Identifier
-
Mp00311:
Plane partitions
—to partition⟶
Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤ
Values
[[1]] => [1] => [1] => 1
[[1],[1]] => [1,1] => [2] => 2
[[2]] => [2] => [1,1] => 1
[[1,1]] => [2] => [1,1] => 1
[[1],[1],[1]] => [1,1,1] => [2,1] => 3
[[2],[1]] => [2,1] => [3] => 2
[[1,1],[1]] => [2,1] => [3] => 2
[[3]] => [3] => [1,1,1] => 1
[[2,1]] => [3] => [1,1,1] => 1
[[1,1,1]] => [3] => [1,1,1] => 1
[[1],[1],[1],[1]] => [1,1,1,1] => [3,1] => 4
[[2],[1],[1]] => [2,1,1] => [2,2] => 3
[[2],[2]] => [2,2] => [4] => 2
[[1,1],[1],[1]] => [2,1,1] => [2,2] => 3
[[1,1],[1,1]] => [2,2] => [4] => 2
[[3],[1]] => [3,1] => [2,1,1] => 2
[[2,1],[1]] => [3,1] => [2,1,1] => 2
[[1,1,1],[1]] => [3,1] => [2,1,1] => 2
[[4]] => [4] => [1,1,1,1] => 1
[[3,1]] => [4] => [1,1,1,1] => 1
[[2,2]] => [4] => [1,1,1,1] => 1
[[2,1,1]] => [4] => [1,1,1,1] => 1
[[1,1,1,1]] => [4] => [1,1,1,1] => 1
[[1],[1],[1],[1],[1]] => [1,1,1,1,1] => [3,2] => 5
[[2],[1],[1],[1]] => [2,1,1,1] => [3,1,1] => 4
[[2],[2],[1]] => [2,2,1] => [2,2,1] => 3
[[1,1],[1],[1],[1]] => [2,1,1,1] => [3,1,1] => 4
[[1,1],[1,1],[1]] => [2,2,1] => [2,2,1] => 3
[[3],[1],[1]] => [3,1,1] => [4,1] => 3
[[3],[2]] => [3,2] => [5] => 2
[[2,1],[1],[1]] => [3,1,1] => [4,1] => 3
[[2,1],[2]] => [3,2] => [5] => 2
[[2,1],[1,1]] => [3,2] => [5] => 2
[[1,1,1],[1],[1]] => [3,1,1] => [4,1] => 3
[[1,1,1],[1,1]] => [3,2] => [5] => 2
[[4],[1]] => [4,1] => [2,1,1,1] => 2
[[3,1],[1]] => [4,1] => [2,1,1,1] => 2
[[2,2],[1]] => [4,1] => [2,1,1,1] => 2
[[2,1,1],[1]] => [4,1] => [2,1,1,1] => 2
[[1,1,1,1],[1]] => [4,1] => [2,1,1,1] => 2
[[5]] => [5] => [1,1,1,1,1] => 1
[[4,1]] => [5] => [1,1,1,1,1] => 1
[[3,2]] => [5] => [1,1,1,1,1] => 1
[[3,1,1]] => [5] => [1,1,1,1,1] => 1
[[2,2,1]] => [5] => [1,1,1,1,1] => 1
[[2,1,1,1]] => [5] => [1,1,1,1,1] => 1
[[1,1,1,1,1]] => [5] => [1,1,1,1,1] => 1
[[1],[1],[1],[1],[1],[1]] => [1,1,1,1,1,1] => [3,2,1] => 6
[[2],[1],[1],[1],[1]] => [2,1,1,1,1] => [4,2] => 5
[[2],[2],[1],[1]] => [2,2,1,1] => [4,1,1] => 4
[[2],[2],[2]] => [2,2,2] => [2,2,2] => 3
[[1,1],[1],[1],[1],[1]] => [2,1,1,1,1] => [4,2] => 5
[[1,1],[1,1],[1],[1]] => [2,2,1,1] => [4,1,1] => 4
[[1,1],[1,1],[1,1]] => [2,2,2] => [2,2,2] => 3
[[3],[1],[1],[1]] => [3,1,1,1] => [3,3] => 4
[[3],[2],[1]] => [3,2,1] => [5,1] => 3
[[3],[3]] => [3,3] => [6] => 2
[[2,1],[1],[1],[1]] => [3,1,1,1] => [3,3] => 4
[[2,1],[2],[1]] => [3,2,1] => [5,1] => 3
[[2,1],[1,1],[1]] => [3,2,1] => [5,1] => 3
[[2,1],[2,1]] => [3,3] => [6] => 2
[[1,1,1],[1],[1],[1]] => [3,1,1,1] => [3,3] => 4
[[1,1,1],[1,1],[1]] => [3,2,1] => [5,1] => 3
[[1,1,1],[1,1,1]] => [3,3] => [6] => 2
[[4],[1],[1]] => [4,1,1] => [3,1,1,1] => 3
[[4],[2]] => [4,2] => [2,2,1,1] => 2
[[3,1],[1],[1]] => [4,1,1] => [3,1,1,1] => 3
[[3,1],[2]] => [4,2] => [2,2,1,1] => 2
[[3,1],[1,1]] => [4,2] => [2,2,1,1] => 2
[[2,2],[1],[1]] => [4,1,1] => [3,1,1,1] => 3
[[2,2],[2]] => [4,2] => [2,2,1,1] => 2
[[2,2],[1,1]] => [4,2] => [2,2,1,1] => 2
[[2,1,1],[1],[1]] => [4,1,1] => [3,1,1,1] => 3
[[2,1,1],[2]] => [4,2] => [2,2,1,1] => 2
[[2,1,1],[1,1]] => [4,2] => [2,2,1,1] => 2
[[1,1,1,1],[1],[1]] => [4,1,1] => [3,1,1,1] => 3
[[1,1,1,1],[1,1]] => [4,2] => [2,2,1,1] => 2
[[5],[1]] => [5,1] => [2,1,1,1,1] => 2
[[4,1],[1]] => [5,1] => [2,1,1,1,1] => 2
[[3,2],[1]] => [5,1] => [2,1,1,1,1] => 2
[[3,1,1],[1]] => [5,1] => [2,1,1,1,1] => 2
[[2,2,1],[1]] => [5,1] => [2,1,1,1,1] => 2
[[2,1,1,1],[1]] => [5,1] => [2,1,1,1,1] => 2
[[1,1,1,1,1],[1]] => [5,1] => [2,1,1,1,1] => 2
[[6]] => [6] => [1,1,1,1,1,1] => 1
[[5,1]] => [6] => [1,1,1,1,1,1] => 1
[[4,2]] => [6] => [1,1,1,1,1,1] => 1
[[4,1,1]] => [6] => [1,1,1,1,1,1] => 1
[[3,3]] => [6] => [1,1,1,1,1,1] => 1
[[3,2,1]] => [6] => [1,1,1,1,1,1] => 1
[[3,1,1,1]] => [6] => [1,1,1,1,1,1] => 1
[[2,2,2]] => [6] => [1,1,1,1,1,1] => 1
[[2,2,1,1]] => [6] => [1,1,1,1,1,1] => 1
[[2,1,1,1,1]] => [6] => [1,1,1,1,1,1] => 1
[[1,1,1,1,1,1]] => [6] => [1,1,1,1,1,1] => 1
[[1],[1],[1],[1],[1],[1],[1]] => [1,1,1,1,1,1,1] => [4,2,1] => 7
[[2],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1] => [3,3,1] => 6
[[2],[2],[1],[1],[1]] => [2,2,1,1,1] => [4,3] => 5
[[2],[2],[2],[1]] => [2,2,2,1] => [4,1,1,1] => 4
[[1,1],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1] => [3,3,1] => 6
[[1,1],[1,1],[1],[1],[1]] => [2,2,1,1,1] => [4,3] => 5
>>> Load all 1123 entries. <<<[[1,1],[1,1],[1,1],[1]] => [2,2,2,1] => [4,1,1,1] => 4
[[3],[1],[1],[1],[1]] => [3,1,1,1,1] => [3,2,2] => 5
[[3],[2],[1],[1]] => [3,2,1,1] => [5,2] => 4
[[3],[2],[2]] => [3,2,2] => [2,2,2,1] => 3
[[3],[3],[1]] => [3,3,1] => [6,1] => 3
[[2,1],[1],[1],[1],[1]] => [3,1,1,1,1] => [3,2,2] => 5
[[2,1],[2],[1],[1]] => [3,2,1,1] => [5,2] => 4
[[2,1],[2],[2]] => [3,2,2] => [2,2,2,1] => 3
[[2,1],[1,1],[1],[1]] => [3,2,1,1] => [5,2] => 4
[[2,1],[1,1],[1,1]] => [3,2,2] => [2,2,2,1] => 3
[[2,1],[2,1],[1]] => [3,3,1] => [6,1] => 3
[[1,1,1],[1],[1],[1],[1]] => [3,1,1,1,1] => [3,2,2] => 5
[[1,1,1],[1,1],[1],[1]] => [3,2,1,1] => [5,2] => 4
[[1,1,1],[1,1],[1,1]] => [3,2,2] => [2,2,2,1] => 3
[[1,1,1],[1,1,1],[1]] => [3,3,1] => [6,1] => 3
[[4],[1],[1],[1]] => [4,1,1,1] => [3,2,1,1] => 4
[[4],[2],[1]] => [4,2,1] => [5,1,1] => 3
[[4],[3]] => [4,3] => [7] => 2
[[3,1],[1],[1],[1]] => [4,1,1,1] => [3,2,1,1] => 4
[[3,1],[2],[1]] => [4,2,1] => [5,1,1] => 3
[[3,1],[1,1],[1]] => [4,2,1] => [5,1,1] => 3
[[3,1],[3]] => [4,3] => [7] => 2
[[3,1],[2,1]] => [4,3] => [7] => 2
[[2,2],[1],[1],[1]] => [4,1,1,1] => [3,2,1,1] => 4
[[2,2],[2],[1]] => [4,2,1] => [5,1,1] => 3
[[2,2],[1,1],[1]] => [4,2,1] => [5,1,1] => 3
[[2,2],[2,1]] => [4,3] => [7] => 2
[[2,1,1],[1],[1],[1]] => [4,1,1,1] => [3,2,1,1] => 4
[[2,1,1],[2],[1]] => [4,2,1] => [5,1,1] => 3
[[2,1,1],[1,1],[1]] => [4,2,1] => [5,1,1] => 3
[[2,1,1],[2,1]] => [4,3] => [7] => 2
[[2,1,1],[1,1,1]] => [4,3] => [7] => 2
[[1,1,1,1],[1],[1],[1]] => [4,1,1,1] => [3,2,1,1] => 4
[[1,1,1,1],[1,1],[1]] => [4,2,1] => [5,1,1] => 3
[[1,1,1,1],[1,1,1]] => [4,3] => [7] => 2
[[5],[1],[1]] => [5,1,1] => [3,1,1,1,1] => 3
[[5],[2]] => [5,2] => [2,2,1,1,1] => 2
[[4,1],[1],[1]] => [5,1,1] => [3,1,1,1,1] => 3
[[4,1],[2]] => [5,2] => [2,2,1,1,1] => 2
[[4,1],[1,1]] => [5,2] => [2,2,1,1,1] => 2
[[3,2],[1],[1]] => [5,1,1] => [3,1,1,1,1] => 3
[[3,2],[2]] => [5,2] => [2,2,1,1,1] => 2
[[3,2],[1,1]] => [5,2] => [2,2,1,1,1] => 2
[[3,1,1],[1],[1]] => [5,1,1] => [3,1,1,1,1] => 3
[[3,1,1],[2]] => [5,2] => [2,2,1,1,1] => 2
[[3,1,1],[1,1]] => [5,2] => [2,2,1,1,1] => 2
[[2,2,1],[1],[1]] => [5,1,1] => [3,1,1,1,1] => 3
[[2,2,1],[2]] => [5,2] => [2,2,1,1,1] => 2
[[2,2,1],[1,1]] => [5,2] => [2,2,1,1,1] => 2
[[2,1,1,1],[1],[1]] => [5,1,1] => [3,1,1,1,1] => 3
[[2,1,1,1],[2]] => [5,2] => [2,2,1,1,1] => 2
[[2,1,1,1],[1,1]] => [5,2] => [2,2,1,1,1] => 2
[[1,1,1,1,1],[1],[1]] => [5,1,1] => [3,1,1,1,1] => 3
[[1,1,1,1,1],[1,1]] => [5,2] => [2,2,1,1,1] => 2
[[6],[1]] => [6,1] => [2,1,1,1,1,1] => 2
[[5,1],[1]] => [6,1] => [2,1,1,1,1,1] => 2
[[4,2],[1]] => [6,1] => [2,1,1,1,1,1] => 2
[[4,1,1],[1]] => [6,1] => [2,1,1,1,1,1] => 2
[[3,3],[1]] => [6,1] => [2,1,1,1,1,1] => 2
[[3,2,1],[1]] => [6,1] => [2,1,1,1,1,1] => 2
[[3,1,1,1],[1]] => [6,1] => [2,1,1,1,1,1] => 2
[[2,2,2],[1]] => [6,1] => [2,1,1,1,1,1] => 2
[[2,2,1,1],[1]] => [6,1] => [2,1,1,1,1,1] => 2
[[2,1,1,1,1],[1]] => [6,1] => [2,1,1,1,1,1] => 2
[[1,1,1,1,1,1],[1]] => [6,1] => [2,1,1,1,1,1] => 2
[[7]] => [7] => [1,1,1,1,1,1,1] => 1
[[6,1]] => [7] => [1,1,1,1,1,1,1] => 1
[[5,2]] => [7] => [1,1,1,1,1,1,1] => 1
[[5,1,1]] => [7] => [1,1,1,1,1,1,1] => 1
[[4,3]] => [7] => [1,1,1,1,1,1,1] => 1
[[4,2,1]] => [7] => [1,1,1,1,1,1,1] => 1
[[4,1,1,1]] => [7] => [1,1,1,1,1,1,1] => 1
[[3,3,1]] => [7] => [1,1,1,1,1,1,1] => 1
[[3,2,2]] => [7] => [1,1,1,1,1,1,1] => 1
[[3,2,1,1]] => [7] => [1,1,1,1,1,1,1] => 1
[[3,1,1,1,1]] => [7] => [1,1,1,1,1,1,1] => 1
[[2,2,2,1]] => [7] => [1,1,1,1,1,1,1] => 1
[[2,2,1,1,1]] => [7] => [1,1,1,1,1,1,1] => 1
[[2,1,1,1,1,1]] => [7] => [1,1,1,1,1,1,1] => 1
[[1,1,1,1,1,1,1]] => [7] => [1,1,1,1,1,1,1] => 1
[[1],[1],[1],[1],[1],[1],[1],[1]] => [1,1,1,1,1,1,1,1] => [4,3,1] => 8
[[2],[1],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1,1] => [4,2,2] => 7
[[2],[2],[1],[1],[1],[1]] => [2,2,1,1,1,1] => [3,3,2] => 6
[[2],[2],[2],[1],[1]] => [2,2,2,1,1] => [5,3] => 5
[[2],[2],[2],[2]] => [2,2,2,2] => [5,1,1,1] => 4
[[1,1],[1],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1,1] => [4,2,2] => 7
[[1,1],[1,1],[1],[1],[1],[1]] => [2,2,1,1,1,1] => [3,3,2] => 6
[[1,1],[1,1],[1,1],[1],[1]] => [2,2,2,1,1] => [5,3] => 5
[[1,1],[1,1],[1,1],[1,1]] => [2,2,2,2] => [5,1,1,1] => 4
[[3],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1] => [4,2,1,1] => 6
[[3],[2],[1],[1],[1]] => [3,2,1,1,1] => [3,3,1,1] => 5
[[3],[2],[2],[1]] => [3,2,2,1] => [3,2,2,1] => 4
[[3],[3],[1],[1]] => [3,3,1,1] => [6,2] => 4
[[3],[3],[2]] => [3,3,2] => [2,2,2,2] => 3
[[2,1],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1] => [4,2,1,1] => 6
[[2,1],[2],[1],[1],[1]] => [3,2,1,1,1] => [3,3,1,1] => 5
[[2,1],[2],[2],[1]] => [3,2,2,1] => [3,2,2,1] => 4
[[2,1],[1,1],[1],[1],[1]] => [3,2,1,1,1] => [3,3,1,1] => 5
[[2,1],[1,1],[1,1],[1]] => [3,2,2,1] => [3,2,2,1] => 4
[[2,1],[2,1],[1],[1]] => [3,3,1,1] => [6,2] => 4
[[2,1],[2,1],[2]] => [3,3,2] => [2,2,2,2] => 3
[[2,1],[2,1],[1,1]] => [3,3,2] => [2,2,2,2] => 3
[[1,1,1],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1] => [4,2,1,1] => 6
[[1,1,1],[1,1],[1],[1],[1]] => [3,2,1,1,1] => [3,3,1,1] => 5
[[1,1,1],[1,1],[1,1],[1]] => [3,2,2,1] => [3,2,2,1] => 4
[[1,1,1],[1,1,1],[1],[1]] => [3,3,1,1] => [6,2] => 4
[[1,1,1],[1,1,1],[1,1]] => [3,3,2] => [2,2,2,2] => 3
[[4],[1],[1],[1],[1]] => [4,1,1,1,1] => [5,2,1] => 5
[[4],[2],[1],[1]] => [4,2,1,1] => [4,4] => 4
[[4],[2],[2]] => [4,2,2] => [6,1,1] => 3
[[4],[3],[1]] => [4,3,1] => [7,1] => 3
[[4],[4]] => [4,4] => [8] => 2
[[3,1],[1],[1],[1],[1]] => [4,1,1,1,1] => [5,2,1] => 5
[[3,1],[2],[1],[1]] => [4,2,1,1] => [4,4] => 4
[[3,1],[2],[2]] => [4,2,2] => [6,1,1] => 3
[[3,1],[1,1],[1],[1]] => [4,2,1,1] => [4,4] => 4
[[3,1],[1,1],[1,1]] => [4,2,2] => [6,1,1] => 3
[[3,1],[3],[1]] => [4,3,1] => [7,1] => 3
[[3,1],[2,1],[1]] => [4,3,1] => [7,1] => 3
[[3,1],[3,1]] => [4,4] => [8] => 2
[[2,2],[1],[1],[1],[1]] => [4,1,1,1,1] => [5,2,1] => 5
[[2,2],[2],[1],[1]] => [4,2,1,1] => [4,4] => 4
[[2,2],[2],[2]] => [4,2,2] => [6,1,1] => 3
[[2,2],[1,1],[1],[1]] => [4,2,1,1] => [4,4] => 4
[[2,2],[1,1],[1,1]] => [4,2,2] => [6,1,1] => 3
[[2,2],[2,1],[1]] => [4,3,1] => [7,1] => 3
[[2,2],[2,2]] => [4,4] => [8] => 2
[[2,1,1],[1],[1],[1],[1]] => [4,1,1,1,1] => [5,2,1] => 5
[[2,1,1],[2],[1],[1]] => [4,2,1,1] => [4,4] => 4
[[2,1,1],[2],[2]] => [4,2,2] => [6,1,1] => 3
[[2,1,1],[1,1],[1],[1]] => [4,2,1,1] => [4,4] => 4
[[2,1,1],[1,1],[1,1]] => [4,2,2] => [6,1,1] => 3
[[2,1,1],[2,1],[1]] => [4,3,1] => [7,1] => 3
[[2,1,1],[1,1,1],[1]] => [4,3,1] => [7,1] => 3
[[2,1,1],[2,1,1]] => [4,4] => [8] => 2
[[1,1,1,1],[1],[1],[1],[1]] => [4,1,1,1,1] => [5,2,1] => 5
[[1,1,1,1],[1,1],[1],[1]] => [4,2,1,1] => [4,4] => 4
[[1,1,1,1],[1,1],[1,1]] => [4,2,2] => [6,1,1] => 3
[[1,1,1,1],[1,1,1],[1]] => [4,3,1] => [7,1] => 3
[[1,1,1,1],[1,1,1,1]] => [4,4] => [8] => 2
[[5],[1],[1],[1]] => [5,1,1,1] => [3,2,1,1,1] => 4
[[5],[2],[1]] => [5,2,1] => [4,1,1,1,1] => 3
[[5],[3]] => [5,3] => [2,2,2,1,1] => 2
[[4,1],[1],[1],[1]] => [5,1,1,1] => [3,2,1,1,1] => 4
[[4,1],[2],[1]] => [5,2,1] => [4,1,1,1,1] => 3
[[4,1],[1,1],[1]] => [5,2,1] => [4,1,1,1,1] => 3
[[4,1],[3]] => [5,3] => [2,2,2,1,1] => 2
[[4,1],[2,1]] => [5,3] => [2,2,2,1,1] => 2
[[3,2],[1],[1],[1]] => [5,1,1,1] => [3,2,1,1,1] => 4
[[3,2],[2],[1]] => [5,2,1] => [4,1,1,1,1] => 3
[[3,2],[1,1],[1]] => [5,2,1] => [4,1,1,1,1] => 3
[[3,2],[3]] => [5,3] => [2,2,2,1,1] => 2
[[3,2],[2,1]] => [5,3] => [2,2,2,1,1] => 2
[[3,1,1],[1],[1],[1]] => [5,1,1,1] => [3,2,1,1,1] => 4
[[3,1,1],[2],[1]] => [5,2,1] => [4,1,1,1,1] => 3
[[3,1,1],[1,1],[1]] => [5,2,1] => [4,1,1,1,1] => 3
[[3,1,1],[3]] => [5,3] => [2,2,2,1,1] => 2
[[3,1,1],[2,1]] => [5,3] => [2,2,2,1,1] => 2
[[3,1,1],[1,1,1]] => [5,3] => [2,2,2,1,1] => 2
[[2,2,1],[1],[1],[1]] => [5,1,1,1] => [3,2,1,1,1] => 4
[[2,2,1],[2],[1]] => [5,2,1] => [4,1,1,1,1] => 3
[[2,2,1],[1,1],[1]] => [5,2,1] => [4,1,1,1,1] => 3
[[2,2,1],[2,1]] => [5,3] => [2,2,2,1,1] => 2
[[2,2,1],[1,1,1]] => [5,3] => [2,2,2,1,1] => 2
[[2,1,1,1],[1],[1],[1]] => [5,1,1,1] => [3,2,1,1,1] => 4
[[2,1,1,1],[2],[1]] => [5,2,1] => [4,1,1,1,1] => 3
[[2,1,1,1],[1,1],[1]] => [5,2,1] => [4,1,1,1,1] => 3
[[2,1,1,1],[2,1]] => [5,3] => [2,2,2,1,1] => 2
[[2,1,1,1],[1,1,1]] => [5,3] => [2,2,2,1,1] => 2
[[1,1,1,1,1],[1],[1],[1]] => [5,1,1,1] => [3,2,1,1,1] => 4
[[1,1,1,1,1],[1,1],[1]] => [5,2,1] => [4,1,1,1,1] => 3
[[1,1,1,1,1],[1,1,1]] => [5,3] => [2,2,2,1,1] => 2
[[6],[1],[1]] => [6,1,1] => [3,1,1,1,1,1] => 3
[[6],[2]] => [6,2] => [2,2,1,1,1,1] => 2
[[5,1],[1],[1]] => [6,1,1] => [3,1,1,1,1,1] => 3
[[5,1],[2]] => [6,2] => [2,2,1,1,1,1] => 2
[[5,1],[1,1]] => [6,2] => [2,2,1,1,1,1] => 2
[[4,2],[1],[1]] => [6,1,1] => [3,1,1,1,1,1] => 3
[[4,2],[2]] => [6,2] => [2,2,1,1,1,1] => 2
[[4,2],[1,1]] => [6,2] => [2,2,1,1,1,1] => 2
[[4,1,1],[1],[1]] => [6,1,1] => [3,1,1,1,1,1] => 3
[[4,1,1],[2]] => [6,2] => [2,2,1,1,1,1] => 2
[[4,1,1],[1,1]] => [6,2] => [2,2,1,1,1,1] => 2
[[3,3],[1],[1]] => [6,1,1] => [3,1,1,1,1,1] => 3
[[3,3],[2]] => [6,2] => [2,2,1,1,1,1] => 2
[[3,3],[1,1]] => [6,2] => [2,2,1,1,1,1] => 2
[[3,2,1],[1],[1]] => [6,1,1] => [3,1,1,1,1,1] => 3
[[3,2,1],[2]] => [6,2] => [2,2,1,1,1,1] => 2
[[3,2,1],[1,1]] => [6,2] => [2,2,1,1,1,1] => 2
[[3,1,1,1],[1],[1]] => [6,1,1] => [3,1,1,1,1,1] => 3
[[3,1,1,1],[2]] => [6,2] => [2,2,1,1,1,1] => 2
[[3,1,1,1],[1,1]] => [6,2] => [2,2,1,1,1,1] => 2
[[2,2,2],[1],[1]] => [6,1,1] => [3,1,1,1,1,1] => 3
[[2,2,2],[2]] => [6,2] => [2,2,1,1,1,1] => 2
[[2,2,2],[1,1]] => [6,2] => [2,2,1,1,1,1] => 2
[[2,2,1,1],[1],[1]] => [6,1,1] => [3,1,1,1,1,1] => 3
[[2,2,1,1],[2]] => [6,2] => [2,2,1,1,1,1] => 2
[[2,2,1,1],[1,1]] => [6,2] => [2,2,1,1,1,1] => 2
[[2,1,1,1,1],[1],[1]] => [6,1,1] => [3,1,1,1,1,1] => 3
[[2,1,1,1,1],[2]] => [6,2] => [2,2,1,1,1,1] => 2
[[2,1,1,1,1],[1,1]] => [6,2] => [2,2,1,1,1,1] => 2
[[1,1,1,1,1,1],[1],[1]] => [6,1,1] => [3,1,1,1,1,1] => 3
[[1,1,1,1,1,1],[1,1]] => [6,2] => [2,2,1,1,1,1] => 2
[[7],[1]] => [7,1] => [2,1,1,1,1,1,1] => 2
[[6,1],[1]] => [7,1] => [2,1,1,1,1,1,1] => 2
[[5,2],[1]] => [7,1] => [2,1,1,1,1,1,1] => 2
[[5,1,1],[1]] => [7,1] => [2,1,1,1,1,1,1] => 2
[[4,3],[1]] => [7,1] => [2,1,1,1,1,1,1] => 2
[[4,2,1],[1]] => [7,1] => [2,1,1,1,1,1,1] => 2
[[4,1,1,1],[1]] => [7,1] => [2,1,1,1,1,1,1] => 2
[[3,3,1],[1]] => [7,1] => [2,1,1,1,1,1,1] => 2
[[3,2,2],[1]] => [7,1] => [2,1,1,1,1,1,1] => 2
[[3,2,1,1],[1]] => [7,1] => [2,1,1,1,1,1,1] => 2
[[3,1,1,1,1],[1]] => [7,1] => [2,1,1,1,1,1,1] => 2
[[2,2,2,1],[1]] => [7,1] => [2,1,1,1,1,1,1] => 2
[[2,2,1,1,1],[1]] => [7,1] => [2,1,1,1,1,1,1] => 2
[[2,1,1,1,1,1],[1]] => [7,1] => [2,1,1,1,1,1,1] => 2
[[1,1,1,1,1,1,1],[1]] => [7,1] => [2,1,1,1,1,1,1] => 2
[[8]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[7,1]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[6,2]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[6,1,1]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[5,3]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[5,2,1]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[5,1,1,1]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[4,4]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[4,3,1]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[4,2,2]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[4,2,1,1]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[4,1,1,1,1]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[3,3,2]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[3,3,1,1]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[3,2,2,1]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[3,2,1,1,1]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[3,1,1,1,1,1]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[2,2,2,2]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[2,2,2,1,1]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[2,2,1,1,1,1]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[2,1,1,1,1,1,1]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[1,1,1,1,1,1,1,1]] => [8] => [1,1,1,1,1,1,1,1] => 1
[[1],[1],[1],[1],[1],[1],[1],[1],[1]] => [1,1,1,1,1,1,1,1,1] => [4,3,2] => 9
[[2],[1],[1],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1,1,1] => [4,3,1,1] => 8
[[2],[2],[1],[1],[1],[1],[1]] => [2,2,1,1,1,1,1] => [4,2,2,1] => 7
[[2],[2],[2],[1],[1],[1]] => [2,2,2,1,1,1] => [3,3,2,1] => 6
[[2],[2],[2],[2],[1]] => [2,2,2,2,1] => [5,4] => 5
[[1,1],[1],[1],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1,1,1] => [4,3,1,1] => 8
[[1,1],[1,1],[1],[1],[1],[1],[1]] => [2,2,1,1,1,1,1] => [4,2,2,1] => 7
[[1,1],[1,1],[1,1],[1],[1],[1]] => [2,2,2,1,1,1] => [3,3,2,1] => 6
[[1,1],[1,1],[1,1],[1,1],[1]] => [2,2,2,2,1] => [5,4] => 5
[[3],[1],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1,1] => [5,3,1] => 7
[[3],[2],[1],[1],[1],[1]] => [3,2,1,1,1,1] => [5,2,2] => 6
[[3],[2],[2],[1],[1]] => [3,2,2,1,1] => [3,3,3] => 5
[[3],[2],[2],[2]] => [3,2,2,2] => [5,1,1,1,1] => 4
[[3],[3],[1],[1],[1]] => [3,3,1,1,1] => [5,2,1,1] => 5
[[3],[3],[2],[1]] => [3,3,2,1] => [3,2,2,2] => 4
[[3],[3],[3]] => [3,3,3] => [2,2,2,2,1] => 3
[[2,1],[1],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1,1] => [5,3,1] => 7
[[2,1],[2],[1],[1],[1],[1]] => [3,2,1,1,1,1] => [5,2,2] => 6
[[2,1],[2],[2],[1],[1]] => [3,2,2,1,1] => [3,3,3] => 5
[[2,1],[2],[2],[2]] => [3,2,2,2] => [5,1,1,1,1] => 4
[[2,1],[1,1],[1],[1],[1],[1]] => [3,2,1,1,1,1] => [5,2,2] => 6
[[2,1],[1,1],[1,1],[1],[1]] => [3,2,2,1,1] => [3,3,3] => 5
[[2,1],[1,1],[1,1],[1,1]] => [3,2,2,2] => [5,1,1,1,1] => 4
[[2,1],[2,1],[1],[1],[1]] => [3,3,1,1,1] => [5,2,1,1] => 5
[[2,1],[2,1],[2],[1]] => [3,3,2,1] => [3,2,2,2] => 4
[[2,1],[2,1],[1,1],[1]] => [3,3,2,1] => [3,2,2,2] => 4
[[2,1],[2,1],[2,1]] => [3,3,3] => [2,2,2,2,1] => 3
[[1,1,1],[1],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1,1] => [5,3,1] => 7
[[1,1,1],[1,1],[1],[1],[1],[1]] => [3,2,1,1,1,1] => [5,2,2] => 6
[[1,1,1],[1,1],[1,1],[1],[1]] => [3,2,2,1,1] => [3,3,3] => 5
[[1,1,1],[1,1],[1,1],[1,1]] => [3,2,2,2] => [5,1,1,1,1] => 4
[[1,1,1],[1,1,1],[1],[1],[1]] => [3,3,1,1,1] => [5,2,1,1] => 5
[[1,1,1],[1,1,1],[1,1],[1]] => [3,3,2,1] => [3,2,2,2] => 4
[[1,1,1],[1,1,1],[1,1,1]] => [3,3,3] => [2,2,2,2,1] => 3
[[4],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1] => [4,4,1] => 6
[[4],[2],[1],[1],[1]] => [4,2,1,1,1] => [6,2,1] => 5
[[4],[2],[2],[1]] => [4,2,2,1] => [6,3] => 4
[[4],[3],[1],[1]] => [4,3,1,1] => [7,2] => 4
[[4],[3],[2]] => [4,3,2] => [7,1,1] => 3
[[4],[4],[1]] => [4,4,1] => [8,1] => 3
[[3,1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1] => [4,4,1] => 6
[[3,1],[2],[1],[1],[1]] => [4,2,1,1,1] => [6,2,1] => 5
[[3,1],[2],[2],[1]] => [4,2,2,1] => [6,3] => 4
[[3,1],[1,1],[1],[1],[1]] => [4,2,1,1,1] => [6,2,1] => 5
[[3,1],[1,1],[1,1],[1]] => [4,2,2,1] => [6,3] => 4
[[3,1],[3],[1],[1]] => [4,3,1,1] => [7,2] => 4
[[3,1],[3],[2]] => [4,3,2] => [7,1,1] => 3
[[3,1],[2,1],[1],[1]] => [4,3,1,1] => [7,2] => 4
[[3,1],[2,1],[2]] => [4,3,2] => [7,1,1] => 3
[[3,1],[2,1],[1,1]] => [4,3,2] => [7,1,1] => 3
[[3,1],[3,1],[1]] => [4,4,1] => [8,1] => 3
[[2,2],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1] => [4,4,1] => 6
[[2,2],[2],[1],[1],[1]] => [4,2,1,1,1] => [6,2,1] => 5
[[2,2],[2],[2],[1]] => [4,2,2,1] => [6,3] => 4
[[2,2],[1,1],[1],[1],[1]] => [4,2,1,1,1] => [6,2,1] => 5
[[2,2],[1,1],[1,1],[1]] => [4,2,2,1] => [6,3] => 4
[[2,2],[2,1],[1],[1]] => [4,3,1,1] => [7,2] => 4
[[2,2],[2,1],[2]] => [4,3,2] => [7,1,1] => 3
[[2,2],[2,1],[1,1]] => [4,3,2] => [7,1,1] => 3
[[2,2],[2,2],[1]] => [4,4,1] => [8,1] => 3
[[2,1,1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1] => [4,4,1] => 6
[[2,1,1],[2],[1],[1],[1]] => [4,2,1,1,1] => [6,2,1] => 5
[[2,1,1],[2],[2],[1]] => [4,2,2,1] => [6,3] => 4
[[2,1,1],[1,1],[1],[1],[1]] => [4,2,1,1,1] => [6,2,1] => 5
[[2,1,1],[1,1],[1,1],[1]] => [4,2,2,1] => [6,3] => 4
[[2,1,1],[2,1],[1],[1]] => [4,3,1,1] => [7,2] => 4
[[2,1,1],[2,1],[2]] => [4,3,2] => [7,1,1] => 3
[[2,1,1],[2,1],[1,1]] => [4,3,2] => [7,1,1] => 3
[[2,1,1],[1,1,1],[1],[1]] => [4,3,1,1] => [7,2] => 4
[[2,1,1],[1,1,1],[1,1]] => [4,3,2] => [7,1,1] => 3
[[2,1,1],[2,1,1],[1]] => [4,4,1] => [8,1] => 3
[[1,1,1,1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1] => [4,4,1] => 6
[[1,1,1,1],[1,1],[1],[1],[1]] => [4,2,1,1,1] => [6,2,1] => 5
[[1,1,1,1],[1,1],[1,1],[1]] => [4,2,2,1] => [6,3] => 4
[[1,1,1,1],[1,1,1],[1],[1]] => [4,3,1,1] => [7,2] => 4
[[1,1,1,1],[1,1,1],[1,1]] => [4,3,2] => [7,1,1] => 3
[[1,1,1,1],[1,1,1,1],[1]] => [4,4,1] => [8,1] => 3
[[5],[1],[1],[1],[1]] => [5,1,1,1,1] => [4,2,1,1,1] => 5
[[5],[2],[1],[1]] => [5,2,1,1] => [3,3,1,1,1] => 4
[[5],[2],[2]] => [5,2,2] => [6,1,1,1] => 3
[[5],[3],[1]] => [5,3,1] => [3,2,2,1,1] => 3
[[5],[4]] => [5,4] => [9] => 2
[[4,1],[1],[1],[1],[1]] => [5,1,1,1,1] => [4,2,1,1,1] => 5
[[4,1],[2],[1],[1]] => [5,2,1,1] => [3,3,1,1,1] => 4
[[4,1],[2],[2]] => [5,2,2] => [6,1,1,1] => 3
[[4,1],[1,1],[1],[1]] => [5,2,1,1] => [3,3,1,1,1] => 4
[[4,1],[1,1],[1,1]] => [5,2,2] => [6,1,1,1] => 3
[[4,1],[3],[1]] => [5,3,1] => [3,2,2,1,1] => 3
[[4,1],[2,1],[1]] => [5,3,1] => [3,2,2,1,1] => 3
[[4,1],[4]] => [5,4] => [9] => 2
[[4,1],[3,1]] => [5,4] => [9] => 2
[[3,2],[1],[1],[1],[1]] => [5,1,1,1,1] => [4,2,1,1,1] => 5
[[3,2],[2],[1],[1]] => [5,2,1,1] => [3,3,1,1,1] => 4
[[3,2],[2],[2]] => [5,2,2] => [6,1,1,1] => 3
[[3,2],[1,1],[1],[1]] => [5,2,1,1] => [3,3,1,1,1] => 4
[[3,2],[1,1],[1,1]] => [5,2,2] => [6,1,1,1] => 3
[[3,2],[3],[1]] => [5,3,1] => [3,2,2,1,1] => 3
[[3,2],[2,1],[1]] => [5,3,1] => [3,2,2,1,1] => 3
[[3,2],[3,1]] => [5,4] => [9] => 2
[[3,2],[2,2]] => [5,4] => [9] => 2
[[3,1,1],[1],[1],[1],[1]] => [5,1,1,1,1] => [4,2,1,1,1] => 5
[[3,1,1],[2],[1],[1]] => [5,2,1,1] => [3,3,1,1,1] => 4
[[3,1,1],[2],[2]] => [5,2,2] => [6,1,1,1] => 3
[[3,1,1],[1,1],[1],[1]] => [5,2,1,1] => [3,3,1,1,1] => 4
[[3,1,1],[1,1],[1,1]] => [5,2,2] => [6,1,1,1] => 3
[[3,1,1],[3],[1]] => [5,3,1] => [3,2,2,1,1] => 3
[[3,1,1],[2,1],[1]] => [5,3,1] => [3,2,2,1,1] => 3
[[3,1,1],[1,1,1],[1]] => [5,3,1] => [3,2,2,1,1] => 3
[[3,1,1],[3,1]] => [5,4] => [9] => 2
[[3,1,1],[2,1,1]] => [5,4] => [9] => 2
[[2,2,1],[1],[1],[1],[1]] => [5,1,1,1,1] => [4,2,1,1,1] => 5
[[2,2,1],[2],[1],[1]] => [5,2,1,1] => [3,3,1,1,1] => 4
[[2,2,1],[2],[2]] => [5,2,2] => [6,1,1,1] => 3
[[2,2,1],[1,1],[1],[1]] => [5,2,1,1] => [3,3,1,1,1] => 4
[[2,2,1],[1,1],[1,1]] => [5,2,2] => [6,1,1,1] => 3
[[2,2,1],[2,1],[1]] => [5,3,1] => [3,2,2,1,1] => 3
[[2,2,1],[1,1,1],[1]] => [5,3,1] => [3,2,2,1,1] => 3
[[2,2,1],[2,2]] => [5,4] => [9] => 2
[[2,2,1],[2,1,1]] => [5,4] => [9] => 2
[[2,1,1,1],[1],[1],[1],[1]] => [5,1,1,1,1] => [4,2,1,1,1] => 5
[[2,1,1,1],[2],[1],[1]] => [5,2,1,1] => [3,3,1,1,1] => 4
[[2,1,1,1],[2],[2]] => [5,2,2] => [6,1,1,1] => 3
[[2,1,1,1],[1,1],[1],[1]] => [5,2,1,1] => [3,3,1,1,1] => 4
[[2,1,1,1],[1,1],[1,1]] => [5,2,2] => [6,1,1,1] => 3
[[2,1,1,1],[2,1],[1]] => [5,3,1] => [3,2,2,1,1] => 3
[[2,1,1,1],[1,1,1],[1]] => [5,3,1] => [3,2,2,1,1] => 3
[[2,1,1,1],[2,1,1]] => [5,4] => [9] => 2
[[2,1,1,1],[1,1,1,1]] => [5,4] => [9] => 2
[[1,1,1,1,1],[1],[1],[1],[1]] => [5,1,1,1,1] => [4,2,1,1,1] => 5
[[1,1,1,1,1],[1,1],[1],[1]] => [5,2,1,1] => [3,3,1,1,1] => 4
[[1,1,1,1,1],[1,1],[1,1]] => [5,2,2] => [6,1,1,1] => 3
[[1,1,1,1,1],[1,1,1],[1]] => [5,3,1] => [3,2,2,1,1] => 3
[[1,1,1,1,1],[1,1,1,1]] => [5,4] => [9] => 2
[[6],[1],[1],[1]] => [6,1,1,1] => [3,2,1,1,1,1] => 4
[[6],[2],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[6],[3]] => [6,3] => [2,2,2,1,1,1] => 2
[[5,1],[1],[1],[1]] => [6,1,1,1] => [3,2,1,1,1,1] => 4
[[5,1],[2],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[5,1],[1,1],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[5,1],[3]] => [6,3] => [2,2,2,1,1,1] => 2
[[5,1],[2,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[4,2],[1],[1],[1]] => [6,1,1,1] => [3,2,1,1,1,1] => 4
[[4,2],[2],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[4,2],[1,1],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[4,2],[3]] => [6,3] => [2,2,2,1,1,1] => 2
[[4,2],[2,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[4,1,1],[1],[1],[1]] => [6,1,1,1] => [3,2,1,1,1,1] => 4
[[4,1,1],[2],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[4,1,1],[1,1],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[4,1,1],[3]] => [6,3] => [2,2,2,1,1,1] => 2
[[4,1,1],[2,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[4,1,1],[1,1,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[3,3],[1],[1],[1]] => [6,1,1,1] => [3,2,1,1,1,1] => 4
[[3,3],[2],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[3,3],[1,1],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[3,3],[3]] => [6,3] => [2,2,2,1,1,1] => 2
[[3,3],[2,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[3,2,1],[1],[1],[1]] => [6,1,1,1] => [3,2,1,1,1,1] => 4
[[3,2,1],[2],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[3,2,1],[1,1],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[3,2,1],[3]] => [6,3] => [2,2,2,1,1,1] => 2
[[3,2,1],[2,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[3,2,1],[1,1,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[3,1,1,1],[1],[1],[1]] => [6,1,1,1] => [3,2,1,1,1,1] => 4
[[3,1,1,1],[2],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[3,1,1,1],[1,1],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[3,1,1,1],[3]] => [6,3] => [2,2,2,1,1,1] => 2
[[3,1,1,1],[2,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[3,1,1,1],[1,1,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[2,2,2],[1],[1],[1]] => [6,1,1,1] => [3,2,1,1,1,1] => 4
[[2,2,2],[2],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[2,2,2],[1,1],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[2,2,2],[2,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[2,2,2],[1,1,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[2,2,1,1],[1],[1],[1]] => [6,1,1,1] => [3,2,1,1,1,1] => 4
[[2,2,1,1],[2],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[2,2,1,1],[1,1],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[2,2,1,1],[2,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[2,2,1,1],[1,1,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[2,1,1,1,1],[1],[1],[1]] => [6,1,1,1] => [3,2,1,1,1,1] => 4
[[2,1,1,1,1],[2],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[2,1,1,1,1],[1,1],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[2,1,1,1,1],[2,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[2,1,1,1,1],[1,1,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[1,1,1,1,1,1],[1],[1],[1]] => [6,1,1,1] => [3,2,1,1,1,1] => 4
[[1,1,1,1,1,1],[1,1],[1]] => [6,2,1] => [4,1,1,1,1,1] => 3
[[1,1,1,1,1,1],[1,1,1]] => [6,3] => [2,2,2,1,1,1] => 2
[[7],[1],[1]] => [7,1,1] => [3,1,1,1,1,1,1] => 3
[[7],[2]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[6,1],[1],[1]] => [7,1,1] => [3,1,1,1,1,1,1] => 3
[[6,1],[2]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[6,1],[1,1]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[5,2],[1],[1]] => [7,1,1] => [3,1,1,1,1,1,1] => 3
[[5,2],[2]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[5,2],[1,1]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[5,1,1],[1],[1]] => [7,1,1] => [3,1,1,1,1,1,1] => 3
[[5,1,1],[2]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[5,1,1],[1,1]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[4,3],[1],[1]] => [7,1,1] => [3,1,1,1,1,1,1] => 3
[[4,3],[2]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[4,3],[1,1]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[4,2,1],[1],[1]] => [7,1,1] => [3,1,1,1,1,1,1] => 3
[[4,2,1],[2]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[4,2,1],[1,1]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[4,1,1,1],[1],[1]] => [7,1,1] => [3,1,1,1,1,1,1] => 3
[[4,1,1,1],[2]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[4,1,1,1],[1,1]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[3,3,1],[1],[1]] => [7,1,1] => [3,1,1,1,1,1,1] => 3
[[3,3,1],[2]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[3,3,1],[1,1]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[3,2,2],[1],[1]] => [7,1,1] => [3,1,1,1,1,1,1] => 3
[[3,2,2],[2]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[3,2,2],[1,1]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[3,2,1,1],[1],[1]] => [7,1,1] => [3,1,1,1,1,1,1] => 3
[[3,2,1,1],[2]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[3,2,1,1],[1,1]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[3,1,1,1,1],[1],[1]] => [7,1,1] => [3,1,1,1,1,1,1] => 3
[[3,1,1,1,1],[2]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[3,1,1,1,1],[1,1]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[2,2,2,1],[1],[1]] => [7,1,1] => [3,1,1,1,1,1,1] => 3
[[2,2,2,1],[2]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[2,2,2,1],[1,1]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[2,2,1,1,1],[1],[1]] => [7,1,1] => [3,1,1,1,1,1,1] => 3
[[2,2,1,1,1],[2]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[2,2,1,1,1],[1,1]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[2,1,1,1,1,1],[1],[1]] => [7,1,1] => [3,1,1,1,1,1,1] => 3
[[2,1,1,1,1,1],[2]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[2,1,1,1,1,1],[1,1]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[1,1,1,1,1,1,1],[1],[1]] => [7,1,1] => [3,1,1,1,1,1,1] => 3
[[1,1,1,1,1,1,1],[1,1]] => [7,2] => [2,2,1,1,1,1,1] => 2
[[8],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[7,1],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[6,2],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[6,1,1],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[5,3],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[5,2,1],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[5,1,1,1],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[4,4],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[4,3,1],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[4,2,2],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[4,2,1,1],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[4,1,1,1,1],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[3,3,2],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[3,3,1,1],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[3,2,2,1],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[3,2,1,1,1],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[3,1,1,1,1,1],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[2,2,2,2],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[2,2,2,1,1],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[2,2,1,1,1,1],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[2,1,1,1,1,1,1],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[1,1,1,1,1,1,1,1],[1]] => [8,1] => [2,1,1,1,1,1,1,1] => 2
[[9]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[8,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[7,2]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[7,1,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[6,3]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[6,2,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[6,1,1,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[5,4]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[5,3,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[5,2,2]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[5,2,1,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[5,1,1,1,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[4,4,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[4,3,2]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[4,3,1,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[4,2,2,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[4,2,1,1,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[4,1,1,1,1,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[3,3,3]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[3,3,2,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[3,3,1,1,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[3,2,2,2]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[3,2,2,1,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[3,2,1,1,1,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[3,1,1,1,1,1,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[2,2,2,2,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[2,2,2,1,1,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[2,2,1,1,1,1,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[2,1,1,1,1,1,1,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[1,1,1,1,1,1,1,1,1]] => [9] => [1,1,1,1,1,1,1,1,1] => 1
[[1],[1],[1],[1],[1],[1],[1],[1],[1],[1]] => [1,1,1,1,1,1,1,1,1,1] => [4,3,2,1] => 10
[[2],[1],[1],[1],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1,1,1,1] => [5,3,2] => 9
[[2],[2],[1],[1],[1],[1],[1],[1]] => [2,2,1,1,1,1,1,1] => [5,3,1,1] => 8
[[2],[2],[2],[1],[1],[1],[1]] => [2,2,2,1,1,1,1] => [5,2,2,1] => 7
[[2],[2],[2],[2],[1],[1]] => [2,2,2,2,1,1] => [3,3,3,1] => 6
[[2],[2],[2],[2],[2]] => [2,2,2,2,2] => [6,4] => 5
[[1,1],[1],[1],[1],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1,1,1,1] => [5,3,2] => 9
[[1,1],[1,1],[1],[1],[1],[1],[1],[1]] => [2,2,1,1,1,1,1,1] => [5,3,1,1] => 8
[[1,1],[1,1],[1,1],[1],[1],[1],[1]] => [2,2,2,1,1,1,1] => [5,2,2,1] => 7
[[1,1],[1,1],[1,1],[1,1],[1],[1]] => [2,2,2,2,1,1] => [3,3,3,1] => 6
[[1,1],[1,1],[1,1],[1,1],[1,1]] => [2,2,2,2,2] => [6,4] => 5
[[3],[1],[1],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1,1,1] => [4,4,2] => 8
[[3],[2],[1],[1],[1],[1],[1]] => [3,2,1,1,1,1,1] => [4,4,1,1] => 7
[[3],[2],[2],[1],[1],[1]] => [3,2,2,1,1,1] => [4,2,2,2] => 6
[[3],[2],[2],[2],[1]] => [3,2,2,2,1] => [5,2,1,1,1] => 5
[[3],[3],[1],[1],[1],[1]] => [3,3,1,1,1,1] => [5,4,1] => 6
[[3],[3],[2],[1],[1]] => [3,3,2,1,1] => [6,2,2] => 5
[[3],[3],[2],[2]] => [3,3,2,2] => [6,1,1,1,1] => 4
[[3],[3],[3],[1]] => [3,3,3,1] => [3,2,2,2,1] => 4
[[2,1],[1],[1],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1,1,1] => [4,4,2] => 8
[[2,1],[2],[1],[1],[1],[1],[1]] => [3,2,1,1,1,1,1] => [4,4,1,1] => 7
[[2,1],[2],[2],[1],[1],[1]] => [3,2,2,1,1,1] => [4,2,2,2] => 6
[[2,1],[2],[2],[2],[1]] => [3,2,2,2,1] => [5,2,1,1,1] => 5
[[2,1],[1,1],[1],[1],[1],[1],[1]] => [3,2,1,1,1,1,1] => [4,4,1,1] => 7
[[2,1],[1,1],[1,1],[1],[1],[1]] => [3,2,2,1,1,1] => [4,2,2,2] => 6
[[2,1],[1,1],[1,1],[1,1],[1]] => [3,2,2,2,1] => [5,2,1,1,1] => 5
[[2,1],[2,1],[1],[1],[1],[1]] => [3,3,1,1,1,1] => [5,4,1] => 6
[[2,1],[2,1],[2],[1],[1]] => [3,3,2,1,1] => [6,2,2] => 5
[[2,1],[2,1],[2],[2]] => [3,3,2,2] => [6,1,1,1,1] => 4
[[2,1],[2,1],[1,1],[1],[1]] => [3,3,2,1,1] => [6,2,2] => 5
[[2,1],[2,1],[1,1],[1,1]] => [3,3,2,2] => [6,1,1,1,1] => 4
[[2,1],[2,1],[2,1],[1]] => [3,3,3,1] => [3,2,2,2,1] => 4
[[1,1,1],[1],[1],[1],[1],[1],[1],[1]] => [3,1,1,1,1,1,1,1] => [4,4,2] => 8
[[1,1,1],[1,1],[1],[1],[1],[1],[1]] => [3,2,1,1,1,1,1] => [4,4,1,1] => 7
[[1,1,1],[1,1],[1,1],[1],[1],[1]] => [3,2,2,1,1,1] => [4,2,2,2] => 6
[[1,1,1],[1,1],[1,1],[1,1],[1]] => [3,2,2,2,1] => [5,2,1,1,1] => 5
[[1,1,1],[1,1,1],[1],[1],[1],[1]] => [3,3,1,1,1,1] => [5,4,1] => 6
[[1,1,1],[1,1,1],[1,1],[1],[1]] => [3,3,2,1,1] => [6,2,2] => 5
[[1,1,1],[1,1,1],[1,1],[1,1]] => [3,3,2,2] => [6,1,1,1,1] => 4
[[1,1,1],[1,1,1],[1,1,1],[1]] => [3,3,3,1] => [3,2,2,2,1] => 4
[[4],[1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1,1] => [4,3,3] => 7
[[4],[2],[1],[1],[1],[1]] => [4,2,1,1,1,1] => [6,3,1] => 6
[[4],[2],[2],[1],[1]] => [4,2,2,1,1] => [3,3,2,2] => 5
[[4],[2],[2],[2]] => [4,2,2,2] => [5,5] => 4
[[4],[3],[1],[1],[1]] => [4,3,1,1,1] => [7,2,1] => 5
[[4],[3],[2],[1]] => [4,3,2,1] => [7,3] => 4
[[4],[3],[3]] => [4,3,3] => [2,2,2,2,2] => 3
[[4],[4],[1],[1]] => [4,4,1,1] => [8,2] => 4
[[4],[4],[2]] => [4,4,2] => [8,1,1] => 3
[[3,1],[1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1,1] => [4,3,3] => 7
[[3,1],[2],[1],[1],[1],[1]] => [4,2,1,1,1,1] => [6,3,1] => 6
[[3,1],[2],[2],[1],[1]] => [4,2,2,1,1] => [3,3,2,2] => 5
[[3,1],[2],[2],[2]] => [4,2,2,2] => [5,5] => 4
[[3,1],[1,1],[1],[1],[1],[1]] => [4,2,1,1,1,1] => [6,3,1] => 6
[[3,1],[1,1],[1,1],[1],[1]] => [4,2,2,1,1] => [3,3,2,2] => 5
[[3,1],[1,1],[1,1],[1,1]] => [4,2,2,2] => [5,5] => 4
[[3,1],[3],[1],[1],[1]] => [4,3,1,1,1] => [7,2,1] => 5
[[3,1],[3],[2],[1]] => [4,3,2,1] => [7,3] => 4
[[3,1],[3],[3]] => [4,3,3] => [2,2,2,2,2] => 3
[[3,1],[2,1],[1],[1],[1]] => [4,3,1,1,1] => [7,2,1] => 5
[[3,1],[2,1],[2],[1]] => [4,3,2,1] => [7,3] => 4
[[3,1],[2,1],[1,1],[1]] => [4,3,2,1] => [7,3] => 4
[[3,1],[2,1],[2,1]] => [4,3,3] => [2,2,2,2,2] => 3
[[3,1],[3,1],[1],[1]] => [4,4,1,1] => [8,2] => 4
[[3,1],[3,1],[2]] => [4,4,2] => [8,1,1] => 3
[[3,1],[3,1],[1,1]] => [4,4,2] => [8,1,1] => 3
[[2,2],[1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1,1] => [4,3,3] => 7
[[2,2],[2],[1],[1],[1],[1]] => [4,2,1,1,1,1] => [6,3,1] => 6
[[2,2],[2],[2],[1],[1]] => [4,2,2,1,1] => [3,3,2,2] => 5
[[2,2],[2],[2],[2]] => [4,2,2,2] => [5,5] => 4
[[2,2],[1,1],[1],[1],[1],[1]] => [4,2,1,1,1,1] => [6,3,1] => 6
[[2,2],[1,1],[1,1],[1],[1]] => [4,2,2,1,1] => [3,3,2,2] => 5
[[2,2],[1,1],[1,1],[1,1]] => [4,2,2,2] => [5,5] => 4
[[2,2],[2,1],[1],[1],[1]] => [4,3,1,1,1] => [7,2,1] => 5
[[2,2],[2,1],[2],[1]] => [4,3,2,1] => [7,3] => 4
[[2,2],[2,1],[1,1],[1]] => [4,3,2,1] => [7,3] => 4
[[2,2],[2,1],[2,1]] => [4,3,3] => [2,2,2,2,2] => 3
[[2,2],[2,2],[1],[1]] => [4,4,1,1] => [8,2] => 4
[[2,2],[2,2],[2]] => [4,4,2] => [8,1,1] => 3
[[2,2],[2,2],[1,1]] => [4,4,2] => [8,1,1] => 3
[[2,1,1],[1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1,1] => [4,3,3] => 7
[[2,1,1],[2],[1],[1],[1],[1]] => [4,2,1,1,1,1] => [6,3,1] => 6
[[2,1,1],[2],[2],[1],[1]] => [4,2,2,1,1] => [3,3,2,2] => 5
[[2,1,1],[2],[2],[2]] => [4,2,2,2] => [5,5] => 4
[[2,1,1],[1,1],[1],[1],[1],[1]] => [4,2,1,1,1,1] => [6,3,1] => 6
[[2,1,1],[1,1],[1,1],[1],[1]] => [4,2,2,1,1] => [3,3,2,2] => 5
[[2,1,1],[1,1],[1,1],[1,1]] => [4,2,2,2] => [5,5] => 4
[[2,1,1],[2,1],[1],[1],[1]] => [4,3,1,1,1] => [7,2,1] => 5
[[2,1,1],[2,1],[2],[1]] => [4,3,2,1] => [7,3] => 4
[[2,1,1],[2,1],[1,1],[1]] => [4,3,2,1] => [7,3] => 4
[[2,1,1],[2,1],[2,1]] => [4,3,3] => [2,2,2,2,2] => 3
[[2,1,1],[1,1,1],[1],[1],[1]] => [4,3,1,1,1] => [7,2,1] => 5
[[2,1,1],[1,1,1],[1,1],[1]] => [4,3,2,1] => [7,3] => 4
[[2,1,1],[1,1,1],[1,1,1]] => [4,3,3] => [2,2,2,2,2] => 3
[[2,1,1],[2,1,1],[1],[1]] => [4,4,1,1] => [8,2] => 4
[[2,1,1],[2,1,1],[2]] => [4,4,2] => [8,1,1] => 3
[[2,1,1],[2,1,1],[1,1]] => [4,4,2] => [8,1,1] => 3
[[1,1,1,1],[1],[1],[1],[1],[1],[1]] => [4,1,1,1,1,1,1] => [4,3,3] => 7
[[1,1,1,1],[1,1],[1],[1],[1],[1]] => [4,2,1,1,1,1] => [6,3,1] => 6
[[1,1,1,1],[1,1],[1,1],[1],[1]] => [4,2,2,1,1] => [3,3,2,2] => 5
[[1,1,1,1],[1,1],[1,1],[1,1]] => [4,2,2,2] => [5,5] => 4
[[1,1,1,1],[1,1,1],[1],[1],[1]] => [4,3,1,1,1] => [7,2,1] => 5
[[1,1,1,1],[1,1,1],[1,1],[1]] => [4,3,2,1] => [7,3] => 4
[[1,1,1,1],[1,1,1],[1,1,1]] => [4,3,3] => [2,2,2,2,2] => 3
[[1,1,1,1],[1,1,1,1],[1],[1]] => [4,4,1,1] => [8,2] => 4
[[1,1,1,1],[1,1,1,1],[1,1]] => [4,4,2] => [8,1,1] => 3
[[5],[1],[1],[1],[1],[1]] => [5,1,1,1,1,1] => [4,3,1,1,1] => 6
[[5],[2],[1],[1],[1]] => [5,2,1,1,1] => [4,2,2,1,1] => 5
[[5],[2],[2],[1]] => [5,2,2,1] => [3,3,2,1,1] => 4
[[5],[3],[1],[1]] => [5,3,1,1] => [6,2,1,1] => 4
[[5],[3],[2]] => [5,3,2] => [7,1,1,1] => 3
[[5],[4],[1]] => [5,4,1] => [9,1] => 3
[[5],[5]] => [5,5] => [10] => 2
[[4,1],[1],[1],[1],[1],[1]] => [5,1,1,1,1,1] => [4,3,1,1,1] => 6
[[4,1],[2],[1],[1],[1]] => [5,2,1,1,1] => [4,2,2,1,1] => 5
[[4,1],[2],[2],[1]] => [5,2,2,1] => [3,3,2,1,1] => 4
[[4,1],[1,1],[1],[1],[1]] => [5,2,1,1,1] => [4,2,2,1,1] => 5
[[4,1],[1,1],[1,1],[1]] => [5,2,2,1] => [3,3,2,1,1] => 4
[[4,1],[3],[1],[1]] => [5,3,1,1] => [6,2,1,1] => 4
[[4,1],[3],[2]] => [5,3,2] => [7,1,1,1] => 3
[[4,1],[2,1],[1],[1]] => [5,3,1,1] => [6,2,1,1] => 4
[[4,1],[2,1],[2]] => [5,3,2] => [7,1,1,1] => 3
[[4,1],[2,1],[1,1]] => [5,3,2] => [7,1,1,1] => 3
[[4,1],[4],[1]] => [5,4,1] => [9,1] => 3
[[4,1],[3,1],[1]] => [5,4,1] => [9,1] => 3
[[4,1],[4,1]] => [5,5] => [10] => 2
[[3,2],[1],[1],[1],[1],[1]] => [5,1,1,1,1,1] => [4,3,1,1,1] => 6
[[3,2],[2],[1],[1],[1]] => [5,2,1,1,1] => [4,2,2,1,1] => 5
[[3,2],[2],[2],[1]] => [5,2,2,1] => [3,3,2,1,1] => 4
[[3,2],[1,1],[1],[1],[1]] => [5,2,1,1,1] => [4,2,2,1,1] => 5
[[3,2],[1,1],[1,1],[1]] => [5,2,2,1] => [3,3,2,1,1] => 4
[[3,2],[3],[1],[1]] => [5,3,1,1] => [6,2,1,1] => 4
[[3,2],[3],[2]] => [5,3,2] => [7,1,1,1] => 3
[[3,2],[2,1],[1],[1]] => [5,3,1,1] => [6,2,1,1] => 4
[[3,2],[2,1],[2]] => [5,3,2] => [7,1,1,1] => 3
[[3,2],[2,1],[1,1]] => [5,3,2] => [7,1,1,1] => 3
[[3,2],[3,1],[1]] => [5,4,1] => [9,1] => 3
[[3,2],[2,2],[1]] => [5,4,1] => [9,1] => 3
[[3,2],[3,2]] => [5,5] => [10] => 2
[[3,1,1],[1],[1],[1],[1],[1]] => [5,1,1,1,1,1] => [4,3,1,1,1] => 6
[[3,1,1],[2],[1],[1],[1]] => [5,2,1,1,1] => [4,2,2,1,1] => 5
[[3,1,1],[2],[2],[1]] => [5,2,2,1] => [3,3,2,1,1] => 4
[[3,1,1],[1,1],[1],[1],[1]] => [5,2,1,1,1] => [4,2,2,1,1] => 5
[[3,1,1],[1,1],[1,1],[1]] => [5,2,2,1] => [3,3,2,1,1] => 4
[[3,1,1],[3],[1],[1]] => [5,3,1,1] => [6,2,1,1] => 4
[[3,1,1],[3],[2]] => [5,3,2] => [7,1,1,1] => 3
[[3,1,1],[2,1],[1],[1]] => [5,3,1,1] => [6,2,1,1] => 4
[[3,1,1],[2,1],[2]] => [5,3,2] => [7,1,1,1] => 3
[[3,1,1],[2,1],[1,1]] => [5,3,2] => [7,1,1,1] => 3
[[3,1,1],[1,1,1],[1],[1]] => [5,3,1,1] => [6,2,1,1] => 4
[[3,1,1],[1,1,1],[1,1]] => [5,3,2] => [7,1,1,1] => 3
[[3,1,1],[3,1],[1]] => [5,4,1] => [9,1] => 3
[[3,1,1],[2,1,1],[1]] => [5,4,1] => [9,1] => 3
[[3,1,1],[3,1,1]] => [5,5] => [10] => 2
[[2,2,1],[1],[1],[1],[1],[1]] => [5,1,1,1,1,1] => [4,3,1,1,1] => 6
[[2,2,1],[2],[1],[1],[1]] => [5,2,1,1,1] => [4,2,2,1,1] => 5
[[2,2,1],[2],[2],[1]] => [5,2,2,1] => [3,3,2,1,1] => 4
[[2,2,1],[1,1],[1],[1],[1]] => [5,2,1,1,1] => [4,2,2,1,1] => 5
[[2,2,1],[1,1],[1,1],[1]] => [5,2,2,1] => [3,3,2,1,1] => 4
[[2,2,1],[2,1],[1],[1]] => [5,3,1,1] => [6,2,1,1] => 4
[[2,2,1],[2,1],[2]] => [5,3,2] => [7,1,1,1] => 3
[[2,2,1],[2,1],[1,1]] => [5,3,2] => [7,1,1,1] => 3
[[2,2,1],[1,1,1],[1],[1]] => [5,3,1,1] => [6,2,1,1] => 4
[[2,2,1],[1,1,1],[1,1]] => [5,3,2] => [7,1,1,1] => 3
[[2,2,1],[2,2],[1]] => [5,4,1] => [9,1] => 3
[[2,2,1],[2,1,1],[1]] => [5,4,1] => [9,1] => 3
[[2,2,1],[2,2,1]] => [5,5] => [10] => 2
[[2,1,1,1],[1],[1],[1],[1],[1]] => [5,1,1,1,1,1] => [4,3,1,1,1] => 6
[[2,1,1,1],[2],[1],[1],[1]] => [5,2,1,1,1] => [4,2,2,1,1] => 5
[[2,1,1,1],[2],[2],[1]] => [5,2,2,1] => [3,3,2,1,1] => 4
[[2,1,1,1],[1,1],[1],[1],[1]] => [5,2,1,1,1] => [4,2,2,1,1] => 5
[[2,1,1,1],[1,1],[1,1],[1]] => [5,2,2,1] => [3,3,2,1,1] => 4
[[2,1,1,1],[2,1],[1],[1]] => [5,3,1,1] => [6,2,1,1] => 4
[[2,1,1,1],[2,1],[2]] => [5,3,2] => [7,1,1,1] => 3
[[2,1,1,1],[2,1],[1,1]] => [5,3,2] => [7,1,1,1] => 3
[[2,1,1,1],[1,1,1],[1],[1]] => [5,3,1,1] => [6,2,1,1] => 4
[[2,1,1,1],[1,1,1],[1,1]] => [5,3,2] => [7,1,1,1] => 3
[[2,1,1,1],[2,1,1],[1]] => [5,4,1] => [9,1] => 3
[[2,1,1,1],[1,1,1,1],[1]] => [5,4,1] => [9,1] => 3
[[2,1,1,1],[2,1,1,1]] => [5,5] => [10] => 2
[[1,1,1,1,1],[1],[1],[1],[1],[1]] => [5,1,1,1,1,1] => [4,3,1,1,1] => 6
[[1,1,1,1,1],[1,1],[1],[1],[1]] => [5,2,1,1,1] => [4,2,2,1,1] => 5
[[1,1,1,1,1],[1,1],[1,1],[1]] => [5,2,2,1] => [3,3,2,1,1] => 4
[[1,1,1,1,1],[1,1,1],[1],[1]] => [5,3,1,1] => [6,2,1,1] => 4
[[1,1,1,1,1],[1,1,1],[1,1]] => [5,3,2] => [7,1,1,1] => 3
[[1,1,1,1,1],[1,1,1,1],[1]] => [5,4,1] => [9,1] => 3
[[1,1,1,1,1],[1,1,1,1,1]] => [5,5] => [10] => 2
[[6],[1],[1],[1],[1]] => [6,1,1,1,1] => [4,2,1,1,1,1] => 5
[[6],[2],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[6],[2],[2]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[6],[3],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[6],[4]] => [6,4] => [2,2,2,2,1,1] => 2
[[5,1],[1],[1],[1],[1]] => [6,1,1,1,1] => [4,2,1,1,1,1] => 5
[[5,1],[2],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[5,1],[2],[2]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[5,1],[1,1],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[5,1],[1,1],[1,1]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[5,1],[3],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[5,1],[2,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[5,1],[4]] => [6,4] => [2,2,2,2,1,1] => 2
[[5,1],[3,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[4,2],[1],[1],[1],[1]] => [6,1,1,1,1] => [4,2,1,1,1,1] => 5
[[4,2],[2],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[4,2],[2],[2]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[4,2],[1,1],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[4,2],[1,1],[1,1]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[4,2],[3],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[4,2],[2,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[4,2],[4]] => [6,4] => [2,2,2,2,1,1] => 2
[[4,2],[3,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[4,2],[2,2]] => [6,4] => [2,2,2,2,1,1] => 2
[[4,1,1],[1],[1],[1],[1]] => [6,1,1,1,1] => [4,2,1,1,1,1] => 5
[[4,1,1],[2],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[4,1,1],[2],[2]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[4,1,1],[1,1],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[4,1,1],[1,1],[1,1]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[4,1,1],[3],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[4,1,1],[2,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[4,1,1],[1,1,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[4,1,1],[4]] => [6,4] => [2,2,2,2,1,1] => 2
[[4,1,1],[3,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[4,1,1],[2,1,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[3,3],[1],[1],[1],[1]] => [6,1,1,1,1] => [4,2,1,1,1,1] => 5
[[3,3],[2],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[3,3],[2],[2]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[3,3],[1,1],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[3,3],[1,1],[1,1]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[3,3],[3],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[3,3],[2,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[3,3],[3,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[3,3],[2,2]] => [6,4] => [2,2,2,2,1,1] => 2
[[3,2,1],[1],[1],[1],[1]] => [6,1,1,1,1] => [4,2,1,1,1,1] => 5
[[3,2,1],[2],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[3,2,1],[2],[2]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[3,2,1],[1,1],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[3,2,1],[1,1],[1,1]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[3,2,1],[3],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[3,2,1],[2,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[3,2,1],[1,1,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[3,2,1],[3,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[3,2,1],[2,2]] => [6,4] => [2,2,2,2,1,1] => 2
[[3,2,1],[2,1,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[3,1,1,1],[1],[1],[1],[1]] => [6,1,1,1,1] => [4,2,1,1,1,1] => 5
[[3,1,1,1],[2],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[3,1,1,1],[2],[2]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[3,1,1,1],[1,1],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[3,1,1,1],[1,1],[1,1]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[3,1,1,1],[3],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[3,1,1,1],[2,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[3,1,1,1],[1,1,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[3,1,1,1],[3,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[3,1,1,1],[2,1,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[3,1,1,1],[1,1,1,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[2,2,2],[1],[1],[1],[1]] => [6,1,1,1,1] => [4,2,1,1,1,1] => 5
[[2,2,2],[2],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[2,2,2],[2],[2]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[2,2,2],[1,1],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[2,2,2],[1,1],[1,1]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[2,2,2],[2,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[2,2,2],[1,1,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[2,2,2],[2,2]] => [6,4] => [2,2,2,2,1,1] => 2
[[2,2,2],[2,1,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[2,2,1,1],[1],[1],[1],[1]] => [6,1,1,1,1] => [4,2,1,1,1,1] => 5
[[2,2,1,1],[2],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[2,2,1,1],[2],[2]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[2,2,1,1],[1,1],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[2,2,1,1],[1,1],[1,1]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[2,2,1,1],[2,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[2,2,1,1],[1,1,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[2,2,1,1],[2,2]] => [6,4] => [2,2,2,2,1,1] => 2
[[2,2,1,1],[2,1,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[2,2,1,1],[1,1,1,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[2,1,1,1,1],[1],[1],[1],[1]] => [6,1,1,1,1] => [4,2,1,1,1,1] => 5
[[2,1,1,1,1],[2],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[2,1,1,1,1],[2],[2]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[2,1,1,1,1],[1,1],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[2,1,1,1,1],[1,1],[1,1]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[2,1,1,1,1],[2,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[2,1,1,1,1],[1,1,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[2,1,1,1,1],[2,1,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[2,1,1,1,1],[1,1,1,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[1,1,1,1,1,1],[1],[1],[1],[1]] => [6,1,1,1,1] => [4,2,1,1,1,1] => 5
[[1,1,1,1,1,1],[1,1],[1],[1]] => [6,2,1,1] => [3,3,1,1,1,1] => 4
[[1,1,1,1,1,1],[1,1],[1,1]] => [6,2,2] => [5,1,1,1,1,1] => 3
[[1,1,1,1,1,1],[1,1,1],[1]] => [6,3,1] => [3,2,2,1,1,1] => 3
[[1,1,1,1,1,1],[1,1,1,1]] => [6,4] => [2,2,2,2,1,1] => 2
[[7],[1],[1],[1]] => [7,1,1,1] => [3,2,1,1,1,1,1] => 4
[[7],[2],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[7],[3]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[6,1],[1],[1],[1]] => [7,1,1,1] => [3,2,1,1,1,1,1] => 4
[[6,1],[2],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[6,1],[1,1],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[6,1],[3]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[6,1],[2,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[5,2],[1],[1],[1]] => [7,1,1,1] => [3,2,1,1,1,1,1] => 4
[[5,2],[2],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[5,2],[1,1],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[5,2],[3]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[5,2],[2,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[5,1,1],[1],[1],[1]] => [7,1,1,1] => [3,2,1,1,1,1,1] => 4
[[5,1,1],[2],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[5,1,1],[1,1],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[5,1,1],[3]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[5,1,1],[2,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[5,1,1],[1,1,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[4,3],[1],[1],[1]] => [7,1,1,1] => [3,2,1,1,1,1,1] => 4
[[4,3],[2],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[4,3],[1,1],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[4,3],[3]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[4,3],[2,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[4,2,1],[1],[1],[1]] => [7,1,1,1] => [3,2,1,1,1,1,1] => 4
[[4,2,1],[2],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[4,2,1],[1,1],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[4,2,1],[3]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[4,2,1],[2,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[4,2,1],[1,1,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[4,1,1,1],[1],[1],[1]] => [7,1,1,1] => [3,2,1,1,1,1,1] => 4
[[4,1,1,1],[2],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[4,1,1,1],[1,1],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[4,1,1,1],[3]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[4,1,1,1],[2,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[4,1,1,1],[1,1,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[3,3,1],[1],[1],[1]] => [7,1,1,1] => [3,2,1,1,1,1,1] => 4
[[3,3,1],[2],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[3,3,1],[1,1],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[3,3,1],[3]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[3,3,1],[2,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[3,3,1],[1,1,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[3,2,2],[1],[1],[1]] => [7,1,1,1] => [3,2,1,1,1,1,1] => 4
[[3,2,2],[2],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[3,2,2],[1,1],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[3,2,2],[3]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[3,2,2],[2,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[3,2,2],[1,1,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[3,2,1,1],[1],[1],[1]] => [7,1,1,1] => [3,2,1,1,1,1,1] => 4
[[3,2,1,1],[2],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[3,2,1,1],[1,1],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[3,2,1,1],[3]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[3,2,1,1],[2,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[3,2,1,1],[1,1,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[3,1,1,1,1],[1],[1],[1]] => [7,1,1,1] => [3,2,1,1,1,1,1] => 4
[[3,1,1,1,1],[2],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[3,1,1,1,1],[1,1],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[3,1,1,1,1],[3]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[3,1,1,1,1],[2,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[3,1,1,1,1],[1,1,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[2,2,2,1],[1],[1],[1]] => [7,1,1,1] => [3,2,1,1,1,1,1] => 4
[[2,2,2,1],[2],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[2,2,2,1],[1,1],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[2,2,2,1],[2,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[2,2,2,1],[1,1,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[2,2,1,1,1],[1],[1],[1]] => [7,1,1,1] => [3,2,1,1,1,1,1] => 4
[[2,2,1,1,1],[2],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[2,2,1,1,1],[1,1],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[2,2,1,1,1],[2,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[2,2,1,1,1],[1,1,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[2,1,1,1,1,1],[1],[1],[1]] => [7,1,1,1] => [3,2,1,1,1,1,1] => 4
[[2,1,1,1,1,1],[2],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[2,1,1,1,1,1],[1,1],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[2,1,1,1,1,1],[2,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[2,1,1,1,1,1],[1,1,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[1,1,1,1,1,1,1],[1],[1],[1]] => [7,1,1,1] => [3,2,1,1,1,1,1] => 4
[[1,1,1,1,1,1,1],[1,1],[1]] => [7,2,1] => [4,1,1,1,1,1,1] => 3
[[1,1,1,1,1,1,1],[1,1,1]] => [7,3] => [2,2,2,1,1,1,1] => 2
[[8],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[8],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[7,1],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[7,1],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[7,1],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[6,2],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[6,2],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[6,2],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[6,1,1],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[6,1,1],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[6,1,1],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[5,3],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[5,3],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[5,3],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[5,2,1],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[5,2,1],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[5,2,1],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[5,1,1,1],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[5,1,1,1],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[5,1,1,1],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[4,4],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[4,4],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[4,4],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[4,3,1],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[4,3,1],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[4,3,1],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[4,2,2],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[4,2,2],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[4,2,2],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[4,2,1,1],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[4,2,1,1],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[4,2,1,1],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[4,1,1,1,1],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[4,1,1,1,1],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[4,1,1,1,1],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[3,3,2],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[3,3,2],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[3,3,2],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[3,3,1,1],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[3,3,1,1],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[3,3,1,1],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[3,2,2,1],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[3,2,2,1],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[3,2,2,1],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[3,2,1,1,1],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[3,2,1,1,1],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[3,2,1,1,1],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[3,1,1,1,1,1],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[3,1,1,1,1,1],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[3,1,1,1,1,1],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[2,2,2,2],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[2,2,2,2],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[2,2,2,2],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[2,2,2,1,1],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[2,2,2,1,1],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[2,2,2,1,1],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[2,2,1,1,1,1],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[2,2,1,1,1,1],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[2,2,1,1,1,1],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[2,1,1,1,1,1,1],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[2,1,1,1,1,1,1],[2]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[2,1,1,1,1,1,1],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[1,1,1,1,1,1,1,1],[1],[1]] => [8,1,1] => [3,1,1,1,1,1,1,1] => 3
[[1,1,1,1,1,1,1,1],[1,1]] => [8,2] => [2,2,1,1,1,1,1,1] => 2
[[9],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[8,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[7,2],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[7,1,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[6,3],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[6,2,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[6,1,1,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[5,4],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[5,3,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[5,2,2],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[5,2,1,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[5,1,1,1,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[4,4,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[4,3,2],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[4,3,1,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[4,2,2,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[4,2,1,1,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[4,1,1,1,1,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[3,3,3],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[3,3,2,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[3,3,1,1,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[3,2,2,2],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[3,2,2,1,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[3,2,1,1,1,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[3,1,1,1,1,1,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[2,2,2,2,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[2,2,2,1,1,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[2,2,1,1,1,1,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[2,1,1,1,1,1,1,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[1,1,1,1,1,1,1,1,1],[1]] => [9,1] => [2,1,1,1,1,1,1,1,1] => 2
[[10]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[9,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[8,2]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[8,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[7,3]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[7,2,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[7,1,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[6,4]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[6,3,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[6,2,2]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[6,2,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[6,1,1,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[5,5]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[5,4,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[5,3,2]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[5,3,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[5,2,2,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[5,2,1,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[5,1,1,1,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[4,4,2]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[4,4,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[4,3,3]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[4,3,2,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[4,3,1,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[4,2,2,2]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[4,2,2,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[4,2,1,1,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[4,1,1,1,1,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[3,3,3,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[3,3,2,2]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[3,3,2,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[3,3,1,1,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[3,2,2,2,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[3,2,2,1,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[3,2,1,1,1,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[3,1,1,1,1,1,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[2,2,2,2,2]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[2,2,2,2,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[2,2,2,1,1,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[2,2,1,1,1,1,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[2,1,1,1,1,1,1,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[[1,1,1,1,1,1,1,1,1,1]] => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
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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:
2,1 3,2,1 5,5,2,1 7,9,5,2,1 11,18,11,5,2,1 15,30,22,11,5,2,1 22,53,42,24,11,5,2,1 30,85,78,46,24,11,5,2,1 42,139,138,90,48,24,11,5,2,1
$F_{1} = q$
$F_{2} = 2\ q + q^{2}$
$F_{3} = 3\ q + 2\ q^{2} + q^{3}$
$F_{4} = 5\ q + 5\ q^{2} + 2\ q^{3} + q^{4}$
$F_{5} = 7\ q + 9\ q^{2} + 5\ q^{3} + 2\ q^{4} + q^{5}$
$F_{6} = 11\ q + 18\ q^{2} + 11\ q^{3} + 5\ q^{4} + 2\ q^{5} + q^{6}$
$F_{7} = 15\ q + 30\ q^{2} + 22\ q^{3} + 11\ q^{4} + 5\ q^{5} + 2\ q^{6} + q^{7}$
$F_{8} = 22\ q + 53\ q^{2} + 42\ q^{3} + 24\ q^{4} + 11\ q^{5} + 5\ q^{6} + 2\ q^{7} + q^{8}$
$F_{9} = 30\ q + 85\ q^{2} + 78\ q^{3} + 46\ q^{4} + 24\ q^{5} + 11\ q^{6} + 5\ q^{7} + 2\ q^{8} + q^{9}$
$F_{10} = 42\ q + 139\ q^{2} + 138\ q^{3} + 90\ q^{4} + 48\ q^{5} + 24\ q^{6} + 11\ q^{7} + 5\ q^{8} + 2\ q^{9} + q^{10}$
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Map
Loehr-Warrington
Description
Return a partition whose diagonal inversion number is the length of the preimage.
Map
to partition
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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