,0] => [[5,4],[2]] => [2] => [1,0,1,0] => 1

[1,1,0,1,0,1,1,0,1,0,0,0] => [[4,4,4],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,0,1,0,1,1,1,0,0,0,0] => [[5,5],[2]] => [2] => [1,0,1,0] => 1

[1,1,0,1,1,0,0,0,1,0,1,0] => [[3,3,3,3],[2,2,1]] => [2,2,1] => [1,1,1,0,0,1,0,0] => 2

[1,1,0,1,1,0,0,0,1,1,0,0] => [[4,3,3],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,0,1,1,0,0,1,0,0,1,0] => [[4,4,3],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 3

[1,1,0,1,1,0,0,1,0,1,0,0] => [[5,3],[1]] => [1] => [1,0] => 0

[1,1,0,1,1,0,0,1,1,0,0,0] => [[4,4,3],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,0,1,1,0,1,0,0,0,1,0] => [[3,3,3,3],[2,1,1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2

[1,1,0,1,1,0,1,0,0,1,0,0] => [[4,3,3],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,0,1,1,0,1,0,1,0,0,0] => [[3,3,3,3],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1

[1,1,0,1,1,0,1,1,0,0,0,0] => [[4,4,3],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,0,1,1,1,0,0,0,0,1,0] => [[4,4,4],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 3

[1,1,0,1,1,1,0,0,0,1,0,0] => [[5,4],[1]] => [1] => [1,0] => 0

[1,1,0,1,1,1,0,0,1,0,0,0] => [[4,4,4],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,0,1,1,1,0,1,0,0,0,0] => [[5,5],[1]] => [1] => [1,0] => 0

[1,1,0,1,1,1,1,0,0,0,0,0] => [[4,4,4],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,1,0,0,0,1,0,1,0,1,0] => [[2,2,2,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1

>>> Load all 471 entries. <<<

[1,1,1,0,0,0,1,0,1,1,0,0] => [[3,2,2,2],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,1,0,0,0,1,1,0,0,1,0] => [[3,3,2,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,1,0,0,0,1,1,0,1,0,0] => [[4,2,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,0,0,1,1,1,0,0,0] => [[3,3,2,2],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,1,0,0,1,0,0,1,0,1,0] => [[3,3,3,2],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,1,0,0,1,0,0,1,1,0,0] => [[4,3,2],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,0,0,1,0,1,0,0,1,0] => [[4,4,2],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,1,0,0,1,0,1,1,0,0,0] => [[4,4,2],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,0,0,1,1,0,0,0,1,0] => [[3,3,3,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,1,0,0,1,1,0,0,1,0,0] => [[4,3,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,0,1,1,0,1,0,0,0] => [[3,3,3,2],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,1,0,0,1,1,1,0,0,0,0] => [[4,4,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,1,0,0,0,1,0,1,0] => [[2,2,2,2,2],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,1,0,1,0,0,0,1,1,0,0] => [[3,2,2,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,1,0,0,1,0,0,1,0] => [[3,3,2,2],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,0,1,0,0,1,1,0,0,0] => [[3,3,2,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,1,0,1,0,0,0,1,0] => [[2,2,2,2,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,1,1,0,0,0,0,1,0] => [[3,3,3,2],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,0,1,1,0,0,1,0,0,0] => [[3,3,3,2],[1]] => [1] => [1,0] => 0

[1,1,1,1,0,0,0,0,1,0,1,0] => [[3,3,3,3],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,1,1,0,0,0,0,1,1,0,0] => [[4,3,3],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,1,0,0,0,1,0,0,1,0] => [[4,4,3],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,1,1,0,0,0,1,1,0,0,0] => [[4,4,3],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,1,0,0,1,0,0,0,1,0] => [[3,3,3,3],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,1,1,0,0,1,0,0,1,0,0] => [[4,3,3],[1]] => [1] => [1,0] => 0

[1,1,1,1,0,0,1,0,1,0,0,0] => [[3,3,3,3],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,1,1,0,0,1,1,0,0,0,0] => [[4,4,3],[1]] => [1] => [1,0] => 0

[1,1,1,1,0,1,0,0,0,0,1,0] => [[4,4,4],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,1,1,0,1,0,0,1,0,0,0] => [[4,4,4],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,1,0,1,1,0,0,0,0,0] => [[4,4,4],[1]] => [1] => [1,0] => 0

[1,1,1,1,1,0,0,0,0,0,1,0] => [[3,3,3,3],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,1,1,0,0,0,1,0,0,0] => [[3,3,3,3],[1]] => [1] => [1,0] => 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] => 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] => [1,1,0,0] => 0

[1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [[3,2,1,1,1],[1]] => [1] => [1,0] => 0

[1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [[3,3,1,1,1],[2]] => [2] => [1,0,1,0] => 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] => [1,1,0,1,0,0] => 1

[1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [[3,2,2,1,1],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[3,3,2,1,1],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,0,1,0,1,1,0,0,1,1,0,1,0,0] => [[4,2,1,1],[1]] => [1] => [1,0] => 0

[1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [[3,3,3,1,1],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [[4,3,1,1],[2]] => [2] => [1,0,1,0] => 1

[1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [[4,4,1,1],[3]] => [3] => [1,0,1,0,1,0] => 3

[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] => [1,1,0,1,0,1,0,0] => 3

[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] => [1,1,0,1,0,0] => 1

[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] => [1,0,1,1,0,1,0,0] => 2

[1,0,1,1,0,0,1,0,1,1,0,1,0,0] => [[4,2,2,1],[1,1]] => [1,1] => [1,1,0,0] => 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] => [1,1,1,0,0,1,0,0] => 2

[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[4,3,2,1],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [[4,4,2,1],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 3

[1,0,1,1,0,0,1,1,0,1,0,1,0,0] => [[5,2,1],[1]] => [1] => [1,0] => 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] => [1,1,1,1,0,0,0,0] => 0

[1,0,1,1,0,1,0,0,1,0,1,1,0,0] => [[4,3,3,1],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,0,1,1,0,1,0,0,1,1,0,0,1,0] => [[4,4,3,1],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1

[1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [[5,3,1],[2]] => [2] => [1,0,1,0] => 1

[1,0,1,1,0,1,0,1,0,0,1,0,1,0] => [[4,4,4,1],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 1

[1,0,1,1,0,1,0,1,0,0,1,1,0,0] => [[5,4,1],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [[5,5,1],[4]] => [4] => [1,0,1,0,1,0,1,0] => 6

[1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [[4,4,4,1],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [[4,4,3,1],[1]] => [1] => [1,0] => 0

[1,0,1,1,1,1,0,1,0,0,1,0,0,0] => [[4,4,4,1],[2]] => [2] => [1,0,1,0] => 1

[1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [[4,4,4,1],[1]] => [1] => [1,0] => 0

[1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [[3,3,3,3,1],[1]] => [1] => [1,0] => 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] => [1,1,0,1,0,1,0,1,0,0] => 6

[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] => [1,1,0,1,0,1,0,0] => 3

[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] => [1,0,1,1,0,1,0,1,0,0] => 4

[1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [[4,2,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1

[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] => [1,1,1,0,0,1,0,1,0,0] => 5

[1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[4,3,2,2],[2,1,1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2

[1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [[4,4,2,2],[3,1,1]] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 4

[1,1,0,0,1,0,1,1,0,1,0,1,0,0] => [[5,2,2],[1,1]] => [1,1] => [1,1,0,0] => 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] => [1,1,1,1,0,0,0,1,0,0] => 2

[1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[4,3,3,2],[2,2,1]] => [2,2,1] => [1,1,1,0,0,1,0,0] => 2

[1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[4,4,3,2],[3,2,1]] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 3

[1,1,0,0,1,1,0,0,1,1,0,1,0,0] => [[5,3,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 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] => [1,1,1,0,1,0,0,1,0,0] => 4

[1,1,0,0,1,1,0,1,0,0,1,1,0,0] => [[5,4,2],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 3

[1,1,0,0,1,1,0,1,0,1,0,0,1,0] => [[5,5,2],[4,1]] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 6

[1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [[6,2],[1]] => [1] => [1,0] => 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] => [1,1,0,1,0,0] => 1

[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] => [1,1,1,1,0,1,0,0,0,0] => 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] => [1,1,1,1,0,0,0,0] => 0

[1,1,0,1,0,0,1,0,1,1,0,0,1,0] => [[4,4,3,3],[3,2,2]] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 1

[1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [[5,3,3],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,0,1,0,0,1,1,0,0,1,0,1,0] => [[4,4,4,3],[3,3,2]] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => 1

[1,1,0,1,0,0,1,1,0,0,1,1,0,0] => [[5,4,3],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1

[1,1,0,1,0,0,1,1,0,1,0,0,1,0] => [[5,5,3],[4,2]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 3

[1,1,0,1,0,0,1,1,0,1,0,1,0,0] => [[6,3],[2]] => [2] => [1,0,1,0] => 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] => [1,1,1,1,1,0,0,0,0,0] => 0

[1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [[5,4,4],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 1

[1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [[5,5,4],[4,3]] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 2

[1,1,0,1,0,1,0,0,1,1,0,1,0,0] => [[6,4],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,0,1,0,1,0,1,0,0,1,0,1,0] => [[5,5,5],[4,4]] => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 3

[1,1,0,1,0,1,0,1,0,0,1,1,0,0] => [[6,5],[4]] => [4] => [1,0,1,0,1,0,1,0] => 6

[1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [[6,6],[5]] => [5] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [[6,6],[4]] => [4] => [1,0,1,0,1,0,1,0] => 6

[1,1,0,1,0,1,0,1,1,0,0,0,1,0] => [[5,5,5],[4,3]] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 2

[1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [[6,5],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [[5,5,5],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 1

[1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [[6,6],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,0,1,0,1,1,0,0,0,1,0,1,0] => [[4,4,4,4],[3,3,2]] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => 1

[1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [[6,4],[2]] => [2] => [1,0,1,0] => 1

[1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [[5,5,4],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1

[1,1,0,1,0,1,1,0,1,0,0,0,1,0] => [[4,4,4,4],[3,2,2]] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 1

[1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [[5,4,4],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [[4,4,4,4],[2,2,2]] => [2,2,2] => [1,1,1,1,0,0,0,0] => 0

[1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [[5,5,4],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,0,1,0,1,1,1,0,0,0,0,1,0] => [[5,5,5],[4,2]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 3

[1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [[6,5],[2]] => [2] => [1,0,1,0] => 1

[1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [[5,5,5],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1

[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [[6,6],[2]] => [2] => [1,0,1,0] => 1

[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [[5,5,5],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[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] => [1,1,1,1,0,0,0,1,0,0] => 2

[1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [[6,3],[1]] => [1] => [1,0] => 0

[1,1,0,1,1,0,0,1,0,1,1,0,0,0] => [[5,5,3],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 3

[1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [[5,4,3],[2,1]] => [2,1] => [1,0,1,1,0,0] => 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] => [1,1,1,0,0,1,0,0] => 2

[1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [[5,5,3],[2,1]] => [2,1] => [1,0,1,1,0,0] => 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] => [1,1,1,0,0,1,0,1,0,0] => 5

[1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [[5,3,3],[1,1]] => [1,1] => [1,1,0,0] => 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] => [1,0,1,1,0,1,0,0] => 2

[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] => [1,0,1,1,0,1,0,1,0,0] => 4

[1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [[4,3,3,3],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1

[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] => [1,1,0,1,0,1,0,0] => 3

[1,1,0,1,1,0,1,0,1,1,0,0,0,0] => [[4,4,3,3],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1

[1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [[5,4,3],[1,1]] => [1,1] => [1,1,0,0] => 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] => [1,0,1,1,0,1,0,0] => 2

[1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [[5,5,3],[1,1]] => [1,1] => [1,1,0,0] => 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] => [1,1,0,1,0,0] => 1

[1,1,0,1,1,1,0,0,0,0,1,0,1,0] => [[4,4,4,4],[3,3,1]] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 4

[1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [[6,4],[1]] => [1] => [1,0] => 0

[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [[5,5,4],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 3

[1,1,0,1,1,1,0,0,1,0,0,0,1,0] => [[4,4,4,4],[3,2,1]] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 3

[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [[5,4,4],[2,1]] => [2,1] => [1,0,1,1,0,0] => 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] => [1,1,1,0,0,1,0,0] => 2

[1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [[5,5,4],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,0,1,1,1,0,1,0,0,0,0,1,0] => [[5,5,5],[4,1]] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 6

[1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [[6,5],[1]] => [1] => [1,0] => 0

[1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [[5,5,5],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 3

[1,1,0,1,1,1,0,1,0,1,0,0,0,0] => [[6,6],[1]] => [1] => [1,0] => 0

[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [[5,5,5],[2,1]] => [2,1] => [1,0,1,1,0,0] => 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] => [1,0,1,0,1,1,0,1,0,0] => 4

[1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [[5,4,4],[1,1]] => [1,1] => [1,1,0,0] => 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] => [1,0,1,1,0,1,0,0] => 2

[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [[5,5,4],[1,1]] => [1,1] => [1,1,0,0] => 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] => [1,1,0,1,0,0] => 1

[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [[5,5,5],[1,1]] => [1,1] => [1,1,0,0] => 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] => [1,1,0,1,0,1,0,0] => 3

[1,1,1,0,0,1,0,1,0,1,1,0,0,0] => [[5,5,2],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,1,0,0,1,0,1,1,0,0,1,0,0] => [[5,4,2],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,0,0,1,0,1,1,0,1,0,0,0] => [[4,4,4,2],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [[5,5,2],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,0,0,1,1,0,0,1,0,1,0,0] => [[5,3,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,0,1,1,0,0,1,1,0,0,0] => [[4,4,3,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,1,0,0,1,1,0,1,0,0,1,0,0] => [[4,3,3,2],[1,1]] => [1,1] => [1,1,0,0] => 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] => [1,1,0,1,0,0] => 1

[1,1,1,0,0,1,1,0,1,1,0,0,0,0] => [[4,4,3,2],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [[5,4,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [[4,4,4,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,1,0,0,1,1,1,0,1,0,0,0,0] => [[5,5,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [[4,4,4,2],[1,1]] => [1,1] => [1,1,0,0] => 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] => [1,1,0,1,0,0] => 1

[1,1,1,0,1,0,0,1,0,1,1,0,0,0] => [[4,4,2,2],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,0,1,0,0,1,1,0,0,1,0,0] => [[4,3,2,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,1,0,0,1,1,0,1,0,0,0] => [[3,3,3,2,2],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,1,0,1,0,0,1,1,1,0,0,0,0] => [[4,4,2,2],[1]] => [1] => [1,0] => 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] => [1,1,0,0] => 0

[1,1,1,0,1,0,1,0,0,1,1,0,0,0] => [[3,3,2,2,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,1,0,1,1,0,0,1,0,0,0] => [[3,3,3,2,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,1,1,0,0,0,1,1,0,0,0] => [[4,4,3,2],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,0,1,1,0,0,1,0,0,1,0,0] => [[4,3,3,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,1,1,0,0,1,0,1,0,0,0] => [[3,3,3,3,2],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,1,0,1,1,0,0,1,1,0,0,0,0] => [[4,4,3,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,1,1,0,1,0,0,0,0,1,0] => [[4,4,4,2],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,1,0,1,1,0,1,0,0,1,0,0,0] => [[4,4,4,2],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [[4,4,4,2],[1]] => [1] => [1,0] => 0

[1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [[3,3,3,3,2],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [[3,3,3,3,2],[1]] => [1] => [1,0] => 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] => [1,1,1,1,0,0,0,0] => 0

[1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [[5,5,3],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [[5,4,3],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,1,0,0,0,1,1,0,1,0,0,0] => [[4,4,4,3],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [[5,5,3],[2]] => [2] => [1,0,1,0] => 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] => [1,1,1,0,0,1,0,0] => 2

[1,1,1,1,0,0,1,0,0,1,0,1,0,0] => [[5,3,3],[1]] => [1] => [1,0] => 0

[1,1,1,1,0,0,1,0,0,1,1,0,0,0] => [[4,4,3,3],[2,1]] => [2,1] => [1,0,1,1,0,0] => 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] => [1,0,1,1,0,1,0,0] => 2

[1,1,1,1,0,0,1,0,1,0,0,1,0,0] => [[4,3,3,3],[1,1]] => [1,1] => [1,1,0,0] => 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] => [1,1,0,1,0,0] => 1

[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [[4,4,3,3],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [[4,4,4,3],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 3

[1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [[5,4,3],[1]] => [1] => [1,0] => 0

[1,1,1,1,0,0,1,1,0,0,1,0,0,0] => [[4,4,4,3],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [[5,5,3],[1]] => [1] => [1,0] => 0

[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [[4,4,4,3],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,1,1,0,1,0,0,0,0,1,0,1,0] => [[4,4,4,4],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 1

[1,1,1,1,0,1,0,0,0,1,1,0,0,0] => [[5,5,4],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [[4,4,4,4],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1

[1,1,1,1,0,1,0,0,1,0,0,1,0,0] => [[5,4,4],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,1,0,1,0,0,1,0,1,0,0,0] => [[4,4,4,4],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [[5,5,4],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [[5,5,5],[4]] => [4] => [1,0,1,0,1,0,1,0] => 6

[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [[5,5,5],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [[5,5,5],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [[4,4,4,4],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 3

[1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [[5,4,4],[1]] => [1] => [1,0] => 0

[1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [[4,4,4,4],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [[5,5,4],[1]] => [1] => [1,0] => 0

[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [[4,4,4,4],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [[5,5,5],[1]] => [1] => [1,0] => 0

[1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [[3,3,3,3,3],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [[4,3,3,3],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,1,1,0,0,0,0,1,0,0,1,0] => [[4,4,3,3],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [[4,4,3,3],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [[3,3,3,3,3],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1

[1,1,1,1,1,0,0,0,1,0,0,1,0,0] => [[4,3,3,3],[1]] => [1] => [1,0] => 0

[1,1,1,1,1,0,0,0,1,0,1,0,0,0] => [[3,3,3,3,3],[1,1]] => [1,1] => [1,1,0,0] => 0

[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [[4,4,3,3],[1]] => [1] => [1,0] => 0

[1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [[4,4,4,3],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,1,1,1,0,0,1,0,0,1,0,0,0] => [[4,4,4,3],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [[4,4,4,3],[1]] => [1] => [1,0] => 0

[1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [[3,3,3,3,3],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [[3,3,3,3,3],[1]] => [1] => [1,0] => 0

[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [[4,4,4,4],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [[4,4,4,4],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [[4,4,4,4],[1]] => [1] => [1,0] => 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] => [1,0] => 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] => [1,1,0,0] => 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] => [1,0] => 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] => [1,1,0,1,0,0] => 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]] => [1,1] => [1,1,0,0] => 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] => [1,0,1,1,0,0] => 1

[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [[4,3,1,1,1],[2]] => [2] => [1,0,1,0] => 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] => [1,1,0,1,0,1,0,0] => 3

[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] => [1,1,0,1,0,0] => 1

[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] => [1,0,1,1,0,1,0,0] => 2

[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] => [1,1,1,0,0,1,0,0] => 2

[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] => [1,0,1,1,0,0] => 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] => [1,0,1,0,1,1,0,0] => 3

[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] => [1,1,1,0,0,0] => 0

[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] => [1,0,1,1,1,0,0,0] => 1

[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [[5,3,1,1],[2]] => [2] => [1,0,1,0] => 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] => [1,1,0,1,0,1,0,1,0,0] => 6

[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] => [1,1,0,1,0,1,0,0] => 3

[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] => [1,0,1,1,0,1,0,1,0,0] => 4

[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] => [1,1,1,0,0,1,0,1,0,0] => 5

[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] => [1,0,1,1,0,1,0,0] => 2

[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] => [1,0,1,0,1,1,0,1,0,0] => 4

[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] => [1,1,1,1,0,0,0,1,0,0] => 2

[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] => [1,1,1,0,0,1,0,0] => 2

[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] => [1,0,1,1,1,0,0,1,0,0] => 3

[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0] => [[5,3,2,1],[2,1]] => [2,1] => [1,0,1,1,0,0] => 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] => [1,1,1,0,1,0,0,1,0,0] => 4

[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0] => [[5,4,2,1],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 3

[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] => [1,1,1,1,0,0,0,0] => 0

[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] => [1,0,1,1,1,1,0,0,0,0] => 1

[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [[5,3,3,1],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0

[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] => [1,1,1,0,1,1,0,0,0,0] => 1

[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0] => [[5,4,3,1],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1

[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0] => [[5,5,3,1],[4,2]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 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] => [1,1,1,1,1,0,0,0,0,0] => 0

[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [[5,4,4,1],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 1

[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [[5,5,4,1],[4,3]] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 2

[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0] => [[6,4,1],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [[5,5,5,1],[1]] => [1] => [1,0] => 0

[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [[4,4,4,4,1],[1]] => [1] => [1,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 10

[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] => [1,1,0,1,0,1,0,1,0,0] => 6

[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] => [1,0,1,1,0,1,0,1,0,1,0,0] => 7

[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] => [1,1,1,0,0,1,0,1,0,1,0,0] => 9

[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] => [1,0,1,1,0,1,0,1,0,0] => 4

[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] => [1,0,1,0,1,1,0,1,0,1,0,0] => 6

[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] => [1,1,1,1,0,0,0,1,0,1,0,0] => 5

[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] => [1,1,1,0,0,1,0,1,0,0] => 5

[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] => [1,0,1,1,1,0,0,1,0,1,0,0] => 6

[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] => [1,0,1,1,0,1,0,0] => 2

[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] => [1,1,1,0,1,0,0,1,0,1,0,0] => 8

[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] => [1,0,1,0,1,1,0,1,0,0] => 4

[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] => [1,1,1,1,0,1,0,0,0,1,0,0] => 3

[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] => [1,1,1,1,0,0,0,1,0,0] => 2

[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] => [1,0,1,1,1,1,0,0,0,1,0,0] => 3

[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] => [1,1,1,0,0,1,0,0] => 2

[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] => [1,1,1,0,1,1,0,0,0,1,0,0] => 4

[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] => [1,0,1,1,1,0,0,1,0,0] => 3

[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] => [1,0,1,0,1,1,1,0,0,1,0,0] => 5

[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] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2

[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] => [1,1,1,0,1,0,0,1,0,0] => 4

[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] => [1,0,1,1,1,0,1,0,0,1,0,0] => 5

[1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0] => [[6,4,2],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 3

[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] => [1,1,1,0,1,0,1,0,0,1,0,0] => 7

[1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0] => [[6,5,2],[4,1]] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 6

[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] => [1,1,1,1,0,1,0,1,0,0,0,0] => 3

[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] => [1,0,1,1,1,1,0,1,0,0,0,0] => 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] => [1,1,1,1,0,0,0,0] => 0

[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] => [1,1,1,0,1,1,0,1,0,0,0,0] => 2

[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] => [1,0,1,1,1,1,0,0,0,0] => 1

[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] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3

[1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[4,4,4,4,3],[3,3,3,2]] => [3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => 2

[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] => [1,1,1,0,1,1,0,0,0,0] => 1

[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] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2

[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0] => [[6,4,3],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1

[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] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3

[1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0] => [[6,5,3],[4,2]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 3

[1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0] => [[6,6,3],[5,2]] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 6

[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0] => [[4,4,4,4,4],[3,3,3,3]] => [3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[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] => [1,1,1,1,1,0,0,0,0,0] => 0

[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] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1

[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0] => [[6,4,4],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 1

[1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0] => [[5,5,5,4],[4,4,3]] => [4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => 1

[1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0] => [[6,5,4],[4,3]] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 2

[1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0] => [[6,6,4],[5,3]] => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 4

[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0] => [[7,4],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0] => [[5,5,5,5],[4,4,4]] => [4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => 1

[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0] => [[6,6,6],[5,5]] => [5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => 6

[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [[7,7],[6]] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 15

[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [[7,7],[5]] => [5] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0] => [[6,6,6],[5,4]] => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 4

[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0] => [[6,6,6],[4,4]] => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 3

[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0] => [[7,7],[4]] => [4] => [1,0,1,0,1,0,1,0] => 6

[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0] => [[5,5,5,5],[4,4,3]] => [4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => 1

[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] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1

[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] => [1,1,1,1,1,0,0,0,0,0] => 0

[1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0] => [[6,6,6],[5,3]] => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 4

[1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0] => [[4,4,4,4,4],[3,3,3,2]] => [3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => 2

[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] => [1,1,1,0,1,1,0,1,0,0,0,0] => 2

[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] => [1,0,1,1,1,1,0,1,0,0,0,0] => 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] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3

[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] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2

[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] => [1,1,1,0,1,1,0,0,0,0] => 1

[1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0] => [[6,6,6],[4,2]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 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] => [1,0,1,0,1,1,1,1,0,0,0,0] => 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] => [1,0,1,1,1,1,0,0,0,0] => 1

[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] => [1,1,1,1,0,1,0,0,0,1,0,0] => 3

[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] => [1,1,1,1,0,0,0,1,0,1,0,0] => 5

[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] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2

[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] => [1,1,1,0,1,1,0,0,0,1,0,0] => 4

[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] => [1,0,1,1,1,1,0,0,0,1,0,0] => 3

[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] => [1,1,1,0,1,0,1,0,0,1,0,0] => 7

[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] => [1,0,1,1,1,0,1,0,0,1,0,0] => 5

[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] => [1,1,1,0,1,0,0,1,0,0] => 4

[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] => [1,0,1,0,1,1,1,0,0,1,0,0] => 5

[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] => [1,0,1,1,1,0,0,1,0,0] => 3

[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] => [1,1,1,0,1,0,0,1,0,1,0,0] => 8

[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] => [1,0,1,1,1,0,0,1,0,1,0,0] => 6

[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] => [1,1,0,1,0,1,0,1,0,0] => 6

[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] => [1,1,0,1,0,1,0,0] => 3

[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] => [1,1,1,1,0,1,0,0,0,0] => 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] => [1,1,1,0,0,1,0,1,0,0] => 5

[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] => [1,1,1,1,1,0,0,0,0,0] => 0

[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] => [1,1,1,0,1,1,0,0,0,0] => 1

[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] => [1,0,1,1,1,1,0,0,0,0] => 1

[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] => [1,1,1,0,1,0,0,1,0,0] => 4

[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] => [1,0,1,1,1,0,0,1,0,0] => 3

[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0] => [[5,5,5,5],[4,1]] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 6

[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0] => [[3,3,3,3,3,3],[2]] => [2] => [1,0,1,0] => 1

[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0] => [[4,4,4,4,3],[3]] => [3] => [1,0,1,0,1,0] => 3

[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => [[5,5,5,4],[4]] => [4] => [1,0,1,0,1,0,1,0] => 6

[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] => [1,0,1,0,1,1,0,0] => 3

[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => [[5,5,5,5],[4]] => [4] => [1,0,1,0,1,0,1,0] => 6

[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [[4,4,4,4,4],[3]] => [3] => [1,0,1,0,1,0] => 3

[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] => [1,0,1,0,1,0,1,0] => 6

[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] => [1,0,1,0,1,0,1,0] => 6

[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] => [1,0,1,0,1,0] => 3

[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] => [1,0,1,0,1,0,1,0,1,0] => 10

[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] => [1,0,1,0,1,0,1,0,1,0] => 10

[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] => [1,0,1,0,1,0,1,0] => 6

[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] => [1,0,1,0,1,0,1,1,0,0] => 6

[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] => [1,0] => 0

[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [[8,8],[6]] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 15

[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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 10

search for individual values

searching the database for the individual values of this statistic

Description

The dinv of a Dyck path.
Let

$a=(a_1,\ldots,a_n)$ be the area sequence of a Dyck path

$D$ (see St000012The area of a Dyck path.).
The dinv statistic of

$D$ is

$\operatorname{dinv}(D) = \# \big\{ i < j : a_i-a_j \in \{ 0,1 \} \big\}.$
Equivalently,

$\operatorname{dinv}(D)$ is also equal to the number of boxes in the partition above

$D$ whose arm length is one larger or equal to its leg length.
There is a recursive definition of the

$(\operatorname{area},\operatorname{dinv})$ pair of statistics, see [2].
Let

$a=(0,a_2,\ldots,a_r,0,a_{r+2},\ldots,a_n)$ be the area sequence of the Dyck path

$D$ with

$a_i > 0$ for

$2\leq i\leq r$ (so that the path touches the diagonal for the first time after

$r$ steps). Assume that

$D$ has

$v$ entries where

$a_i=0$ . Let

$D'$ be the path with the area sequence

$(0,a_{r+2},\ldots,a_n,a_2-1,a_3-1,\ldots,a_r-1)$ , then the statistics are related by

$(\operatorname{area}(D),\operatorname{dinv}(D)) = (\operatorname{area}(D')+r-1,\operatorname{dinv}(D')+v-1).$

Map

inner shape

Description

The inner shape of a skew partition.

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

Map

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

$\gamma(D)$ .