Identifier
-
Mp00186:
Skew partitions
—dominating partition⟶
Integer partitions
St000143: Integer partitions ⟶ ℤ
Values
[[1],[]] => [1] => 0
[[2],[]] => [2] => 0
[[1,1],[]] => [1,1] => 1
[[2,1],[1]] => [2] => 0
[[3],[]] => [3] => 0
[[2,1],[]] => [2,1] => 0
[[3,1],[1]] => [3] => 0
[[2,2],[1]] => [2,1] => 0
[[3,2],[2]] => [3] => 0
[[1,1,1],[]] => [1,1,1] => 1
[[2,2,1],[1,1]] => [2,1] => 0
[[2,1,1],[1]] => [2,1] => 0
[[3,2,1],[2,1]] => [3] => 0
[[4],[]] => [4] => 0
[[3,1],[]] => [3,1] => 0
[[4,1],[1]] => [4] => 0
[[2,2],[]] => [2,2] => 2
[[3,2],[1]] => [3,1] => 0
[[4,2],[2]] => [4] => 0
[[2,1,1],[]] => [2,1,1] => 1
[[3,2,1],[1,1]] => [3,1] => 0
[[3,1,1],[1]] => [3,1] => 0
[[4,2,1],[2,1]] => [4] => 0
[[3,3],[2]] => [3,1] => 0
[[4,3],[3]] => [4] => 0
[[2,2,1],[1]] => [2,2] => 2
[[3,3,1],[2,1]] => [3,1] => 0
[[3,2,1],[2]] => [3,1] => 0
[[4,3,1],[3,1]] => [4] => 0
[[2,2,2],[1,1]] => [2,1,1] => 1
[[3,3,2],[2,2]] => [3,1] => 0
[[3,2,2],[2,1]] => [3,1] => 0
[[4,3,2],[3,2]] => [4] => 0
[[1,1,1,1],[]] => [1,1,1,1] => 1
[[2,2,2,1],[1,1,1]] => [2,1,1] => 1
[[2,2,1,1],[1,1]] => [2,2] => 2
[[3,3,2,1],[2,2,1]] => [3,1] => 0
[[2,1,1,1],[1]] => [2,1,1] => 1
[[3,2,2,1],[2,1,1]] => [3,1] => 0
[[3,2,1,1],[2,1]] => [3,1] => 0
[[4,3,2,1],[3,2,1]] => [4] => 0
[[5],[]] => [5] => 0
[[4,1],[]] => [4,1] => 0
[[5,1],[1]] => [5] => 0
[[3,2],[]] => [3,2] => 0
[[4,2],[1]] => [4,1] => 0
[[5,2],[2]] => [5] => 0
[[3,1,1],[]] => [3,1,1] => 1
[[4,2,1],[1,1]] => [4,1] => 0
[[4,1,1],[1]] => [4,1] => 0
[[5,2,1],[2,1]] => [5] => 0
[[3,3],[1]] => [3,2] => 0
[[4,3],[2]] => [4,1] => 0
[[5,3],[3]] => [5] => 0
[[2,2,1],[]] => [2,2,1] => 2
[[3,3,1],[1,1]] => [3,2] => 0
[[3,2,1],[1]] => [3,2] => 0
[[4,3,1],[2,1]] => [4,1] => 0
[[4,2,1],[2]] => [4,1] => 0
[[5,3,1],[3,1]] => [5] => 0
[[3,2,2],[1,1]] => [3,1,1] => 1
[[4,3,2],[2,2]] => [4,1] => 0
[[4,2,2],[2,1]] => [4,1] => 0
[[5,3,2],[3,2]] => [5] => 0
[[2,1,1,1],[]] => [2,1,1,1] => 1
[[3,2,2,1],[1,1,1]] => [3,1,1] => 1
[[3,2,1,1],[1,1]] => [3,2] => 0
[[4,3,2,1],[2,2,1]] => [4,1] => 0
[[3,1,1,1],[1]] => [3,1,1] => 1
[[4,2,2,1],[2,1,1]] => [4,1] => 0
[[4,2,1,1],[2,1]] => [4,1] => 0
[[5,3,2,1],[3,2,1]] => [5] => 0
[[4,4],[3]] => [4,1] => 0
[[5,4],[4]] => [5] => 0
[[3,3,1],[2]] => [3,2] => 0
[[4,4,1],[3,1]] => [4,1] => 0
[[4,3,1],[3]] => [4,1] => 0
[[5,4,1],[4,1]] => [5] => 0
[[2,2,2],[1]] => [2,2,1] => 2
[[3,3,2],[2,1]] => [3,2] => 0
[[4,4,2],[3,2]] => [4,1] => 0
[[3,2,2],[2]] => [3,2] => 0
[[4,3,2],[3,1]] => [4,1] => 0
[[5,4,2],[4,2]] => [5] => 0
[[2,2,1,1],[1]] => [2,2,1] => 2
[[3,3,2,1],[2,1,1]] => [3,2] => 0
[[3,3,1,1],[2,1]] => [3,2] => 0
[[4,4,2,1],[3,2,1]] => [4,1] => 0
[[3,2,1,1],[2]] => [3,1,1] => 1
[[4,3,2,1],[3,1,1]] => [4,1] => 0
[[4,3,1,1],[3,1]] => [4,1] => 0
[[5,4,2,1],[4,2,1]] => [5] => 0
[[3,3,3],[2,2]] => [3,1,1] => 1
[[4,4,3],[3,3]] => [4,1] => 0
[[4,3,3],[3,2]] => [4,1] => 0
[[5,4,3],[4,3]] => [5] => 0
[[2,2,2,1],[1,1]] => [2,2,1] => 2
[[3,3,3,1],[2,2,1]] => [3,1,1] => 1
[[3,3,2,1],[2,2]] => [3,2] => 0
[[4,4,3,1],[3,3,1]] => [4,1] => 0
[[3,2,2,1],[2,1]] => [3,2] => 0
>>> Load all 1492 entries. <<<[[4,3,3,1],[3,2,1]] => [4,1] => 0
[[4,3,2,1],[3,2]] => [4,1] => 0
[[5,4,3,1],[4,3,1]] => [5] => 0
[[2,2,2,2],[1,1,1]] => [2,1,1,1] => 1
[[3,3,3,2],[2,2,2]] => [3,1,1] => 1
[[3,3,2,2],[2,2,1]] => [3,2] => 0
[[4,4,3,2],[3,3,2]] => [4,1] => 0
[[3,2,2,2],[2,1,1]] => [3,1,1] => 1
[[4,3,3,2],[3,2,2]] => [4,1] => 0
[[4,3,2,2],[3,2,1]] => [4,1] => 0
[[5,4,3,2],[4,3,2]] => [5] => 0
[[1,1,1,1,1],[]] => [1,1,1,1,1] => 1
[[2,2,2,2,1],[1,1,1,1]] => [2,1,1,1] => 1
[[2,2,2,1,1],[1,1,1]] => [2,2,1] => 2
[[3,3,3,2,1],[2,2,2,1]] => [3,1,1] => 1
[[2,2,1,1,1],[1,1]] => [2,2,1] => 2
[[3,3,2,2,1],[2,2,1,1]] => [3,2] => 0
[[3,3,2,1,1],[2,2,1]] => [3,2] => 0
[[4,4,3,2,1],[3,3,2,1]] => [4,1] => 0
[[2,1,1,1,1],[1]] => [2,1,1,1] => 1
[[3,2,2,2,1],[2,1,1,1]] => [3,1,1] => 1
[[3,2,2,1,1],[2,1,1]] => [3,2] => 0
[[4,3,3,2,1],[3,2,2,1]] => [4,1] => 0
[[3,2,1,1,1],[2,1]] => [3,1,1] => 1
[[4,3,2,2,1],[3,2,1,1]] => [4,1] => 0
[[4,3,2,1,1],[3,2,1]] => [4,1] => 0
[[5,4,3,2,1],[4,3,2,1]] => [5] => 0
[[6],[]] => [6] => 0
[[5,1],[]] => [5,1] => 0
[[6,1],[1]] => [6] => 0
[[4,2],[]] => [4,2] => 0
[[5,2],[1]] => [5,1] => 0
[[6,2],[2]] => [6] => 0
[[4,1,1],[]] => [4,1,1] => 1
[[5,2,1],[1,1]] => [5,1] => 0
[[5,1,1],[1]] => [5,1] => 0
[[6,2,1],[2,1]] => [6] => 0
[[3,3],[]] => [3,3] => 3
[[4,3],[1]] => [4,2] => 0
[[5,3],[2]] => [5,1] => 0
[[6,3],[3]] => [6] => 0
[[3,2,1],[]] => [3,2,1] => 0
[[4,3,1],[1,1]] => [4,2] => 0
[[4,2,1],[1]] => [4,2] => 0
[[5,3,1],[2,1]] => [5,1] => 0
[[5,2,1],[2]] => [5,1] => 0
[[6,3,1],[3,1]] => [6] => 0
[[4,2,2],[1,1]] => [4,1,1] => 1
[[5,3,2],[2,2]] => [5,1] => 0
[[5,2,2],[2,1]] => [5,1] => 0
[[6,3,2],[3,2]] => [6] => 0
[[3,1,1,1],[]] => [3,1,1,1] => 1
[[4,2,2,1],[1,1,1]] => [4,1,1] => 1
[[4,2,1,1],[1,1]] => [4,2] => 0
[[5,3,2,1],[2,2,1]] => [5,1] => 0
[[4,1,1,1],[1]] => [4,1,1] => 1
[[5,2,2,1],[2,1,1]] => [5,1] => 0
[[5,2,1,1],[2,1]] => [5,1] => 0
[[6,3,2,1],[3,2,1]] => [6] => 0
[[4,4],[2]] => [4,2] => 0
[[5,4],[3]] => [5,1] => 0
[[6,4],[4]] => [6] => 0
[[3,3,1],[1]] => [3,3] => 3
[[4,4,1],[2,1]] => [4,2] => 0
[[4,3,1],[2]] => [4,2] => 0
[[5,4,1],[3,1]] => [5,1] => 0
[[5,3,1],[3]] => [5,1] => 0
[[6,4,1],[4,1]] => [6] => 0
[[2,2,2],[]] => [2,2,2] => 2
[[3,3,2],[1,1]] => [3,2,1] => 0
[[4,4,2],[2,2]] => [4,2] => 0
[[3,2,2],[1]] => [3,2,1] => 0
[[4,3,2],[2,1]] => [4,2] => 0
[[5,4,2],[3,2]] => [5,1] => 0
[[4,2,2],[2]] => [4,2] => 0
[[5,3,2],[3,1]] => [5,1] => 0
[[6,4,2],[4,2]] => [6] => 0
[[2,2,1,1],[]] => [2,2,1,1] => 2
[[3,3,2,1],[1,1,1]] => [3,2,1] => 0
[[3,3,1,1],[1,1]] => [3,3] => 3
[[4,4,2,1],[2,2,1]] => [4,2] => 0
[[3,2,1,1],[1]] => [3,2,1] => 0
[[4,3,2,1],[2,1,1]] => [4,2] => 0
[[4,3,1,1],[2,1]] => [4,2] => 0
[[5,4,2,1],[3,2,1]] => [5,1] => 0
[[4,2,1,1],[2]] => [4,1,1] => 1
[[5,3,2,1],[3,1,1]] => [5,1] => 0
[[5,3,1,1],[3,1]] => [5,1] => 0
[[6,4,2,1],[4,2,1]] => [6] => 0
[[4,3,3],[2,2]] => [4,1,1] => 1
[[5,4,3],[3,3]] => [5,1] => 0
[[5,3,3],[3,2]] => [5,1] => 0
[[6,4,3],[4,3]] => [6] => 0
[[3,2,2,1],[1,1]] => [3,2,1] => 0
[[4,3,3,1],[2,2,1]] => [4,1,1] => 1
[[4,3,2,1],[2,2]] => [4,2] => 0
[[5,4,3,1],[3,3,1]] => [5,1] => 0
[[4,2,2,1],[2,1]] => [4,2] => 0
[[5,3,3,1],[3,2,1]] => [5,1] => 0
[[5,3,2,1],[3,2]] => [5,1] => 0
[[6,4,3,1],[4,3,1]] => [6] => 0
[[3,2,2,2],[1,1,1]] => [3,1,1,1] => 1
[[4,3,3,2],[2,2,2]] => [4,1,1] => 1
[[4,3,2,2],[2,2,1]] => [4,2] => 0
[[5,4,3,2],[3,3,2]] => [5,1] => 0
[[4,2,2,2],[2,1,1]] => [4,1,1] => 1
[[5,3,3,2],[3,2,2]] => [5,1] => 0
[[5,3,2,2],[3,2,1]] => [5,1] => 0
[[6,4,3,2],[4,3,2]] => [6] => 0
[[2,1,1,1,1],[]] => [2,1,1,1,1] => 1
[[3,2,2,2,1],[1,1,1,1]] => [3,1,1,1] => 1
[[3,2,2,1,1],[1,1,1]] => [3,2,1] => 0
[[4,3,3,2,1],[2,2,2,1]] => [4,1,1] => 1
[[3,2,1,1,1],[1,1]] => [3,2,1] => 0
[[4,3,2,2,1],[2,2,1,1]] => [4,2] => 0
[[4,3,2,1,1],[2,2,1]] => [4,2] => 0
[[5,4,3,2,1],[3,3,2,1]] => [5,1] => 0
[[3,1,1,1,1],[1]] => [3,1,1,1] => 1
[[4,2,2,2,1],[2,1,1,1]] => [4,1,1] => 1
[[4,2,2,1,1],[2,1,1]] => [4,2] => 0
[[5,3,3,2,1],[3,2,2,1]] => [5,1] => 0
[[4,2,1,1,1],[2,1]] => [4,1,1] => 1
[[5,3,2,2,1],[3,2,1,1]] => [5,1] => 0
[[5,3,2,1,1],[3,2,1]] => [5,1] => 0
[[6,4,3,2,1],[4,3,2,1]] => [6] => 0
[[5,5],[4]] => [5,1] => 0
[[6,5],[5]] => [6] => 0
[[4,4,1],[3]] => [4,2] => 0
[[5,5,1],[4,1]] => [5,1] => 0
[[5,4,1],[4]] => [5,1] => 0
[[6,5,1],[5,1]] => [6] => 0
[[3,3,2],[2]] => [3,3] => 3
[[4,4,2],[3,1]] => [4,2] => 0
[[5,5,2],[4,2]] => [5,1] => 0
[[4,3,2],[3]] => [4,2] => 0
[[5,4,2],[4,1]] => [5,1] => 0
[[6,5,2],[5,2]] => [6] => 0
[[3,3,1,1],[2]] => [3,2,1] => 0
[[4,4,2,1],[3,1,1]] => [4,2] => 0
[[4,4,1,1],[3,1]] => [4,2] => 0
[[5,5,2,1],[4,2,1]] => [5,1] => 0
[[4,3,1,1],[3]] => [4,1,1] => 1
[[5,4,2,1],[4,1,1]] => [5,1] => 0
[[5,4,1,1],[4,1]] => [5,1] => 0
[[6,5,2,1],[5,2,1]] => [6] => 0
[[3,3,3],[2,1]] => [3,2,1] => 0
[[4,4,3],[3,2]] => [4,2] => 0
[[5,5,3],[4,3]] => [5,1] => 0
[[4,3,3],[3,1]] => [4,2] => 0
[[5,4,3],[4,2]] => [5,1] => 0
[[6,5,3],[5,3]] => [6] => 0
[[2,2,2,1],[1]] => [2,2,2] => 2
[[3,3,3,1],[2,1,1]] => [3,2,1] => 0
[[3,3,2,1],[2,1]] => [3,3] => 3
[[4,4,3,1],[3,2,1]] => [4,2] => 0
[[4,4,2,1],[3,2]] => [4,2] => 0
[[5,5,3,1],[4,3,1]] => [5,1] => 0
[[3,2,2,1],[2]] => [3,2,1] => 0
[[4,3,3,1],[3,1,1]] => [4,2] => 0
[[4,3,2,1],[3,1]] => [4,2] => 0
[[5,4,3,1],[4,2,1]] => [5,1] => 0
[[5,4,2,1],[4,2]] => [5,1] => 0
[[6,5,3,1],[5,3,1]] => [6] => 0
[[3,3,2,2],[2,1,1]] => [3,2,1] => 0
[[4,4,3,2],[3,2,2]] => [4,2] => 0
[[4,4,2,2],[3,2,1]] => [4,2] => 0
[[5,5,3,2],[4,3,2]] => [5,1] => 0
[[4,3,2,2],[3,1,1]] => [4,1,1] => 1
[[5,4,3,2],[4,2,2]] => [5,1] => 0
[[5,4,2,2],[4,2,1]] => [5,1] => 0
[[6,5,3,2],[5,3,2]] => [6] => 0
[[2,2,1,1,1],[1]] => [2,2,1,1] => 2
[[3,3,2,2,1],[2,1,1,1]] => [3,2,1] => 0
[[3,3,2,1,1],[2,1,1]] => [3,3] => 3
[[4,4,3,2,1],[3,2,2,1]] => [4,2] => 0
[[3,3,1,1,1],[2,1]] => [3,2,1] => 0
[[4,4,2,2,1],[3,2,1,1]] => [4,2] => 0
[[4,4,2,1,1],[3,2,1]] => [4,2] => 0
[[5,5,3,2,1],[4,3,2,1]] => [5,1] => 0
[[3,2,1,1,1],[2]] => [3,1,1,1] => 1
[[4,3,2,2,1],[3,1,1,1]] => [4,1,1] => 1
[[4,3,2,1,1],[3,1,1]] => [4,2] => 0
[[5,4,3,2,1],[4,2,2,1]] => [5,1] => 0
[[4,3,1,1,1],[3,1]] => [4,1,1] => 1
[[5,4,2,2,1],[4,2,1,1]] => [5,1] => 0
[[5,4,2,1,1],[4,2,1]] => [5,1] => 0
[[6,5,3,2,1],[5,3,2,1]] => [6] => 0
[[4,4,4],[3,3]] => [4,1,1] => 1
[[5,5,4],[4,4]] => [5,1] => 0
[[5,4,4],[4,3]] => [5,1] => 0
[[6,5,4],[5,4]] => [6] => 0
[[3,3,3,1],[2,2]] => [3,2,1] => 0
[[4,4,4,1],[3,3,1]] => [4,1,1] => 1
[[4,4,3,1],[3,3]] => [4,2] => 0
[[5,5,4,1],[4,4,1]] => [5,1] => 0
[[4,3,3,1],[3,2]] => [4,2] => 0
[[5,4,4,1],[4,3,1]] => [5,1] => 0
[[5,4,3,1],[4,3]] => [5,1] => 0
[[6,5,4,1],[5,4,1]] => [6] => 0
[[2,2,2,2],[1,1]] => [2,2,1,1] => 2
[[3,3,3,2],[2,2,1]] => [3,2,1] => 0
[[4,4,4,2],[3,3,2]] => [4,1,1] => 1
[[3,3,2,2],[2,2]] => [3,3] => 3
[[4,4,3,2],[3,3,1]] => [4,2] => 0
[[5,5,4,2],[4,4,2]] => [5,1] => 0
[[3,2,2,2],[2,1]] => [3,2,1] => 0
[[4,3,3,2],[3,2,1]] => [4,2] => 0
[[5,4,4,2],[4,3,2]] => [5,1] => 0
[[4,3,2,2],[3,2]] => [4,2] => 0
[[5,4,3,2],[4,3,1]] => [5,1] => 0
[[6,5,4,2],[5,4,2]] => [6] => 0
[[2,2,2,1,1],[1,1]] => [2,2,2] => 2
[[3,3,3,2,1],[2,2,1,1]] => [3,2,1] => 0
[[3,3,3,1,1],[2,2,1]] => [3,2,1] => 0
[[4,4,4,2,1],[3,3,2,1]] => [4,1,1] => 1
[[3,3,2,1,1],[2,2]] => [3,2,1] => 0
[[4,4,3,2,1],[3,3,1,1]] => [4,2] => 0
[[4,4,3,1,1],[3,3,1]] => [4,2] => 0
[[5,5,4,2,1],[4,4,2,1]] => [5,1] => 0
[[3,2,2,1,1],[2,1]] => [3,2,1] => 0
[[4,3,3,2,1],[3,2,1,1]] => [4,2] => 0
[[4,3,3,1,1],[3,2,1]] => [4,2] => 0
[[5,4,4,2,1],[4,3,2,1]] => [5,1] => 0
[[4,3,2,1,1],[3,2]] => [4,1,1] => 1
[[5,4,3,2,1],[4,3,1,1]] => [5,1] => 0
[[5,4,3,1,1],[4,3,1]] => [5,1] => 0
[[6,5,4,2,1],[5,4,2,1]] => [6] => 0
[[3,3,3,3],[2,2,2]] => [3,1,1,1] => 1
[[4,4,4,3],[3,3,3]] => [4,1,1] => 1
[[4,4,3,3],[3,3,2]] => [4,2] => 0
[[5,5,4,3],[4,4,3]] => [5,1] => 0
[[4,3,3,3],[3,2,2]] => [4,1,1] => 1
[[5,4,4,3],[4,3,3]] => [5,1] => 0
[[5,4,3,3],[4,3,2]] => [5,1] => 0
[[6,5,4,3],[5,4,3]] => [6] => 0
[[2,2,2,2,1],[1,1,1]] => [2,2,1,1] => 2
[[3,3,3,3,1],[2,2,2,1]] => [3,1,1,1] => 1
[[3,3,3,2,1],[2,2,2]] => [3,2,1] => 0
[[4,4,4,3,1],[3,3,3,1]] => [4,1,1] => 1
[[3,3,2,2,1],[2,2,1]] => [3,3] => 3
[[4,4,3,3,1],[3,3,2,1]] => [4,2] => 0
[[4,4,3,2,1],[3,3,2]] => [4,2] => 0
[[5,5,4,3,1],[4,4,3,1]] => [5,1] => 0
[[3,2,2,2,1],[2,1,1]] => [3,2,1] => 0
[[4,3,3,3,1],[3,2,2,1]] => [4,1,1] => 1
[[4,3,3,2,1],[3,2,2]] => [4,2] => 0
[[5,4,4,3,1],[4,3,3,1]] => [5,1] => 0
[[4,3,2,2,1],[3,2,1]] => [4,2] => 0
[[5,4,3,3,1],[4,3,2,1]] => [5,1] => 0
[[5,4,3,2,1],[4,3,2]] => [5,1] => 0
[[6,5,4,3,1],[5,4,3,1]] => [6] => 0
[[2,2,2,2,2],[1,1,1,1]] => [2,1,1,1,1] => 1
[[3,3,3,3,2],[2,2,2,2]] => [3,1,1,1] => 1
[[3,3,3,2,2],[2,2,2,1]] => [3,2,1] => 0
[[4,4,4,3,2],[3,3,3,2]] => [4,1,1] => 1
[[3,3,2,2,2],[2,2,1,1]] => [3,2,1] => 0
[[4,4,3,3,2],[3,3,2,2]] => [4,2] => 0
[[4,4,3,2,2],[3,3,2,1]] => [4,2] => 0
[[5,5,4,3,2],[4,4,3,2]] => [5,1] => 0
[[3,2,2,2,2],[2,1,1,1]] => [3,1,1,1] => 1
[[4,3,3,3,2],[3,2,2,2]] => [4,1,1] => 1
[[4,3,3,2,2],[3,2,2,1]] => [4,2] => 0
[[5,4,4,3,2],[4,3,3,2]] => [5,1] => 0
[[4,3,2,2,2],[3,2,1,1]] => [4,1,1] => 1
[[5,4,3,3,2],[4,3,2,2]] => [5,1] => 0
[[5,4,3,2,2],[4,3,2,1]] => [5,1] => 0
[[6,5,4,3,2],[5,4,3,2]] => [6] => 0
[[1,1,1,1,1,1],[]] => [1,1,1,1,1,1] => 1
[[2,2,2,2,2,1],[1,1,1,1,1]] => [2,1,1,1,1] => 1
[[2,2,2,2,1,1],[1,1,1,1]] => [2,2,1,1] => 2
[[3,3,3,3,2,1],[2,2,2,2,1]] => [3,1,1,1] => 1
[[2,2,2,1,1,1],[1,1,1]] => [2,2,2] => 2
[[3,3,3,2,2,1],[2,2,2,1,1]] => [3,2,1] => 0
[[3,3,3,2,1,1],[2,2,2,1]] => [3,2,1] => 0
[[4,4,4,3,2,1],[3,3,3,2,1]] => [4,1,1] => 1
[[2,2,1,1,1,1],[1,1]] => [2,2,1,1] => 2
[[3,3,2,2,2,1],[2,2,1,1,1]] => [3,2,1] => 0
[[3,3,2,2,1,1],[2,2,1,1]] => [3,3] => 3
[[4,4,3,3,2,1],[3,3,2,2,1]] => [4,2] => 0
[[3,3,2,1,1,1],[2,2,1]] => [3,2,1] => 0
[[4,4,3,2,2,1],[3,3,2,1,1]] => [4,2] => 0
[[4,4,3,2,1,1],[3,3,2,1]] => [4,2] => 0
[[5,5,4,3,2,1],[4,4,3,2,1]] => [5,1] => 0
[[2,1,1,1,1,1],[1]] => [2,1,1,1,1] => 1
[[3,2,2,2,2,1],[2,1,1,1,1]] => [3,1,1,1] => 1
[[3,2,2,2,1,1],[2,1,1,1]] => [3,2,1] => 0
[[4,3,3,3,2,1],[3,2,2,2,1]] => [4,1,1] => 1
[[3,2,2,1,1,1],[2,1,1]] => [3,2,1] => 0
[[4,3,3,2,2,1],[3,2,2,1,1]] => [4,2] => 0
[[4,3,3,2,1,1],[3,2,2,1]] => [4,2] => 0
[[5,4,4,3,2,1],[4,3,3,2,1]] => [5,1] => 0
[[3,2,1,1,1,1],[2,1]] => [3,1,1,1] => 1
[[4,3,2,2,2,1],[3,2,1,1,1]] => [4,1,1] => 1
[[4,3,2,2,1,1],[3,2,1,1]] => [4,2] => 0
[[5,4,3,3,2,1],[4,3,2,2,1]] => [5,1] => 0
[[4,3,2,1,1,1],[3,2,1]] => [4,1,1] => 1
[[5,4,3,2,2,1],[4,3,2,1,1]] => [5,1] => 0
[[5,4,3,2,1,1],[4,3,2,1]] => [5,1] => 0
[[6,5,4,3,2,1],[5,4,3,2,1]] => [6] => 0
[[7],[]] => [7] => 0
[[6,1],[]] => [6,1] => 0
[[7,1],[1]] => [7] => 0
[[5,2],[]] => [5,2] => 0
[[6,2],[1]] => [6,1] => 0
[[7,2],[2]] => [7] => 0
[[5,1,1],[]] => [5,1,1] => 1
[[6,2,1],[1,1]] => [6,1] => 0
[[6,1,1],[1]] => [6,1] => 0
[[7,2,1],[2,1]] => [7] => 0
[[4,3],[]] => [4,3] => 0
[[5,3],[1]] => [5,2] => 0
[[6,3],[2]] => [6,1] => 0
[[7,3],[3]] => [7] => 0
[[4,2,1],[]] => [4,2,1] => 0
[[5,3,1],[1,1]] => [5,2] => 0
[[5,2,1],[1]] => [5,2] => 0
[[6,3,1],[2,1]] => [6,1] => 0
[[6,2,1],[2]] => [6,1] => 0
[[7,3,1],[3,1]] => [7] => 0
[[5,2,2],[1,1]] => [5,1,1] => 1
[[6,3,2],[2,2]] => [6,1] => 0
[[6,2,2],[2,1]] => [6,1] => 0
[[7,3,2],[3,2]] => [7] => 0
[[4,1,1,1],[]] => [4,1,1,1] => 1
[[5,2,2,1],[1,1,1]] => [5,1,1] => 1
[[5,2,1,1],[1,1]] => [5,2] => 0
[[6,3,2,1],[2,2,1]] => [6,1] => 0
[[5,1,1,1],[1]] => [5,1,1] => 1
[[6,2,2,1],[2,1,1]] => [6,1] => 0
[[6,2,1,1],[2,1]] => [6,1] => 0
[[7,3,2,1],[3,2,1]] => [7] => 0
[[4,4],[1]] => [4,3] => 0
[[5,4],[2]] => [5,2] => 0
[[6,4],[3]] => [6,1] => 0
[[7,4],[4]] => [7] => 0
[[3,3,1],[]] => [3,3,1] => 3
[[4,4,1],[1,1]] => [4,3] => 0
[[4,3,1],[1]] => [4,3] => 0
[[5,4,1],[2,1]] => [5,2] => 0
[[5,3,1],[2]] => [5,2] => 0
[[6,4,1],[3,1]] => [6,1] => 0
[[6,3,1],[3]] => [6,1] => 0
[[7,4,1],[4,1]] => [7] => 0
[[3,2,2],[]] => [3,2,2] => 2
[[4,3,2],[1,1]] => [4,2,1] => 0
[[5,4,2],[2,2]] => [5,2] => 0
[[4,2,2],[1]] => [4,2,1] => 0
[[5,3,2],[2,1]] => [5,2] => 0
[[6,4,2],[3,2]] => [6,1] => 0
[[5,2,2],[2]] => [5,2] => 0
[[6,3,2],[3,1]] => [6,1] => 0
[[7,4,2],[4,2]] => [7] => 0
[[3,2,1,1],[]] => [3,2,1,1] => 1
[[4,3,2,1],[1,1,1]] => [4,2,1] => 0
[[4,3,1,1],[1,1]] => [4,3] => 0
[[5,4,2,1],[2,2,1]] => [5,2] => 0
[[4,2,1,1],[1]] => [4,2,1] => 0
[[5,3,2,1],[2,1,1]] => [5,2] => 0
[[5,3,1,1],[2,1]] => [5,2] => 0
[[6,4,2,1],[3,2,1]] => [6,1] => 0
[[5,2,1,1],[2]] => [5,1,1] => 1
[[6,3,2,1],[3,1,1]] => [6,1] => 0
[[6,3,1,1],[3,1]] => [6,1] => 0
[[7,4,2,1],[4,2,1]] => [7] => 0
[[5,3,3],[2,2]] => [5,1,1] => 1
[[6,4,3],[3,3]] => [6,1] => 0
[[6,3,3],[3,2]] => [6,1] => 0
[[7,4,3],[4,3]] => [7] => 0
[[4,2,2,1],[1,1]] => [4,2,1] => 0
[[5,3,3,1],[2,2,1]] => [5,1,1] => 1
[[5,3,2,1],[2,2]] => [5,2] => 0
[[6,4,3,1],[3,3,1]] => [6,1] => 0
[[5,2,2,1],[2,1]] => [5,2] => 0
[[6,3,3,1],[3,2,1]] => [6,1] => 0
[[6,3,2,1],[3,2]] => [6,1] => 0
[[7,4,3,1],[4,3,1]] => [7] => 0
[[4,2,2,2],[1,1,1]] => [4,1,1,1] => 1
[[5,3,3,2],[2,2,2]] => [5,1,1] => 1
[[5,3,2,2],[2,2,1]] => [5,2] => 0
[[6,4,3,2],[3,3,2]] => [6,1] => 0
[[5,2,2,2],[2,1,1]] => [5,1,1] => 1
[[6,3,3,2],[3,2,2]] => [6,1] => 0
[[6,3,2,2],[3,2,1]] => [6,1] => 0
[[7,4,3,2],[4,3,2]] => [7] => 0
[[3,1,1,1,1],[]] => [3,1,1,1,1] => 1
[[4,2,2,2,1],[1,1,1,1]] => [4,1,1,1] => 1
[[4,2,2,1,1],[1,1,1]] => [4,2,1] => 0
[[5,3,3,2,1],[2,2,2,1]] => [5,1,1] => 1
[[4,2,1,1,1],[1,1]] => [4,2,1] => 0
[[5,3,2,2,1],[2,2,1,1]] => [5,2] => 0
[[5,3,2,1,1],[2,2,1]] => [5,2] => 0
[[6,4,3,2,1],[3,3,2,1]] => [6,1] => 0
[[4,1,1,1,1],[1]] => [4,1,1,1] => 1
[[5,2,2,2,1],[2,1,1,1]] => [5,1,1] => 1
[[5,2,2,1,1],[2,1,1]] => [5,2] => 0
[[6,3,3,2,1],[3,2,2,1]] => [6,1] => 0
[[5,2,1,1,1],[2,1]] => [5,1,1] => 1
[[6,3,2,2,1],[3,2,1,1]] => [6,1] => 0
[[6,3,2,1,1],[3,2,1]] => [6,1] => 0
[[7,4,3,2,1],[4,3,2,1]] => [7] => 0
[[5,5],[3]] => [5,2] => 0
[[6,5],[4]] => [6,1] => 0
[[7,5],[5]] => [7] => 0
[[4,4,1],[2]] => [4,3] => 0
[[5,5,1],[3,1]] => [5,2] => 0
[[5,4,1],[3]] => [5,2] => 0
[[6,5,1],[4,1]] => [6,1] => 0
[[6,4,1],[4]] => [6,1] => 0
[[7,5,1],[5,1]] => [7] => 0
[[3,3,2],[1]] => [3,3,1] => 3
[[4,4,2],[2,1]] => [4,3] => 0
[[5,5,2],[3,2]] => [5,2] => 0
[[4,3,2],[2]] => [4,3] => 0
[[5,4,2],[3,1]] => [5,2] => 0
[[6,5,2],[4,2]] => [6,1] => 0
[[5,3,2],[3]] => [5,2] => 0
[[6,4,2],[4,1]] => [6,1] => 0
[[7,5,2],[5,2]] => [7] => 0
[[3,3,1,1],[1]] => [3,3,1] => 3
[[4,4,2,1],[2,1,1]] => [4,3] => 0
[[4,4,1,1],[2,1]] => [4,3] => 0
[[5,5,2,1],[3,2,1]] => [5,2] => 0
[[4,3,1,1],[2]] => [4,2,1] => 0
[[5,4,2,1],[3,1,1]] => [5,2] => 0
[[5,4,1,1],[3,1]] => [5,2] => 0
[[6,5,2,1],[4,2,1]] => [6,1] => 0
[[5,3,1,1],[3]] => [5,1,1] => 1
[[6,4,2,1],[4,1,1]] => [6,1] => 0
[[6,4,1,1],[4,1]] => [6,1] => 0
[[7,5,2,1],[5,2,1]] => [7] => 0
[[3,3,3],[1,1]] => [3,2,2] => 2
[[4,4,3],[2,2]] => [4,2,1] => 0
[[5,5,3],[3,3]] => [5,2] => 0
[[4,3,3],[2,1]] => [4,2,1] => 0
[[5,4,3],[3,2]] => [5,2] => 0
[[6,5,3],[4,3]] => [6,1] => 0
[[5,3,3],[3,1]] => [5,2] => 0
[[6,4,3],[4,2]] => [6,1] => 0
[[7,5,3],[5,3]] => [7] => 0
[[2,2,2,1],[]] => [2,2,2,1] => 2
[[3,3,3,1],[1,1,1]] => [3,2,2] => 2
[[3,3,2,1],[1,1]] => [3,3,1] => 3
[[4,4,3,1],[2,2,1]] => [4,2,1] => 0
[[4,4,2,1],[2,2]] => [4,3] => 0
[[5,5,3,1],[3,3,1]] => [5,2] => 0
[[3,2,2,1],[1]] => [3,2,2] => 2
[[4,3,3,1],[2,1,1]] => [4,2,1] => 0
[[4,3,2,1],[2,1]] => [4,3] => 0
[[5,4,3,1],[3,2,1]] => [5,2] => 0
[[5,4,2,1],[3,2]] => [5,2] => 0
[[6,5,3,1],[4,3,1]] => [6,1] => 0
[[4,2,2,1],[2]] => [4,2,1] => 0
[[5,3,3,1],[3,1,1]] => [5,2] => 0
[[5,3,2,1],[3,1]] => [5,2] => 0
[[6,4,3,1],[4,2,1]] => [6,1] => 0
[[6,4,2,1],[4,2]] => [6,1] => 0
[[7,5,3,1],[5,3,1]] => [7] => 0
[[3,3,2,2],[1,1,1]] => [3,2,1,1] => 1
[[4,4,3,2],[2,2,2]] => [4,2,1] => 0
[[4,4,2,2],[2,2,1]] => [4,3] => 0
[[5,5,3,2],[3,3,2]] => [5,2] => 0
[[4,3,2,2],[2,1,1]] => [4,2,1] => 0
[[5,4,3,2],[3,2,2]] => [5,2] => 0
[[5,4,2,2],[3,2,1]] => [5,2] => 0
[[6,5,3,2],[4,3,2]] => [6,1] => 0
[[5,3,2,2],[3,1,1]] => [5,1,1] => 1
[[6,4,3,2],[4,2,2]] => [6,1] => 0
[[6,4,2,2],[4,2,1]] => [6,1] => 0
[[7,5,3,2],[5,3,2]] => [7] => 0
[[2,2,1,1,1],[]] => [2,2,1,1,1] => 2
[[3,3,2,2,1],[1,1,1,1]] => [3,2,1,1] => 1
[[3,3,2,1,1],[1,1,1]] => [3,3,1] => 3
[[4,4,3,2,1],[2,2,2,1]] => [4,2,1] => 0
[[3,3,1,1,1],[1,1]] => [3,3,1] => 3
[[4,4,2,2,1],[2,2,1,1]] => [4,3] => 0
[[4,4,2,1,1],[2,2,1]] => [4,3] => 0
[[5,5,3,2,1],[3,3,2,1]] => [5,2] => 0
[[3,2,1,1,1],[1]] => [3,2,1,1] => 1
[[4,3,2,2,1],[2,1,1,1]] => [4,2,1] => 0
[[4,3,2,1,1],[2,1,1]] => [4,3] => 0
[[5,4,3,2,1],[3,2,2,1]] => [5,2] => 0
[[4,3,1,1,1],[2,1]] => [4,2,1] => 0
[[5,4,2,2,1],[3,2,1,1]] => [5,2] => 0
[[5,4,2,1,1],[3,2,1]] => [5,2] => 0
[[6,5,3,2,1],[4,3,2,1]] => [6,1] => 0
[[4,2,1,1,1],[2]] => [4,1,1,1] => 1
[[5,3,2,2,1],[3,1,1,1]] => [5,1,1] => 1
[[5,3,2,1,1],[3,1,1]] => [5,2] => 0
[[6,4,3,2,1],[4,2,2,1]] => [6,1] => 0
[[5,3,1,1,1],[3,1]] => [5,1,1] => 1
[[6,4,2,2,1],[4,2,1,1]] => [6,1] => 0
[[6,4,2,1,1],[4,2,1]] => [6,1] => 0
[[7,5,3,2,1],[5,3,2,1]] => [7] => 0
[[5,4,4],[3,3]] => [5,1,1] => 1
[[6,5,4],[4,4]] => [6,1] => 0
[[6,4,4],[4,3]] => [6,1] => 0
[[7,5,4],[5,4]] => [7] => 0
[[4,3,3,1],[2,2]] => [4,2,1] => 0
[[5,4,4,1],[3,3,1]] => [5,1,1] => 1
[[5,4,3,1],[3,3]] => [5,2] => 0
[[6,5,4,1],[4,4,1]] => [6,1] => 0
[[5,3,3,1],[3,2]] => [5,2] => 0
[[6,4,4,1],[4,3,1]] => [6,1] => 0
[[6,4,3,1],[4,3]] => [6,1] => 0
[[7,5,4,1],[5,4,1]] => [7] => 0
[[3,2,2,2],[1,1]] => [3,2,1,1] => 1
[[4,3,3,2],[2,2,1]] => [4,2,1] => 0
[[5,4,4,2],[3,3,2]] => [5,1,1] => 1
[[4,3,2,2],[2,2]] => [4,3] => 0
[[5,4,3,2],[3,3,1]] => [5,2] => 0
[[6,5,4,2],[4,4,2]] => [6,1] => 0
[[4,2,2,2],[2,1]] => [4,2,1] => 0
[[5,3,3,2],[3,2,1]] => [5,2] => 0
[[6,4,4,2],[4,3,2]] => [6,1] => 0
[[5,3,2,2],[3,2]] => [5,2] => 0
[[6,4,3,2],[4,3,1]] => [6,1] => 0
[[7,5,4,2],[5,4,2]] => [7] => 0
[[3,2,2,1,1],[1,1]] => [3,2,2] => 2
[[4,3,3,2,1],[2,2,1,1]] => [4,2,1] => 0
[[4,3,3,1,1],[2,2,1]] => [4,2,1] => 0
[[5,4,4,2,1],[3,3,2,1]] => [5,1,1] => 1
[[4,3,2,1,1],[2,2]] => [4,2,1] => 0
[[5,4,3,2,1],[3,3,1,1]] => [5,2] => 0
[[5,4,3,1,1],[3,3,1]] => [5,2] => 0
[[6,5,4,2,1],[4,4,2,1]] => [6,1] => 0
[[4,2,2,1,1],[2,1]] => [4,2,1] => 0
[[5,3,3,2,1],[3,2,1,1]] => [5,2] => 0
[[5,3,3,1,1],[3,2,1]] => [5,2] => 0
[[6,4,4,2,1],[4,3,2,1]] => [6,1] => 0
[[5,3,2,1,1],[3,2]] => [5,1,1] => 1
[[6,4,3,2,1],[4,3,1,1]] => [6,1] => 0
[[6,4,3,1,1],[4,3,1]] => [6,1] => 0
[[7,5,4,2,1],[5,4,2,1]] => [7] => 0
[[4,3,3,3],[2,2,2]] => [4,1,1,1] => 1
[[5,4,4,3],[3,3,3]] => [5,1,1] => 1
[[5,4,3,3],[3,3,2]] => [5,2] => 0
[[6,5,4,3],[4,4,3]] => [6,1] => 0
[[5,3,3,3],[3,2,2]] => [5,1,1] => 1
[[6,4,4,3],[4,3,3]] => [6,1] => 0
[[6,4,3,3],[4,3,2]] => [6,1] => 0
[[7,5,4,3],[5,4,3]] => [7] => 0
[[3,2,2,2,1],[1,1,1]] => [3,2,1,1] => 1
[[4,3,3,3,1],[2,2,2,1]] => [4,1,1,1] => 1
[[4,3,3,2,1],[2,2,2]] => [4,2,1] => 0
[[5,4,4,3,1],[3,3,3,1]] => [5,1,1] => 1
[[4,3,2,2,1],[2,2,1]] => [4,3] => 0
[[5,4,3,3,1],[3,3,2,1]] => [5,2] => 0
[[5,4,3,2,1],[3,3,2]] => [5,2] => 0
[[6,5,4,3,1],[4,4,3,1]] => [6,1] => 0
[[4,2,2,2,1],[2,1,1]] => [4,2,1] => 0
[[5,3,3,3,1],[3,2,2,1]] => [5,1,1] => 1
[[5,3,3,2,1],[3,2,2]] => [5,2] => 0
[[6,4,4,3,1],[4,3,3,1]] => [6,1] => 0
[[5,3,2,2,1],[3,2,1]] => [5,2] => 0
[[6,4,3,3,1],[4,3,2,1]] => [6,1] => 0
[[6,4,3,2,1],[4,3,2]] => [6,1] => 0
[[7,5,4,3,1],[5,4,3,1]] => [7] => 0
[[3,2,2,2,2],[1,1,1,1]] => [3,1,1,1,1] => 1
[[4,3,3,3,2],[2,2,2,2]] => [4,1,1,1] => 1
[[4,3,3,2,2],[2,2,2,1]] => [4,2,1] => 0
[[5,4,4,3,2],[3,3,3,2]] => [5,1,1] => 1
[[4,3,2,2,2],[2,2,1,1]] => [4,2,1] => 0
[[5,4,3,3,2],[3,3,2,2]] => [5,2] => 0
[[5,4,3,2,2],[3,3,2,1]] => [5,2] => 0
[[6,5,4,3,2],[4,4,3,2]] => [6,1] => 0
[[4,2,2,2,2],[2,1,1,1]] => [4,1,1,1] => 1
[[5,3,3,3,2],[3,2,2,2]] => [5,1,1] => 1
[[5,3,3,2,2],[3,2,2,1]] => [5,2] => 0
[[6,4,4,3,2],[4,3,3,2]] => [6,1] => 0
[[5,3,2,2,2],[3,2,1,1]] => [5,1,1] => 1
[[6,4,3,3,2],[4,3,2,2]] => [6,1] => 0
[[6,4,3,2,2],[4,3,2,1]] => [6,1] => 0
[[7,5,4,3,2],[5,4,3,2]] => [7] => 0
[[2,1,1,1,1,1],[]] => [2,1,1,1,1,1] => 1
[[3,2,2,2,2,1],[1,1,1,1,1]] => [3,1,1,1,1] => 1
[[3,2,2,2,1,1],[1,1,1,1]] => [3,2,1,1] => 1
[[4,3,3,3,2,1],[2,2,2,2,1]] => [4,1,1,1] => 1
[[3,2,2,1,1,1],[1,1,1]] => [3,2,2] => 2
[[4,3,3,2,2,1],[2,2,2,1,1]] => [4,2,1] => 0
[[4,3,3,2,1,1],[2,2,2,1]] => [4,2,1] => 0
[[5,4,4,3,2,1],[3,3,3,2,1]] => [5,1,1] => 1
[[3,2,1,1,1,1],[1,1]] => [3,2,1,1] => 1
[[4,3,2,2,2,1],[2,2,1,1,1]] => [4,2,1] => 0
[[4,3,2,2,1,1],[2,2,1,1]] => [4,3] => 0
[[5,4,3,3,2,1],[3,3,2,2,1]] => [5,2] => 0
[[4,3,2,1,1,1],[2,2,1]] => [4,2,1] => 0
[[5,4,3,2,2,1],[3,3,2,1,1]] => [5,2] => 0
[[5,4,3,2,1,1],[3,3,2,1]] => [5,2] => 0
[[6,5,4,3,2,1],[4,4,3,2,1]] => [6,1] => 0
[[3,1,1,1,1,1],[1]] => [3,1,1,1,1] => 1
[[4,2,2,2,2,1],[2,1,1,1,1]] => [4,1,1,1] => 1
[[4,2,2,2,1,1],[2,1,1,1]] => [4,2,1] => 0
[[5,3,3,3,2,1],[3,2,2,2,1]] => [5,1,1] => 1
[[4,2,2,1,1,1],[2,1,1]] => [4,2,1] => 0
[[5,3,3,2,2,1],[3,2,2,1,1]] => [5,2] => 0
[[5,3,3,2,1,1],[3,2,2,1]] => [5,2] => 0
[[6,4,4,3,2,1],[4,3,3,2,1]] => [6,1] => 0
[[4,2,1,1,1,1],[2,1]] => [4,1,1,1] => 1
[[5,3,2,2,2,1],[3,2,1,1,1]] => [5,1,1] => 1
[[5,3,2,2,1,1],[3,2,1,1]] => [5,2] => 0
[[6,4,3,3,2,1],[4,3,2,2,1]] => [6,1] => 0
[[5,3,2,1,1,1],[3,2,1]] => [5,1,1] => 1
[[6,4,3,2,2,1],[4,3,2,1,1]] => [6,1] => 0
[[6,4,3,2,1,1],[4,3,2,1]] => [6,1] => 0
[[7,5,4,3,2,1],[5,4,3,2,1]] => [7] => 0
[[6,6],[5]] => [6,1] => 0
[[7,6],[6]] => [7] => 0
[[5,5,1],[4]] => [5,2] => 0
[[6,6,1],[5,1]] => [6,1] => 0
[[6,5,1],[5]] => [6,1] => 0
[[7,6,1],[6,1]] => [7] => 0
[[4,4,2],[3]] => [4,3] => 0
[[5,5,2],[4,1]] => [5,2] => 0
[[6,6,2],[5,2]] => [6,1] => 0
[[5,4,2],[4]] => [5,2] => 0
[[6,5,2],[5,1]] => [6,1] => 0
[[7,6,2],[6,2]] => [7] => 0
[[4,4,1,1],[3]] => [4,2,1] => 0
[[5,5,2,1],[4,1,1]] => [5,2] => 0
[[5,5,1,1],[4,1]] => [5,2] => 0
[[6,6,2,1],[5,2,1]] => [6,1] => 0
[[5,4,1,1],[4]] => [5,1,1] => 1
[[6,5,2,1],[5,1,1]] => [6,1] => 0
[[6,5,1,1],[5,1]] => [6,1] => 0
[[7,6,2,1],[6,2,1]] => [7] => 0
[[3,3,3],[2]] => [3,3,1] => 3
[[4,4,3],[3,1]] => [4,3] => 0
[[5,5,3],[4,2]] => [5,2] => 0
[[6,6,3],[5,3]] => [6,1] => 0
[[4,3,3],[3]] => [4,3] => 0
[[5,4,3],[4,1]] => [5,2] => 0
[[6,5,3],[5,2]] => [6,1] => 0
[[7,6,3],[6,3]] => [7] => 0
[[3,3,2,1],[2]] => [3,3,1] => 3
[[4,4,3,1],[3,1,1]] => [4,3] => 0
[[4,4,2,1],[3,1]] => [4,3] => 0
[[5,5,3,1],[4,2,1]] => [5,2] => 0
[[5,5,2,1],[4,2]] => [5,2] => 0
[[6,6,3,1],[5,3,1]] => [6,1] => 0
[[4,3,2,1],[3]] => [4,2,1] => 0
[[5,4,3,1],[4,1,1]] => [5,2] => 0
[[5,4,2,1],[4,1]] => [5,2] => 0
[[6,5,3,1],[5,2,1]] => [6,1] => 0
[[6,5,2,1],[5,2]] => [6,1] => 0
[[7,6,3,1],[6,3,1]] => [7] => 0
[[4,4,2,2],[3,1,1]] => [4,2,1] => 0
[[5,5,3,2],[4,2,2]] => [5,2] => 0
[[5,5,2,2],[4,2,1]] => [5,2] => 0
[[6,6,3,2],[5,3,2]] => [6,1] => 0
[[5,4,2,2],[4,1,1]] => [5,1,1] => 1
[[6,5,3,2],[5,2,2]] => [6,1] => 0
[[6,5,2,2],[5,2,1]] => [6,1] => 0
[[7,6,3,2],[6,3,2]] => [7] => 0
[[3,3,1,1,1],[2]] => [3,2,1,1] => 1
[[4,4,2,2,1],[3,1,1,1]] => [4,2,1] => 0
[[4,4,2,1,1],[3,1,1]] => [4,3] => 0
[[5,5,3,2,1],[4,2,2,1]] => [5,2] => 0
[[4,4,1,1,1],[3,1]] => [4,2,1] => 0
[[5,5,2,2,1],[4,2,1,1]] => [5,2] => 0
[[5,5,2,1,1],[4,2,1]] => [5,2] => 0
[[6,6,3,2,1],[5,3,2,1]] => [6,1] => 0
[[4,3,1,1,1],[3]] => [4,1,1,1] => 1
[[5,4,2,2,1],[4,1,1,1]] => [5,1,1] => 1
[[5,4,2,1,1],[4,1,1]] => [5,2] => 0
[[6,5,3,2,1],[5,2,2,1]] => [6,1] => 0
[[5,4,1,1,1],[4,1]] => [5,1,1] => 1
[[6,5,2,2,1],[5,2,1,1]] => [6,1] => 0
[[6,5,2,1,1],[5,2,1]] => [6,1] => 0
[[7,6,3,2,1],[6,3,2,1]] => [7] => 0
[[4,4,4],[3,2]] => [4,2,1] => 0
[[5,5,4],[4,3]] => [5,2] => 0
[[6,6,4],[5,4]] => [6,1] => 0
[[5,4,4],[4,2]] => [5,2] => 0
[[6,5,4],[5,3]] => [6,1] => 0
[[7,6,4],[6,4]] => [7] => 0
[[3,3,3,1],[2,1]] => [3,3,1] => 3
[[4,4,4,1],[3,2,1]] => [4,2,1] => 0
[[4,4,3,1],[3,2]] => [4,3] => 0
[[5,5,4,1],[4,3,1]] => [5,2] => 0
[[5,5,3,1],[4,3]] => [5,2] => 0
[[6,6,4,1],[5,4,1]] => [6,1] => 0
[[4,3,3,1],[3,1]] => [4,3] => 0
[[5,4,4,1],[4,2,1]] => [5,2] => 0
[[5,4,3,1],[4,2]] => [5,2] => 0
[[6,5,4,1],[5,3,1]] => [6,1] => 0
[[6,5,3,1],[5,3]] => [6,1] => 0
[[7,6,4,1],[6,4,1]] => [7] => 0
[[2,2,2,2],[1]] => [2,2,2,1] => 2
[[3,3,3,2],[2,1,1]] => [3,2,2] => 2
[[4,4,4,2],[3,2,2]] => [4,2,1] => 0
[[3,3,2,2],[2,1]] => [3,3,1] => 3
[[4,4,3,2],[3,2,1]] => [4,3] => 0
[[5,5,4,2],[4,3,2]] => [5,2] => 0
[[4,4,2,2],[3,2]] => [4,3] => 0
[[5,5,3,2],[4,3,1]] => [5,2] => 0
[[6,6,4,2],[5,4,2]] => [6,1] => 0
[[3,2,2,2],[2]] => [3,2,2] => 2
[[4,3,3,2],[3,1,1]] => [4,2,1] => 0
[[5,4,4,2],[4,2,2]] => [5,2] => 0
[[4,3,2,2],[3,1]] => [4,2,1] => 0
[[5,4,3,2],[4,2,1]] => [5,2] => 0
[[6,5,4,2],[5,3,2]] => [6,1] => 0
[[5,4,2,2],[4,2]] => [5,2] => 0
[[6,5,3,2],[5,3,1]] => [6,1] => 0
[[7,6,4,2],[6,4,2]] => [7] => 0
[[2,2,2,1,1],[1]] => [2,2,2,1] => 2
[[3,3,3,2,1],[2,1,1,1]] => [3,2,2] => 2
[[3,3,3,1,1],[2,1,1]] => [3,3,1] => 3
[[4,4,4,2,1],[3,2,2,1]] => [4,2,1] => 0
[[3,3,2,1,1],[2,1]] => [3,3,1] => 3
[[4,4,3,2,1],[3,2,1,1]] => [4,3] => 0
[[4,4,3,1,1],[3,2,1]] => [4,3] => 0
[[5,5,4,2,1],[4,3,2,1]] => [5,2] => 0
[[4,4,2,1,1],[3,2]] => [4,2,1] => 0
[[5,5,3,2,1],[4,3,1,1]] => [5,2] => 0
[[5,5,3,1,1],[4,3,1]] => [5,2] => 0
[[6,6,4,2,1],[5,4,2,1]] => [6,1] => 0
[[3,2,2,1,1],[2]] => [3,2,1,1] => 1
[[4,3,3,2,1],[3,1,1,1]] => [4,2,1] => 0
[[4,3,3,1,1],[3,1,1]] => [4,3] => 0
[[5,4,4,2,1],[4,2,2,1]] => [5,2] => 0
[[4,3,2,1,1],[3,1]] => [4,2,1] => 0
[[5,4,3,2,1],[4,2,1,1]] => [5,2] => 0
[[5,4,3,1,1],[4,2,1]] => [5,2] => 0
[[6,5,4,2,1],[5,3,2,1]] => [6,1] => 0
[[5,4,2,1,1],[4,2]] => [5,1,1] => 1
[[6,5,3,2,1],[5,3,1,1]] => [6,1] => 0
[[6,5,3,1,1],[5,3,1]] => [6,1] => 0
[[7,6,4,2,1],[6,4,2,1]] => [7] => 0
[[4,4,3,3],[3,2,2]] => [4,2,1] => 0
[[5,5,4,3],[4,3,3]] => [5,2] => 0
[[5,5,3,3],[4,3,2]] => [5,2] => 0
[[6,6,4,3],[5,4,3]] => [6,1] => 0
[[5,4,3,3],[4,2,2]] => [5,1,1] => 1
[[6,5,4,3],[5,3,3]] => [6,1] => 0
[[6,5,3,3],[5,3,2]] => [6,1] => 0
[[7,6,4,3],[6,4,3]] => [7] => 0
[[3,3,2,2,1],[2,1,1]] => [3,3,1] => 3
[[4,4,3,3,1],[3,2,2,1]] => [4,2,1] => 0
[[4,4,3,2,1],[3,2,2]] => [4,3] => 0
[[5,5,4,3,1],[4,3,3,1]] => [5,2] => 0
[[4,4,2,2,1],[3,2,1]] => [4,3] => 0
[[5,5,3,3,1],[4,3,2,1]] => [5,2] => 0
[[5,5,3,2,1],[4,3,2]] => [5,2] => 0
[[6,6,4,3,1],[5,4,3,1]] => [6,1] => 0
[[4,3,2,2,1],[3,1,1]] => [4,2,1] => 0
[[5,4,3,3,1],[4,2,2,1]] => [5,1,1] => 1
[[5,4,3,2,1],[4,2,2]] => [5,2] => 0
[[6,5,4,3,1],[5,3,3,1]] => [6,1] => 0
[[5,4,2,2,1],[4,2,1]] => [5,2] => 0
[[6,5,3,3,1],[5,3,2,1]] => [6,1] => 0
[[6,5,3,2,1],[5,3,2]] => [6,1] => 0
[[7,6,4,3,1],[6,4,3,1]] => [7] => 0
[[3,3,2,2,2],[2,1,1,1]] => [3,2,1,1] => 1
[[4,4,3,3,2],[3,2,2,2]] => [4,2,1] => 0
[[4,4,3,2,2],[3,2,2,1]] => [4,3] => 0
[[5,5,4,3,2],[4,3,3,2]] => [5,2] => 0
[[4,4,2,2,2],[3,2,1,1]] => [4,2,1] => 0
[[5,5,3,3,2],[4,3,2,2]] => [5,2] => 0
[[5,5,3,2,2],[4,3,2,1]] => [5,2] => 0
[[6,6,4,3,2],[5,4,3,2]] => [6,1] => 0
[[4,3,2,2,2],[3,1,1,1]] => [4,1,1,1] => 1
[[5,4,3,3,2],[4,2,2,2]] => [5,1,1] => 1
[[5,4,3,2,2],[4,2,2,1]] => [5,2] => 0
[[6,5,4,3,2],[5,3,3,2]] => [6,1] => 0
[[5,4,2,2,2],[4,2,1,1]] => [5,1,1] => 1
[[6,5,3,3,2],[5,3,2,2]] => [6,1] => 0
[[6,5,3,2,2],[5,3,2,1]] => [6,1] => 0
[[7,6,4,3,2],[6,4,3,2]] => [7] => 0
[[2,2,1,1,1,1],[1]] => [2,2,1,1,1] => 2
[[3,3,2,2,2,1],[2,1,1,1,1]] => [3,2,1,1] => 1
[[3,3,2,2,1,1],[2,1,1,1]] => [3,3,1] => 3
[[4,4,3,3,2,1],[3,2,2,2,1]] => [4,2,1] => 0
[[3,3,2,1,1,1],[2,1,1]] => [3,3,1] => 3
[[4,4,3,2,2,1],[3,2,2,1,1]] => [4,3] => 0
[[4,4,3,2,1,1],[3,2,2,1]] => [4,3] => 0
[[5,5,4,3,2,1],[4,3,3,2,1]] => [5,2] => 0
[[3,3,1,1,1,1],[2,1]] => [3,2,1,1] => 1
[[4,4,2,2,2,1],[3,2,1,1,1]] => [4,2,1] => 0
[[4,4,2,2,1,1],[3,2,1,1]] => [4,3] => 0
[[5,5,3,3,2,1],[4,3,2,2,1]] => [5,2] => 0
[[4,4,2,1,1,1],[3,2,1]] => [4,2,1] => 0
[[5,5,3,2,2,1],[4,3,2,1,1]] => [5,2] => 0
[[5,5,3,2,1,1],[4,3,2,1]] => [5,2] => 0
[[6,6,4,3,2,1],[5,4,3,2,1]] => [6,1] => 0
[[3,2,1,1,1,1],[2]] => [3,1,1,1,1] => 1
[[4,3,2,2,2,1],[3,1,1,1,1]] => [4,1,1,1] => 1
[[4,3,2,2,1,1],[3,1,1,1]] => [4,2,1] => 0
[[5,4,3,3,2,1],[4,2,2,2,1]] => [5,1,1] => 1
[[4,3,2,1,1,1],[3,1,1]] => [4,2,1] => 0
[[5,4,3,2,2,1],[4,2,2,1,1]] => [5,2] => 0
[[5,4,3,2,1,1],[4,2,2,1]] => [5,2] => 0
[[6,5,4,3,2,1],[5,3,3,2,1]] => [6,1] => 0
[[4,3,1,1,1,1],[3,1]] => [4,1,1,1] => 1
[[5,4,2,2,2,1],[4,2,1,1,1]] => [5,1,1] => 1
[[5,4,2,2,1,1],[4,2,1,1]] => [5,2] => 0
[[6,5,3,3,2,1],[5,3,2,2,1]] => [6,1] => 0
[[5,4,2,1,1,1],[4,2,1]] => [5,1,1] => 1
[[6,5,3,2,2,1],[5,3,2,1,1]] => [6,1] => 0
[[6,5,3,2,1,1],[5,3,2,1]] => [6,1] => 0
[[7,6,4,3,2,1],[6,4,3,2,1]] => [7] => 0
[[5,5,5],[4,4]] => [5,1,1] => 1
[[6,6,5],[5,5]] => [6,1] => 0
[[6,5,5],[5,4]] => [6,1] => 0
[[7,6,5],[6,5]] => [7] => 0
[[4,4,4,1],[3,3]] => [4,2,1] => 0
[[5,5,5,1],[4,4,1]] => [5,1,1] => 1
[[5,5,4,1],[4,4]] => [5,2] => 0
[[6,6,5,1],[5,5,1]] => [6,1] => 0
[[5,4,4,1],[4,3]] => [5,2] => 0
[[6,5,5,1],[5,4,1]] => [6,1] => 0
[[6,5,4,1],[5,4]] => [6,1] => 0
[[7,6,5,1],[6,5,1]] => [7] => 0
[[3,3,3,2],[2,2]] => [3,3,1] => 3
[[4,4,4,2],[3,3,1]] => [4,2,1] => 0
[[5,5,5,2],[4,4,2]] => [5,1,1] => 1
[[4,4,3,2],[3,3]] => [4,3] => 0
[[5,5,4,2],[4,4,1]] => [5,2] => 0
[[6,6,5,2],[5,5,2]] => [6,1] => 0
[[4,3,3,2],[3,2]] => [4,3] => 0
[[5,4,4,2],[4,3,1]] => [5,2] => 0
[[6,5,5,2],[5,4,2]] => [6,1] => 0
[[5,4,3,2],[4,3]] => [5,2] => 0
[[6,5,4,2],[5,4,1]] => [6,1] => 0
[[7,6,5,2],[6,5,2]] => [7] => 0
[[3,3,3,1,1],[2,2]] => [3,2,2] => 2
[[4,4,4,2,1],[3,3,1,1]] => [4,2,1] => 0
[[4,4,4,1,1],[3,3,1]] => [4,2,1] => 0
[[5,5,5,2,1],[4,4,2,1]] => [5,1,1] => 1
[[4,4,3,1,1],[3,3]] => [4,2,1] => 0
[[5,5,4,2,1],[4,4,1,1]] => [5,2] => 0
[[5,5,4,1,1],[4,4,1]] => [5,2] => 0
[[6,6,5,2,1],[5,5,2,1]] => [6,1] => 0
[[4,3,3,1,1],[3,2]] => [4,2,1] => 0
[[5,4,4,2,1],[4,3,1,1]] => [5,2] => 0
[[5,4,4,1,1],[4,3,1]] => [5,2] => 0
[[6,5,5,2,1],[5,4,2,1]] => [6,1] => 0
[[5,4,3,1,1],[4,3]] => [5,1,1] => 1
[[6,5,4,2,1],[5,4,1,1]] => [6,1] => 0
[[6,5,4,1,1],[5,4,1]] => [6,1] => 0
[[7,6,5,2,1],[6,5,2,1]] => [7] => 0
[[3,3,3,3],[2,2,1]] => [3,2,1,1] => 1
[[4,4,4,3],[3,3,2]] => [4,2,1] => 0
[[5,5,5,3],[4,4,3]] => [5,1,1] => 1
[[4,4,3,3],[3,3,1]] => [4,3] => 0
[[5,5,4,3],[4,4,2]] => [5,2] => 0
[[6,6,5,3],[5,5,3]] => [6,1] => 0
[[4,3,3,3],[3,2,1]] => [4,2,1] => 0
[[5,4,4,3],[4,3,2]] => [5,2] => 0
[[6,5,5,3],[5,4,3]] => [6,1] => 0
[[5,4,3,3],[4,3,1]] => [5,2] => 0
[[6,5,4,3],[5,4,2]] => [6,1] => 0
[[7,6,5,3],[6,5,3]] => [7] => 0
[[2,2,2,2,1],[1,1]] => [2,2,2,1] => 2
[[3,3,3,3,1],[2,2,1,1]] => [3,2,1,1] => 1
[[3,3,3,2,1],[2,2,1]] => [3,3,1] => 3
[[4,4,4,3,1],[3,3,2,1]] => [4,2,1] => 0
[[4,4,4,2,1],[3,3,2]] => [4,2,1] => 0
[[5,5,5,3,1],[4,4,3,1]] => [5,1,1] => 1
[[3,3,2,2,1],[2,2]] => [3,3,1] => 3
[[4,4,3,3,1],[3,3,1,1]] => [4,3] => 0
[[4,4,3,2,1],[3,3,1]] => [4,3] => 0
[[5,5,4,3,1],[4,4,2,1]] => [5,2] => 0
[[5,5,4,2,1],[4,4,2]] => [5,2] => 0
[[6,6,5,3,1],[5,5,3,1]] => [6,1] => 0
[[3,2,2,2,1],[2,1]] => [3,2,2] => 2
[[4,3,3,3,1],[3,2,1,1]] => [4,2,1] => 0
[[4,3,3,2,1],[3,2,1]] => [4,3] => 0
[[5,4,4,3,1],[4,3,2,1]] => [5,2] => 0
[[5,4,4,2,1],[4,3,2]] => [5,2] => 0
[[6,5,5,3,1],[5,4,3,1]] => [6,1] => 0
[[4,3,2,2,1],[3,2]] => [4,2,1] => 0
[[5,4,3,3,1],[4,3,1,1]] => [5,2] => 0
[[5,4,3,2,1],[4,3,1]] => [5,2] => 0
[[6,5,4,3,1],[5,4,2,1]] => [6,1] => 0
[[6,5,4,2,1],[5,4,2]] => [6,1] => 0
[[7,6,5,3,1],[6,5,3,1]] => [7] => 0
[[3,3,3,2,2],[2,2,1,1]] => [3,2,2] => 2
[[4,4,4,3,2],[3,3,2,2]] => [4,2,1] => 0
[[4,4,4,2,2],[3,3,2,1]] => [4,2,1] => 0
[[5,5,5,3,2],[4,4,3,2]] => [5,1,1] => 1
[[4,4,3,2,2],[3,3,1,1]] => [4,2,1] => 0
[[5,5,4,3,2],[4,4,2,2]] => [5,2] => 0
[[5,5,4,2,2],[4,4,2,1]] => [5,2] => 0
[[6,6,5,3,2],[5,5,3,2]] => [6,1] => 0
[[4,3,3,2,2],[3,2,1,1]] => [4,2,1] => 0
[[5,4,4,3,2],[4,3,2,2]] => [5,2] => 0
[[5,4,4,2,2],[4,3,2,1]] => [5,2] => 0
[[6,5,5,3,2],[5,4,3,2]] => [6,1] => 0
[[5,4,3,2,2],[4,3,1,1]] => [5,1,1] => 1
[[6,5,4,3,2],[5,4,2,2]] => [6,1] => 0
[[6,5,4,2,2],[5,4,2,1]] => [6,1] => 0
[[7,6,5,3,2],[6,5,3,2]] => [7] => 0
[[2,2,2,1,1,1],[1,1]] => [2,2,2,1] => 2
[[3,3,3,2,2,1],[2,2,1,1,1]] => [3,2,2] => 2
[[3,3,3,2,1,1],[2,2,1,1]] => [3,3,1] => 3
[[4,4,4,3,2,1],[3,3,2,2,1]] => [4,2,1] => 0
[[3,3,3,1,1,1],[2,2,1]] => [3,2,2] => 2
[[4,4,4,2,2,1],[3,3,2,1,1]] => [4,2,1] => 0
[[4,4,4,2,1,1],[3,3,2,1]] => [4,2,1] => 0
[[5,5,5,3,2,1],[4,4,3,2,1]] => [5,1,1] => 1
[[3,3,2,1,1,1],[2,2]] => [3,2,1,1] => 1
[[4,4,3,2,2,1],[3,3,1,1,1]] => [4,2,1] => 0
[[4,4,3,2,1,1],[3,3,1,1]] => [4,3] => 0
[[5,5,4,3,2,1],[4,4,2,2,1]] => [5,2] => 0
[[4,4,3,1,1,1],[3,3,1]] => [4,2,1] => 0
[[5,5,4,2,2,1],[4,4,2,1,1]] => [5,2] => 0
[[5,5,4,2,1,1],[4,4,2,1]] => [5,2] => 0
[[6,6,5,3,2,1],[5,5,3,2,1]] => [6,1] => 0
[[3,2,2,1,1,1],[2,1]] => [3,2,1,1] => 1
[[4,3,3,2,2,1],[3,2,1,1,1]] => [4,2,1] => 0
[[4,3,3,2,1,1],[3,2,1,1]] => [4,3] => 0
[[5,4,4,3,2,1],[4,3,2,2,1]] => [5,2] => 0
[[4,3,3,1,1,1],[3,2,1]] => [4,2,1] => 0
[[5,4,4,2,2,1],[4,3,2,1,1]] => [5,2] => 0
[[5,4,4,2,1,1],[4,3,2,1]] => [5,2] => 0
[[6,5,5,3,2,1],[5,4,3,2,1]] => [6,1] => 0
[[4,3,2,1,1,1],[3,2]] => [4,1,1,1] => 1
[[5,4,3,2,2,1],[4,3,1,1,1]] => [5,1,1] => 1
[[5,4,3,2,1,1],[4,3,1,1]] => [5,2] => 0
[[6,5,4,3,2,1],[5,4,2,2,1]] => [6,1] => 0
[[5,4,3,1,1,1],[4,3,1]] => [5,1,1] => 1
[[6,5,4,2,2,1],[5,4,2,1,1]] => [6,1] => 0
[[6,5,4,2,1,1],[5,4,2,1]] => [6,1] => 0
[[7,6,5,3,2,1],[6,5,3,2,1]] => [7] => 0
[[4,4,4,4],[3,3,3]] => [4,1,1,1] => 1
[[5,5,5,4],[4,4,4]] => [5,1,1] => 1
[[5,5,4,4],[4,4,3]] => [5,2] => 0
[[6,6,5,4],[5,5,4]] => [6,1] => 0
[[5,4,4,4],[4,3,3]] => [5,1,1] => 1
[[6,5,5,4],[5,4,4]] => [6,1] => 0
[[6,5,4,4],[5,4,3]] => [6,1] => 0
[[7,6,5,4],[6,5,4]] => [7] => 0
[[3,3,3,3,1],[2,2,2]] => [3,2,1,1] => 1
[[4,4,4,4,1],[3,3,3,1]] => [4,1,1,1] => 1
[[4,4,4,3,1],[3,3,3]] => [4,2,1] => 0
[[5,5,5,4,1],[4,4,4,1]] => [5,1,1] => 1
[[4,4,3,3,1],[3,3,2]] => [4,3] => 0
[[5,5,4,4,1],[4,4,3,1]] => [5,2] => 0
[[5,5,4,3,1],[4,4,3]] => [5,2] => 0
[[6,6,5,4,1],[5,5,4,1]] => [6,1] => 0
[[4,3,3,3,1],[3,2,2]] => [4,2,1] => 0
[[5,4,4,4,1],[4,3,3,1]] => [5,1,1] => 1
[[5,4,4,3,1],[4,3,3]] => [5,2] => 0
[[6,5,5,4,1],[5,4,4,1]] => [6,1] => 0
[[5,4,3,3,1],[4,3,2]] => [5,2] => 0
[[6,5,4,4,1],[5,4,3,1]] => [6,1] => 0
[[6,5,4,3,1],[5,4,3]] => [6,1] => 0
[[7,6,5,4,1],[6,5,4,1]] => [7] => 0
[[2,2,2,2,2],[1,1,1]] => [2,2,1,1,1] => 2
[[3,3,3,3,2],[2,2,2,1]] => [3,2,1,1] => 1
[[4,4,4,4,2],[3,3,3,2]] => [4,1,1,1] => 1
[[3,3,3,2,2],[2,2,2]] => [3,3,1] => 3
[[4,4,4,3,2],[3,3,3,1]] => [4,2,1] => 0
[[5,5,5,4,2],[4,4,4,2]] => [5,1,1] => 1
[[3,3,2,2,2],[2,2,1]] => [3,3,1] => 3
[[4,4,3,3,2],[3,3,2,1]] => [4,3] => 0
[[5,5,4,4,2],[4,4,3,2]] => [5,2] => 0
[[4,4,3,2,2],[3,3,2]] => [4,3] => 0
[[5,5,4,3,2],[4,4,3,1]] => [5,2] => 0
[[6,6,5,4,2],[5,5,4,2]] => [6,1] => 0
[[3,2,2,2,2],[2,1,1]] => [3,2,1,1] => 1
[[4,3,3,3,2],[3,2,2,1]] => [4,2,1] => 0
[[5,4,4,4,2],[4,3,3,2]] => [5,1,1] => 1
[[4,3,3,2,2],[3,2,2]] => [4,3] => 0
[[5,4,4,3,2],[4,3,3,1]] => [5,2] => 0
[[6,5,5,4,2],[5,4,4,2]] => [6,1] => 0
[[4,3,2,2,2],[3,2,1]] => [4,2,1] => 0
[[5,4,3,3,2],[4,3,2,1]] => [5,2] => 0
[[6,5,4,4,2],[5,4,3,2]] => [6,1] => 0
[[5,4,3,2,2],[4,3,2]] => [5,2] => 0
[[6,5,4,3,2],[5,4,3,1]] => [6,1] => 0
[[7,6,5,4,2],[6,5,4,2]] => [7] => 0
[[2,2,2,2,1,1],[1,1,1]] => [2,2,2,1] => 2
[[3,3,3,3,2,1],[2,2,2,1,1]] => [3,2,1,1] => 1
[[3,3,3,3,1,1],[2,2,2,1]] => [3,2,1,1] => 1
[[4,4,4,4,2,1],[3,3,3,2,1]] => [4,1,1,1] => 1
[[3,3,3,2,1,1],[2,2,2]] => [3,2,2] => 2
[[4,4,4,3,2,1],[3,3,3,1,1]] => [4,2,1] => 0
[[4,4,4,3,1,1],[3,3,3,1]] => [4,2,1] => 0
[[5,5,5,4,2,1],[4,4,4,2,1]] => [5,1,1] => 1
[[3,3,2,2,1,1],[2,2,1]] => [3,3,1] => 3
[[4,4,3,3,2,1],[3,3,2,1,1]] => [4,3] => 0
[[4,4,3,3,1,1],[3,3,2,1]] => [4,3] => 0
[[5,5,4,4,2,1],[4,4,3,2,1]] => [5,2] => 0
[[4,4,3,2,1,1],[3,3,2]] => [4,2,1] => 0
[[5,5,4,3,2,1],[4,4,3,1,1]] => [5,2] => 0
[[5,5,4,3,1,1],[4,4,3,1]] => [5,2] => 0
[[6,6,5,4,2,1],[5,5,4,2,1]] => [6,1] => 0
[[3,2,2,2,1,1],[2,1,1]] => [3,2,2] => 2
[[4,3,3,3,2,1],[3,2,2,1,1]] => [4,2,1] => 0
[[4,3,3,3,1,1],[3,2,2,1]] => [4,2,1] => 0
[[5,4,4,4,2,1],[4,3,3,2,1]] => [5,1,1] => 1
[[4,3,3,2,1,1],[3,2,2]] => [4,2,1] => 0
[[5,4,4,3,2,1],[4,3,3,1,1]] => [5,2] => 0
[[5,4,4,3,1,1],[4,3,3,1]] => [5,2] => 0
[[6,5,5,4,2,1],[5,4,4,2,1]] => [6,1] => 0
[[4,3,2,2,1,1],[3,2,1]] => [4,2,1] => 0
[[5,4,3,3,2,1],[4,3,2,1,1]] => [5,2] => 0
[[5,4,3,3,1,1],[4,3,2,1]] => [5,2] => 0
[[6,5,4,4,2,1],[5,4,3,2,1]] => [6,1] => 0
[[5,4,3,2,1,1],[4,3,2]] => [5,1,1] => 1
[[6,5,4,3,2,1],[5,4,3,1,1]] => [6,1] => 0
[[6,5,4,3,1,1],[5,4,3,1]] => [6,1] => 0
[[7,6,5,4,2,1],[6,5,4,2,1]] => [7] => 0
[[3,3,3,3,3],[2,2,2,2]] => [3,1,1,1,1] => 1
[[4,4,4,4,3],[3,3,3,3]] => [4,1,1,1] => 1
[[4,4,4,3,3],[3,3,3,2]] => [4,2,1] => 0
[[5,5,5,4,3],[4,4,4,3]] => [5,1,1] => 1
[[4,4,3,3,3],[3,3,2,2]] => [4,2,1] => 0
[[5,5,4,4,3],[4,4,3,3]] => [5,2] => 0
[[5,5,4,3,3],[4,4,3,2]] => [5,2] => 0
[[6,6,5,4,3],[5,5,4,3]] => [6,1] => 0
[[4,3,3,3,3],[3,2,2,2]] => [4,1,1,1] => 1
[[5,4,4,4,3],[4,3,3,3]] => [5,1,1] => 1
[[5,4,4,3,3],[4,3,3,2]] => [5,2] => 0
[[6,5,5,4,3],[5,4,4,3]] => [6,1] => 0
[[5,4,3,3,3],[4,3,2,2]] => [5,1,1] => 1
[[6,5,4,4,3],[5,4,3,3]] => [6,1] => 0
[[6,5,4,3,3],[5,4,3,2]] => [6,1] => 0
[[7,6,5,4,3],[6,5,4,3]] => [7] => 0
[[2,2,2,2,2,1],[1,1,1,1]] => [2,2,1,1,1] => 2
[[3,3,3,3,3,1],[2,2,2,2,1]] => [3,1,1,1,1] => 1
[[3,3,3,3,2,1],[2,2,2,2]] => [3,2,1,1] => 1
[[4,4,4,4,3,1],[3,3,3,3,1]] => [4,1,1,1] => 1
[[3,3,3,2,2,1],[2,2,2,1]] => [3,3,1] => 3
[[4,4,4,3,3,1],[3,3,3,2,1]] => [4,2,1] => 0
[[4,4,4,3,2,1],[3,3,3,2]] => [4,2,1] => 0
[[5,5,5,4,3,1],[4,4,4,3,1]] => [5,1,1] => 1
[[3,3,2,2,2,1],[2,2,1,1]] => [3,3,1] => 3
[[4,4,3,3,3,1],[3,3,2,2,1]] => [4,2,1] => 0
[[4,4,3,3,2,1],[3,3,2,2]] => [4,3] => 0
[[5,5,4,4,3,1],[4,4,3,3,1]] => [5,2] => 0
[[4,4,3,2,2,1],[3,3,2,1]] => [4,3] => 0
[[5,5,4,3,3,1],[4,4,3,2,1]] => [5,2] => 0
[[5,5,4,3,2,1],[4,4,3,2]] => [5,2] => 0
[[6,6,5,4,3,1],[5,5,4,3,1]] => [6,1] => 0
[[3,2,2,2,2,1],[2,1,1,1]] => [3,2,1,1] => 1
[[4,3,3,3,3,1],[3,2,2,2,1]] => [4,1,1,1] => 1
[[4,3,3,3,2,1],[3,2,2,2]] => [4,2,1] => 0
[[5,4,4,4,3,1],[4,3,3,3,1]] => [5,1,1] => 1
[[4,3,3,2,2,1],[3,2,2,1]] => [4,3] => 0
[[5,4,4,3,3,1],[4,3,3,2,1]] => [5,2] => 0
[[5,4,4,3,2,1],[4,3,3,2]] => [5,2] => 0
[[6,5,5,4,3,1],[5,4,4,3,1]] => [6,1] => 0
[[4,3,2,2,2,1],[3,2,1,1]] => [4,2,1] => 0
[[5,4,3,3,3,1],[4,3,2,2,1]] => [5,1,1] => 1
[[5,4,3,3,2,1],[4,3,2,2]] => [5,2] => 0
[[6,5,4,4,3,1],[5,4,3,3,1]] => [6,1] => 0
[[5,4,3,2,2,1],[4,3,2,1]] => [5,2] => 0
[[6,5,4,3,3,1],[5,4,3,2,1]] => [6,1] => 0
[[6,5,4,3,2,1],[5,4,3,2]] => [6,1] => 0
[[7,6,5,4,3,1],[6,5,4,3,1]] => [7] => 0
[[2,2,2,2,2,2],[1,1,1,1,1]] => [2,1,1,1,1,1] => 1
[[3,3,3,3,3,2],[2,2,2,2,2]] => [3,1,1,1,1] => 1
[[3,3,3,3,2,2],[2,2,2,2,1]] => [3,2,1,1] => 1
[[4,4,4,4,3,2],[3,3,3,3,2]] => [4,1,1,1] => 1
[[3,3,3,2,2,2],[2,2,2,1,1]] => [3,2,2] => 2
[[4,4,4,3,3,2],[3,3,3,2,2]] => [4,2,1] => 0
[[4,4,4,3,2,2],[3,3,3,2,1]] => [4,2,1] => 0
[[5,5,5,4,3,2],[4,4,4,3,2]] => [5,1,1] => 1
[[3,3,2,2,2,2],[2,2,1,1,1]] => [3,2,1,1] => 1
[[4,4,3,3,3,2],[3,3,2,2,2]] => [4,2,1] => 0
[[4,4,3,3,2,2],[3,3,2,2,1]] => [4,3] => 0
[[5,5,4,4,3,2],[4,4,3,3,2]] => [5,2] => 0
[[4,4,3,2,2,2],[3,3,2,1,1]] => [4,2,1] => 0
[[5,5,4,3,3,2],[4,4,3,2,2]] => [5,2] => 0
[[5,5,4,3,2,2],[4,4,3,2,1]] => [5,2] => 0
[[6,6,5,4,3,2],[5,5,4,3,2]] => [6,1] => 0
[[3,2,2,2,2,2],[2,1,1,1,1]] => [3,1,1,1,1] => 1
[[4,3,3,3,3,2],[3,2,2,2,2]] => [4,1,1,1] => 1
[[4,3,3,3,2,2],[3,2,2,2,1]] => [4,2,1] => 0
[[5,4,4,4,3,2],[4,3,3,3,2]] => [5,1,1] => 1
[[4,3,3,2,2,2],[3,2,2,1,1]] => [4,2,1] => 0
[[5,4,4,3,3,2],[4,3,3,2,2]] => [5,2] => 0
[[5,4,4,3,2,2],[4,3,3,2,1]] => [5,2] => 0
[[6,5,5,4,3,2],[5,4,4,3,2]] => [6,1] => 0
[[4,3,2,2,2,2],[3,2,1,1,1]] => [4,1,1,1] => 1
[[5,4,3,3,3,2],[4,3,2,2,2]] => [5,1,1] => 1
[[5,4,3,3,2,2],[4,3,2,2,1]] => [5,2] => 0
[[6,5,4,4,3,2],[5,4,3,3,2]] => [6,1] => 0
[[5,4,3,2,2,2],[4,3,2,1,1]] => [5,1,1] => 1
[[6,5,4,3,3,2],[5,4,3,2,2]] => [6,1] => 0
[[6,5,4,3,2,2],[5,4,3,2,1]] => [6,1] => 0
[[7,6,5,4,3,2],[6,5,4,3,2]] => [7] => 0
[[1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1] => 1
[[2,2,2,2,2,2,1],[1,1,1,1,1,1]] => [2,1,1,1,1,1] => 1
[[2,2,2,2,2,1,1],[1,1,1,1,1]] => [2,2,1,1,1] => 2
[[3,3,3,3,3,2,1],[2,2,2,2,2,1]] => [3,1,1,1,1] => 1
[[2,2,2,2,1,1,1],[1,1,1,1]] => [2,2,2,1] => 2
[[3,3,3,3,2,2,1],[2,2,2,2,1,1]] => [3,2,1,1] => 1
[[3,3,3,3,2,1,1],[2,2,2,2,1]] => [3,2,1,1] => 1
[[4,4,4,4,3,2,1],[3,3,3,3,2,1]] => [4,1,1,1] => 1
[[2,2,2,1,1,1,1],[1,1,1]] => [2,2,2,1] => 2
[[3,3,3,2,2,2,1],[2,2,2,1,1,1]] => [3,2,2] => 2
[[3,3,3,2,2,1,1],[2,2,2,1,1]] => [3,3,1] => 3
[[4,4,4,3,3,2,1],[3,3,3,2,2,1]] => [4,2,1] => 0
[[3,3,3,2,1,1,1],[2,2,2,1]] => [3,2,2] => 2
[[4,4,4,3,2,2,1],[3,3,3,2,1,1]] => [4,2,1] => 0
[[4,4,4,3,2,1,1],[3,3,3,2,1]] => [4,2,1] => 0
[[5,5,5,4,3,2,1],[4,4,4,3,2,1]] => [5,1,1] => 1
[[2,2,1,1,1,1,1],[1,1]] => [2,2,1,1,1] => 2
[[3,3,2,2,2,2,1],[2,2,1,1,1,1]] => [3,2,1,1] => 1
[[3,3,2,2,2,1,1],[2,2,1,1,1]] => [3,3,1] => 3
[[4,4,3,3,3,2,1],[3,3,2,2,2,1]] => [4,2,1] => 0
[[3,3,2,2,1,1,1],[2,2,1,1]] => [3,3,1] => 3
[[4,4,3,3,2,2,1],[3,3,2,2,1,1]] => [4,3] => 0
[[4,4,3,3,2,1,1],[3,3,2,2,1]] => [4,3] => 0
[[5,5,4,4,3,2,1],[4,4,3,3,2,1]] => [5,2] => 0
[[3,3,2,1,1,1,1],[2,2,1]] => [3,2,1,1] => 1
[[4,4,3,2,2,2,1],[3,3,2,1,1,1]] => [4,2,1] => 0
[[4,4,3,2,2,1,1],[3,3,2,1,1]] => [4,3] => 0
[[5,5,4,3,3,2,1],[4,4,3,2,2,1]] => [5,2] => 0
[[4,4,3,2,1,1,1],[3,3,2,1]] => [4,2,1] => 0
[[5,5,4,3,2,2,1],[4,4,3,2,1,1]] => [5,2] => 0
[[5,5,4,3,2,1,1],[4,4,3,2,1]] => [5,2] => 0
[[6,6,5,4,3,2,1],[5,5,4,3,2,1]] => [6,1] => 0
[[2,1,1,1,1,1,1],[1]] => [2,1,1,1,1,1] => 1
[[3,2,2,2,2,2,1],[2,1,1,1,1,1]] => [3,1,1,1,1] => 1
[[3,2,2,2,2,1,1],[2,1,1,1,1]] => [3,2,1,1] => 1
[[4,3,3,3,3,2,1],[3,2,2,2,2,1]] => [4,1,1,1] => 1
[[3,2,2,2,1,1,1],[2,1,1,1]] => [3,2,2] => 2
[[4,3,3,3,2,2,1],[3,2,2,2,1,1]] => [4,2,1] => 0
[[4,3,3,3,2,1,1],[3,2,2,2,1]] => [4,2,1] => 0
[[5,4,4,4,3,2,1],[4,3,3,3,2,1]] => [5,1,1] => 1
[[3,2,2,1,1,1,1],[2,1,1]] => [3,2,1,1] => 1
[[4,3,3,2,2,2,1],[3,2,2,1,1,1]] => [4,2,1] => 0
[[4,3,3,2,2,1,1],[3,2,2,1,1]] => [4,3] => 0
[[5,4,4,3,3,2,1],[4,3,3,2,2,1]] => [5,2] => 0
[[4,3,3,2,1,1,1],[3,2,2,1]] => [4,2,1] => 0
[[5,4,4,3,2,2,1],[4,3,3,2,1,1]] => [5,2] => 0
[[5,4,4,3,2,1,1],[4,3,3,2,1]] => [5,2] => 0
[[6,5,5,4,3,2,1],[5,4,4,3,2,1]] => [6,1] => 0
[[3,2,1,1,1,1,1],[2,1]] => [3,1,1,1,1] => 1
[[4,3,2,2,2,2,1],[3,2,1,1,1,1]] => [4,1,1,1] => 1
[[4,3,2,2,2,1,1],[3,2,1,1,1]] => [4,2,1] => 0
[[5,4,3,3,3,2,1],[4,3,2,2,2,1]] => [5,1,1] => 1
[[4,3,2,2,1,1,1],[3,2,1,1]] => [4,2,1] => 0
[[5,4,3,3,2,2,1],[4,3,2,2,1,1]] => [5,2] => 0
[[5,4,3,3,2,1,1],[4,3,2,2,1]] => [5,2] => 0
[[6,5,4,4,3,2,1],[5,4,3,3,2,1]] => [6,1] => 0
[[4,3,2,1,1,1,1],[3,2,1]] => [4,1,1,1] => 1
[[5,4,3,2,2,2,1],[4,3,2,1,1,1]] => [5,1,1] => 1
[[5,4,3,2,2,1,1],[4,3,2,1,1]] => [5,2] => 0
[[6,5,4,3,3,2,1],[5,4,3,2,2,1]] => [6,1] => 0
[[5,4,3,2,1,1,1],[4,3,2,1]] => [5,1,1] => 1
[[6,5,4,3,2,2,1],[5,4,3,2,1,1]] => [6,1] => 0
[[6,5,4,3,2,1,1],[5,4,3,2,1]] => [6,1] => 0
[[7,6,5,4,3,2,1],[6,5,4,3,2,1]] => [7] => 0
[[8],[]] => [8] => 0
[[7,1],[]] => [7,1] => 0
[[6,2],[]] => [6,2] => 0
[[6,1,1],[]] => [6,1,1] => 1
[[5,3],[]] => [5,3] => 0
[[5,2,1],[]] => [5,2,1] => 0
[[5,1,1,1],[]] => [5,1,1,1] => 1
[[4,4],[]] => [4,4] => 4
[[4,3,1],[]] => [4,3,1] => 0
[[4,2,2],[]] => [4,2,2] => 2
[[4,2,1,1],[]] => [4,2,1,1] => 1
[[4,1,1,1,1],[]] => [4,1,1,1,1] => 1
[[3,3,2],[]] => [3,3,2] => 3
[[3,3,1,1],[]] => [3,3,1,1] => 3
[[3,2,2,1],[]] => [3,2,2,1] => 2
[[3,2,1,1,1],[]] => [3,2,1,1,1] => 1
[[3,1,1,1,1,1],[]] => [3,1,1,1,1,1] => 1
[[2,2,2,2],[]] => [2,2,2,2] => 2
[[2,2,2,1,1],[]] => [2,2,2,1,1] => 2
[[2,2,1,1,1,1],[]] => [2,2,1,1,1,1] => 2
[[2,1,1,1,1,1,1],[]] => [2,1,1,1,1,1,1] => 1
[[1,1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1,1] => 1
[[9],[]] => [9] => 0
[[8,1],[]] => [8,1] => 0
[[7,2],[]] => [7,2] => 0
[[7,1,1],[]] => [7,1,1] => 1
[[6,3],[]] => [6,3] => 0
[[6,2,1],[]] => [6,2,1] => 0
[[6,1,1,1],[]] => [6,1,1,1] => 1
[[5,4],[]] => [5,4] => 0
[[5,3,1],[]] => [5,3,1] => 0
[[5,2,2],[]] => [5,2,2] => 2
[[5,2,1,1],[]] => [5,2,1,1] => 1
[[5,1,1,1,1],[]] => [5,1,1,1,1] => 1
[[4,4,1],[]] => [4,4,1] => 4
[[4,3,2],[]] => [4,3,2] => 0
[[4,3,1,1],[]] => [4,3,1,1] => 1
[[4,2,2,1],[]] => [4,2,2,1] => 2
[[4,2,1,1,1],[]] => [4,2,1,1,1] => 1
[[4,1,1,1,1,1],[]] => [4,1,1,1,1,1] => 1
[[3,3,3],[]] => [3,3,3] => 3
[[3,3,2,1],[]] => [3,3,2,1] => 3
[[3,3,1,1,1],[]] => [3,3,1,1,1] => 3
[[3,2,2,2],[]] => [3,2,2,2] => 2
[[3,2,2,1,1],[]] => [3,2,2,1,1] => 2
[[3,2,1,1,1,1],[]] => [3,2,1,1,1,1] => 1
[[3,1,1,1,1,1,1],[]] => [3,1,1,1,1,1,1] => 1
[[2,2,2,2,1],[]] => [2,2,2,2,1] => 2
[[2,2,2,1,1,1],[]] => [2,2,2,1,1,1] => 2
[[2,2,1,1,1,1,1],[]] => [2,2,1,1,1,1,1] => 2
[[2,1,1,1,1,1,1,1],[]] => [2,1,1,1,1,1,1,1] => 1
[[1,1,1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1,1,1] => 1
[[10],[]] => [10] => 0
[[9,1],[]] => [9,1] => 0
[[8,2],[]] => [8,2] => 0
[[8,1,1],[]] => [8,1,1] => 1
[[7,3],[]] => [7,3] => 0
[[7,2,1],[]] => [7,2,1] => 0
[[7,1,1,1],[]] => [7,1,1,1] => 1
[[6,4],[]] => [6,4] => 0
[[6,3,1],[]] => [6,3,1] => 0
[[6,2,2],[]] => [6,2,2] => 2
[[6,2,1,1],[]] => [6,2,1,1] => 1
[[6,1,1,1,1],[]] => [6,1,1,1,1] => 1
[[5,5],[]] => [5,5] => 5
[[5,4,1],[]] => [5,4,1] => 0
[[5,3,2],[]] => [5,3,2] => 0
[[5,3,1,1],[]] => [5,3,1,1] => 1
[[5,2,2,1],[]] => [5,2,2,1] => 2
[[5,2,1,1,1],[]] => [5,2,1,1,1] => 1
[[5,1,1,1,1,1],[]] => [5,1,1,1,1,1] => 1
[[4,4,2],[]] => [4,4,2] => 4
[[4,4,1,1],[]] => [4,4,1,1] => 4
[[4,3,3],[]] => [4,3,3] => 3
[[4,3,2,1],[]] => [4,3,2,1] => 0
[[4,3,1,1,1],[]] => [4,3,1,1,1] => 1
[[4,2,2,2],[]] => [4,2,2,2] => 2
[[4,2,2,1,1],[]] => [4,2,2,1,1] => 2
[[4,2,1,1,1,1],[]] => [4,2,1,1,1,1] => 1
[[4,1,1,1,1,1,1],[]] => [4,1,1,1,1,1,1] => 1
[[3,3,3,1],[]] => [3,3,3,1] => 3
[[3,3,2,2],[]] => [3,3,2,2] => 3
[[3,3,2,1,1],[]] => [3,3,2,1,1] => 3
[[3,3,1,1,1,1],[]] => [3,3,1,1,1,1] => 3
[[3,2,2,2,1],[]] => [3,2,2,2,1] => 2
[[3,2,2,1,1,1],[]] => [3,2,2,1,1,1] => 2
[[3,2,1,1,1,1,1],[]] => [3,2,1,1,1,1,1] => 1
[[3,1,1,1,1,1,1,1],[]] => [3,1,1,1,1,1,1,1] => 1
[[2,2,2,2,2],[]] => [2,2,2,2,2] => 2
[[2,2,2,2,1,1],[]] => [2,2,2,2,1,1] => 2
[[2,2,2,1,1,1,1],[]] => [2,2,2,1,1,1,1] => 2
[[2,2,1,1,1,1,1,1],[]] => [2,2,1,1,1,1,1,1] => 2
[[2,1,1,1,1,1,1,1,1],[]] => [2,1,1,1,1,1,1,1,1] => 1
[[1,1,1,1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1,1,1,1] => 1
[[5,4,2],[]] => [5,4,2] => 0
[[5,3,3],[]] => [5,3,3] => 3
[[4,4,3],[]] => [4,4,3] => 4
[[4,4,2,1],[]] => [4,4,2,1] => 4
[[4,3,3,1],[]] => [4,3,3,1] => 3
[[4,3,2,2],[]] => [4,3,2,2] => 2
[[3,3,3,2],[]] => [3,3,3,2] => 3
[[5,4,3],[]] => [5,4,3] => 0
[[4,4,3,1],[]] => [4,4,3,1] => 4
[[4,4,2,2],[]] => [4,4,2,2] => 4
[[4,3,3,2],[]] => [4,3,3,2] => 3
[[3,3,3,2,1],[]] => [3,3,3,2,1] => 3
[[4,4,3,2],[]] => [4,4,3,2] => 4
[[3,3,3],[1]] => [3,3,2] => 3
[[3,3,3,1],[1,1]] => [3,3,2] => 3
[[4,4,1],[1]] => [4,4] => 4
[[3,3,2,1],[1]] => [3,3,2] => 3
[[2,2,2,2,1],[1]] => [2,2,2,2] => 2
[[3,3,3,1],[2]] => [3,3,2] => 3
[[3,3,3,1],[1]] => [3,3,3] => 3
[[3,3,3,2],[1,1,1]] => [3,2,2,1] => 2
[[4,4,2],[1,1]] => [4,3,1] => 0
[[4,4,4],[2,2]] => [4,2,2] => 2
[[5,5],[2]] => [5,3] => 0
[[4,4,3],[2,1]] => [4,3,1] => 0
[[3,3,3,3],[2,1,1]] => [3,2,2,1] => 2
[[4,3,3],[1,1]] => [4,2,2] => 2
[[3,3,3,3],[1,1,1]] => [3,2,2,2] => 2
[[4,4,3],[1,1]] => [4,3,2] => 0
[[4,4,4],[3,1]] => [4,3,1] => 0
[[5,4],[1]] => [5,3] => 0
[[4,4,4],[2,1]] => [4,3,2] => 0
[[5,5],[1]] => [5,4] => 0
[[4,4,4],[1,1]] => [4,3,3] => 3
[[3,3,2,2],[1,1]] => [3,3,1,1] => 3
[[4,4,2],[2]] => [4,4] => 4
[[3,3,3,2],[2,1]] => [3,3,2] => 3
[[4,3,2],[1]] => [4,3,1] => 0
[[3,3,3,2],[1,1]] => [3,3,2,1] => 3
[[4,4,2],[1]] => [4,4,1] => 4
[[2,2,2,2,2],[1,1]] => [2,2,2,1,1] => 2
[[3,2,2,2],[1]] => [3,2,2,1] => 2
[[3,3,2,2],[2]] => [3,3,2] => 3
[[3,3,2,2],[1]] => [3,3,2,1] => 3
[[2,2,2,2,2],[1]] => [2,2,2,2,1] => 2
[[3,3,3,2],[2]] => [3,3,3] => 3
[[3,3,3,2],[1]] => [3,3,3,1] => 3
[[3,3,3,3],[2,2]] => [3,3,1,1] => 3
[[4,3,3],[2]] => [4,3,1] => 0
[[4,4,3],[3]] => [4,4] => 4
[[4,4,3],[2]] => [4,4,1] => 4
[[3,3,3,3],[2,1]] => [3,3,2,1] => 3
[[4,3,3],[1]] => [4,3,2] => 0
[[3,3,3,3],[1,1]] => [3,3,2,2] => 3
[[4,4,3],[1]] => [4,4,2] => 4
[[4,4,4],[3]] => [4,4,1] => 4
[[4,4,4],[2]] => [4,4,2] => 4
[[4,4,4],[1]] => [4,4,3] => 4
[[3,3,3,3],[2]] => [3,3,3,1] => 3
[[3,3,3,3],[1]] => [3,3,3,2] => 3
[[4,4,4,1],[1,1]] => [4,4,3] => 4
[[4,4,3,1],[1]] => [4,4,3] => 4
[[4,4,4,1],[2]] => [4,4,3] => 4
[[4,4,4,2],[1,1,1]] => [4,3,3,1] => 3
[[6,6],[4]] => [6,2] => 0
[[6,5],[3]] => [6,2] => 0
[[5,5,5],[3,3]] => [5,2,2] => 2
[[6,6],[3]] => [6,3] => 0
[[6,4],[2]] => [6,2] => 0
[[5,5,4],[3,2]] => [5,3,1] => 0
[[5,4,4],[2,2]] => [5,2,2] => 2
[[4,4,4,4],[2,2,2]] => [4,2,2,2] => 2
[[5,5,4],[2,2]] => [5,3,2] => 0
[[6,5],[2]] => [6,3] => 0
[[5,5,5],[3,2]] => [5,3,2] => 0
[[6,6],[2]] => [6,4] => 0
[[5,5,5],[2,2]] => [5,3,3] => 3
[[6,3],[1]] => [6,2] => 0
[[5,5,3],[3,1]] => [5,4] => 0
[[5,4,3],[2,1]] => [5,3,1] => 0
[[4,4,4,3],[2,2,1]] => [4,3,2,1] => 0
[[5,5,3],[2,1]] => [5,4,1] => 0
[[5,3,3],[1,1]] => [5,2,2] => 2
[[4,4,3,3],[2,1,1]] => [4,3,2,1] => 0
[[4,3,3,3],[1,1,1]] => [4,2,2,2] => 2
[[4,4,3,3],[1,1,1]] => [4,3,2,2] => 2
[[5,4,3],[1,1]] => [5,3,2] => 0
[[4,4,4,3],[2,1,1]] => [4,3,3,1] => 3
[[5,5,3],[1,1]] => [5,4,2] => 0
[[4,4,4,3],[1,1,1]] => [4,3,3,2] => 3
[[6,4],[1]] => [6,3] => 0
[[5,5,4],[3,1]] => [5,4,1] => 0
[[5,4,4],[2,1]] => [5,3,2] => 0
[[4,4,4,4],[2,2,1]] => [4,3,2,2] => 2
[[5,5,4],[2,1]] => [5,4,2] => 0
[[6,5],[1]] => [6,4] => 0
[[5,5,5],[3,1]] => [5,4,2] => 0
[[5,5,5],[2,1]] => [5,4,3] => 0
[[4,4,4,4],[3,1,1]] => [4,3,3,1] => 3
[[5,4,4],[1,1]] => [5,3,3] => 3
[[4,4,4,4],[2,1,1]] => [4,3,3,2] => 3
[[5,5,4],[1,1]] => [5,4,3] => 0
[[5,5,2],[3]] => [5,4] => 0
[[5,4,2],[2]] => [5,4] => 0
[[4,4,4,2],[2,2]] => [4,4,2] => 4
[[5,5,2],[2]] => [5,5] => 5
[[5,3,2],[1]] => [5,3,1] => 0
[[4,4,3,2],[2,1]] => [4,4,2] => 4
[[4,3,3,2],[1,1]] => [4,3,2,1] => 0
[[3,3,3,3,2],[1,1,1]] => [3,3,2,2,1] => 3
[[4,4,3,2],[1,1]] => [4,4,2,1] => 4
[[5,4,2],[1]] => [5,4,1] => 0
[[4,4,4,2],[2,1]] => [4,4,3] => 4
[[4,4,4,2],[1,1]] => [4,4,3,1] => 4
[[4,4,2,2],[2]] => [4,4,2] => 4
[[4,3,2,2],[1]] => [4,3,2,1] => 0
[[3,3,3,2,2],[1,1]] => [3,3,3,1,1] => 3
[[4,4,2,2],[1]] => [4,4,2,1] => 4
[[3,3,2,2,2],[1]] => [3,3,2,2,1] => 3
[[3,3,3,2,2],[1]] => [3,3,3,2,1] => 3
[[4,4,3,2],[2]] => [4,4,3] => 4
[[4,3,3,2],[1]] => [4,3,3,1] => 3
[[3,3,3,3,2],[1,1]] => [3,3,3,2,1] => 3
[[4,4,3,2],[1]] => [4,4,3,1] => 4
[[4,4,4,2],[3]] => [4,4,3] => 4
[[5,5,3],[3]] => [5,5] => 5
[[5,4,3],[2]] => [5,4,1] => 0
[[4,4,4,3],[2,2]] => [4,4,2,1] => 4
[[5,3,3],[1]] => [5,3,2] => 0
[[4,4,3,3],[2,1]] => [4,4,2,1] => 4
[[4,3,3,3],[1,1]] => [4,3,2,2] => 2
[[4,4,3,3],[1,1]] => [4,4,2,2] => 4
[[4,4,4,3],[3,1]] => [4,4,3] => 4
[[5,4,3],[1]] => [5,4,2] => 0
[[4,4,4,3],[2,1]] => [4,4,3,1] => 4
[[4,4,4,3],[1,1]] => [4,4,3,2] => 4
[[4,4,4,4],[3,2]] => [4,4,2,1] => 4
[[5,4,4],[2]] => [5,4,2] => 0
[[4,4,4,4],[2,2]] => [4,4,2,2] => 4
[[4,4,4,4],[3,1]] => [4,4,3,1] => 4
[[5,4,4],[1]] => [5,4,3] => 0
[[4,4,4,4],[2,1]] => [4,4,3,2] => 4
[[4,3,3,3],[2]] => [4,3,3,1] => 3
[[4,4,3,3],[3]] => [4,4,3] => 4
[[4,4,3,3],[2]] => [4,4,3,1] => 4
[[3,3,3,3,3],[2,1]] => [3,3,3,2,1] => 3
[[4,3,3,3],[1]] => [4,3,3,2] => 3
[[4,4,3,3],[1]] => [4,4,3,2] => 4
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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 8,1 20,5,3 64,17,6 204,49,10,9 628,161,34,27
$F_{1} = 1$
$F_{2} = 2 + q$
$F_{3} = 8 + q$
$F_{4} = 20 + 5\ q + 3\ q^{2}$
$F_{5} = 64 + 17\ q + 6\ q^{2}$
$F_{6} = 204 + 49\ q + 10\ q^{2} + 9\ q^{3}$
$F_{7} = 628 + 161\ q + 34\ q^{2} + 27\ q^{3}$
Description
The largest repeated part of a partition.
If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
Map
dominating partition
Description
The dominating partition in the Schur expansion.
Consider the expansion of the skew Schur function $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu, \nu} s_\nu$ as a linear combination of straight Schur functions.
It is shown in [1] that the partitions $\nu$ with $c^\lambda_{\mu, \nu} > 0$ form a sublattice of the dominance order and that its top element is the conjugate of the partition formed by sorting the column lengths of $\lambda / \mu$ into decreasing order.
This map returns the largest partition $\nu$ in dominance order for which $c^\lambda_{\mu, \nu}$ is positive.
For example,
$$ s_{331/2} = s_{311} + s_{32}, $$
and the partition $32$ dominates $311$.
Consider the expansion of the skew Schur function $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu, \nu} s_\nu$ as a linear combination of straight Schur functions.
It is shown in [1] that the partitions $\nu$ with $c^\lambda_{\mu, \nu} > 0$ form a sublattice of the dominance order and that its top element is the conjugate of the partition formed by sorting the column lengths of $\lambda / \mu$ into decreasing order.
This map returns the largest partition $\nu$ in dominance order for which $c^\lambda_{\mu, \nu}$ is positive.
For example,
$$ s_{331/2} = s_{311} + s_{32}, $$
and the partition $32$ dominates $311$.
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