Identifier
-
Mp00311:
Plane partitions
—to partition⟶
Integer partitions
St000147: Integer partitions ⟶ ℤ
Values
[[1]] => [1] => 1
[[1],[1]] => [1,1] => 1
[[2]] => [2] => 2
[[1,1]] => [2] => 2
[[1],[1],[1]] => [1,1,1] => 1
[[2],[1]] => [2,1] => 2
[[1,1],[1]] => [2,1] => 2
[[3]] => [3] => 3
[[2,1]] => [3] => 3
[[1,1,1]] => [3] => 3
[[1],[1],[1],[1]] => [1,1,1,1] => 1
[[2],[1],[1]] => [2,1,1] => 2
[[2],[2]] => [2,2] => 2
[[1,1],[1],[1]] => [2,1,1] => 2
[[1,1],[1,1]] => [2,2] => 2
[[3],[1]] => [3,1] => 3
[[2,1],[1]] => [3,1] => 3
[[1,1,1],[1]] => [3,1] => 3
[[4]] => [4] => 4
[[3,1]] => [4] => 4
[[2,2]] => [4] => 4
[[2,1,1]] => [4] => 4
[[1,1,1,1]] => [4] => 4
[[1],[1],[1],[1],[1]] => [1,1,1,1,1] => 1
[[2],[1],[1],[1]] => [2,1,1,1] => 2
[[2],[2],[1]] => [2,2,1] => 2
[[1,1],[1],[1],[1]] => [2,1,1,1] => 2
[[1,1],[1,1],[1]] => [2,2,1] => 2
[[3],[1],[1]] => [3,1,1] => 3
[[3],[2]] => [3,2] => 3
[[2,1],[1],[1]] => [3,1,1] => 3
[[2,1],[2]] => [3,2] => 3
[[2,1],[1,1]] => [3,2] => 3
[[1,1,1],[1],[1]] => [3,1,1] => 3
[[1,1,1],[1,1]] => [3,2] => 3
[[4],[1]] => [4,1] => 4
[[3,1],[1]] => [4,1] => 4
[[2,2],[1]] => [4,1] => 4
[[2,1,1],[1]] => [4,1] => 4
[[1,1,1,1],[1]] => [4,1] => 4
[[5]] => [5] => 5
[[4,1]] => [5] => 5
[[3,2]] => [5] => 5
[[3,1,1]] => [5] => 5
[[2,2,1]] => [5] => 5
[[2,1,1,1]] => [5] => 5
[[1,1,1,1,1]] => [5] => 5
[[1],[1],[1],[1],[1],[1]] => [1,1,1,1,1,1] => 1
[[2],[1],[1],[1],[1]] => [2,1,1,1,1] => 2
[[2],[2],[1],[1]] => [2,2,1,1] => 2
[[2],[2],[2]] => [2,2,2] => 2
[[1,1],[1],[1],[1],[1]] => [2,1,1,1,1] => 2
[[1,1],[1,1],[1],[1]] => [2,2,1,1] => 2
[[1,1],[1,1],[1,1]] => [2,2,2] => 2
[[3],[1],[1],[1]] => [3,1,1,1] => 3
[[3],[2],[1]] => [3,2,1] => 3
[[3],[3]] => [3,3] => 3
[[2,1],[1],[1],[1]] => [3,1,1,1] => 3
[[2,1],[2],[1]] => [3,2,1] => 3
[[2,1],[1,1],[1]] => [3,2,1] => 3
[[2,1],[2,1]] => [3,3] => 3
[[1,1,1],[1],[1],[1]] => [3,1,1,1] => 3
[[1,1,1],[1,1],[1]] => [3,2,1] => 3
[[1,1,1],[1,1,1]] => [3,3] => 3
[[4],[1],[1]] => [4,1,1] => 4
[[4],[2]] => [4,2] => 4
[[3,1],[1],[1]] => [4,1,1] => 4
[[3,1],[2]] => [4,2] => 4
[[3,1],[1,1]] => [4,2] => 4
[[2,2],[1],[1]] => [4,1,1] => 4
[[2,2],[2]] => [4,2] => 4
[[2,2],[1,1]] => [4,2] => 4
[[2,1,1],[1],[1]] => [4,1,1] => 4
[[2,1,1],[2]] => [4,2] => 4
[[2,1,1],[1,1]] => [4,2] => 4
[[1,1,1,1],[1],[1]] => [4,1,1] => 4
[[1,1,1,1],[1,1]] => [4,2] => 4
[[5],[1]] => [5,1] => 5
[[4,1],[1]] => [5,1] => 5
[[3,2],[1]] => [5,1] => 5
[[3,1,1],[1]] => [5,1] => 5
[[2,2,1],[1]] => [5,1] => 5
[[2,1,1,1],[1]] => [5,1] => 5
[[1,1,1,1,1],[1]] => [5,1] => 5
[[6]] => [6] => 6
[[5,1]] => [6] => 6
[[4,2]] => [6] => 6
[[4,1,1]] => [6] => 6
[[3,3]] => [6] => 6
[[3,2,1]] => [6] => 6
[[3,1,1,1]] => [6] => 6
[[2,2,2]] => [6] => 6
[[2,2,1,1]] => [6] => 6
[[2,1,1,1,1]] => [6] => 6
[[1,1,1,1,1,1]] => [6] => 6
[[1],[1],[1],[1],[1],[1],[1]] => [1,1,1,1,1,1,1] => 1
[[2],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1] => 2
[[2],[2],[1],[1],[1]] => [2,2,1,1,1] => 2
[[2],[2],[2],[1]] => [2,2,2,1] => 2
[[1,1],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1] => 2
[[1,1],[1,1],[1],[1],[1]] => [2,2,1,1,1] => 2
>>> Load all 1123 entries. <<<[[1,1],[1,1],[1,1],[1]] => [2,2,2,1] => 2
[[3],[1],[1],[1],[1]] => [3,1,1,1,1] => 3
[[3],[2],[1],[1]] => [3,2,1,1] => 3
[[3],[2],[2]] => [3,2,2] => 3
[[3],[3],[1]] => [3,3,1] => 3
[[2,1],[1],[1],[1],[1]] => [3,1,1,1,1] => 3
[[2,1],[2],[1],[1]] => [3,2,1,1] => 3
[[2,1],[2],[2]] => [3,2,2] => 3
[[2,1],[1,1],[1],[1]] => [3,2,1,1] => 3
[[2,1],[1,1],[1,1]] => [3,2,2] => 3
[[2,1],[2,1],[1]] => [3,3,1] => 3
[[1,1,1],[1],[1],[1],[1]] => [3,1,1,1,1] => 3
[[1,1,1],[1,1],[1],[1]] => [3,2,1,1] => 3
[[1,1,1],[1,1],[1,1]] => [3,2,2] => 3
[[1,1,1],[1,1,1],[1]] => [3,3,1] => 3
[[4],[1],[1],[1]] => [4,1,1,1] => 4
[[4],[2],[1]] => [4,2,1] => 4
[[4],[3]] => [4,3] => 4
[[3,1],[1],[1],[1]] => [4,1,1,1] => 4
[[3,1],[2],[1]] => [4,2,1] => 4
[[3,1],[1,1],[1]] => [4,2,1] => 4
[[3,1],[3]] => [4,3] => 4
[[3,1],[2,1]] => [4,3] => 4
[[2,2],[1],[1],[1]] => [4,1,1,1] => 4
[[2,2],[2],[1]] => [4,2,1] => 4
[[2,2],[1,1],[1]] => [4,2,1] => 4
[[2,2],[2,1]] => [4,3] => 4
[[2,1,1],[1],[1],[1]] => [4,1,1,1] => 4
[[2,1,1],[2],[1]] => [4,2,1] => 4
[[2,1,1],[1,1],[1]] => [4,2,1] => 4
[[2,1,1],[2,1]] => [4,3] => 4
[[2,1,1],[1,1,1]] => [4,3] => 4
[[1,1,1,1],[1],[1],[1]] => [4,1,1,1] => 4
[[1,1,1,1],[1,1],[1]] => [4,2,1] => 4
[[1,1,1,1],[1,1,1]] => [4,3] => 4
[[5],[1],[1]] => [5,1,1] => 5
[[5],[2]] => [5,2] => 5
[[4,1],[1],[1]] => [5,1,1] => 5
[[4,1],[2]] => [5,2] => 5
[[4,1],[1,1]] => [5,2] => 5
[[3,2],[1],[1]] => [5,1,1] => 5
[[3,2],[2]] => [5,2] => 5
[[3,2],[1,1]] => [5,2] => 5
[[3,1,1],[1],[1]] => [5,1,1] => 5
[[3,1,1],[2]] => [5,2] => 5
[[3,1,1],[1,1]] => [5,2] => 5
[[2,2,1],[1],[1]] => [5,1,1] => 5
[[2,2,1],[2]] => [5,2] => 5
[[2,2,1],[1,1]] => [5,2] => 5
[[2,1,1,1],[1],[1]] => [5,1,1] => 5
[[2,1,1,1],[2]] => [5,2] => 5
[[2,1,1,1],[1,1]] => [5,2] => 5
[[1,1,1,1,1],[1],[1]] => [5,1,1] => 5
[[1,1,1,1,1],[1,1]] => [5,2] => 5
[[6],[1]] => [6,1] => 6
[[5,1],[1]] => [6,1] => 6
[[4,2],[1]] => [6,1] => 6
[[4,1,1],[1]] => [6,1] => 6
[[3,3],[1]] => [6,1] => 6
[[3,2,1],[1]] => [6,1] => 6
[[3,1,1,1],[1]] => [6,1] => 6
[[2,2,2],[1]] => [6,1] => 6
[[2,2,1,1],[1]] => [6,1] => 6
[[2,1,1,1,1],[1]] => [6,1] => 6
[[1,1,1,1,1,1],[1]] => [6,1] => 6
[[7]] => [7] => 7
[[6,1]] => [7] => 7
[[5,2]] => [7] => 7
[[5,1,1]] => [7] => 7
[[4,3]] => [7] => 7
[[4,2,1]] => [7] => 7
[[4,1,1,1]] => [7] => 7
[[3,3,1]] => [7] => 7
[[3,2,2]] => [7] => 7
[[3,2,1,1]] => [7] => 7
[[3,1,1,1,1]] => [7] => 7
[[2,2,2,1]] => [7] => 7
[[2,2,1,1,1]] => [7] => 7
[[2,1,1,1,1,1]] => [7] => 7
[[1,1,1,1,1,1,1]] => [7] => 7
[[1],[1],[1],[1],[1],[1],[1],[1]] => [1,1,1,1,1,1,1,1] => 1
[[2],[1],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1,1] => 2
[[2],[2],[1],[1],[1],[1]] => [2,2,1,1,1,1] => 2
[[2],[2],[2],[1],[1]] => [2,2,2,1,1] => 2
[[2],[2],[2],[2]] => [2,2,2,2] => 2
[[1,1],[1],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1,1] => 2
[[1,1],[1,1],[1],[1],[1],[1]] => [2,2,1,1,1,1] => 2
[[1,1],[1,1],[1,1],[1],[1]] => [2,2,2,1,1] => 2
[[1,1],[1,1],[1,1],[1,1]] => [2,2,2,2] => 2
[[3],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1] => 3
[[3],[2],[1],[1],[1]] => [3,2,1,1,1] => 3
[[3],[2],[2],[1]] => [3,2,2,1] => 3
[[3],[3],[1],[1]] => [3,3,1,1] => 3
[[3],[3],[2]] => [3,3,2] => 3
[[2,1],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1] => 3
[[2,1],[2],[1],[1],[1]] => [3,2,1,1,1] => 3
[[2,1],[2],[2],[1]] => [3,2,2,1] => 3
[[2,1],[1,1],[1],[1],[1]] => [3,2,1,1,1] => 3
[[2,1],[1,1],[1,1],[1]] => [3,2,2,1] => 3
[[2,1],[2,1],[1],[1]] => [3,3,1,1] => 3
[[2,1],[2,1],[2]] => [3,3,2] => 3
[[2,1],[2,1],[1,1]] => [3,3,2] => 3
[[1,1,1],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1] => 3
[[1,1,1],[1,1],[1],[1],[1]] => [3,2,1,1,1] => 3
[[1,1,1],[1,1],[1,1],[1]] => [3,2,2,1] => 3
[[1,1,1],[1,1,1],[1],[1]] => [3,3,1,1] => 3
[[1,1,1],[1,1,1],[1,1]] => [3,3,2] => 3
[[4],[1],[1],[1],[1]] => [4,1,1,1,1] => 4
[[4],[2],[1],[1]] => [4,2,1,1] => 4
[[4],[2],[2]] => [4,2,2] => 4
[[4],[3],[1]] => [4,3,1] => 4
[[4],[4]] => [4,4] => 4
[[3,1],[1],[1],[1],[1]] => [4,1,1,1,1] => 4
[[3,1],[2],[1],[1]] => [4,2,1,1] => 4
[[3,1],[2],[2]] => [4,2,2] => 4
[[3,1],[1,1],[1],[1]] => [4,2,1,1] => 4
[[3,1],[1,1],[1,1]] => [4,2,2] => 4
[[3,1],[3],[1]] => [4,3,1] => 4
[[3,1],[2,1],[1]] => [4,3,1] => 4
[[3,1],[3,1]] => [4,4] => 4
[[2,2],[1],[1],[1],[1]] => [4,1,1,1,1] => 4
[[2,2],[2],[1],[1]] => [4,2,1,1] => 4
[[2,2],[2],[2]] => [4,2,2] => 4
[[2,2],[1,1],[1],[1]] => [4,2,1,1] => 4
[[2,2],[1,1],[1,1]] => [4,2,2] => 4
[[2,2],[2,1],[1]] => [4,3,1] => 4
[[2,2],[2,2]] => [4,4] => 4
[[2,1,1],[1],[1],[1],[1]] => [4,1,1,1,1] => 4
[[2,1,1],[2],[1],[1]] => [4,2,1,1] => 4
[[2,1,1],[2],[2]] => [4,2,2] => 4
[[2,1,1],[1,1],[1],[1]] => [4,2,1,1] => 4
[[2,1,1],[1,1],[1,1]] => [4,2,2] => 4
[[2,1,1],[2,1],[1]] => [4,3,1] => 4
[[2,1,1],[1,1,1],[1]] => [4,3,1] => 4
[[2,1,1],[2,1,1]] => [4,4] => 4
[[1,1,1,1],[1],[1],[1],[1]] => [4,1,1,1,1] => 4
[[1,1,1,1],[1,1],[1],[1]] => [4,2,1,1] => 4
[[1,1,1,1],[1,1],[1,1]] => [4,2,2] => 4
[[1,1,1,1],[1,1,1],[1]] => [4,3,1] => 4
[[1,1,1,1],[1,1,1,1]] => [4,4] => 4
[[5],[1],[1],[1]] => [5,1,1,1] => 5
[[5],[2],[1]] => [5,2,1] => 5
[[5],[3]] => [5,3] => 5
[[4,1],[1],[1],[1]] => [5,1,1,1] => 5
[[4,1],[2],[1]] => [5,2,1] => 5
[[4,1],[1,1],[1]] => [5,2,1] => 5
[[4,1],[3]] => [5,3] => 5
[[4,1],[2,1]] => [5,3] => 5
[[3,2],[1],[1],[1]] => [5,1,1,1] => 5
[[3,2],[2],[1]] => [5,2,1] => 5
[[3,2],[1,1],[1]] => [5,2,1] => 5
[[3,2],[3]] => [5,3] => 5
[[3,2],[2,1]] => [5,3] => 5
[[3,1,1],[1],[1],[1]] => [5,1,1,1] => 5
[[3,1,1],[2],[1]] => [5,2,1] => 5
[[3,1,1],[1,1],[1]] => [5,2,1] => 5
[[3,1,1],[3]] => [5,3] => 5
[[3,1,1],[2,1]] => [5,3] => 5
[[3,1,1],[1,1,1]] => [5,3] => 5
[[2,2,1],[1],[1],[1]] => [5,1,1,1] => 5
[[2,2,1],[2],[1]] => [5,2,1] => 5
[[2,2,1],[1,1],[1]] => [5,2,1] => 5
[[2,2,1],[2,1]] => [5,3] => 5
[[2,2,1],[1,1,1]] => [5,3] => 5
[[2,1,1,1],[1],[1],[1]] => [5,1,1,1] => 5
[[2,1,1,1],[2],[1]] => [5,2,1] => 5
[[2,1,1,1],[1,1],[1]] => [5,2,1] => 5
[[2,1,1,1],[2,1]] => [5,3] => 5
[[2,1,1,1],[1,1,1]] => [5,3] => 5
[[1,1,1,1,1],[1],[1],[1]] => [5,1,1,1] => 5
[[1,1,1,1,1],[1,1],[1]] => [5,2,1] => 5
[[1,1,1,1,1],[1,1,1]] => [5,3] => 5
[[6],[1],[1]] => [6,1,1] => 6
[[6],[2]] => [6,2] => 6
[[5,1],[1],[1]] => [6,1,1] => 6
[[5,1],[2]] => [6,2] => 6
[[5,1],[1,1]] => [6,2] => 6
[[4,2],[1],[1]] => [6,1,1] => 6
[[4,2],[2]] => [6,2] => 6
[[4,2],[1,1]] => [6,2] => 6
[[4,1,1],[1],[1]] => [6,1,1] => 6
[[4,1,1],[2]] => [6,2] => 6
[[4,1,1],[1,1]] => [6,2] => 6
[[3,3],[1],[1]] => [6,1,1] => 6
[[3,3],[2]] => [6,2] => 6
[[3,3],[1,1]] => [6,2] => 6
[[3,2,1],[1],[1]] => [6,1,1] => 6
[[3,2,1],[2]] => [6,2] => 6
[[3,2,1],[1,1]] => [6,2] => 6
[[3,1,1,1],[1],[1]] => [6,1,1] => 6
[[3,1,1,1],[2]] => [6,2] => 6
[[3,1,1,1],[1,1]] => [6,2] => 6
[[2,2,2],[1],[1]] => [6,1,1] => 6
[[2,2,2],[2]] => [6,2] => 6
[[2,2,2],[1,1]] => [6,2] => 6
[[2,2,1,1],[1],[1]] => [6,1,1] => 6
[[2,2,1,1],[2]] => [6,2] => 6
[[2,2,1,1],[1,1]] => [6,2] => 6
[[2,1,1,1,1],[1],[1]] => [6,1,1] => 6
[[2,1,1,1,1],[2]] => [6,2] => 6
[[2,1,1,1,1],[1,1]] => [6,2] => 6
[[1,1,1,1,1,1],[1],[1]] => [6,1,1] => 6
[[1,1,1,1,1,1],[1,1]] => [6,2] => 6
[[7],[1]] => [7,1] => 7
[[6,1],[1]] => [7,1] => 7
[[5,2],[1]] => [7,1] => 7
[[5,1,1],[1]] => [7,1] => 7
[[4,3],[1]] => [7,1] => 7
[[4,2,1],[1]] => [7,1] => 7
[[4,1,1,1],[1]] => [7,1] => 7
[[3,3,1],[1]] => [7,1] => 7
[[3,2,2],[1]] => [7,1] => 7
[[3,2,1,1],[1]] => [7,1] => 7
[[3,1,1,1,1],[1]] => [7,1] => 7
[[2,2,2,1],[1]] => [7,1] => 7
[[2,2,1,1,1],[1]] => [7,1] => 7
[[2,1,1,1,1,1],[1]] => [7,1] => 7
[[1,1,1,1,1,1,1],[1]] => [7,1] => 7
[[8]] => [8] => 8
[[7,1]] => [8] => 8
[[6,2]] => [8] => 8
[[6,1,1]] => [8] => 8
[[5,3]] => [8] => 8
[[5,2,1]] => [8] => 8
[[5,1,1,1]] => [8] => 8
[[4,4]] => [8] => 8
[[4,3,1]] => [8] => 8
[[4,2,2]] => [8] => 8
[[4,2,1,1]] => [8] => 8
[[4,1,1,1,1]] => [8] => 8
[[3,3,2]] => [8] => 8
[[3,3,1,1]] => [8] => 8
[[3,2,2,1]] => [8] => 8
[[3,2,1,1,1]] => [8] => 8
[[3,1,1,1,1,1]] => [8] => 8
[[2,2,2,2]] => [8] => 8
[[2,2,2,1,1]] => [8] => 8
[[2,2,1,1,1,1]] => [8] => 8
[[2,1,1,1,1,1,1]] => [8] => 8
[[1,1,1,1,1,1,1,1]] => [8] => 8
[[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]] => [2,1,1,1,1,1,1,1] => 2
[[2],[2],[1],[1],[1],[1],[1]] => [2,2,1,1,1,1,1] => 2
[[2],[2],[2],[1],[1],[1]] => [2,2,2,1,1,1] => 2
[[2],[2],[2],[2],[1]] => [2,2,2,2,1] => 2
[[1,1],[1],[1],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1,1,1] => 2
[[1,1],[1,1],[1],[1],[1],[1],[1]] => [2,2,1,1,1,1,1] => 2
[[1,1],[1,1],[1,1],[1],[1],[1]] => [2,2,2,1,1,1] => 2
[[1,1],[1,1],[1,1],[1,1],[1]] => [2,2,2,2,1] => 2
[[3],[1],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1,1] => 3
[[3],[2],[1],[1],[1],[1]] => [3,2,1,1,1,1] => 3
[[3],[2],[2],[1],[1]] => [3,2,2,1,1] => 3
[[3],[2],[2],[2]] => [3,2,2,2] => 3
[[3],[3],[1],[1],[1]] => [3,3,1,1,1] => 3
[[3],[3],[2],[1]] => [3,3,2,1] => 3
[[3],[3],[3]] => [3,3,3] => 3
[[2,1],[1],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1,1] => 3
[[2,1],[2],[1],[1],[1],[1]] => [3,2,1,1,1,1] => 3
[[2,1],[2],[2],[1],[1]] => [3,2,2,1,1] => 3
[[2,1],[2],[2],[2]] => [3,2,2,2] => 3
[[2,1],[1,1],[1],[1],[1],[1]] => [3,2,1,1,1,1] => 3
[[2,1],[1,1],[1,1],[1],[1]] => [3,2,2,1,1] => 3
[[2,1],[1,1],[1,1],[1,1]] => [3,2,2,2] => 3
[[2,1],[2,1],[1],[1],[1]] => [3,3,1,1,1] => 3
[[2,1],[2,1],[2],[1]] => [3,3,2,1] => 3
[[2,1],[2,1],[1,1],[1]] => [3,3,2,1] => 3
[[2,1],[2,1],[2,1]] => [3,3,3] => 3
[[1,1,1],[1],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1,1] => 3
[[1,1,1],[1,1],[1],[1],[1],[1]] => [3,2,1,1,1,1] => 3
[[1,1,1],[1,1],[1,1],[1],[1]] => [3,2,2,1,1] => 3
[[1,1,1],[1,1],[1,1],[1,1]] => [3,2,2,2] => 3
[[1,1,1],[1,1,1],[1],[1],[1]] => [3,3,1,1,1] => 3
[[1,1,1],[1,1,1],[1,1],[1]] => [3,3,2,1] => 3
[[1,1,1],[1,1,1],[1,1,1]] => [3,3,3] => 3
[[4],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1] => 4
[[4],[2],[1],[1],[1]] => [4,2,1,1,1] => 4
[[4],[2],[2],[1]] => [4,2,2,1] => 4
[[4],[3],[1],[1]] => [4,3,1,1] => 4
[[4],[3],[2]] => [4,3,2] => 4
[[4],[4],[1]] => [4,4,1] => 4
[[3,1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1] => 4
[[3,1],[2],[1],[1],[1]] => [4,2,1,1,1] => 4
[[3,1],[2],[2],[1]] => [4,2,2,1] => 4
[[3,1],[1,1],[1],[1],[1]] => [4,2,1,1,1] => 4
[[3,1],[1,1],[1,1],[1]] => [4,2,2,1] => 4
[[3,1],[3],[1],[1]] => [4,3,1,1] => 4
[[3,1],[3],[2]] => [4,3,2] => 4
[[3,1],[2,1],[1],[1]] => [4,3,1,1] => 4
[[3,1],[2,1],[2]] => [4,3,2] => 4
[[3,1],[2,1],[1,1]] => [4,3,2] => 4
[[3,1],[3,1],[1]] => [4,4,1] => 4
[[2,2],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1] => 4
[[2,2],[2],[1],[1],[1]] => [4,2,1,1,1] => 4
[[2,2],[2],[2],[1]] => [4,2,2,1] => 4
[[2,2],[1,1],[1],[1],[1]] => [4,2,1,1,1] => 4
[[2,2],[1,1],[1,1],[1]] => [4,2,2,1] => 4
[[2,2],[2,1],[1],[1]] => [4,3,1,1] => 4
[[2,2],[2,1],[2]] => [4,3,2] => 4
[[2,2],[2,1],[1,1]] => [4,3,2] => 4
[[2,2],[2,2],[1]] => [4,4,1] => 4
[[2,1,1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1] => 4
[[2,1,1],[2],[1],[1],[1]] => [4,2,1,1,1] => 4
[[2,1,1],[2],[2],[1]] => [4,2,2,1] => 4
[[2,1,1],[1,1],[1],[1],[1]] => [4,2,1,1,1] => 4
[[2,1,1],[1,1],[1,1],[1]] => [4,2,2,1] => 4
[[2,1,1],[2,1],[1],[1]] => [4,3,1,1] => 4
[[2,1,1],[2,1],[2]] => [4,3,2] => 4
[[2,1,1],[2,1],[1,1]] => [4,3,2] => 4
[[2,1,1],[1,1,1],[1],[1]] => [4,3,1,1] => 4
[[2,1,1],[1,1,1],[1,1]] => [4,3,2] => 4
[[2,1,1],[2,1,1],[1]] => [4,4,1] => 4
[[1,1,1,1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1] => 4
[[1,1,1,1],[1,1],[1],[1],[1]] => [4,2,1,1,1] => 4
[[1,1,1,1],[1,1],[1,1],[1]] => [4,2,2,1] => 4
[[1,1,1,1],[1,1,1],[1],[1]] => [4,3,1,1] => 4
[[1,1,1,1],[1,1,1],[1,1]] => [4,3,2] => 4
[[1,1,1,1],[1,1,1,1],[1]] => [4,4,1] => 4
[[5],[1],[1],[1],[1]] => [5,1,1,1,1] => 5
[[5],[2],[1],[1]] => [5,2,1,1] => 5
[[5],[2],[2]] => [5,2,2] => 5
[[5],[3],[1]] => [5,3,1] => 5
[[5],[4]] => [5,4] => 5
[[4,1],[1],[1],[1],[1]] => [5,1,1,1,1] => 5
[[4,1],[2],[1],[1]] => [5,2,1,1] => 5
[[4,1],[2],[2]] => [5,2,2] => 5
[[4,1],[1,1],[1],[1]] => [5,2,1,1] => 5
[[4,1],[1,1],[1,1]] => [5,2,2] => 5
[[4,1],[3],[1]] => [5,3,1] => 5
[[4,1],[2,1],[1]] => [5,3,1] => 5
[[4,1],[4]] => [5,4] => 5
[[4,1],[3,1]] => [5,4] => 5
[[3,2],[1],[1],[1],[1]] => [5,1,1,1,1] => 5
[[3,2],[2],[1],[1]] => [5,2,1,1] => 5
[[3,2],[2],[2]] => [5,2,2] => 5
[[3,2],[1,1],[1],[1]] => [5,2,1,1] => 5
[[3,2],[1,1],[1,1]] => [5,2,2] => 5
[[3,2],[3],[1]] => [5,3,1] => 5
[[3,2],[2,1],[1]] => [5,3,1] => 5
[[3,2],[3,1]] => [5,4] => 5
[[3,2],[2,2]] => [5,4] => 5
[[3,1,1],[1],[1],[1],[1]] => [5,1,1,1,1] => 5
[[3,1,1],[2],[1],[1]] => [5,2,1,1] => 5
[[3,1,1],[2],[2]] => [5,2,2] => 5
[[3,1,1],[1,1],[1],[1]] => [5,2,1,1] => 5
[[3,1,1],[1,1],[1,1]] => [5,2,2] => 5
[[3,1,1],[3],[1]] => [5,3,1] => 5
[[3,1,1],[2,1],[1]] => [5,3,1] => 5
[[3,1,1],[1,1,1],[1]] => [5,3,1] => 5
[[3,1,1],[3,1]] => [5,4] => 5
[[3,1,1],[2,1,1]] => [5,4] => 5
[[2,2,1],[1],[1],[1],[1]] => [5,1,1,1,1] => 5
[[2,2,1],[2],[1],[1]] => [5,2,1,1] => 5
[[2,2,1],[2],[2]] => [5,2,2] => 5
[[2,2,1],[1,1],[1],[1]] => [5,2,1,1] => 5
[[2,2,1],[1,1],[1,1]] => [5,2,2] => 5
[[2,2,1],[2,1],[1]] => [5,3,1] => 5
[[2,2,1],[1,1,1],[1]] => [5,3,1] => 5
[[2,2,1],[2,2]] => [5,4] => 5
[[2,2,1],[2,1,1]] => [5,4] => 5
[[2,1,1,1],[1],[1],[1],[1]] => [5,1,1,1,1] => 5
[[2,1,1,1],[2],[1],[1]] => [5,2,1,1] => 5
[[2,1,1,1],[2],[2]] => [5,2,2] => 5
[[2,1,1,1],[1,1],[1],[1]] => [5,2,1,1] => 5
[[2,1,1,1],[1,1],[1,1]] => [5,2,2] => 5
[[2,1,1,1],[2,1],[1]] => [5,3,1] => 5
[[2,1,1,1],[1,1,1],[1]] => [5,3,1] => 5
[[2,1,1,1],[2,1,1]] => [5,4] => 5
[[2,1,1,1],[1,1,1,1]] => [5,4] => 5
[[1,1,1,1,1],[1],[1],[1],[1]] => [5,1,1,1,1] => 5
[[1,1,1,1,1],[1,1],[1],[1]] => [5,2,1,1] => 5
[[1,1,1,1,1],[1,1],[1,1]] => [5,2,2] => 5
[[1,1,1,1,1],[1,1,1],[1]] => [5,3,1] => 5
[[1,1,1,1,1],[1,1,1,1]] => [5,4] => 5
[[6],[1],[1],[1]] => [6,1,1,1] => 6
[[6],[2],[1]] => [6,2,1] => 6
[[6],[3]] => [6,3] => 6
[[5,1],[1],[1],[1]] => [6,1,1,1] => 6
[[5,1],[2],[1]] => [6,2,1] => 6
[[5,1],[1,1],[1]] => [6,2,1] => 6
[[5,1],[3]] => [6,3] => 6
[[5,1],[2,1]] => [6,3] => 6
[[4,2],[1],[1],[1]] => [6,1,1,1] => 6
[[4,2],[2],[1]] => [6,2,1] => 6
[[4,2],[1,1],[1]] => [6,2,1] => 6
[[4,2],[3]] => [6,3] => 6
[[4,2],[2,1]] => [6,3] => 6
[[4,1,1],[1],[1],[1]] => [6,1,1,1] => 6
[[4,1,1],[2],[1]] => [6,2,1] => 6
[[4,1,1],[1,1],[1]] => [6,2,1] => 6
[[4,1,1],[3]] => [6,3] => 6
[[4,1,1],[2,1]] => [6,3] => 6
[[4,1,1],[1,1,1]] => [6,3] => 6
[[3,3],[1],[1],[1]] => [6,1,1,1] => 6
[[3,3],[2],[1]] => [6,2,1] => 6
[[3,3],[1,1],[1]] => [6,2,1] => 6
[[3,3],[3]] => [6,3] => 6
[[3,3],[2,1]] => [6,3] => 6
[[3,2,1],[1],[1],[1]] => [6,1,1,1] => 6
[[3,2,1],[2],[1]] => [6,2,1] => 6
[[3,2,1],[1,1],[1]] => [6,2,1] => 6
[[3,2,1],[3]] => [6,3] => 6
[[3,2,1],[2,1]] => [6,3] => 6
[[3,2,1],[1,1,1]] => [6,3] => 6
[[3,1,1,1],[1],[1],[1]] => [6,1,1,1] => 6
[[3,1,1,1],[2],[1]] => [6,2,1] => 6
[[3,1,1,1],[1,1],[1]] => [6,2,1] => 6
[[3,1,1,1],[3]] => [6,3] => 6
[[3,1,1,1],[2,1]] => [6,3] => 6
[[3,1,1,1],[1,1,1]] => [6,3] => 6
[[2,2,2],[1],[1],[1]] => [6,1,1,1] => 6
[[2,2,2],[2],[1]] => [6,2,1] => 6
[[2,2,2],[1,1],[1]] => [6,2,1] => 6
[[2,2,2],[2,1]] => [6,3] => 6
[[2,2,2],[1,1,1]] => [6,3] => 6
[[2,2,1,1],[1],[1],[1]] => [6,1,1,1] => 6
[[2,2,1,1],[2],[1]] => [6,2,1] => 6
[[2,2,1,1],[1,1],[1]] => [6,2,1] => 6
[[2,2,1,1],[2,1]] => [6,3] => 6
[[2,2,1,1],[1,1,1]] => [6,3] => 6
[[2,1,1,1,1],[1],[1],[1]] => [6,1,1,1] => 6
[[2,1,1,1,1],[2],[1]] => [6,2,1] => 6
[[2,1,1,1,1],[1,1],[1]] => [6,2,1] => 6
[[2,1,1,1,1],[2,1]] => [6,3] => 6
[[2,1,1,1,1],[1,1,1]] => [6,3] => 6
[[1,1,1,1,1,1],[1],[1],[1]] => [6,1,1,1] => 6
[[1,1,1,1,1,1],[1,1],[1]] => [6,2,1] => 6
[[1,1,1,1,1,1],[1,1,1]] => [6,3] => 6
[[7],[1],[1]] => [7,1,1] => 7
[[7],[2]] => [7,2] => 7
[[6,1],[1],[1]] => [7,1,1] => 7
[[6,1],[2]] => [7,2] => 7
[[6,1],[1,1]] => [7,2] => 7
[[5,2],[1],[1]] => [7,1,1] => 7
[[5,2],[2]] => [7,2] => 7
[[5,2],[1,1]] => [7,2] => 7
[[5,1,1],[1],[1]] => [7,1,1] => 7
[[5,1,1],[2]] => [7,2] => 7
[[5,1,1],[1,1]] => [7,2] => 7
[[4,3],[1],[1]] => [7,1,1] => 7
[[4,3],[2]] => [7,2] => 7
[[4,3],[1,1]] => [7,2] => 7
[[4,2,1],[1],[1]] => [7,1,1] => 7
[[4,2,1],[2]] => [7,2] => 7
[[4,2,1],[1,1]] => [7,2] => 7
[[4,1,1,1],[1],[1]] => [7,1,1] => 7
[[4,1,1,1],[2]] => [7,2] => 7
[[4,1,1,1],[1,1]] => [7,2] => 7
[[3,3,1],[1],[1]] => [7,1,1] => 7
[[3,3,1],[2]] => [7,2] => 7
[[3,3,1],[1,1]] => [7,2] => 7
[[3,2,2],[1],[1]] => [7,1,1] => 7
[[3,2,2],[2]] => [7,2] => 7
[[3,2,2],[1,1]] => [7,2] => 7
[[3,2,1,1],[1],[1]] => [7,1,1] => 7
[[3,2,1,1],[2]] => [7,2] => 7
[[3,2,1,1],[1,1]] => [7,2] => 7
[[3,1,1,1,1],[1],[1]] => [7,1,1] => 7
[[3,1,1,1,1],[2]] => [7,2] => 7
[[3,1,1,1,1],[1,1]] => [7,2] => 7
[[2,2,2,1],[1],[1]] => [7,1,1] => 7
[[2,2,2,1],[2]] => [7,2] => 7
[[2,2,2,1],[1,1]] => [7,2] => 7
[[2,2,1,1,1],[1],[1]] => [7,1,1] => 7
[[2,2,1,1,1],[2]] => [7,2] => 7
[[2,2,1,1,1],[1,1]] => [7,2] => 7
[[2,1,1,1,1,1],[1],[1]] => [7,1,1] => 7
[[2,1,1,1,1,1],[2]] => [7,2] => 7
[[2,1,1,1,1,1],[1,1]] => [7,2] => 7
[[1,1,1,1,1,1,1],[1],[1]] => [7,1,1] => 7
[[1,1,1,1,1,1,1],[1,1]] => [7,2] => 7
[[8],[1]] => [8,1] => 8
[[7,1],[1]] => [8,1] => 8
[[6,2],[1]] => [8,1] => 8
[[6,1,1],[1]] => [8,1] => 8
[[5,3],[1]] => [8,1] => 8
[[5,2,1],[1]] => [8,1] => 8
[[5,1,1,1],[1]] => [8,1] => 8
[[4,4],[1]] => [8,1] => 8
[[4,3,1],[1]] => [8,1] => 8
[[4,2,2],[1]] => [8,1] => 8
[[4,2,1,1],[1]] => [8,1] => 8
[[4,1,1,1,1],[1]] => [8,1] => 8
[[3,3,2],[1]] => [8,1] => 8
[[3,3,1,1],[1]] => [8,1] => 8
[[3,2,2,1],[1]] => [8,1] => 8
[[3,2,1,1,1],[1]] => [8,1] => 8
[[3,1,1,1,1,1],[1]] => [8,1] => 8
[[2,2,2,2],[1]] => [8,1] => 8
[[2,2,2,1,1],[1]] => [8,1] => 8
[[2,2,1,1,1,1],[1]] => [8,1] => 8
[[2,1,1,1,1,1,1],[1]] => [8,1] => 8
[[1,1,1,1,1,1,1,1],[1]] => [8,1] => 8
[[9]] => [9] => 9
[[8,1]] => [9] => 9
[[7,2]] => [9] => 9
[[7,1,1]] => [9] => 9
[[6,3]] => [9] => 9
[[6,2,1]] => [9] => 9
[[6,1,1,1]] => [9] => 9
[[5,4]] => [9] => 9
[[5,3,1]] => [9] => 9
[[5,2,2]] => [9] => 9
[[5,2,1,1]] => [9] => 9
[[5,1,1,1,1]] => [9] => 9
[[4,4,1]] => [9] => 9
[[4,3,2]] => [9] => 9
[[4,3,1,1]] => [9] => 9
[[4,2,2,1]] => [9] => 9
[[4,2,1,1,1]] => [9] => 9
[[4,1,1,1,1,1]] => [9] => 9
[[3,3,3]] => [9] => 9
[[3,3,2,1]] => [9] => 9
[[3,3,1,1,1]] => [9] => 9
[[3,2,2,2]] => [9] => 9
[[3,2,2,1,1]] => [9] => 9
[[3,2,1,1,1,1]] => [9] => 9
[[3,1,1,1,1,1,1]] => [9] => 9
[[2,2,2,2,1]] => [9] => 9
[[2,2,2,1,1,1]] => [9] => 9
[[2,2,1,1,1,1,1]] => [9] => 9
[[2,1,1,1,1,1,1,1]] => [9] => 9
[[1,1,1,1,1,1,1,1,1]] => [9] => 9
[[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]] => [2,1,1,1,1,1,1,1,1] => 2
[[2],[2],[1],[1],[1],[1],[1],[1]] => [2,2,1,1,1,1,1,1] => 2
[[2],[2],[2],[1],[1],[1],[1]] => [2,2,2,1,1,1,1] => 2
[[2],[2],[2],[2],[1],[1]] => [2,2,2,2,1,1] => 2
[[2],[2],[2],[2],[2]] => [2,2,2,2,2] => 2
[[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],[1],[1],[1]] => [2,2,1,1,1,1,1,1] => 2
[[1,1],[1,1],[1,1],[1],[1],[1],[1]] => [2,2,2,1,1,1,1] => 2
[[1,1],[1,1],[1,1],[1,1],[1],[1]] => [2,2,2,2,1,1] => 2
[[1,1],[1,1],[1,1],[1,1],[1,1]] => [2,2,2,2,2] => 2
[[3],[1],[1],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1,1,1] => 3
[[3],[2],[1],[1],[1],[1],[1]] => [3,2,1,1,1,1,1] => 3
[[3],[2],[2],[1],[1],[1]] => [3,2,2,1,1,1] => 3
[[3],[2],[2],[2],[1]] => [3,2,2,2,1] => 3
[[3],[3],[1],[1],[1],[1]] => [3,3,1,1,1,1] => 3
[[3],[3],[2],[1],[1]] => [3,3,2,1,1] => 3
[[3],[3],[2],[2]] => [3,3,2,2] => 3
[[3],[3],[3],[1]] => [3,3,3,1] => 3
[[2,1],[1],[1],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1,1,1] => 3
[[2,1],[2],[1],[1],[1],[1],[1]] => [3,2,1,1,1,1,1] => 3
[[2,1],[2],[2],[1],[1],[1]] => [3,2,2,1,1,1] => 3
[[2,1],[2],[2],[2],[1]] => [3,2,2,2,1] => 3
[[2,1],[1,1],[1],[1],[1],[1],[1]] => [3,2,1,1,1,1,1] => 3
[[2,1],[1,1],[1,1],[1],[1],[1]] => [3,2,2,1,1,1] => 3
[[2,1],[1,1],[1,1],[1,1],[1]] => [3,2,2,2,1] => 3
[[2,1],[2,1],[1],[1],[1],[1]] => [3,3,1,1,1,1] => 3
[[2,1],[2,1],[2],[1],[1]] => [3,3,2,1,1] => 3
[[2,1],[2,1],[2],[2]] => [3,3,2,2] => 3
[[2,1],[2,1],[1,1],[1],[1]] => [3,3,2,1,1] => 3
[[2,1],[2,1],[1,1],[1,1]] => [3,3,2,2] => 3
[[2,1],[2,1],[2,1],[1]] => [3,3,3,1] => 3
[[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],[1],[1],[1],[1]] => [3,2,1,1,1,1,1] => 3
[[1,1,1],[1,1],[1,1],[1],[1],[1]] => [3,2,2,1,1,1] => 3
[[1,1,1],[1,1],[1,1],[1,1],[1]] => [3,2,2,2,1] => 3
[[1,1,1],[1,1,1],[1],[1],[1],[1]] => [3,3,1,1,1,1] => 3
[[1,1,1],[1,1,1],[1,1],[1],[1]] => [3,3,2,1,1] => 3
[[1,1,1],[1,1,1],[1,1],[1,1]] => [3,3,2,2] => 3
[[1,1,1],[1,1,1],[1,1,1],[1]] => [3,3,3,1] => 3
[[4],[1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1,1] => 4
[[4],[2],[1],[1],[1],[1]] => [4,2,1,1,1,1] => 4
[[4],[2],[2],[1],[1]] => [4,2,2,1,1] => 4
[[4],[2],[2],[2]] => [4,2,2,2] => 4
[[4],[3],[1],[1],[1]] => [4,3,1,1,1] => 4
[[4],[3],[2],[1]] => [4,3,2,1] => 4
[[4],[3],[3]] => [4,3,3] => 4
[[4],[4],[1],[1]] => [4,4,1,1] => 4
[[4],[4],[2]] => [4,4,2] => 4
[[3,1],[1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1,1] => 4
[[3,1],[2],[1],[1],[1],[1]] => [4,2,1,1,1,1] => 4
[[3,1],[2],[2],[1],[1]] => [4,2,2,1,1] => 4
[[3,1],[2],[2],[2]] => [4,2,2,2] => 4
[[3,1],[1,1],[1],[1],[1],[1]] => [4,2,1,1,1,1] => 4
[[3,1],[1,1],[1,1],[1],[1]] => [4,2,2,1,1] => 4
[[3,1],[1,1],[1,1],[1,1]] => [4,2,2,2] => 4
[[3,1],[3],[1],[1],[1]] => [4,3,1,1,1] => 4
[[3,1],[3],[2],[1]] => [4,3,2,1] => 4
[[3,1],[3],[3]] => [4,3,3] => 4
[[3,1],[2,1],[1],[1],[1]] => [4,3,1,1,1] => 4
[[3,1],[2,1],[2],[1]] => [4,3,2,1] => 4
[[3,1],[2,1],[1,1],[1]] => [4,3,2,1] => 4
[[3,1],[2,1],[2,1]] => [4,3,3] => 4
[[3,1],[3,1],[1],[1]] => [4,4,1,1] => 4
[[3,1],[3,1],[2]] => [4,4,2] => 4
[[3,1],[3,1],[1,1]] => [4,4,2] => 4
[[2,2],[1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1,1] => 4
[[2,2],[2],[1],[1],[1],[1]] => [4,2,1,1,1,1] => 4
[[2,2],[2],[2],[1],[1]] => [4,2,2,1,1] => 4
[[2,2],[2],[2],[2]] => [4,2,2,2] => 4
[[2,2],[1,1],[1],[1],[1],[1]] => [4,2,1,1,1,1] => 4
[[2,2],[1,1],[1,1],[1],[1]] => [4,2,2,1,1] => 4
[[2,2],[1,1],[1,1],[1,1]] => [4,2,2,2] => 4
[[2,2],[2,1],[1],[1],[1]] => [4,3,1,1,1] => 4
[[2,2],[2,1],[2],[1]] => [4,3,2,1] => 4
[[2,2],[2,1],[1,1],[1]] => [4,3,2,1] => 4
[[2,2],[2,1],[2,1]] => [4,3,3] => 4
[[2,2],[2,2],[1],[1]] => [4,4,1,1] => 4
[[2,2],[2,2],[2]] => [4,4,2] => 4
[[2,2],[2,2],[1,1]] => [4,4,2] => 4
[[2,1,1],[1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1,1] => 4
[[2,1,1],[2],[1],[1],[1],[1]] => [4,2,1,1,1,1] => 4
[[2,1,1],[2],[2],[1],[1]] => [4,2,2,1,1] => 4
[[2,1,1],[2],[2],[2]] => [4,2,2,2] => 4
[[2,1,1],[1,1],[1],[1],[1],[1]] => [4,2,1,1,1,1] => 4
[[2,1,1],[1,1],[1,1],[1],[1]] => [4,2,2,1,1] => 4
[[2,1,1],[1,1],[1,1],[1,1]] => [4,2,2,2] => 4
[[2,1,1],[2,1],[1],[1],[1]] => [4,3,1,1,1] => 4
[[2,1,1],[2,1],[2],[1]] => [4,3,2,1] => 4
[[2,1,1],[2,1],[1,1],[1]] => [4,3,2,1] => 4
[[2,1,1],[2,1],[2,1]] => [4,3,3] => 4
[[2,1,1],[1,1,1],[1],[1],[1]] => [4,3,1,1,1] => 4
[[2,1,1],[1,1,1],[1,1],[1]] => [4,3,2,1] => 4
[[2,1,1],[1,1,1],[1,1,1]] => [4,3,3] => 4
[[2,1,1],[2,1,1],[1],[1]] => [4,4,1,1] => 4
[[2,1,1],[2,1,1],[2]] => [4,4,2] => 4
[[2,1,1],[2,1,1],[1,1]] => [4,4,2] => 4
[[1,1,1,1],[1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1,1] => 4
[[1,1,1,1],[1,1],[1],[1],[1],[1]] => [4,2,1,1,1,1] => 4
[[1,1,1,1],[1,1],[1,1],[1],[1]] => [4,2,2,1,1] => 4
[[1,1,1,1],[1,1],[1,1],[1,1]] => [4,2,2,2] => 4
[[1,1,1,1],[1,1,1],[1],[1],[1]] => [4,3,1,1,1] => 4
[[1,1,1,1],[1,1,1],[1,1],[1]] => [4,3,2,1] => 4
[[1,1,1,1],[1,1,1],[1,1,1]] => [4,3,3] => 4
[[1,1,1,1],[1,1,1,1],[1],[1]] => [4,4,1,1] => 4
[[1,1,1,1],[1,1,1,1],[1,1]] => [4,4,2] => 4
[[5],[1],[1],[1],[1],[1]] => [5,1,1,1,1,1] => 5
[[5],[2],[1],[1],[1]] => [5,2,1,1,1] => 5
[[5],[2],[2],[1]] => [5,2,2,1] => 5
[[5],[3],[1],[1]] => [5,3,1,1] => 5
[[5],[3],[2]] => [5,3,2] => 5
[[5],[4],[1]] => [5,4,1] => 5
[[5],[5]] => [5,5] => 5
[[4,1],[1],[1],[1],[1],[1]] => [5,1,1,1,1,1] => 5
[[4,1],[2],[1],[1],[1]] => [5,2,1,1,1] => 5
[[4,1],[2],[2],[1]] => [5,2,2,1] => 5
[[4,1],[1,1],[1],[1],[1]] => [5,2,1,1,1] => 5
[[4,1],[1,1],[1,1],[1]] => [5,2,2,1] => 5
[[4,1],[3],[1],[1]] => [5,3,1,1] => 5
[[4,1],[3],[2]] => [5,3,2] => 5
[[4,1],[2,1],[1],[1]] => [5,3,1,1] => 5
[[4,1],[2,1],[2]] => [5,3,2] => 5
[[4,1],[2,1],[1,1]] => [5,3,2] => 5
[[4,1],[4],[1]] => [5,4,1] => 5
[[4,1],[3,1],[1]] => [5,4,1] => 5
[[4,1],[4,1]] => [5,5] => 5
[[3,2],[1],[1],[1],[1],[1]] => [5,1,1,1,1,1] => 5
[[3,2],[2],[1],[1],[1]] => [5,2,1,1,1] => 5
[[3,2],[2],[2],[1]] => [5,2,2,1] => 5
[[3,2],[1,1],[1],[1],[1]] => [5,2,1,1,1] => 5
[[3,2],[1,1],[1,1],[1]] => [5,2,2,1] => 5
[[3,2],[3],[1],[1]] => [5,3,1,1] => 5
[[3,2],[3],[2]] => [5,3,2] => 5
[[3,2],[2,1],[1],[1]] => [5,3,1,1] => 5
[[3,2],[2,1],[2]] => [5,3,2] => 5
[[3,2],[2,1],[1,1]] => [5,3,2] => 5
[[3,2],[3,1],[1]] => [5,4,1] => 5
[[3,2],[2,2],[1]] => [5,4,1] => 5
[[3,2],[3,2]] => [5,5] => 5
[[3,1,1],[1],[1],[1],[1],[1]] => [5,1,1,1,1,1] => 5
[[3,1,1],[2],[1],[1],[1]] => [5,2,1,1,1] => 5
[[3,1,1],[2],[2],[1]] => [5,2,2,1] => 5
[[3,1,1],[1,1],[1],[1],[1]] => [5,2,1,1,1] => 5
[[3,1,1],[1,1],[1,1],[1]] => [5,2,2,1] => 5
[[3,1,1],[3],[1],[1]] => [5,3,1,1] => 5
[[3,1,1],[3],[2]] => [5,3,2] => 5
[[3,1,1],[2,1],[1],[1]] => [5,3,1,1] => 5
[[3,1,1],[2,1],[2]] => [5,3,2] => 5
[[3,1,1],[2,1],[1,1]] => [5,3,2] => 5
[[3,1,1],[1,1,1],[1],[1]] => [5,3,1,1] => 5
[[3,1,1],[1,1,1],[1,1]] => [5,3,2] => 5
[[3,1,1],[3,1],[1]] => [5,4,1] => 5
[[3,1,1],[2,1,1],[1]] => [5,4,1] => 5
[[3,1,1],[3,1,1]] => [5,5] => 5
[[2,2,1],[1],[1],[1],[1],[1]] => [5,1,1,1,1,1] => 5
[[2,2,1],[2],[1],[1],[1]] => [5,2,1,1,1] => 5
[[2,2,1],[2],[2],[1]] => [5,2,2,1] => 5
[[2,2,1],[1,1],[1],[1],[1]] => [5,2,1,1,1] => 5
[[2,2,1],[1,1],[1,1],[1]] => [5,2,2,1] => 5
[[2,2,1],[2,1],[1],[1]] => [5,3,1,1] => 5
[[2,2,1],[2,1],[2]] => [5,3,2] => 5
[[2,2,1],[2,1],[1,1]] => [5,3,2] => 5
[[2,2,1],[1,1,1],[1],[1]] => [5,3,1,1] => 5
[[2,2,1],[1,1,1],[1,1]] => [5,3,2] => 5
[[2,2,1],[2,2],[1]] => [5,4,1] => 5
[[2,2,1],[2,1,1],[1]] => [5,4,1] => 5
[[2,2,1],[2,2,1]] => [5,5] => 5
[[2,1,1,1],[1],[1],[1],[1],[1]] => [5,1,1,1,1,1] => 5
[[2,1,1,1],[2],[1],[1],[1]] => [5,2,1,1,1] => 5
[[2,1,1,1],[2],[2],[1]] => [5,2,2,1] => 5
[[2,1,1,1],[1,1],[1],[1],[1]] => [5,2,1,1,1] => 5
[[2,1,1,1],[1,1],[1,1],[1]] => [5,2,2,1] => 5
[[2,1,1,1],[2,1],[1],[1]] => [5,3,1,1] => 5
[[2,1,1,1],[2,1],[2]] => [5,3,2] => 5
[[2,1,1,1],[2,1],[1,1]] => [5,3,2] => 5
[[2,1,1,1],[1,1,1],[1],[1]] => [5,3,1,1] => 5
[[2,1,1,1],[1,1,1],[1,1]] => [5,3,2] => 5
[[2,1,1,1],[2,1,1],[1]] => [5,4,1] => 5
[[2,1,1,1],[1,1,1,1],[1]] => [5,4,1] => 5
[[2,1,1,1],[2,1,1,1]] => [5,5] => 5
[[1,1,1,1,1],[1],[1],[1],[1],[1]] => [5,1,1,1,1,1] => 5
[[1,1,1,1,1],[1,1],[1],[1],[1]] => [5,2,1,1,1] => 5
[[1,1,1,1,1],[1,1],[1,1],[1]] => [5,2,2,1] => 5
[[1,1,1,1,1],[1,1,1],[1],[1]] => [5,3,1,1] => 5
[[1,1,1,1,1],[1,1,1],[1,1]] => [5,3,2] => 5
[[1,1,1,1,1],[1,1,1,1],[1]] => [5,4,1] => 5
[[1,1,1,1,1],[1,1,1,1,1]] => [5,5] => 5
[[6],[1],[1],[1],[1]] => [6,1,1,1,1] => 6
[[6],[2],[1],[1]] => [6,2,1,1] => 6
[[6],[2],[2]] => [6,2,2] => 6
[[6],[3],[1]] => [6,3,1] => 6
[[6],[4]] => [6,4] => 6
[[5,1],[1],[1],[1],[1]] => [6,1,1,1,1] => 6
[[5,1],[2],[1],[1]] => [6,2,1,1] => 6
[[5,1],[2],[2]] => [6,2,2] => 6
[[5,1],[1,1],[1],[1]] => [6,2,1,1] => 6
[[5,1],[1,1],[1,1]] => [6,2,2] => 6
[[5,1],[3],[1]] => [6,3,1] => 6
[[5,1],[2,1],[1]] => [6,3,1] => 6
[[5,1],[4]] => [6,4] => 6
[[5,1],[3,1]] => [6,4] => 6
[[4,2],[1],[1],[1],[1]] => [6,1,1,1,1] => 6
[[4,2],[2],[1],[1]] => [6,2,1,1] => 6
[[4,2],[2],[2]] => [6,2,2] => 6
[[4,2],[1,1],[1],[1]] => [6,2,1,1] => 6
[[4,2],[1,1],[1,1]] => [6,2,2] => 6
[[4,2],[3],[1]] => [6,3,1] => 6
[[4,2],[2,1],[1]] => [6,3,1] => 6
[[4,2],[4]] => [6,4] => 6
[[4,2],[3,1]] => [6,4] => 6
[[4,2],[2,2]] => [6,4] => 6
[[4,1,1],[1],[1],[1],[1]] => [6,1,1,1,1] => 6
[[4,1,1],[2],[1],[1]] => [6,2,1,1] => 6
[[4,1,1],[2],[2]] => [6,2,2] => 6
[[4,1,1],[1,1],[1],[1]] => [6,2,1,1] => 6
[[4,1,1],[1,1],[1,1]] => [6,2,2] => 6
[[4,1,1],[3],[1]] => [6,3,1] => 6
[[4,1,1],[2,1],[1]] => [6,3,1] => 6
[[4,1,1],[1,1,1],[1]] => [6,3,1] => 6
[[4,1,1],[4]] => [6,4] => 6
[[4,1,1],[3,1]] => [6,4] => 6
[[4,1,1],[2,1,1]] => [6,4] => 6
[[3,3],[1],[1],[1],[1]] => [6,1,1,1,1] => 6
[[3,3],[2],[1],[1]] => [6,2,1,1] => 6
[[3,3],[2],[2]] => [6,2,2] => 6
[[3,3],[1,1],[1],[1]] => [6,2,1,1] => 6
[[3,3],[1,1],[1,1]] => [6,2,2] => 6
[[3,3],[3],[1]] => [6,3,1] => 6
[[3,3],[2,1],[1]] => [6,3,1] => 6
[[3,3],[3,1]] => [6,4] => 6
[[3,3],[2,2]] => [6,4] => 6
[[3,2,1],[1],[1],[1],[1]] => [6,1,1,1,1] => 6
[[3,2,1],[2],[1],[1]] => [6,2,1,1] => 6
[[3,2,1],[2],[2]] => [6,2,2] => 6
[[3,2,1],[1,1],[1],[1]] => [6,2,1,1] => 6
[[3,2,1],[1,1],[1,1]] => [6,2,2] => 6
[[3,2,1],[3],[1]] => [6,3,1] => 6
[[3,2,1],[2,1],[1]] => [6,3,1] => 6
[[3,2,1],[1,1,1],[1]] => [6,3,1] => 6
[[3,2,1],[3,1]] => [6,4] => 6
[[3,2,1],[2,2]] => [6,4] => 6
[[3,2,1],[2,1,1]] => [6,4] => 6
[[3,1,1,1],[1],[1],[1],[1]] => [6,1,1,1,1] => 6
[[3,1,1,1],[2],[1],[1]] => [6,2,1,1] => 6
[[3,1,1,1],[2],[2]] => [6,2,2] => 6
[[3,1,1,1],[1,1],[1],[1]] => [6,2,1,1] => 6
[[3,1,1,1],[1,1],[1,1]] => [6,2,2] => 6
[[3,1,1,1],[3],[1]] => [6,3,1] => 6
[[3,1,1,1],[2,1],[1]] => [6,3,1] => 6
[[3,1,1,1],[1,1,1],[1]] => [6,3,1] => 6
[[3,1,1,1],[3,1]] => [6,4] => 6
[[3,1,1,1],[2,1,1]] => [6,4] => 6
[[3,1,1,1],[1,1,1,1]] => [6,4] => 6
[[2,2,2],[1],[1],[1],[1]] => [6,1,1,1,1] => 6
[[2,2,2],[2],[1],[1]] => [6,2,1,1] => 6
[[2,2,2],[2],[2]] => [6,2,2] => 6
[[2,2,2],[1,1],[1],[1]] => [6,2,1,1] => 6
[[2,2,2],[1,1],[1,1]] => [6,2,2] => 6
[[2,2,2],[2,1],[1]] => [6,3,1] => 6
[[2,2,2],[1,1,1],[1]] => [6,3,1] => 6
[[2,2,2],[2,2]] => [6,4] => 6
[[2,2,2],[2,1,1]] => [6,4] => 6
[[2,2,1,1],[1],[1],[1],[1]] => [6,1,1,1,1] => 6
[[2,2,1,1],[2],[1],[1]] => [6,2,1,1] => 6
[[2,2,1,1],[2],[2]] => [6,2,2] => 6
[[2,2,1,1],[1,1],[1],[1]] => [6,2,1,1] => 6
[[2,2,1,1],[1,1],[1,1]] => [6,2,2] => 6
[[2,2,1,1],[2,1],[1]] => [6,3,1] => 6
[[2,2,1,1],[1,1,1],[1]] => [6,3,1] => 6
[[2,2,1,1],[2,2]] => [6,4] => 6
[[2,2,1,1],[2,1,1]] => [6,4] => 6
[[2,2,1,1],[1,1,1,1]] => [6,4] => 6
[[2,1,1,1,1],[1],[1],[1],[1]] => [6,1,1,1,1] => 6
[[2,1,1,1,1],[2],[1],[1]] => [6,2,1,1] => 6
[[2,1,1,1,1],[2],[2]] => [6,2,2] => 6
[[2,1,1,1,1],[1,1],[1],[1]] => [6,2,1,1] => 6
[[2,1,1,1,1],[1,1],[1,1]] => [6,2,2] => 6
[[2,1,1,1,1],[2,1],[1]] => [6,3,1] => 6
[[2,1,1,1,1],[1,1,1],[1]] => [6,3,1] => 6
[[2,1,1,1,1],[2,1,1]] => [6,4] => 6
[[2,1,1,1,1],[1,1,1,1]] => [6,4] => 6
[[1,1,1,1,1,1],[1],[1],[1],[1]] => [6,1,1,1,1] => 6
[[1,1,1,1,1,1],[1,1],[1],[1]] => [6,2,1,1] => 6
[[1,1,1,1,1,1],[1,1],[1,1]] => [6,2,2] => 6
[[1,1,1,1,1,1],[1,1,1],[1]] => [6,3,1] => 6
[[1,1,1,1,1,1],[1,1,1,1]] => [6,4] => 6
[[7],[1],[1],[1]] => [7,1,1,1] => 7
[[7],[2],[1]] => [7,2,1] => 7
[[7],[3]] => [7,3] => 7
[[6,1],[1],[1],[1]] => [7,1,1,1] => 7
[[6,1],[2],[1]] => [7,2,1] => 7
[[6,1],[1,1],[1]] => [7,2,1] => 7
[[6,1],[3]] => [7,3] => 7
[[6,1],[2,1]] => [7,3] => 7
[[5,2],[1],[1],[1]] => [7,1,1,1] => 7
[[5,2],[2],[1]] => [7,2,1] => 7
[[5,2],[1,1],[1]] => [7,2,1] => 7
[[5,2],[3]] => [7,3] => 7
[[5,2],[2,1]] => [7,3] => 7
[[5,1,1],[1],[1],[1]] => [7,1,1,1] => 7
[[5,1,1],[2],[1]] => [7,2,1] => 7
[[5,1,1],[1,1],[1]] => [7,2,1] => 7
[[5,1,1],[3]] => [7,3] => 7
[[5,1,1],[2,1]] => [7,3] => 7
[[5,1,1],[1,1,1]] => [7,3] => 7
[[4,3],[1],[1],[1]] => [7,1,1,1] => 7
[[4,3],[2],[1]] => [7,2,1] => 7
[[4,3],[1,1],[1]] => [7,2,1] => 7
[[4,3],[3]] => [7,3] => 7
[[4,3],[2,1]] => [7,3] => 7
[[4,2,1],[1],[1],[1]] => [7,1,1,1] => 7
[[4,2,1],[2],[1]] => [7,2,1] => 7
[[4,2,1],[1,1],[1]] => [7,2,1] => 7
[[4,2,1],[3]] => [7,3] => 7
[[4,2,1],[2,1]] => [7,3] => 7
[[4,2,1],[1,1,1]] => [7,3] => 7
[[4,1,1,1],[1],[1],[1]] => [7,1,1,1] => 7
[[4,1,1,1],[2],[1]] => [7,2,1] => 7
[[4,1,1,1],[1,1],[1]] => [7,2,1] => 7
[[4,1,1,1],[3]] => [7,3] => 7
[[4,1,1,1],[2,1]] => [7,3] => 7
[[4,1,1,1],[1,1,1]] => [7,3] => 7
[[3,3,1],[1],[1],[1]] => [7,1,1,1] => 7
[[3,3,1],[2],[1]] => [7,2,1] => 7
[[3,3,1],[1,1],[1]] => [7,2,1] => 7
[[3,3,1],[3]] => [7,3] => 7
[[3,3,1],[2,1]] => [7,3] => 7
[[3,3,1],[1,1,1]] => [7,3] => 7
[[3,2,2],[1],[1],[1]] => [7,1,1,1] => 7
[[3,2,2],[2],[1]] => [7,2,1] => 7
[[3,2,2],[1,1],[1]] => [7,2,1] => 7
[[3,2,2],[3]] => [7,3] => 7
[[3,2,2],[2,1]] => [7,3] => 7
[[3,2,2],[1,1,1]] => [7,3] => 7
[[3,2,1,1],[1],[1],[1]] => [7,1,1,1] => 7
[[3,2,1,1],[2],[1]] => [7,2,1] => 7
[[3,2,1,1],[1,1],[1]] => [7,2,1] => 7
[[3,2,1,1],[3]] => [7,3] => 7
[[3,2,1,1],[2,1]] => [7,3] => 7
[[3,2,1,1],[1,1,1]] => [7,3] => 7
[[3,1,1,1,1],[1],[1],[1]] => [7,1,1,1] => 7
[[3,1,1,1,1],[2],[1]] => [7,2,1] => 7
[[3,1,1,1,1],[1,1],[1]] => [7,2,1] => 7
[[3,1,1,1,1],[3]] => [7,3] => 7
[[3,1,1,1,1],[2,1]] => [7,3] => 7
[[3,1,1,1,1],[1,1,1]] => [7,3] => 7
[[2,2,2,1],[1],[1],[1]] => [7,1,1,1] => 7
[[2,2,2,1],[2],[1]] => [7,2,1] => 7
[[2,2,2,1],[1,1],[1]] => [7,2,1] => 7
[[2,2,2,1],[2,1]] => [7,3] => 7
[[2,2,2,1],[1,1,1]] => [7,3] => 7
[[2,2,1,1,1],[1],[1],[1]] => [7,1,1,1] => 7
[[2,2,1,1,1],[2],[1]] => [7,2,1] => 7
[[2,2,1,1,1],[1,1],[1]] => [7,2,1] => 7
[[2,2,1,1,1],[2,1]] => [7,3] => 7
[[2,2,1,1,1],[1,1,1]] => [7,3] => 7
[[2,1,1,1,1,1],[1],[1],[1]] => [7,1,1,1] => 7
[[2,1,1,1,1,1],[2],[1]] => [7,2,1] => 7
[[2,1,1,1,1,1],[1,1],[1]] => [7,2,1] => 7
[[2,1,1,1,1,1],[2,1]] => [7,3] => 7
[[2,1,1,1,1,1],[1,1,1]] => [7,3] => 7
[[1,1,1,1,1,1,1],[1],[1],[1]] => [7,1,1,1] => 7
[[1,1,1,1,1,1,1],[1,1],[1]] => [7,2,1] => 7
[[1,1,1,1,1,1,1],[1,1,1]] => [7,3] => 7
[[8],[1],[1]] => [8,1,1] => 8
[[8],[2]] => [8,2] => 8
[[7,1],[1],[1]] => [8,1,1] => 8
[[7,1],[2]] => [8,2] => 8
[[7,1],[1,1]] => [8,2] => 8
[[6,2],[1],[1]] => [8,1,1] => 8
[[6,2],[2]] => [8,2] => 8
[[6,2],[1,1]] => [8,2] => 8
[[6,1,1],[1],[1]] => [8,1,1] => 8
[[6,1,1],[2]] => [8,2] => 8
[[6,1,1],[1,1]] => [8,2] => 8
[[5,3],[1],[1]] => [8,1,1] => 8
[[5,3],[2]] => [8,2] => 8
[[5,3],[1,1]] => [8,2] => 8
[[5,2,1],[1],[1]] => [8,1,1] => 8
[[5,2,1],[2]] => [8,2] => 8
[[5,2,1],[1,1]] => [8,2] => 8
[[5,1,1,1],[1],[1]] => [8,1,1] => 8
[[5,1,1,1],[2]] => [8,2] => 8
[[5,1,1,1],[1,1]] => [8,2] => 8
[[4,4],[1],[1]] => [8,1,1] => 8
[[4,4],[2]] => [8,2] => 8
[[4,4],[1,1]] => [8,2] => 8
[[4,3,1],[1],[1]] => [8,1,1] => 8
[[4,3,1],[2]] => [8,2] => 8
[[4,3,1],[1,1]] => [8,2] => 8
[[4,2,2],[1],[1]] => [8,1,1] => 8
[[4,2,2],[2]] => [8,2] => 8
[[4,2,2],[1,1]] => [8,2] => 8
[[4,2,1,1],[1],[1]] => [8,1,1] => 8
[[4,2,1,1],[2]] => [8,2] => 8
[[4,2,1,1],[1,1]] => [8,2] => 8
[[4,1,1,1,1],[1],[1]] => [8,1,1] => 8
[[4,1,1,1,1],[2]] => [8,2] => 8
[[4,1,1,1,1],[1,1]] => [8,2] => 8
[[3,3,2],[1],[1]] => [8,1,1] => 8
[[3,3,2],[2]] => [8,2] => 8
[[3,3,2],[1,1]] => [8,2] => 8
[[3,3,1,1],[1],[1]] => [8,1,1] => 8
[[3,3,1,1],[2]] => [8,2] => 8
[[3,3,1,1],[1,1]] => [8,2] => 8
[[3,2,2,1],[1],[1]] => [8,1,1] => 8
[[3,2,2,1],[2]] => [8,2] => 8
[[3,2,2,1],[1,1]] => [8,2] => 8
[[3,2,1,1,1],[1],[1]] => [8,1,1] => 8
[[3,2,1,1,1],[2]] => [8,2] => 8
[[3,2,1,1,1],[1,1]] => [8,2] => 8
[[3,1,1,1,1,1],[1],[1]] => [8,1,1] => 8
[[3,1,1,1,1,1],[2]] => [8,2] => 8
[[3,1,1,1,1,1],[1,1]] => [8,2] => 8
[[2,2,2,2],[1],[1]] => [8,1,1] => 8
[[2,2,2,2],[2]] => [8,2] => 8
[[2,2,2,2],[1,1]] => [8,2] => 8
[[2,2,2,1,1],[1],[1]] => [8,1,1] => 8
[[2,2,2,1,1],[2]] => [8,2] => 8
[[2,2,2,1,1],[1,1]] => [8,2] => 8
[[2,2,1,1,1,1],[1],[1]] => [8,1,1] => 8
[[2,2,1,1,1,1],[2]] => [8,2] => 8
[[2,2,1,1,1,1],[1,1]] => [8,2] => 8
[[2,1,1,1,1,1,1],[1],[1]] => [8,1,1] => 8
[[2,1,1,1,1,1,1],[2]] => [8,2] => 8
[[2,1,1,1,1,1,1],[1,1]] => [8,2] => 8
[[1,1,1,1,1,1,1,1],[1],[1]] => [8,1,1] => 8
[[1,1,1,1,1,1,1,1],[1,1]] => [8,2] => 8
[[9],[1]] => [9,1] => 9
[[8,1],[1]] => [9,1] => 9
[[7,2],[1]] => [9,1] => 9
[[7,1,1],[1]] => [9,1] => 9
[[6,3],[1]] => [9,1] => 9
[[6,2,1],[1]] => [9,1] => 9
[[6,1,1,1],[1]] => [9,1] => 9
[[5,4],[1]] => [9,1] => 9
[[5,3,1],[1]] => [9,1] => 9
[[5,2,2],[1]] => [9,1] => 9
[[5,2,1,1],[1]] => [9,1] => 9
[[5,1,1,1,1],[1]] => [9,1] => 9
[[4,4,1],[1]] => [9,1] => 9
[[4,3,2],[1]] => [9,1] => 9
[[4,3,1,1],[1]] => [9,1] => 9
[[4,2,2,1],[1]] => [9,1] => 9
[[4,2,1,1,1],[1]] => [9,1] => 9
[[4,1,1,1,1,1],[1]] => [9,1] => 9
[[3,3,3],[1]] => [9,1] => 9
[[3,3,2,1],[1]] => [9,1] => 9
[[3,3,1,1,1],[1]] => [9,1] => 9
[[3,2,2,2],[1]] => [9,1] => 9
[[3,2,2,1,1],[1]] => [9,1] => 9
[[3,2,1,1,1,1],[1]] => [9,1] => 9
[[3,1,1,1,1,1,1],[1]] => [9,1] => 9
[[2,2,2,2,1],[1]] => [9,1] => 9
[[2,2,2,1,1,1],[1]] => [9,1] => 9
[[2,2,1,1,1,1,1],[1]] => [9,1] => 9
[[2,1,1,1,1,1,1,1],[1]] => [9,1] => 9
[[1,1,1,1,1,1,1,1,1],[1]] => [9,1] => 9
[[10]] => [10] => 10
[[9,1]] => [10] => 10
[[8,2]] => [10] => 10
[[8,1,1]] => [10] => 10
[[7,3]] => [10] => 10
[[7,2,1]] => [10] => 10
[[7,1,1,1]] => [10] => 10
[[6,4]] => [10] => 10
[[6,3,1]] => [10] => 10
[[6,2,2]] => [10] => 10
[[6,2,1,1]] => [10] => 10
[[6,1,1,1,1]] => [10] => 10
[[5,5]] => [10] => 10
[[5,4,1]] => [10] => 10
[[5,3,2]] => [10] => 10
[[5,3,1,1]] => [10] => 10
[[5,2,2,1]] => [10] => 10
[[5,2,1,1,1]] => [10] => 10
[[5,1,1,1,1,1]] => [10] => 10
[[4,4,2]] => [10] => 10
[[4,4,1,1]] => [10] => 10
[[4,3,3]] => [10] => 10
[[4,3,2,1]] => [10] => 10
[[4,3,1,1,1]] => [10] => 10
[[4,2,2,2]] => [10] => 10
[[4,2,2,1,1]] => [10] => 10
[[4,2,1,1,1,1]] => [10] => 10
[[4,1,1,1,1,1,1]] => [10] => 10
[[3,3,3,1]] => [10] => 10
[[3,3,2,2]] => [10] => 10
[[3,3,2,1,1]] => [10] => 10
[[3,3,1,1,1,1]] => [10] => 10
[[3,2,2,2,1]] => [10] => 10
[[3,2,2,1,1,1]] => [10] => 10
[[3,2,1,1,1,1,1]] => [10] => 10
[[3,1,1,1,1,1,1,1]] => [10] => 10
[[2,2,2,2,2]] => [10] => 10
[[2,2,2,2,1,1]] => [10] => 10
[[2,2,2,1,1,1,1]] => [10] => 10
[[2,2,1,1,1,1,1,1]] => [10] => 10
[[2,1,1,1,1,1,1,1,1]] => [10] => 10
[[1,1,1,1,1,1,1,1,1,1]] => [10] => 10
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Generating function
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Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
1,2 1,2,3 1,4,3,5 1,4,7,5,7 1,6,10,13,7,11 1,6,14,20,19,11,15 1,8,18,33,32,31,15,22 1,8,25,43,56,54,43,22,30 1,10,29,66,81,99,78,64,30,42
$F_{1} = q$
$F_{2} = q + 2\ q^{2}$
$F_{3} = q + 2\ q^{2} + 3\ q^{3}$
$F_{4} = q + 4\ q^{2} + 3\ q^{3} + 5\ q^{4}$
$F_{5} = q + 4\ q^{2} + 7\ q^{3} + 5\ q^{4} + 7\ q^{5}$
$F_{6} = q + 6\ q^{2} + 10\ q^{3} + 13\ q^{4} + 7\ q^{5} + 11\ q^{6}$
$F_{7} = q + 6\ q^{2} + 14\ q^{3} + 20\ q^{4} + 19\ q^{5} + 11\ q^{6} + 15\ q^{7}$
$F_{8} = q + 8\ q^{2} + 18\ q^{3} + 33\ q^{4} + 32\ q^{5} + 31\ q^{6} + 15\ q^{7} + 22\ q^{8}$
$F_{9} = q + 8\ q^{2} + 25\ q^{3} + 43\ q^{4} + 56\ q^{5} + 54\ q^{6} + 43\ q^{7} + 22\ q^{8} + 30\ q^{9}$
$F_{10} = q + 10\ q^{2} + 29\ q^{3} + 66\ q^{4} + 81\ q^{5} + 99\ q^{6} + 78\ q^{7} + 64\ q^{8} + 30\ q^{9} + 42\ q^{10}$
Description
The largest part of an integer partition.
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Description
The underlying integer partition of a plane partition.
This is the partition whose parts are the sums of the individual rows of the plane partition.
This is the partition whose parts are the sums of the individual rows of the plane partition.
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