Identifier
-
Mp00182:
Skew partitions
—outer shape⟶
Integer partitions
St000137: Integer partitions ⟶ ℤ
Values
[[1],[]] => [1] => 1
[[2],[]] => [2] => 0
[[1,1],[]] => [1,1] => 0
[[2,1],[1]] => [2,1] => 1
[[3],[]] => [3] => 1
[[2,1],[]] => [2,1] => 1
[[3,1],[1]] => [3,1] => 0
[[2,2],[1]] => [2,2] => 2
[[3,2],[2]] => [3,2] => 1
[[1,1,1],[]] => [1,1,1] => 1
[[2,2,1],[1,1]] => [2,2,1] => 1
[[2,1,1],[1]] => [2,1,1] => 0
[[3,2,1],[2,1]] => [3,2,1] => 0
[[4],[]] => [4] => 0
[[3,1],[]] => [3,1] => 0
[[4,1],[1]] => [4,1] => 1
[[2,2],[]] => [2,2] => 2
[[3,2],[1]] => [3,2] => 1
[[4,2],[2]] => [4,2] => 0
[[2,1,1],[]] => [2,1,1] => 0
[[3,2,1],[1,1]] => [3,2,1] => 0
[[3,1,1],[1]] => [3,1,1] => 1
[[4,2,1],[2,1]] => [4,2,1] => 1
[[3,3],[2]] => [3,3] => 2
[[4,3],[3]] => [4,3] => 1
[[2,2,1],[1]] => [2,2,1] => 1
[[3,3,1],[2,1]] => [3,3,1] => 3
[[3,2,1],[2]] => [3,2,1] => 0
[[4,3,1],[3,1]] => [4,3,1] => 0
[[2,2,2],[1,1]] => [2,2,2] => 2
[[3,3,2],[2,2]] => [3,3,2] => 0
[[3,2,2],[2,1]] => [3,2,2] => 3
[[4,3,2],[3,2]] => [4,3,2] => 1
[[1,1,1,1],[]] => [1,1,1,1] => 0
[[2,2,2,1],[1,1,1]] => [2,2,2,1] => 1
[[2,2,1,1],[1,1]] => [2,2,1,1] => 0
[[3,3,2,1],[2,2,1]] => [3,3,2,1] => 1
[[2,1,1,1],[1]] => [2,1,1,1] => 1
[[3,2,2,1],[2,1,1]] => [3,2,2,1] => 0
[[3,2,1,1],[2,1]] => [3,2,1,1] => 1
[[4,3,2,1],[3,2,1]] => [4,3,2,1] => 0
[[5],[]] => [5] => 1
[[4,1],[]] => [4,1] => 1
[[5,1],[1]] => [5,1] => 0
[[3,2],[]] => [3,2] => 1
[[4,2],[1]] => [4,2] => 0
[[5,2],[2]] => [5,2] => 1
[[3,1,1],[]] => [3,1,1] => 1
[[4,2,1],[1,1]] => [4,2,1] => 1
[[4,1,1],[1]] => [4,1,1] => 0
[[5,2,1],[2,1]] => [5,2,1] => 0
[[3,3],[1]] => [3,3] => 2
[[4,3],[2]] => [4,3] => 1
[[5,3],[3]] => [5,3] => 0
[[2,2,1],[]] => [2,2,1] => 1
[[3,3,1],[1,1]] => [3,3,1] => 3
[[3,2,1],[1]] => [3,2,1] => 0
[[4,3,1],[2,1]] => [4,3,1] => 0
[[4,2,1],[2]] => [4,2,1] => 1
[[5,3,1],[3,1]] => [5,3,1] => 1
[[3,2,2],[1,1]] => [3,2,2] => 3
[[4,3,2],[2,2]] => [4,3,2] => 1
[[4,2,2],[2,1]] => [4,2,2] => 2
[[5,3,2],[3,2]] => [5,3,2] => 0
[[2,1,1,1],[]] => [2,1,1,1] => 1
[[3,2,2,1],[1,1,1]] => [3,2,2,1] => 0
[[3,2,1,1],[1,1]] => [3,2,1,1] => 1
[[4,3,2,1],[2,2,1]] => [4,3,2,1] => 0
[[3,1,1,1],[1]] => [3,1,1,1] => 0
[[4,2,2,1],[2,1,1]] => [4,2,2,1] => 1
[[4,2,1,1],[2,1]] => [4,2,1,1] => 0
[[5,3,2,1],[3,2,1]] => [5,3,2,1] => 1
[[4,4],[3]] => [4,4] => 0
[[5,4],[4]] => [5,4] => 1
[[3,3,1],[2]] => [3,3,1] => 3
[[4,4,1],[3,1]] => [4,4,1] => 2
[[4,3,1],[3]] => [4,3,1] => 0
[[5,4,1],[4,1]] => [5,4,1] => 0
[[2,2,2],[1]] => [2,2,2] => 2
[[3,3,2],[2,1]] => [3,3,2] => 0
[[4,4,2],[3,2]] => [4,4,2] => 0
[[3,2,2],[2]] => [3,2,2] => 3
[[4,3,2],[3,1]] => [4,3,2] => 1
[[5,4,2],[4,2]] => [5,4,2] => 1
[[2,2,1,1],[1]] => [2,2,1,1] => 0
[[3,3,2,1],[2,1,1]] => [3,3,2,1] => 1
[[3,3,1,1],[2,1]] => [3,3,1,1] => 2
[[4,4,2,1],[3,2,1]] => [4,4,2,1] => 2
[[3,2,1,1],[2]] => [3,2,1,1] => 1
[[4,3,2,1],[3,1,1]] => [4,3,2,1] => 0
[[4,3,1,1],[3,1]] => [4,3,1,1] => 1
[[5,4,2,1],[4,2,1]] => [5,4,2,1] => 0
[[3,3,3],[2,2]] => [3,3,3] => 2
[[4,4,3],[3,3]] => [4,4,3] => 1
[[4,3,3],[3,2]] => [4,3,3] => 3
[[5,4,3],[4,3]] => [5,4,3] => 0
[[2,2,2,1],[1,1]] => [2,2,2,1] => 1
[[3,3,3,1],[2,2,1]] => [3,3,3,1] => 3
[[3,3,2,1],[2,2]] => [3,3,2,1] => 1
[[4,4,3,1],[3,3,1]] => [4,4,3,1] => 0
[[3,2,2,1],[2,1]] => [3,2,2,1] => 0
>>> Load all 1416 entries. <<<[[4,3,3,1],[3,2,1]] => [4,3,3,1] => 2
[[4,3,2,1],[3,2]] => [4,3,2,1] => 0
[[5,4,3,1],[4,3,1]] => [5,4,3,1] => 1
[[2,2,2,2],[1,1,1]] => [2,2,2,2] => 0
[[3,3,3,2],[2,2,2]] => [3,3,3,2] => 1
[[3,3,2,2],[2,2,1]] => [3,3,2,2] => 0
[[4,4,3,2],[3,3,2]] => [4,4,3,2] => 2
[[3,2,2,2],[2,1,1]] => [3,2,2,2] => 2
[[4,3,3,2],[3,2,2]] => [4,3,3,2] => 0
[[4,3,2,2],[3,2,1]] => [4,3,2,2] => 2
[[5,4,3,2],[4,3,2]] => [5,4,3,2] => 0
[[1,1,1,1,1],[]] => [1,1,1,1,1] => 1
[[2,2,2,2,1],[1,1,1,1]] => [2,2,2,2,1] => 1
[[2,2,2,1,1],[1,1,1]] => [2,2,2,1,1] => 0
[[3,3,3,2,1],[2,2,2,1]] => [3,3,3,2,1] => 0
[[2,2,1,1,1],[1,1]] => [2,2,1,1,1] => 1
[[3,3,2,2,1],[2,2,1,1]] => [3,3,2,2,1] => 1
[[3,3,2,1,1],[2,2,1]] => [3,3,2,1,1] => 0
[[4,4,3,2,1],[3,3,2,1]] => [4,4,3,2,1] => 0
[[2,1,1,1,1],[1]] => [2,1,1,1,1] => 0
[[3,2,2,2,1],[2,1,1,1]] => [3,2,2,2,1] => 0
[[3,2,2,1,1],[2,1,1]] => [3,2,2,1,1] => 1
[[4,3,3,2,1],[3,2,2,1]] => [4,3,3,2,1] => 1
[[3,2,1,1,1],[2,1]] => [3,2,1,1,1] => 0
[[4,3,2,2,1],[3,2,1,1]] => [4,3,2,2,1] => 0
[[4,3,2,1,1],[3,2,1]] => [4,3,2,1,1] => 1
[[5,4,3,2,1],[4,3,2,1]] => [5,4,3,2,1] => 1
[[6],[]] => [6] => 0
[[5,1],[]] => [5,1] => 0
[[6,1],[1]] => [6,1] => 1
[[4,2],[]] => [4,2] => 0
[[5,2],[1]] => [5,2] => 1
[[6,2],[2]] => [6,2] => 0
[[4,1,1],[]] => [4,1,1] => 0
[[5,2,1],[1,1]] => [5,2,1] => 0
[[5,1,1],[1]] => [5,1,1] => 1
[[6,2,1],[2,1]] => [6,2,1] => 1
[[3,3],[]] => [3,3] => 2
[[4,3],[1]] => [4,3] => 1
[[5,3],[2]] => [5,3] => 0
[[6,3],[3]] => [6,3] => 1
[[3,2,1],[]] => [3,2,1] => 0
[[4,3,1],[1,1]] => [4,3,1] => 0
[[4,2,1],[1]] => [4,2,1] => 1
[[5,3,1],[2,1]] => [5,3,1] => 1
[[5,2,1],[2]] => [5,2,1] => 0
[[6,3,1],[3,1]] => [6,3,1] => 0
[[4,2,2],[1,1]] => [4,2,2] => 2
[[5,3,2],[2,2]] => [5,3,2] => 0
[[5,2,2],[2,1]] => [5,2,2] => 3
[[6,3,2],[3,2]] => [6,3,2] => 1
[[3,1,1,1],[]] => [3,1,1,1] => 0
[[4,2,2,1],[1,1,1]] => [4,2,2,1] => 1
[[4,2,1,1],[1,1]] => [4,2,1,1] => 0
[[5,3,2,1],[2,2,1]] => [5,3,2,1] => 1
[[4,1,1,1],[1]] => [4,1,1,1] => 1
[[5,2,2,1],[2,1,1]] => [5,2,2,1] => 0
[[5,2,1,1],[2,1]] => [5,2,1,1] => 1
[[6,3,2,1],[3,2,1]] => [6,3,2,1] => 0
[[4,4],[2]] => [4,4] => 0
[[5,4],[3]] => [5,4] => 1
[[6,4],[4]] => [6,4] => 0
[[3,3,1],[1]] => [3,3,1] => 3
[[4,4,1],[2,1]] => [4,4,1] => 2
[[4,3,1],[2]] => [4,3,1] => 0
[[5,4,1],[3,1]] => [5,4,1] => 0
[[5,3,1],[3]] => [5,3,1] => 1
[[6,4,1],[4,1]] => [6,4,1] => 1
[[2,2,2],[]] => [2,2,2] => 2
[[3,3,2],[1,1]] => [3,3,2] => 0
[[4,4,2],[2,2]] => [4,4,2] => 0
[[3,2,2],[1]] => [3,2,2] => 3
[[4,3,2],[2,1]] => [4,3,2] => 1
[[5,4,2],[3,2]] => [5,4,2] => 1
[[4,2,2],[2]] => [4,2,2] => 2
[[5,3,2],[3,1]] => [5,3,2] => 0
[[6,4,2],[4,2]] => [6,4,2] => 0
[[2,2,1,1],[]] => [2,2,1,1] => 0
[[3,3,2,1],[1,1,1]] => [3,3,2,1] => 1
[[3,3,1,1],[1,1]] => [3,3,1,1] => 2
[[4,4,2,1],[2,2,1]] => [4,4,2,1] => 2
[[3,2,1,1],[1]] => [3,2,1,1] => 1
[[4,3,2,1],[2,1,1]] => [4,3,2,1] => 0
[[4,3,1,1],[2,1]] => [4,3,1,1] => 1
[[5,4,2,1],[3,2,1]] => [5,4,2,1] => 0
[[4,2,1,1],[2]] => [4,2,1,1] => 0
[[5,3,2,1],[3,1,1]] => [5,3,2,1] => 1
[[5,3,1,1],[3,1]] => [5,3,1,1] => 0
[[6,4,2,1],[4,2,1]] => [6,4,2,1] => 1
[[4,3,3],[2,2]] => [4,3,3] => 3
[[5,4,3],[3,3]] => [5,4,3] => 0
[[5,3,3],[3,2]] => [5,3,3] => 2
[[6,4,3],[4,3]] => [6,4,3] => 1
[[3,2,2,1],[1,1]] => [3,2,2,1] => 0
[[4,3,3,1],[2,2,1]] => [4,3,3,1] => 2
[[4,3,2,1],[2,2]] => [4,3,2,1] => 0
[[5,4,3,1],[3,3,1]] => [5,4,3,1] => 1
[[4,2,2,1],[2,1]] => [4,2,2,1] => 1
[[5,3,3,1],[3,2,1]] => [5,3,3,1] => 3
[[5,3,2,1],[3,2]] => [5,3,2,1] => 1
[[6,4,3,1],[4,3,1]] => [6,4,3,1] => 0
[[3,2,2,2],[1,1,1]] => [3,2,2,2] => 2
[[4,3,3,2],[2,2,2]] => [4,3,3,2] => 0
[[4,3,2,2],[2,2,1]] => [4,3,2,2] => 2
[[5,4,3,2],[3,3,2]] => [5,4,3,2] => 0
[[4,2,2,2],[2,1,1]] => [4,2,2,2] => 3
[[5,3,3,2],[3,2,2]] => [5,3,3,2] => 1
[[5,3,2,2],[3,2,1]] => [5,3,2,2] => 3
[[6,4,3,2],[4,3,2]] => [6,4,3,2] => 1
[[2,1,1,1,1],[]] => [2,1,1,1,1] => 0
[[3,2,2,2,1],[1,1,1,1]] => [3,2,2,2,1] => 0
[[3,2,2,1,1],[1,1,1]] => [3,2,2,1,1] => 1
[[4,3,3,2,1],[2,2,2,1]] => [4,3,3,2,1] => 1
[[3,2,1,1,1],[1,1]] => [3,2,1,1,1] => 0
[[4,3,2,2,1],[2,2,1,1]] => [4,3,2,2,1] => 0
[[4,3,2,1,1],[2,2,1]] => [4,3,2,1,1] => 1
[[5,4,3,2,1],[3,3,2,1]] => [5,4,3,2,1] => 1
[[3,1,1,1,1],[1]] => [3,1,1,1,1] => 1
[[4,2,2,2,1],[2,1,1,1]] => [4,2,2,2,1] => 1
[[4,2,2,1,1],[2,1,1]] => [4,2,2,1,1] => 0
[[5,3,3,2,1],[3,2,2,1]] => [5,3,3,2,1] => 0
[[4,2,1,1,1],[2,1]] => [4,2,1,1,1] => 1
[[5,3,2,2,1],[3,2,1,1]] => [5,3,2,2,1] => 1
[[5,3,2,1,1],[3,2,1]] => [5,3,2,1,1] => 0
[[6,4,3,2,1],[4,3,2,1]] => [6,4,3,2,1] => 0
[[5,5],[4]] => [5,5] => 0
[[6,5],[5]] => [6,5] => 1
[[4,4,1],[3]] => [4,4,1] => 2
[[5,5,1],[4,1]] => [5,5,1] => 1
[[5,4,1],[4]] => [5,4,1] => 0
[[6,5,1],[5,1]] => [6,5,1] => 0
[[3,3,2],[2]] => [3,3,2] => 0
[[4,4,2],[3,1]] => [4,4,2] => 0
[[5,5,2],[4,2]] => [5,5,2] => 2
[[4,3,2],[3]] => [4,3,2] => 1
[[5,4,2],[4,1]] => [5,4,2] => 1
[[6,5,2],[5,2]] => [6,5,2] => 1
[[3,3,1,1],[2]] => [3,3,1,1] => 2
[[4,4,2,1],[3,1,1]] => [4,4,2,1] => 2
[[4,4,1,1],[3,1]] => [4,4,1,1] => 3
[[5,5,2,1],[4,2,1]] => [5,5,2,1] => 3
[[4,3,1,1],[3]] => [4,3,1,1] => 1
[[5,4,2,1],[4,1,1]] => [5,4,2,1] => 0
[[5,4,1,1],[4,1]] => [5,4,1,1] => 1
[[6,5,2,1],[5,2,1]] => [6,5,2,1] => 0
[[3,3,3],[2,1]] => [3,3,3] => 2
[[4,4,3],[3,2]] => [4,4,3] => 1
[[5,5,3],[4,3]] => [5,5,3] => 1
[[4,3,3],[3,1]] => [4,3,3] => 3
[[5,4,3],[4,2]] => [5,4,3] => 0
[[6,5,3],[5,3]] => [6,5,3] => 0
[[2,2,2,1],[1]] => [2,2,2,1] => 1
[[3,3,3,1],[2,1,1]] => [3,3,3,1] => 3
[[3,3,2,1],[2,1]] => [3,3,2,1] => 1
[[4,4,3,1],[3,2,1]] => [4,4,3,1] => 0
[[4,4,2,1],[3,2]] => [4,4,2,1] => 2
[[5,5,3,1],[4,3,1]] => [5,5,3,1] => 0
[[3,2,2,1],[2]] => [3,2,2,1] => 0
[[4,3,3,1],[3,1,1]] => [4,3,3,1] => 2
[[4,3,2,1],[3,1]] => [4,3,2,1] => 0
[[5,4,3,1],[4,2,1]] => [5,4,3,1] => 1
[[5,4,2,1],[4,2]] => [5,4,2,1] => 0
[[6,5,3,1],[5,3,1]] => [6,5,3,1] => 1
[[3,3,2,2],[2,1,1]] => [3,3,2,2] => 0
[[4,4,3,2],[3,2,2]] => [4,4,3,2] => 2
[[4,4,2,2],[3,2,1]] => [4,4,2,2] => 1
[[5,5,3,2],[4,3,2]] => [5,5,3,2] => 2
[[4,3,2,2],[3,1,1]] => [4,3,2,2] => 2
[[5,4,3,2],[4,2,2]] => [5,4,3,2] => 0
[[5,4,2,2],[4,2,1]] => [5,4,2,2] => 2
[[6,5,3,2],[5,3,2]] => [6,5,3,2] => 0
[[2,2,1,1,1],[1]] => [2,2,1,1,1] => 1
[[3,3,2,2,1],[2,1,1,1]] => [3,3,2,2,1] => 1
[[3,3,2,1,1],[2,1,1]] => [3,3,2,1,1] => 0
[[4,4,3,2,1],[3,2,2,1]] => [4,4,3,2,1] => 0
[[3,3,1,1,1],[2,1]] => [3,3,1,1,1] => 3
[[4,4,2,2,1],[3,2,1,1]] => [4,4,2,2,1] => 2
[[4,4,2,1,1],[3,2,1]] => [4,4,2,1,1] => 3
[[5,5,3,2,1],[4,3,2,1]] => [5,5,3,2,1] => 4
[[3,2,1,1,1],[2]] => [3,2,1,1,1] => 0
[[4,3,2,2,1],[3,1,1,1]] => [4,3,2,2,1] => 0
[[4,3,2,1,1],[3,1,1]] => [4,3,2,1,1] => 1
[[5,4,3,2,1],[4,2,2,1]] => [5,4,3,2,1] => 1
[[4,3,1,1,1],[3,1]] => [4,3,1,1,1] => 0
[[5,4,2,2,1],[4,2,1,1]] => [5,4,2,2,1] => 0
[[5,4,2,1,1],[4,2,1]] => [5,4,2,1,1] => 1
[[6,5,3,2,1],[5,3,2,1]] => [6,5,3,2,1] => 1
[[4,4,4],[3,3]] => [4,4,4] => 0
[[5,5,4],[4,4]] => [5,5,4] => 0
[[5,4,4],[4,3]] => [5,4,4] => 1
[[6,5,4],[5,4]] => [6,5,4] => 1
[[3,3,3,1],[2,2]] => [3,3,3,1] => 3
[[4,4,4,1],[3,3,1]] => [4,4,4,1] => 2
[[4,4,3,1],[3,3]] => [4,4,3,1] => 0
[[5,5,4,1],[4,4,1]] => [5,5,4,1] => 1
[[4,3,3,1],[3,2]] => [4,3,3,1] => 2
[[5,4,4,1],[4,3,1]] => [5,4,4,1] => 3
[[5,4,3,1],[4,3]] => [5,4,3,1] => 1
[[6,5,4,1],[5,4,1]] => [6,5,4,1] => 0
[[2,2,2,2],[1,1]] => [2,2,2,2] => 0
[[3,3,3,2],[2,2,1]] => [3,3,3,2] => 1
[[4,4,4,2],[3,3,2]] => [4,4,4,2] => 0
[[3,3,2,2],[2,2]] => [3,3,2,2] => 0
[[4,4,3,2],[3,3,1]] => [4,4,3,2] => 2
[[5,5,4,2],[4,4,2]] => [5,5,4,2] => 0
[[3,2,2,2],[2,1]] => [3,2,2,2] => 2
[[4,3,3,2],[3,2,1]] => [4,3,3,2] => 0
[[5,4,4,2],[4,3,2]] => [5,4,4,2] => 1
[[4,3,2,2],[3,2]] => [4,3,2,2] => 2
[[5,4,3,2],[4,3,1]] => [5,4,3,2] => 0
[[6,5,4,2],[5,4,2]] => [6,5,4,2] => 1
[[2,2,2,1,1],[1,1]] => [2,2,2,1,1] => 0
[[3,3,3,2,1],[2,2,1,1]] => [3,3,3,2,1] => 0
[[3,3,3,1,1],[2,2,1]] => [3,3,3,1,1] => 2
[[4,4,4,2,1],[3,3,2,1]] => [4,4,4,2,1] => 2
[[3,3,2,1,1],[2,2]] => [3,3,2,1,1] => 0
[[4,4,3,2,1],[3,3,1,1]] => [4,4,3,2,1] => 0
[[4,4,3,1,1],[3,3,1]] => [4,4,3,1,1] => 1
[[5,5,4,2,1],[4,4,2,1]] => [5,5,4,2,1] => 1
[[3,2,2,1,1],[2,1]] => [3,2,2,1,1] => 1
[[4,3,3,2,1],[3,2,1,1]] => [4,3,3,2,1] => 1
[[4,3,3,1,1],[3,2,1]] => [4,3,3,1,1] => 3
[[5,4,4,2,1],[4,3,2,1]] => [5,4,4,2,1] => 3
[[4,3,2,1,1],[3,2]] => [4,3,2,1,1] => 1
[[5,4,3,2,1],[4,3,1,1]] => [5,4,3,2,1] => 1
[[5,4,3,1,1],[4,3,1]] => [5,4,3,1,1] => 0
[[3,3,3,3],[2,2,2]] => [3,3,3,3] => 0
[[4,4,4,3],[3,3,3]] => [4,4,4,3] => 1
[[4,4,3,3],[3,3,2]] => [4,4,3,3] => 0
[[5,5,4,3],[4,4,3]] => [5,5,4,3] => 2
[[4,3,3,3],[3,2,2]] => [4,3,3,3] => 2
[[5,4,4,3],[4,3,3]] => [5,4,4,3] => 0
[[5,4,3,3],[4,3,2]] => [5,4,3,3] => 2
[[2,2,2,2,1],[1,1,1]] => [2,2,2,2,1] => 1
[[3,3,3,3,1],[2,2,2,1]] => [3,3,3,3,1] => 1
[[3,3,3,2,1],[2,2,2]] => [3,3,3,2,1] => 0
[[4,4,4,3,1],[3,3,3,1]] => [4,4,4,3,1] => 0
[[3,3,2,2,1],[2,2,1]] => [3,3,2,2,1] => 1
[[4,4,3,3,1],[3,3,2,1]] => [4,4,3,3,1] => 1
[[4,4,3,2,1],[3,3,2]] => [4,4,3,2,1] => 0
[[3,2,2,2,1],[2,1,1]] => [3,2,2,2,1] => 0
[[4,3,3,3,1],[3,2,2,1]] => [4,3,3,3,1] => 3
[[4,3,3,2,1],[3,2,2]] => [4,3,3,2,1] => 1
[[5,4,4,3,1],[4,3,3,1]] => [5,4,4,3,1] => 1
[[4,3,2,2,1],[3,2,1]] => [4,3,2,2,1] => 0
[[5,4,3,3,1],[4,3,2,1]] => [5,4,3,3,1] => 3
[[5,4,3,2,1],[4,3,2]] => [5,4,3,2,1] => 1
[[2,2,2,2,2],[1,1,1,1]] => [2,2,2,2,2] => 0
[[3,3,3,3,2],[2,2,2,2]] => [3,3,3,3,2] => 0
[[3,3,3,2,2],[2,2,2,1]] => [3,3,3,2,2] => 1
[[4,4,4,3,2],[3,3,3,2]] => [4,4,4,3,2] => 2
[[3,3,2,2,2],[2,2,1,1]] => [3,3,2,2,2] => 2
[[4,4,3,3,2],[3,3,2,2]] => [4,4,3,3,2] => 0
[[4,4,3,2,2],[3,3,2,1]] => [4,4,3,2,2] => 2
[[3,2,2,2,2],[2,1,1,1]] => [3,2,2,2,2] => 1
[[4,3,3,3,2],[3,2,2,2]] => [4,3,3,3,2] => 1
[[4,3,3,2,2],[3,2,2,1]] => [4,3,3,2,2] => 0
[[4,3,2,2,2],[3,2,1,1]] => [4,3,2,2,2] => 3
[[5,4,3,3,2],[4,3,2,2]] => [5,4,3,3,2] => 1
[[5,4,3,2,2],[4,3,2,1]] => [5,4,3,2,2] => 4
[[1,1,1,1,1,1],[]] => [1,1,1,1,1,1] => 0
[[2,2,2,2,2,1],[1,1,1,1,1]] => [2,2,2,2,2,1] => 1
[[2,2,2,2,1,1],[1,1,1,1]] => [2,2,2,2,1,1] => 0
[[3,3,3,3,2,1],[2,2,2,2,1]] => [3,3,3,3,2,1] => 1
[[2,2,2,1,1,1],[1,1,1]] => [2,2,2,1,1,1] => 1
[[3,3,3,2,2,1],[2,2,2,1,1]] => [3,3,3,2,2,1] => 0
[[3,3,3,2,1,1],[2,2,2,1]] => [3,3,3,2,1,1] => 1
[[2,2,1,1,1,1],[1,1]] => [2,2,1,1,1,1] => 0
[[3,3,2,2,2,1],[2,2,1,1,1]] => [3,3,2,2,2,1] => 1
[[3,3,2,2,1,1],[2,2,1,1]] => [3,3,2,2,1,1] => 0
[[4,4,3,3,2,1],[3,3,2,2,1]] => [4,4,3,3,2,1] => 1
[[3,3,2,1,1,1],[2,2,1]] => [3,3,2,1,1,1] => 1
[[4,4,3,2,2,1],[3,3,2,1,1]] => [4,4,3,2,2,1] => 0
[[4,4,3,2,1,1],[3,3,2,1]] => [4,4,3,2,1,1] => 1
[[2,1,1,1,1,1],[1]] => [2,1,1,1,1,1] => 1
[[3,2,2,2,2,1],[2,1,1,1,1]] => [3,2,2,2,2,1] => 0
[[3,2,2,2,1,1],[2,1,1,1]] => [3,2,2,2,1,1] => 1
[[4,3,3,3,2,1],[3,2,2,2,1]] => [4,3,3,3,2,1] => 0
[[3,2,2,1,1,1],[2,1,1]] => [3,2,2,1,1,1] => 0
[[4,3,3,2,2,1],[3,2,2,1,1]] => [4,3,3,2,2,1] => 1
[[4,3,3,2,1,1],[3,2,2,1]] => [4,3,3,2,1,1] => 0
[[3,2,1,1,1,1],[2,1]] => [3,2,1,1,1,1] => 1
[[4,3,2,2,2,1],[3,2,1,1,1]] => [4,3,2,2,2,1] => 0
[[4,3,2,2,1,1],[3,2,1,1]] => [4,3,2,2,1,1] => 1
[[4,3,2,1,1,1],[3,2,1]] => [4,3,2,1,1,1] => 0
[[5,4,3,2,2,1],[4,3,2,1,1]] => [5,4,3,2,2,1] => 1
[[5,4,3,2,1,1],[4,3,2,1]] => [5,4,3,2,1,1] => 0
[[7],[]] => [7] => 1
[[6,1],[]] => [6,1] => 1
[[7,1],[1]] => [7,1] => 0
[[5,2],[]] => [5,2] => 1
[[6,2],[1]] => [6,2] => 0
[[7,2],[2]] => [7,2] => 1
[[5,1,1],[]] => [5,1,1] => 1
[[6,2,1],[1,1]] => [6,2,1] => 1
[[6,1,1],[1]] => [6,1,1] => 0
[[7,2,1],[2,1]] => [7,2,1] => 0
[[4,3],[]] => [4,3] => 1
[[5,3],[1]] => [5,3] => 0
[[6,3],[2]] => [6,3] => 1
[[7,3],[3]] => [7,3] => 0
[[4,2,1],[]] => [4,2,1] => 1
[[5,3,1],[1,1]] => [5,3,1] => 1
[[5,2,1],[1]] => [5,2,1] => 0
[[6,3,1],[2,1]] => [6,3,1] => 0
[[6,2,1],[2]] => [6,2,1] => 1
[[7,3,1],[3,1]] => [7,3,1] => 1
[[5,2,2],[1,1]] => [5,2,2] => 3
[[6,3,2],[2,2]] => [6,3,2] => 1
[[6,2,2],[2,1]] => [6,2,2] => 2
[[7,3,2],[3,2]] => [7,3,2] => 0
[[4,1,1,1],[]] => [4,1,1,1] => 1
[[5,2,2,1],[1,1,1]] => [5,2,2,1] => 0
[[5,2,1,1],[1,1]] => [5,2,1,1] => 1
[[6,3,2,1],[2,2,1]] => [6,3,2,1] => 0
[[5,1,1,1],[1]] => [5,1,1,1] => 0
[[6,2,2,1],[2,1,1]] => [6,2,2,1] => 1
[[6,2,1,1],[2,1]] => [6,2,1,1] => 0
[[7,3,2,1],[3,2,1]] => [7,3,2,1] => 1
[[4,4],[1]] => [4,4] => 0
[[5,4],[2]] => [5,4] => 1
[[6,4],[3]] => [6,4] => 0
[[7,4],[4]] => [7,4] => 1
[[3,3,1],[]] => [3,3,1] => 3
[[4,4,1],[1,1]] => [4,4,1] => 2
[[4,3,1],[1]] => [4,3,1] => 0
[[5,4,1],[2,1]] => [5,4,1] => 0
[[5,3,1],[2]] => [5,3,1] => 1
[[6,4,1],[3,1]] => [6,4,1] => 1
[[6,3,1],[3]] => [6,3,1] => 0
[[7,4,1],[4,1]] => [7,4,1] => 0
[[3,2,2],[]] => [3,2,2] => 3
[[4,3,2],[1,1]] => [4,3,2] => 1
[[5,4,2],[2,2]] => [5,4,2] => 1
[[4,2,2],[1]] => [4,2,2] => 2
[[5,3,2],[2,1]] => [5,3,2] => 0
[[6,4,2],[3,2]] => [6,4,2] => 0
[[5,2,2],[2]] => [5,2,2] => 3
[[6,3,2],[3,1]] => [6,3,2] => 1
[[7,4,2],[4,2]] => [7,4,2] => 1
[[3,2,1,1],[]] => [3,2,1,1] => 1
[[4,3,2,1],[1,1,1]] => [4,3,2,1] => 0
[[4,3,1,1],[1,1]] => [4,3,1,1] => 1
[[5,4,2,1],[2,2,1]] => [5,4,2,1] => 0
[[4,2,1,1],[1]] => [4,2,1,1] => 0
[[5,3,2,1],[2,1,1]] => [5,3,2,1] => 1
[[5,3,1,1],[2,1]] => [5,3,1,1] => 0
[[6,4,2,1],[3,2,1]] => [6,4,2,1] => 1
[[5,2,1,1],[2]] => [5,2,1,1] => 1
[[6,3,2,1],[3,1,1]] => [6,3,2,1] => 0
[[6,3,1,1],[3,1]] => [6,3,1,1] => 1
[[7,4,2,1],[4,2,1]] => [7,4,2,1] => 0
[[5,3,3],[2,2]] => [5,3,3] => 2
[[6,4,3],[3,3]] => [6,4,3] => 1
[[6,3,3],[3,2]] => [6,3,3] => 3
[[7,4,3],[4,3]] => [7,4,3] => 0
[[4,2,2,1],[1,1]] => [4,2,2,1] => 1
[[5,3,3,1],[2,2,1]] => [5,3,3,1] => 3
[[5,3,2,1],[2,2]] => [5,3,2,1] => 1
[[6,4,3,1],[3,3,1]] => [6,4,3,1] => 0
[[5,2,2,1],[2,1]] => [5,2,2,1] => 0
[[6,3,3,1],[3,2,1]] => [6,3,3,1] => 2
[[6,3,2,1],[3,2]] => [6,3,2,1] => 0
[[7,4,3,1],[4,3,1]] => [7,4,3,1] => 1
[[4,2,2,2],[1,1,1]] => [4,2,2,2] => 3
[[5,3,3,2],[2,2,2]] => [5,3,3,2] => 1
[[5,3,2,2],[2,2,1]] => [5,3,2,2] => 3
[[6,4,3,2],[3,3,2]] => [6,4,3,2] => 1
[[5,2,2,2],[2,1,1]] => [5,2,2,2] => 2
[[6,3,3,2],[3,2,2]] => [6,3,3,2] => 0
[[6,3,2,2],[3,2,1]] => [6,3,2,2] => 2
[[7,4,3,2],[4,3,2]] => [7,4,3,2] => 0
[[3,1,1,1,1],[]] => [3,1,1,1,1] => 1
[[4,2,2,2,1],[1,1,1,1]] => [4,2,2,2,1] => 1
[[4,2,2,1,1],[1,1,1]] => [4,2,2,1,1] => 0
[[5,3,3,2,1],[2,2,2,1]] => [5,3,3,2,1] => 0
[[4,2,1,1,1],[1,1]] => [4,2,1,1,1] => 1
[[5,3,2,2,1],[2,2,1,1]] => [5,3,2,2,1] => 1
[[5,3,2,1,1],[2,2,1]] => [5,3,2,1,1] => 0
[[6,4,3,2,1],[3,3,2,1]] => [6,4,3,2,1] => 0
[[4,1,1,1,1],[1]] => [4,1,1,1,1] => 0
[[5,2,2,2,1],[2,1,1,1]] => [5,2,2,2,1] => 0
[[5,2,2,1,1],[2,1,1]] => [5,2,2,1,1] => 1
[[6,3,3,2,1],[3,2,2,1]] => [6,3,3,2,1] => 1
[[5,2,1,1,1],[2,1]] => [5,2,1,1,1] => 0
[[6,3,2,2,1],[3,2,1,1]] => [6,3,2,2,1] => 0
[[6,3,2,1,1],[3,2,1]] => [6,3,2,1,1] => 1
[[7,4,3,2,1],[4,3,2,1]] => [7,4,3,2,1] => 1
[[5,5],[3]] => [5,5] => 0
[[6,5],[4]] => [6,5] => 1
[[7,5],[5]] => [7,5] => 0
[[4,4,1],[2]] => [4,4,1] => 2
[[5,5,1],[3,1]] => [5,5,1] => 1
[[5,4,1],[3]] => [5,4,1] => 0
[[6,5,1],[4,1]] => [6,5,1] => 0
[[6,4,1],[4]] => [6,4,1] => 1
[[7,5,1],[5,1]] => [7,5,1] => 1
[[3,3,2],[1]] => [3,3,2] => 0
[[4,4,2],[2,1]] => [4,4,2] => 0
[[5,5,2],[3,2]] => [5,5,2] => 2
[[4,3,2],[2]] => [4,3,2] => 1
[[5,4,2],[3,1]] => [5,4,2] => 1
[[6,5,2],[4,2]] => [6,5,2] => 1
[[5,3,2],[3]] => [5,3,2] => 0
[[6,4,2],[4,1]] => [6,4,2] => 0
[[7,5,2],[5,2]] => [7,5,2] => 0
[[3,3,1,1],[1]] => [3,3,1,1] => 2
[[4,4,2,1],[2,1,1]] => [4,4,2,1] => 2
[[4,4,1,1],[2,1]] => [4,4,1,1] => 3
[[5,5,2,1],[3,2,1]] => [5,5,2,1] => 3
[[4,3,1,1],[2]] => [4,3,1,1] => 1
[[5,4,2,1],[3,1,1]] => [5,4,2,1] => 0
[[5,4,1,1],[3,1]] => [5,4,1,1] => 1
[[6,5,2,1],[4,2,1]] => [6,5,2,1] => 0
[[5,3,1,1],[3]] => [5,3,1,1] => 0
[[6,4,2,1],[4,1,1]] => [6,4,2,1] => 1
[[6,4,1,1],[4,1]] => [6,4,1,1] => 0
[[7,5,2,1],[5,2,1]] => [7,5,2,1] => 1
[[3,3,3],[1,1]] => [3,3,3] => 2
[[4,4,3],[2,2]] => [4,4,3] => 1
[[5,5,3],[3,3]] => [5,5,3] => 1
[[4,3,3],[2,1]] => [4,3,3] => 3
[[5,4,3],[3,2]] => [5,4,3] => 0
[[6,5,3],[4,3]] => [6,5,3] => 0
[[5,3,3],[3,1]] => [5,3,3] => 2
[[6,4,3],[4,2]] => [6,4,3] => 1
[[7,5,3],[5,3]] => [7,5,3] => 1
[[2,2,2,1],[]] => [2,2,2,1] => 1
[[3,3,3,1],[1,1,1]] => [3,3,3,1] => 3
[[3,3,2,1],[1,1]] => [3,3,2,1] => 1
[[4,4,3,1],[2,2,1]] => [4,4,3,1] => 0
[[4,4,2,1],[2,2]] => [4,4,2,1] => 2
[[5,5,3,1],[3,3,1]] => [5,5,3,1] => 0
[[3,2,2,1],[1]] => [3,2,2,1] => 0
[[4,3,3,1],[2,1,1]] => [4,3,3,1] => 2
[[4,3,2,1],[2,1]] => [4,3,2,1] => 0
[[5,4,3,1],[3,2,1]] => [5,4,3,1] => 1
[[5,4,2,1],[3,2]] => [5,4,2,1] => 0
[[6,5,3,1],[4,3,1]] => [6,5,3,1] => 1
[[4,2,2,1],[2]] => [4,2,2,1] => 1
[[5,3,3,1],[3,1,1]] => [5,3,3,1] => 3
[[5,3,2,1],[3,1]] => [5,3,2,1] => 1
[[6,4,3,1],[4,2,1]] => [6,4,3,1] => 0
[[6,4,2,1],[4,2]] => [6,4,2,1] => 1
[[7,5,3,1],[5,3,1]] => [7,5,3,1] => 0
[[3,3,2,2],[1,1,1]] => [3,3,2,2] => 0
[[4,4,3,2],[2,2,2]] => [4,4,3,2] => 2
[[4,4,2,2],[2,2,1]] => [4,4,2,2] => 1
[[5,5,3,2],[3,3,2]] => [5,5,3,2] => 2
[[4,3,2,2],[2,1,1]] => [4,3,2,2] => 2
[[5,4,3,2],[3,2,2]] => [5,4,3,2] => 0
[[5,4,2,2],[3,2,1]] => [5,4,2,2] => 2
[[6,5,3,2],[4,3,2]] => [6,5,3,2] => 0
[[5,3,2,2],[3,1,1]] => [5,3,2,2] => 3
[[6,4,3,2],[4,2,2]] => [6,4,3,2] => 1
[[6,4,2,2],[4,2,1]] => [6,4,2,2] => 3
[[7,5,3,2],[5,3,2]] => [7,5,3,2] => 1
[[2,2,1,1,1],[]] => [2,2,1,1,1] => 1
[[3,3,2,2,1],[1,1,1,1]] => [3,3,2,2,1] => 1
[[3,3,2,1,1],[1,1,1]] => [3,3,2,1,1] => 0
[[4,4,3,2,1],[2,2,2,1]] => [4,4,3,2,1] => 0
[[3,3,1,1,1],[1,1]] => [3,3,1,1,1] => 3
[[4,4,2,2,1],[2,2,1,1]] => [4,4,2,2,1] => 2
[[4,4,2,1,1],[2,2,1]] => [4,4,2,1,1] => 3
[[5,5,3,2,1],[3,3,2,1]] => [5,5,3,2,1] => 4
[[3,2,1,1,1],[1]] => [3,2,1,1,1] => 0
[[4,3,2,2,1],[2,1,1,1]] => [4,3,2,2,1] => 0
[[4,3,2,1,1],[2,1,1]] => [4,3,2,1,1] => 1
[[5,4,3,2,1],[3,2,2,1]] => [5,4,3,2,1] => 1
[[4,3,1,1,1],[2,1]] => [4,3,1,1,1] => 0
[[5,4,2,2,1],[3,2,1,1]] => [5,4,2,2,1] => 0
[[5,4,2,1,1],[3,2,1]] => [5,4,2,1,1] => 1
[[6,5,3,2,1],[4,3,2,1]] => [6,5,3,2,1] => 1
[[4,2,1,1,1],[2]] => [4,2,1,1,1] => 1
[[5,3,2,2,1],[3,1,1,1]] => [5,3,2,2,1] => 1
[[5,3,2,1,1],[3,1,1]] => [5,3,2,1,1] => 0
[[6,4,3,2,1],[4,2,2,1]] => [6,4,3,2,1] => 0
[[5,3,1,1,1],[3,1]] => [5,3,1,1,1] => 1
[[6,4,2,2,1],[4,2,1,1]] => [6,4,2,2,1] => 1
[[6,4,2,1,1],[4,2,1]] => [6,4,2,1,1] => 0
[[5,4,4],[3,3]] => [5,4,4] => 1
[[6,5,4],[4,4]] => [6,5,4] => 1
[[6,4,4],[4,3]] => [6,4,4] => 0
[[7,5,4],[5,4]] => [7,5,4] => 0
[[4,3,3,1],[2,2]] => [4,3,3,1] => 2
[[5,4,4,1],[3,3,1]] => [5,4,4,1] => 3
[[5,4,3,1],[3,3]] => [5,4,3,1] => 1
[[6,5,4,1],[4,4,1]] => [6,5,4,1] => 0
[[5,3,3,1],[3,2]] => [5,3,3,1] => 3
[[6,4,4,1],[4,3,1]] => [6,4,4,1] => 2
[[6,4,3,1],[4,3]] => [6,4,3,1] => 0
[[7,5,4,1],[5,4,1]] => [7,5,4,1] => 1
[[3,2,2,2],[1,1]] => [3,2,2,2] => 2
[[4,3,3,2],[2,2,1]] => [4,3,3,2] => 0
[[5,4,4,2],[3,3,2]] => [5,4,4,2] => 1
[[4,3,2,2],[2,2]] => [4,3,2,2] => 2
[[5,4,3,2],[3,3,1]] => [5,4,3,2] => 0
[[6,5,4,2],[4,4,2]] => [6,5,4,2] => 1
[[4,2,2,2],[2,1]] => [4,2,2,2] => 3
[[5,3,3,2],[3,2,1]] => [5,3,3,2] => 1
[[6,4,4,2],[4,3,2]] => [6,4,4,2] => 0
[[5,3,2,2],[3,2]] => [5,3,2,2] => 3
[[6,4,3,2],[4,3,1]] => [6,4,3,2] => 1
[[3,2,2,1,1],[1,1]] => [3,2,2,1,1] => 1
[[4,3,3,2,1],[2,2,1,1]] => [4,3,3,2,1] => 1
[[4,3,3,1,1],[2,2,1]] => [4,3,3,1,1] => 3
[[5,4,4,2,1],[3,3,2,1]] => [5,4,4,2,1] => 3
[[4,3,2,1,1],[2,2]] => [4,3,2,1,1] => 1
[[5,4,3,2,1],[3,3,1,1]] => [5,4,3,2,1] => 1
[[5,4,3,1,1],[3,3,1]] => [5,4,3,1,1] => 0
[[4,2,2,1,1],[2,1]] => [4,2,2,1,1] => 0
[[5,3,3,2,1],[3,2,1,1]] => [5,3,3,2,1] => 0
[[5,3,3,1,1],[3,2,1]] => [5,3,3,1,1] => 2
[[6,4,4,2,1],[4,3,2,1]] => [6,4,4,2,1] => 2
[[5,3,2,1,1],[3,2]] => [5,3,2,1,1] => 0
[[6,4,3,2,1],[4,3,1,1]] => [6,4,3,2,1] => 0
[[6,4,3,1,1],[4,3,1]] => [6,4,3,1,1] => 1
[[4,3,3,3],[2,2,2]] => [4,3,3,3] => 2
[[5,4,4,3],[3,3,3]] => [5,4,4,3] => 0
[[5,4,3,3],[3,3,2]] => [5,4,3,3] => 2
[[5,3,3,3],[3,2,2]] => [5,3,3,3] => 3
[[6,4,4,3],[4,3,3]] => [6,4,4,3] => 1
[[6,4,3,3],[4,3,2]] => [6,4,3,3] => 3
[[3,2,2,2,1],[1,1,1]] => [3,2,2,2,1] => 0
[[4,3,3,3,1],[2,2,2,1]] => [4,3,3,3,1] => 3
[[4,3,3,2,1],[2,2,2]] => [4,3,3,2,1] => 1
[[5,4,4,3,1],[3,3,3,1]] => [5,4,4,3,1] => 1
[[4,3,2,2,1],[2,2,1]] => [4,3,2,2,1] => 0
[[5,4,3,3,1],[3,3,2,1]] => [5,4,3,3,1] => 3
[[5,4,3,2,1],[3,3,2]] => [5,4,3,2,1] => 1
[[4,2,2,2,1],[2,1,1]] => [4,2,2,2,1] => 1
[[5,3,3,3,1],[3,2,2,1]] => [5,3,3,3,1] => 2
[[5,3,3,2,1],[3,2,2]] => [5,3,3,2,1] => 0
[[5,3,2,2,1],[3,2,1]] => [5,3,2,2,1] => 1
[[6,4,3,3,1],[4,3,2,1]] => [6,4,3,3,1] => 2
[[6,4,3,2,1],[4,3,2]] => [6,4,3,2,1] => 0
[[3,2,2,2,2],[1,1,1,1]] => [3,2,2,2,2] => 1
[[4,3,3,3,2],[2,2,2,2]] => [4,3,3,3,2] => 1
[[4,3,3,2,2],[2,2,2,1]] => [4,3,3,2,2] => 0
[[4,3,2,2,2],[2,2,1,1]] => [4,3,2,2,2] => 3
[[5,4,3,3,2],[3,3,2,2]] => [5,4,3,3,2] => 1
[[5,4,3,2,2],[3,3,2,1]] => [5,4,3,2,2] => 4
[[4,2,2,2,2],[2,1,1,1]] => [4,2,2,2,2] => 0
[[5,3,3,3,2],[3,2,2,2]] => [5,3,3,3,2] => 0
[[5,3,3,2,2],[3,2,2,1]] => [5,3,3,2,2] => 1
[[5,3,2,2,2],[3,2,1,1]] => [5,3,2,2,2] => 2
[[6,4,3,2,2],[4,3,2,1]] => [6,4,3,2,2] => 3
[[2,1,1,1,1,1],[]] => [2,1,1,1,1,1] => 1
[[3,2,2,2,2,1],[1,1,1,1,1]] => [3,2,2,2,2,1] => 0
[[3,2,2,2,1,1],[1,1,1,1]] => [3,2,2,2,1,1] => 1
[[4,3,3,3,2,1],[2,2,2,2,1]] => [4,3,3,3,2,1] => 0
[[3,2,2,1,1,1],[1,1,1]] => [3,2,2,1,1,1] => 0
[[4,3,3,2,2,1],[2,2,2,1,1]] => [4,3,3,2,2,1] => 1
[[4,3,3,2,1,1],[2,2,2,1]] => [4,3,3,2,1,1] => 0
[[3,2,1,1,1,1],[1,1]] => [3,2,1,1,1,1] => 1
[[4,3,2,2,2,1],[2,2,1,1,1]] => [4,3,2,2,2,1] => 0
[[4,3,2,2,1,1],[2,2,1,1]] => [4,3,2,2,1,1] => 1
[[4,3,2,1,1,1],[2,2,1]] => [4,3,2,1,1,1] => 0
[[5,4,3,2,2,1],[3,3,2,1,1]] => [5,4,3,2,2,1] => 1
[[5,4,3,2,1,1],[3,3,2,1]] => [5,4,3,2,1,1] => 0
[[3,1,1,1,1,1],[1]] => [3,1,1,1,1,1] => 0
[[4,2,2,2,2,1],[2,1,1,1,1]] => [4,2,2,2,2,1] => 1
[[4,2,2,2,1,1],[2,1,1,1]] => [4,2,2,2,1,1] => 0
[[5,3,3,3,2,1],[3,2,2,2,1]] => [5,3,3,3,2,1] => 1
[[4,2,2,1,1,1],[2,1,1]] => [4,2,2,1,1,1] => 1
[[5,3,3,2,2,1],[3,2,2,1,1]] => [5,3,3,2,2,1] => 0
[[5,3,3,2,1,1],[3,2,2,1]] => [5,3,3,2,1,1] => 1
[[4,2,1,1,1,1],[2,1]] => [4,2,1,1,1,1] => 0
[[5,3,2,2,2,1],[3,2,1,1,1]] => [5,3,2,2,2,1] => 1
[[5,3,2,2,1,1],[3,2,1,1]] => [5,3,2,2,1,1] => 0
[[5,3,2,1,1,1],[3,2,1]] => [5,3,2,1,1,1] => 1
[[6,4,3,2,1,1],[4,3,2,1]] => [6,4,3,2,1,1] => 1
[[6,6],[5]] => [6,6] => 2
[[7,6],[6]] => [7,6] => 1
[[5,5,1],[4]] => [5,5,1] => 1
[[6,6,1],[5,1]] => [6,6,1] => 1
[[6,5,1],[5]] => [6,5,1] => 0
[[7,6,1],[6,1]] => [7,6,1] => 0
[[4,4,2],[3]] => [4,4,2] => 0
[[5,5,2],[4,1]] => [5,5,2] => 2
[[6,6,2],[5,2]] => [6,6,2] => 2
[[5,4,2],[4]] => [5,4,2] => 1
[[6,5,2],[5,1]] => [6,5,2] => 1
[[7,6,2],[6,2]] => [7,6,2] => 1
[[4,4,1,1],[3]] => [4,4,1,1] => 3
[[5,5,2,1],[4,1,1]] => [5,5,2,1] => 3
[[5,5,1,1],[4,1]] => [5,5,1,1] => 0
[[6,6,2,1],[5,2,1]] => [6,6,2,1] => 1
[[5,4,1,1],[4]] => [5,4,1,1] => 1
[[6,5,2,1],[5,1,1]] => [6,5,2,1] => 0
[[6,5,1,1],[5,1]] => [6,5,1,1] => 1
[[7,6,2,1],[6,2,1]] => [7,6,2,1] => 0
[[3,3,3],[2]] => [3,3,3] => 2
[[4,4,3],[3,1]] => [4,4,3] => 1
[[5,5,3],[4,2]] => [5,5,3] => 1
[[6,6,3],[5,3]] => [6,6,3] => 3
[[4,3,3],[3]] => [4,3,3] => 3
[[5,4,3],[4,1]] => [5,4,3] => 0
[[6,5,3],[5,2]] => [6,5,3] => 0
[[7,6,3],[6,3]] => [7,6,3] => 0
[[3,3,2,1],[2]] => [3,3,2,1] => 1
[[4,4,3,1],[3,1,1]] => [4,4,3,1] => 0
[[4,4,2,1],[3,1]] => [4,4,2,1] => 2
[[5,5,3,1],[4,2,1]] => [5,5,3,1] => 0
[[5,5,2,1],[4,2]] => [5,5,2,1] => 3
[[6,6,3,1],[5,3,1]] => [6,6,3,1] => 2
[[4,3,2,1],[3]] => [4,3,2,1] => 0
[[5,4,3,1],[4,1,1]] => [5,4,3,1] => 1
[[5,4,2,1],[4,1]] => [5,4,2,1] => 0
[[6,5,3,1],[5,2,1]] => [6,5,3,1] => 1
[[6,5,2,1],[5,2]] => [6,5,2,1] => 0
[[7,6,3,1],[6,3,1]] => [7,6,3,1] => 1
[[4,4,2,2],[3,1,1]] => [4,4,2,2] => 1
[[5,5,3,2],[4,2,2]] => [5,5,3,2] => 2
[[5,5,2,2],[4,2,1]] => [5,5,2,2] => 1
[[6,6,3,2],[5,3,2]] => [6,6,3,2] => 1
[[5,4,2,2],[4,1,1]] => [5,4,2,2] => 2
[[6,5,3,2],[5,2,2]] => [6,5,3,2] => 0
[[6,5,2,2],[5,2,1]] => [6,5,2,2] => 2
[[3,3,1,1,1],[2]] => [3,3,1,1,1] => 3
[[4,4,2,2,1],[3,1,1,1]] => [4,4,2,2,1] => 2
[[4,4,2,1,1],[3,1,1]] => [4,4,2,1,1] => 3
[[5,5,3,2,1],[4,2,2,1]] => [5,5,3,2,1] => 4
[[4,4,1,1,1],[3,1]] => [4,4,1,1,1] => 2
[[5,5,2,2,1],[4,2,1,1]] => [5,5,2,2,1] => 3
[[5,5,2,1,1],[4,2,1]] => [5,5,2,1,1] => 2
[[4,3,1,1,1],[3]] => [4,3,1,1,1] => 0
[[5,4,2,2,1],[4,1,1,1]] => [5,4,2,2,1] => 0
[[5,4,2,1,1],[4,1,1]] => [5,4,2,1,1] => 1
[[6,5,3,2,1],[5,2,2,1]] => [6,5,3,2,1] => 1
[[5,4,1,1,1],[4,1]] => [5,4,1,1,1] => 0
[[6,5,2,2,1],[5,2,1,1]] => [6,5,2,2,1] => 0
[[6,5,2,1,1],[5,2,1]] => [6,5,2,1,1] => 1
[[4,4,4],[3,2]] => [4,4,4] => 0
[[5,5,4],[4,3]] => [5,5,4] => 0
[[6,6,4],[5,4]] => [6,6,4] => 2
[[5,4,4],[4,2]] => [5,4,4] => 1
[[6,5,4],[5,3]] => [6,5,4] => 1
[[7,6,4],[6,4]] => [7,6,4] => 1
[[3,3,3,1],[2,1]] => [3,3,3,1] => 3
[[4,4,4,1],[3,2,1]] => [4,4,4,1] => 2
[[4,4,3,1],[3,2]] => [4,4,3,1] => 0
[[5,5,4,1],[4,3,1]] => [5,5,4,1] => 1
[[5,5,3,1],[4,3]] => [5,5,3,1] => 0
[[6,6,4,1],[5,4,1]] => [6,6,4,1] => 1
[[4,3,3,1],[3,1]] => [4,3,3,1] => 2
[[5,4,4,1],[4,2,1]] => [5,4,4,1] => 3
[[5,4,3,1],[4,2]] => [5,4,3,1] => 1
[[6,5,4,1],[5,3,1]] => [6,5,4,1] => 0
[[6,5,3,1],[5,3]] => [6,5,3,1] => 1
[[2,2,2,2],[1]] => [2,2,2,2] => 0
[[3,3,3,2],[2,1,1]] => [3,3,3,2] => 1
[[4,4,4,2],[3,2,2]] => [4,4,4,2] => 0
[[3,3,2,2],[2,1]] => [3,3,2,2] => 0
[[4,4,3,2],[3,2,1]] => [4,4,3,2] => 2
[[5,5,4,2],[4,3,2]] => [5,5,4,2] => 0
[[4,4,2,2],[3,2]] => [4,4,2,2] => 1
[[5,5,3,2],[4,3,1]] => [5,5,3,2] => 2
[[3,2,2,2],[2]] => [3,2,2,2] => 2
[[4,3,3,2],[3,1,1]] => [4,3,3,2] => 0
[[5,4,4,2],[4,2,2]] => [5,4,4,2] => 1
[[4,3,2,2],[3,1]] => [4,3,2,2] => 2
[[5,4,3,2],[4,2,1]] => [5,4,3,2] => 0
[[6,5,4,2],[5,3,2]] => [6,5,4,2] => 1
[[5,4,2,2],[4,2]] => [5,4,2,2] => 2
[[6,5,3,2],[5,3,1]] => [6,5,3,2] => 0
[[2,2,2,1,1],[1]] => [2,2,2,1,1] => 0
[[3,3,3,2,1],[2,1,1,1]] => [3,3,3,2,1] => 0
[[3,3,3,1,1],[2,1,1]] => [3,3,3,1,1] => 2
[[4,4,4,2,1],[3,2,2,1]] => [4,4,4,2,1] => 2
[[3,3,2,1,1],[2,1]] => [3,3,2,1,1] => 0
[[4,4,3,2,1],[3,2,1,1]] => [4,4,3,2,1] => 0
[[4,4,3,1,1],[3,2,1]] => [4,4,3,1,1] => 1
[[5,5,4,2,1],[4,3,2,1]] => [5,5,4,2,1] => 1
[[4,4,2,1,1],[3,2]] => [4,4,2,1,1] => 3
[[5,5,3,2,1],[4,3,1,1]] => [5,5,3,2,1] => 4
[[5,5,3,1,1],[4,3,1]] => [5,5,3,1,1] => 1
[[3,2,2,1,1],[2]] => [3,2,2,1,1] => 1
[[4,3,3,2,1],[3,1,1,1]] => [4,3,3,2,1] => 1
[[4,3,3,1,1],[3,1,1]] => [4,3,3,1,1] => 3
[[5,4,4,2,1],[4,2,2,1]] => [5,4,4,2,1] => 3
[[4,3,2,1,1],[3,1]] => [4,3,2,1,1] => 1
[[5,4,3,2,1],[4,2,1,1]] => [5,4,3,2,1] => 1
[[5,4,3,1,1],[4,2,1]] => [5,4,3,1,1] => 0
[[5,4,2,1,1],[4,2]] => [5,4,2,1,1] => 1
[[6,5,3,2,1],[5,3,1,1]] => [6,5,3,2,1] => 1
[[6,5,3,1,1],[5,3,1]] => [6,5,3,1,1] => 0
[[4,4,3,3],[3,2,2]] => [4,4,3,3] => 0
[[5,5,4,3],[4,3,3]] => [5,5,4,3] => 2
[[5,5,3,3],[4,3,2]] => [5,5,3,3] => 1
[[5,4,3,3],[4,2,2]] => [5,4,3,3] => 2
[[6,5,3,3],[5,3,2]] => [6,5,3,3] => 2
[[3,3,2,2,1],[2,1,1]] => [3,3,2,2,1] => 1
[[4,4,3,3,1],[3,2,2,1]] => [4,4,3,3,1] => 1
[[4,4,3,2,1],[3,2,2]] => [4,4,3,2,1] => 0
[[4,4,2,2,1],[3,2,1]] => [4,4,2,2,1] => 2
[[5,5,3,3,1],[4,3,2,1]] => [5,5,3,3,1] => 0
[[5,5,3,2,1],[4,3,2]] => [5,5,3,2,1] => 4
[[4,3,2,2,1],[3,1,1]] => [4,3,2,2,1] => 0
[[5,4,3,3,1],[4,2,2,1]] => [5,4,3,3,1] => 3
[[5,4,3,2,1],[4,2,2]] => [5,4,3,2,1] => 1
[[5,4,2,2,1],[4,2,1]] => [5,4,2,2,1] => 0
[[6,5,3,2,1],[5,3,2]] => [6,5,3,2,1] => 1
[[3,3,2,2,2],[2,1,1,1]] => [3,3,2,2,2] => 2
[[4,4,3,3,2],[3,2,2,2]] => [4,4,3,3,2] => 0
[[4,4,3,2,2],[3,2,2,1]] => [4,4,3,2,2] => 2
[[4,4,2,2,2],[3,2,1,1]] => [4,4,2,2,2] => 1
[[5,5,3,2,2],[4,3,2,1]] => [5,5,3,2,2] => 2
[[4,3,2,2,2],[3,1,1,1]] => [4,3,2,2,2] => 3
[[5,4,3,3,2],[4,2,2,2]] => [5,4,3,3,2] => 1
[[5,4,3,2,2],[4,2,2,1]] => [5,4,3,2,2] => 4
[[5,4,2,2,2],[4,2,1,1]] => [5,4,2,2,2] => 3
[[2,2,1,1,1,1],[1]] => [2,2,1,1,1,1] => 0
[[3,3,2,2,2,1],[2,1,1,1,1]] => [3,3,2,2,2,1] => 1
[[3,3,2,2,1,1],[2,1,1,1]] => [3,3,2,2,1,1] => 0
[[4,4,3,3,2,1],[3,2,2,2,1]] => [4,4,3,3,2,1] => 1
[[3,3,2,1,1,1],[2,1,1]] => [3,3,2,1,1,1] => 1
[[4,4,3,2,2,1],[3,2,2,1,1]] => [4,4,3,2,2,1] => 0
[[4,4,3,2,1,1],[3,2,2,1]] => [4,4,3,2,1,1] => 1
[[3,3,1,1,1,1],[2,1]] => [3,3,1,1,1,1] => 2
[[4,4,2,2,2,1],[3,2,1,1,1]] => [4,4,2,2,2,1] => 2
[[4,4,2,2,1,1],[3,2,1,1]] => [4,4,2,2,1,1] => 3
[[4,4,2,1,1,1],[3,2,1]] => [4,4,2,1,1,1] => 2
[[5,5,3,2,1,1],[4,3,2,1]] => [5,5,3,2,1,1] => 3
[[3,2,1,1,1,1],[2]] => [3,2,1,1,1,1] => 1
[[4,3,2,2,2,1],[3,1,1,1,1]] => [4,3,2,2,2,1] => 0
[[4,3,2,2,1,1],[3,1,1,1]] => [4,3,2,2,1,1] => 1
[[4,3,2,1,1,1],[3,1,1]] => [4,3,2,1,1,1] => 0
[[5,4,3,2,2,1],[4,2,2,1,1]] => [5,4,3,2,2,1] => 1
[[5,4,3,2,1,1],[4,2,2,1]] => [5,4,3,2,1,1] => 0
[[4,3,1,1,1,1],[3,1]] => [4,3,1,1,1,1] => 1
[[5,4,2,2,2,1],[4,2,1,1,1]] => [5,4,2,2,2,1] => 0
[[5,4,2,2,1,1],[4,2,1,1]] => [5,4,2,2,1,1] => 1
[[5,4,2,1,1,1],[4,2,1]] => [5,4,2,1,1,1] => 0
[[5,5,5],[4,4]] => [5,5,5] => 1
[[6,6,5],[5,5]] => [6,6,5] => 1
[[6,5,5],[5,4]] => [6,5,5] => 0
[[4,4,4,1],[3,3]] => [4,4,4,1] => 2
[[5,5,5,1],[4,4,1]] => [5,5,5,1] => 0
[[5,5,4,1],[4,4]] => [5,5,4,1] => 1
[[5,4,4,1],[4,3]] => [5,4,4,1] => 3
[[6,5,5,1],[5,4,1]] => [6,5,5,1] => 1
[[6,5,4,1],[5,4]] => [6,5,4,1] => 0
[[3,3,3,2],[2,2]] => [3,3,3,2] => 1
[[4,4,4,2],[3,3,1]] => [4,4,4,2] => 0
[[5,5,5,2],[4,4,2]] => [5,5,5,2] => 2
[[4,4,3,2],[3,3]] => [4,4,3,2] => 2
[[5,5,4,2],[4,4,1]] => [5,5,4,2] => 0
[[4,3,3,2],[3,2]] => [4,3,3,2] => 0
[[5,4,4,2],[4,3,1]] => [5,4,4,2] => 1
[[5,4,3,2],[4,3]] => [5,4,3,2] => 0
[[6,5,4,2],[5,4,1]] => [6,5,4,2] => 1
[[3,3,3,1,1],[2,2]] => [3,3,3,1,1] => 2
[[4,4,4,2,1],[3,3,1,1]] => [4,4,4,2,1] => 2
[[4,4,4,1,1],[3,3,1]] => [4,4,4,1,1] => 3
[[4,4,3,1,1],[3,3]] => [4,4,3,1,1] => 1
[[5,5,4,2,1],[4,4,1,1]] => [5,5,4,2,1] => 1
[[5,5,4,1,1],[4,4,1]] => [5,5,4,1,1] => 0
[[4,3,3,1,1],[3,2]] => [4,3,3,1,1] => 3
[[5,4,4,2,1],[4,3,1,1]] => [5,4,4,2,1] => 3
[[5,4,4,1,1],[4,3,1]] => [5,4,4,1,1] => 2
[[5,4,3,1,1],[4,3]] => [5,4,3,1,1] => 0
[[6,5,4,1,1],[5,4,1]] => [6,5,4,1,1] => 1
[[3,3,3,3],[2,2,1]] => [3,3,3,3] => 0
[[4,4,4,3],[3,3,2]] => [4,4,4,3] => 1
[[4,4,3,3],[3,3,1]] => [4,4,3,3] => 0
[[5,5,4,3],[4,4,2]] => [5,5,4,3] => 2
[[4,3,3,3],[3,2,1]] => [4,3,3,3] => 2
[[5,4,4,3],[4,3,2]] => [5,4,4,3] => 0
[[5,4,3,3],[4,3,1]] => [5,4,3,3] => 2
[[2,2,2,2,1],[1,1]] => [2,2,2,2,1] => 1
[[3,3,3,3,1],[2,2,1,1]] => [3,3,3,3,1] => 1
[[3,3,3,2,1],[2,2,1]] => [3,3,3,2,1] => 0
[[4,4,4,3,1],[3,3,2,1]] => [4,4,4,3,1] => 0
[[4,4,4,2,1],[3,3,2]] => [4,4,4,2,1] => 2
[[3,3,2,2,1],[2,2]] => [3,3,2,2,1] => 1
[[4,4,3,3,1],[3,3,1,1]] => [4,4,3,3,1] => 1
[[4,4,3,2,1],[3,3,1]] => [4,4,3,2,1] => 0
[[5,5,4,2,1],[4,4,2]] => [5,5,4,2,1] => 1
[[3,2,2,2,1],[2,1]] => [3,2,2,2,1] => 0
[[4,3,3,3,1],[3,2,1,1]] => [4,3,3,3,1] => 3
[[4,3,3,2,1],[3,2,1]] => [4,3,3,2,1] => 1
[[5,4,4,3,1],[4,3,2,1]] => [5,4,4,3,1] => 1
[[5,4,4,2,1],[4,3,2]] => [5,4,4,2,1] => 3
[[4,3,2,2,1],[3,2]] => [4,3,2,2,1] => 0
[[5,4,3,3,1],[4,3,1,1]] => [5,4,3,3,1] => 3
[[5,4,3,2,1],[4,3,1]] => [5,4,3,2,1] => 1
[[3,3,3,2,2],[2,2,1,1]] => [3,3,3,2,2] => 1
[[4,4,4,3,2],[3,3,2,2]] => [4,4,4,3,2] => 2
[[4,4,4,2,2],[3,3,2,1]] => [4,4,4,2,2] => 1
[[4,4,3,2,2],[3,3,1,1]] => [4,4,3,2,2] => 2
[[4,3,3,2,2],[3,2,1,1]] => [4,3,3,2,2] => 0
[[5,4,4,2,2],[4,3,2,1]] => [5,4,4,2,2] => 0
[[5,4,3,2,2],[4,3,1,1]] => [5,4,3,2,2] => 4
[[2,2,2,1,1,1],[1,1]] => [2,2,2,1,1,1] => 1
[[3,3,3,2,2,1],[2,2,1,1,1]] => [3,3,3,2,2,1] => 0
[[3,3,3,2,1,1],[2,2,1,1]] => [3,3,3,2,1,1] => 1
[[3,3,3,1,1,1],[2,2,1]] => [3,3,3,1,1,1] => 3
[[4,4,4,2,2,1],[3,3,2,1,1]] => [4,4,4,2,2,1] => 2
[[4,4,4,2,1,1],[3,3,2,1]] => [4,4,4,2,1,1] => 3
[[3,3,2,1,1,1],[2,2]] => [3,3,2,1,1,1] => 1
[[4,4,3,2,2,1],[3,3,1,1,1]] => [4,4,3,2,2,1] => 0
[[4,4,3,2,1,1],[3,3,1,1]] => [4,4,3,2,1,1] => 1
[[4,4,3,1,1,1],[3,3,1]] => [4,4,3,1,1,1] => 0
[[3,2,2,1,1,1],[2,1]] => [3,2,2,1,1,1] => 0
[[4,3,3,2,2,1],[3,2,1,1,1]] => [4,3,3,2,2,1] => 1
[[4,3,3,2,1,1],[3,2,1,1]] => [4,3,3,2,1,1] => 0
[[4,3,3,1,1,1],[3,2,1]] => [4,3,3,1,1,1] => 2
[[5,4,4,2,1,1],[4,3,2,1]] => [5,4,4,2,1,1] => 2
[[4,3,2,1,1,1],[3,2]] => [4,3,2,1,1,1] => 0
[[5,4,3,2,2,1],[4,3,1,1,1]] => [5,4,3,2,2,1] => 1
[[5,4,3,2,1,1],[4,3,1,1]] => [5,4,3,2,1,1] => 0
[[5,4,3,1,1,1],[4,3,1]] => [5,4,3,1,1,1] => 1
[[4,4,4,4],[3,3,3]] => [4,4,4,4] => 0
[[5,4,4,4],[4,3,3]] => [5,4,4,4] => 1
[[3,3,3,3,1],[2,2,2]] => [3,3,3,3,1] => 1
[[4,4,4,4,1],[3,3,3,1]] => [4,4,4,4,1] => 1
[[4,4,4,3,1],[3,3,3]] => [4,4,4,3,1] => 0
[[4,4,3,3,1],[3,3,2]] => [4,4,3,3,1] => 1
[[4,3,3,3,1],[3,2,2]] => [4,3,3,3,1] => 3
[[5,4,4,3,1],[4,3,3]] => [5,4,4,3,1] => 1
[[5,4,3,3,1],[4,3,2]] => [5,4,3,3,1] => 3
[[2,2,2,2,2],[1,1,1]] => [2,2,2,2,2] => 0
[[3,3,3,3,2],[2,2,2,1]] => [3,3,3,3,2] => 0
[[3,3,3,2,2],[2,2,2]] => [3,3,3,2,2] => 1
[[4,4,4,3,2],[3,3,3,1]] => [4,4,4,3,2] => 2
[[3,3,2,2,2],[2,2,1]] => [3,3,2,2,2] => 2
[[4,4,3,3,2],[3,3,2,1]] => [4,4,3,3,2] => 0
[[4,4,3,2,2],[3,3,2]] => [4,4,3,2,2] => 2
[[3,2,2,2,2],[2,1,1]] => [3,2,2,2,2] => 1
[[4,3,3,3,2],[3,2,2,1]] => [4,3,3,3,2] => 1
[[4,3,3,2,2],[3,2,2]] => [4,3,3,2,2] => 0
[[4,3,2,2,2],[3,2,1]] => [4,3,2,2,2] => 3
[[5,4,3,3,2],[4,3,2,1]] => [5,4,3,3,2] => 1
[[5,4,3,2,2],[4,3,2]] => [5,4,3,2,2] => 4
[[2,2,2,2,1,1],[1,1,1]] => [2,2,2,2,1,1] => 0
[[3,3,3,3,2,1],[2,2,2,1,1]] => [3,3,3,3,2,1] => 1
[[3,3,3,3,1,1],[2,2,2,1]] => [3,3,3,3,1,1] => 0
[[3,3,3,2,1,1],[2,2,2]] => [3,3,3,2,1,1] => 1
[[4,4,4,3,1,1],[3,3,3,1]] => [4,4,4,3,1,1] => 1
[[3,3,2,2,1,1],[2,2,1]] => [3,3,2,2,1,1] => 0
[[4,4,3,3,2,1],[3,3,2,1,1]] => [4,4,3,3,2,1] => 1
[[4,4,3,3,1,1],[3,3,2,1]] => [4,4,3,3,1,1] => 0
[[4,4,3,2,1,1],[3,3,2]] => [4,4,3,2,1,1] => 1
[[3,2,2,2,1,1],[2,1,1]] => [3,2,2,2,1,1] => 1
[[4,3,3,3,2,1],[3,2,2,1,1]] => [4,3,3,3,2,1] => 0
[[4,3,3,3,1,1],[3,2,2,1]] => [4,3,3,3,1,1] => 2
[[4,3,3,2,1,1],[3,2,2]] => [4,3,3,2,1,1] => 0
[[4,3,2,2,1,1],[3,2,1]] => [4,3,2,2,1,1] => 1
[[5,4,3,3,1,1],[4,3,2,1]] => [5,4,3,3,1,1] => 2
[[5,4,3,2,1,1],[4,3,2]] => [5,4,3,2,1,1] => 0
[[3,3,3,3,3],[2,2,2,2]] => [3,3,3,3,3] => 1
[[4,4,3,3,3],[3,3,2,2]] => [4,4,3,3,3] => 2
[[4,3,3,3,3],[3,2,2,2]] => [4,3,3,3,3] => 0
[[2,2,2,2,2,1],[1,1,1,1]] => [2,2,2,2,2,1] => 1
[[3,3,3,3,3,1],[2,2,2,2,1]] => [3,3,3,3,3,1] => 0
[[3,3,3,3,2,1],[2,2,2,2]] => [3,3,3,3,2,1] => 1
[[3,3,3,2,2,1],[2,2,2,1]] => [3,3,3,2,2,1] => 0
[[3,3,2,2,2,1],[2,2,1,1]] => [3,3,2,2,2,1] => 1
[[4,4,3,3,2,1],[3,3,2,2]] => [4,4,3,3,2,1] => 1
[[4,4,3,2,2,1],[3,3,2,1]] => [4,4,3,2,2,1] => 0
[[3,2,2,2,2,1],[2,1,1,1]] => [3,2,2,2,2,1] => 0
[[4,3,3,3,3,1],[3,2,2,2,1]] => [4,3,3,3,3,1] => 1
[[4,3,3,3,2,1],[3,2,2,2]] => [4,3,3,3,2,1] => 0
[[4,3,3,2,2,1],[3,2,2,1]] => [4,3,3,2,2,1] => 1
[[4,3,2,2,2,1],[3,2,1,1]] => [4,3,2,2,2,1] => 0
[[5,4,3,2,2,1],[4,3,2,1]] => [5,4,3,2,2,1] => 1
[[2,2,2,2,2,2],[1,1,1,1,1]] => [2,2,2,2,2,2] => 2
[[3,3,3,3,3,2],[2,2,2,2,2]] => [3,3,3,3,3,2] => 1
[[3,3,3,3,2,2],[2,2,2,2,1]] => [3,3,3,3,2,2] => 2
[[3,3,3,2,2,2],[2,2,2,1,1]] => [3,3,3,2,2,2] => 3
[[3,3,2,2,2,2],[2,2,1,1,1]] => [3,3,2,2,2,2] => 2
[[4,4,3,2,2,2],[3,3,2,1,1]] => [4,4,3,2,2,2] => 1
[[3,2,2,2,2,2],[2,1,1,1,1]] => [3,2,2,2,2,2] => 1
[[4,3,3,3,2,2],[3,2,2,2,1]] => [4,3,3,3,2,2] => 1
[[4,3,3,2,2,2],[3,2,2,1,1]] => [4,3,3,2,2,2] => 2
[[4,3,2,2,2,2],[3,2,1,1,1]] => [4,3,2,2,2,2] => 1
[[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]] => [2,2,2,2,2,2,1] => 1
[[2,2,2,2,2,1,1],[1,1,1,1,1]] => [2,2,2,2,2,1,1] => 0
[[2,2,2,2,1,1,1],[1,1,1,1]] => [2,2,2,2,1,1,1] => 1
[[3,3,3,3,2,2,1],[2,2,2,2,1,1]] => [3,3,3,3,2,2,1] => 1
[[3,3,3,3,2,1,1],[2,2,2,2,1]] => [3,3,3,3,2,1,1] => 0
[[2,2,2,1,1,1,1],[1,1,1]] => [2,2,2,1,1,1,1] => 0
[[3,3,3,2,2,2,1],[2,2,2,1,1,1]] => [3,3,3,2,2,2,1] => 0
[[3,3,3,2,2,1,1],[2,2,2,1,1]] => [3,3,3,2,2,1,1] => 1
[[3,3,3,2,1,1,1],[2,2,2,1]] => [3,3,3,2,1,1,1] => 0
[[2,2,1,1,1,1,1],[1,1]] => [2,2,1,1,1,1,1] => 1
[[3,3,2,2,2,2,1],[2,2,1,1,1,1]] => [3,3,2,2,2,2,1] => 1
[[3,3,2,2,2,1,1],[2,2,1,1,1]] => [3,3,2,2,2,1,1] => 0
[[3,3,2,2,1,1,1],[2,2,1,1]] => [3,3,2,2,1,1,1] => 1
[[3,3,2,1,1,1,1],[2,2,1]] => [3,3,2,1,1,1,1] => 0
[[4,4,3,2,2,1,1],[3,3,2,1,1]] => [4,4,3,2,2,1,1] => 1
[[4,4,3,2,1,1,1],[3,3,2,1]] => [4,4,3,2,1,1,1] => 0
[[2,1,1,1,1,1,1],[1]] => [2,1,1,1,1,1,1] => 0
[[3,2,2,2,2,2,1],[2,1,1,1,1,1]] => [3,2,2,2,2,2,1] => 0
[[3,2,2,2,2,1,1],[2,1,1,1,1]] => [3,2,2,2,2,1,1] => 1
[[3,2,2,2,1,1,1],[2,1,1,1]] => [3,2,2,2,1,1,1] => 0
[[4,3,3,3,2,1,1],[3,2,2,2,1]] => [4,3,3,3,2,1,1] => 1
[[3,2,2,1,1,1,1],[2,1,1]] => [3,2,2,1,1,1,1] => 1
[[4,3,3,2,2,2,1],[3,2,2,1,1,1]] => [4,3,3,2,2,2,1] => 1
[[4,3,3,2,2,1,1],[3,2,2,1,1]] => [4,3,3,2,2,1,1] => 0
[[4,3,3,2,1,1,1],[3,2,2,1]] => [4,3,3,2,1,1,1] => 1
[[3,2,1,1,1,1,1],[2,1]] => [3,2,1,1,1,1,1] => 0
[[4,3,2,2,2,2,1],[3,2,1,1,1,1]] => [4,3,2,2,2,2,1] => 0
[[4,3,2,2,2,1,1],[3,2,1,1,1]] => [4,3,2,2,2,1,1] => 1
[[4,3,2,2,1,1,1],[3,2,1,1]] => [4,3,2,2,1,1,1] => 0
[[4,3,2,1,1,1,1],[3,2,1]] => [4,3,2,1,1,1,1] => 1
[[5,4,3,2,1,1,1],[4,3,2,1]] => [5,4,3,2,1,1,1] => 1
[[8],[]] => [8] => 0
[[7,1],[]] => [7,1] => 0
[[6,2],[]] => [6,2] => 0
[[6,1,1],[]] => [6,1,1] => 0
[[5,3],[]] => [5,3] => 0
[[5,2,1],[]] => [5,2,1] => 0
[[5,1,1,1],[]] => [5,1,1,1] => 0
[[4,4],[]] => [4,4] => 0
[[4,3,1],[]] => [4,3,1] => 0
[[4,2,2],[]] => [4,2,2] => 2
[[4,2,1,1],[]] => [4,2,1,1] => 0
[[4,1,1,1,1],[]] => [4,1,1,1,1] => 0
[[3,3,2],[]] => [3,3,2] => 0
[[3,3,1,1],[]] => [3,3,1,1] => 2
[[3,2,2,1],[]] => [3,2,2,1] => 0
[[3,2,1,1,1],[]] => [3,2,1,1,1] => 0
[[3,1,1,1,1,1],[]] => [3,1,1,1,1,1] => 0
[[2,2,2,2],[]] => [2,2,2,2] => 0
[[2,2,2,1,1],[]] => [2,2,2,1,1] => 0
[[2,2,1,1,1,1],[]] => [2,2,1,1,1,1] => 0
[[2,1,1,1,1,1,1],[]] => [2,1,1,1,1,1,1] => 0
[[1,1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1,1] => 0
[[9],[]] => [9] => 1
[[8,1],[]] => [8,1] => 1
[[7,2],[]] => [7,2] => 1
[[7,1,1],[]] => [7,1,1] => 1
[[6,3],[]] => [6,3] => 1
[[6,2,1],[]] => [6,2,1] => 1
[[6,1,1,1],[]] => [6,1,1,1] => 1
[[5,4],[]] => [5,4] => 1
[[5,3,1],[]] => [5,3,1] => 1
[[5,2,2],[]] => [5,2,2] => 3
[[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] => 2
[[4,3,2],[]] => [4,3,2] => 1
[[4,3,1,1],[]] => [4,3,1,1] => 1
[[4,2,2,1],[]] => [4,2,2,1] => 1
[[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] => 2
[[3,3,2,1],[]] => [3,3,2,1] => 1
[[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] => 1
[[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] => 1
[[2,2,2,1,1,1],[]] => [2,2,2,1,1,1] => 1
[[2,2,1,1,1,1,1],[]] => [2,2,1,1,1,1,1] => 1
[[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] => 0
[[7,3],[]] => [7,3] => 0
[[7,2,1],[]] => [7,2,1] => 0
[[7,1,1,1],[]] => [7,1,1,1] => 0
[[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] => 0
[[6,1,1,1,1],[]] => [6,1,1,1,1] => 0
[[5,5],[]] => [5,5] => 0
[[5,4,1],[]] => [5,4,1] => 0
[[5,3,2],[]] => [5,3,2] => 0
[[5,3,1,1],[]] => [5,3,1,1] => 0
[[5,2,2,1],[]] => [5,2,2,1] => 0
[[5,2,1,1,1],[]] => [5,2,1,1,1] => 0
[[5,1,1,1,1,1],[]] => [5,1,1,1,1,1] => 0
[[4,4,2],[]] => [4,4,2] => 0
[[4,4,1,1],[]] => [4,4,1,1] => 3
[[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] => 0
[[4,2,2,2],[]] => [4,2,2,2] => 3
[[4,2,2,1,1],[]] => [4,2,2,1,1] => 0
[[4,2,1,1,1,1],[]] => [4,2,1,1,1,1] => 0
[[4,1,1,1,1,1,1],[]] => [4,1,1,1,1,1,1] => 0
[[3,3,3,1],[]] => [3,3,3,1] => 3
[[3,3,2,2],[]] => [3,3,2,2] => 0
[[3,3,2,1,1],[]] => [3,3,2,1,1] => 0
[[3,3,1,1,1,1],[]] => [3,3,1,1,1,1] => 2
[[3,2,2,2,1],[]] => [3,2,2,2,1] => 0
[[3,2,2,1,1,1],[]] => [3,2,2,1,1,1] => 0
[[3,2,1,1,1,1,1],[]] => [3,2,1,1,1,1,1] => 0
[[3,1,1,1,1,1,1,1],[]] => [3,1,1,1,1,1,1,1] => 0
[[2,2,2,2,2],[]] => [2,2,2,2,2] => 0
[[2,2,2,2,1,1],[]] => [2,2,2,2,1,1] => 0
[[2,2,2,1,1,1,1],[]] => [2,2,2,1,1,1,1] => 0
[[2,2,1,1,1,1,1,1],[]] => [2,2,1,1,1,1,1,1] => 0
[[2,1,1,1,1,1,1,1,1],[]] => [2,1,1,1,1,1,1,1,1] => 0
[[1,1,1,1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1,1,1,1] => 0
[[6,5],[]] => [6,5] => 1
[[5,5,1],[]] => [5,5,1] => 1
[[5,4,2],[]] => [5,4,2] => 1
[[5,3,3],[]] => [5,3,3] => 2
[[4,4,3],[]] => [4,4,3] => 1
[[4,4,2,1],[]] => [4,4,2,1] => 2
[[4,3,3,1],[]] => [4,3,3,1] => 2
[[4,3,2,2],[]] => [4,3,2,2] => 2
[[3,3,3,2],[]] => [3,3,3,2] => 1
[[3,2,2,2,2],[]] => [3,2,2,2,2] => 1
[[6,6],[]] => [6,6] => 2
[[5,5,2],[]] => [5,5,2] => 2
[[5,4,3],[]] => [5,4,3] => 0
[[4,4,4],[]] => [4,4,4] => 0
[[4,4,3,1],[]] => [4,4,3,1] => 0
[[4,4,2,2],[]] => [4,4,2,2] => 1
[[4,3,3,2],[]] => [4,3,3,2] => 0
[[3,3,3,3],[]] => [3,3,3,3] => 0
[[3,3,3,2,1],[]] => [3,3,3,2,1] => 0
[[3,3,2,2,2],[]] => [3,3,2,2,2] => 2
[[2,2,2,2,2,2],[]] => [2,2,2,2,2,2] => 2
[[5,5,3],[]] => [5,5,3] => 1
[[5,4,4],[]] => [5,4,4] => 1
[[4,4,4,1],[]] => [4,4,4,1] => 2
[[4,4,3,2],[]] => [4,4,3,2] => 2
[[4,3,3,3],[]] => [4,3,3,3] => 2
[[3,3,3,3,1],[]] => [3,3,3,3,1] => 1
[[3,3,3,2,2],[]] => [3,3,3,2,2] => 1
[[5,5,4],[]] => [5,5,4] => 0
[[4,4,4,2],[]] => [4,4,4,2] => 0
[[4,4,3,3],[]] => [4,4,3,3] => 0
[[3,3,3,3,2],[]] => [3,3,3,3,2] => 0
[[5,5,5],[]] => [5,5,5] => 1
[[5,5,4,1],[]] => [5,5,4,1] => 1
[[4,4,4,3],[]] => [4,4,4,3] => 1
[[3,3,3,3,3],[]] => [3,3,3,3,3] => 1
[[5,5,5,1],[]] => [5,5,5,1] => 0
[[4,4,4,4],[]] => [4,4,4,4] => 0
[[4,4,4,3,1],[]] => [4,4,4,3,1] => 0
[[3,3,3,3,3,1],[]] => [3,3,3,3,3,1] => 0
[[4,4,4,4,1],[]] => [4,4,4,4,1] => 1
[[2,2,1,1,1,1,1],[1]] => [2,2,1,1,1,1,1] => 1
[[2,2,2,1,1,1,1],[1,1]] => [2,2,2,1,1,1,1] => 0
[[3,2,1,1,1,1],[1]] => [3,2,1,1,1,1] => 1
[[2,2,2,2,1,1,1],[1,1,1]] => [2,2,2,2,1,1,1] => 1
[[3,2,2,1,1,1],[1,1]] => [3,2,2,1,1,1] => 0
[[3,3,2,1,1,1],[2,1]] => [3,3,2,1,1,1] => 1
[[4,3,1,1,1],[2]] => [4,3,1,1,1] => 0
[[2,2,2,2,2,1,1],[1,1,1,1]] => [2,2,2,2,2,1,1] => 0
[[3,2,2,2,1,1],[1,1,1]] => [3,2,2,2,1,1] => 1
[[3,3,2,2,1,1],[2,1,1]] => [3,3,2,2,1,1] => 0
[[3,3,3,2,1,1],[2,2,1]] => [3,3,3,2,1,1] => 1
[[4,3,2,1,1],[2,1]] => [4,3,2,1,1] => 1
[[4,4,2,1,1],[3,1]] => [4,4,2,1,1] => 3
[[4,3,3,1,1],[2,2]] => [4,3,3,1,1] => 3
[[4,4,3,1,1],[3,2]] => [4,4,3,1,1] => 1
[[5,3,1,1],[2]] => [5,3,1,1] => 0
[[2,2,2,2,2,2,1],[1,1,1,1,1]] => [2,2,2,2,2,2,1] => 1
[[3,2,2,2,2,1],[1,1,1,1]] => [3,2,2,2,2,1] => 0
[[3,3,2,2,2,1],[2,1,1,1]] => [3,3,2,2,2,1] => 1
[[3,3,3,2,2,1],[2,2,1,1]] => [3,3,3,2,2,1] => 0
[[4,3,2,2,1],[2,1,1]] => [4,3,2,2,1] => 0
[[4,4,2,2,1],[3,1,1]] => [4,4,2,2,1] => 2
[[3,3,3,3,2,1],[2,2,2,1]] => [3,3,3,3,2,1] => 1
[[4,3,3,2,1],[2,2,1]] => [4,3,3,2,1] => 1
[[4,4,3,2,1],[3,2,1]] => [4,4,3,2,1] => 0
[[5,3,2,1],[2,1]] => [5,3,2,1] => 1
[[4,4,4,2,1],[3,3,1]] => [4,4,4,2,1] => 2
[[5,4,2,1],[3,1]] => [5,4,2,1] => 0
[[4,3,3,3,1],[2,2,2]] => [4,3,3,3,1] => 3
[[4,4,3,3,1],[3,2,2]] => [4,4,3,3,1] => 1
[[5,3,3,1],[2,2]] => [5,3,3,1] => 3
[[4,4,4,3,1],[3,3,2]] => [4,4,4,3,1] => 0
[[5,4,3,1],[3,2]] => [5,4,3,1] => 1
[[5,5,3,1],[4,2]] => [5,5,3,1] => 0
[[4,4,4,4,1],[3,3,3]] => [4,4,4,4,1] => 1
[[5,4,4,1],[3,3]] => [5,4,4,1] => 3
[[5,5,4,1],[4,3]] => [5,5,4,1] => 1
[[6,4,1],[3]] => [6,4,1] => 1
[[2,2,2,2,2,2,2],[1,1,1,1,1,1]] => [2,2,2,2,2,2,2] => 2
[[3,2,2,2,2,2],[1,1,1,1,1]] => [3,2,2,2,2,2] => 1
[[3,3,2,2,2,2],[2,1,1,1,1]] => [3,3,2,2,2,2] => 2
[[3,3,3,2,2,2],[2,2,1,1,1]] => [3,3,3,2,2,2] => 3
[[4,3,2,2,2],[2,1,1,1]] => [4,3,2,2,2] => 3
[[4,4,2,2,2],[3,1,1,1]] => [4,4,2,2,2] => 1
[[3,3,3,3,2,2],[2,2,2,1,1]] => [3,3,3,3,2,2] => 2
[[4,3,3,2,2],[2,2,1,1]] => [4,3,3,2,2] => 0
[[4,4,3,2,2],[3,2,1,1]] => [4,4,3,2,2] => 2
[[5,3,2,2],[2,1,1]] => [5,3,2,2] => 3
[[4,4,4,2,2],[3,3,1,1]] => [4,4,4,2,2] => 1
[[5,4,2,2],[3,1,1]] => [5,4,2,2] => 2
[[3,3,3,3,3,2],[2,2,2,2,1]] => [3,3,3,3,3,2] => 1
[[4,3,3,3,2],[2,2,2,1]] => [4,3,3,3,2] => 1
[[4,4,3,3,2],[3,2,2,1]] => [4,4,3,3,2] => 0
[[5,3,3,2],[2,2,1]] => [5,3,3,2] => 1
[[4,4,4,3,2],[3,3,2,1]] => [4,4,4,3,2] => 2
[[5,4,3,2],[3,2,1]] => [5,4,3,2] => 0
[[5,5,3,2],[4,2,1]] => [5,5,3,2] => 2
[[5,4,4,2],[3,3,1]] => [5,4,4,2] => 1
[[5,5,4,2],[4,3,1]] => [5,5,4,2] => 0
[[6,4,2],[3,1]] => [6,4,2] => 0
[[5,5,5,2],[4,4,1]] => [5,5,5,2] => 2
[[6,5,2],[4,1]] => [6,5,2] => 1
[[4,4,3,3,3],[3,2,2,2]] => [4,4,3,3,3] => 2
[[5,3,3,3],[2,2,2]] => [5,3,3,3] => 3
[[5,4,3,3],[3,2,2]] => [5,4,3,3] => 2
[[5,5,3,3],[4,2,2]] => [5,5,3,3] => 1
[[5,4,4,3],[3,3,2]] => [5,4,4,3] => 0
[[5,5,4,3],[4,3,2]] => [5,5,4,3] => 2
[[6,4,3],[3,2]] => [6,4,3] => 1
[[6,5,3],[4,2]] => [6,5,3] => 0
[[6,6,3],[5,2]] => [6,6,3] => 3
[[5,4,4,4],[3,3,3]] => [5,4,4,4] => 1
[[6,4,4],[3,3]] => [6,4,4] => 0
[[6,5,4],[4,3]] => [6,5,4] => 1
[[6,6,4],[5,3]] => [6,6,4] => 2
[[7,4],[3]] => [7,4] => 1
[[7,7],[6]] => [7,7] => 2
[[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]] => [2,2,2,2,2,2,2,2] => 0
[[8,8],[7]] => [8,8] => 0
[[3,3,3],[1]] => [3,3,3] => 2
[[3,3,3,1],[1,1]] => [3,3,3,1] => 3
[[4,4,1],[1]] => [4,4,1] => 2
[[3,3,2,1],[1]] => [3,3,2,1] => 1
[[2,2,2,2,1],[1]] => [2,2,2,2,1] => 1
[[3,3,3,1],[2]] => [3,3,3,1] => 3
[[3,3,3,1],[1]] => [3,3,3,1] => 3
[[3,3,3,2],[1,1,1]] => [3,3,3,2] => 1
[[4,4,2],[1,1]] => [4,4,2] => 0
[[4,4,4],[2,2]] => [4,4,4] => 0
[[5,5],[2]] => [5,5] => 0
[[4,4,3],[2,1]] => [4,4,3] => 1
[[3,3,3,3],[2,1,1]] => [3,3,3,3] => 0
[[4,3,3],[1,1]] => [4,3,3] => 3
[[3,3,3,3],[1,1,1]] => [3,3,3,3] => 0
[[4,4,3],[1,1]] => [4,4,3] => 1
[[4,4,4],[3,1]] => [4,4,4] => 0
[[5,4],[1]] => [5,4] => 1
[[4,4,4],[2,1]] => [4,4,4] => 0
[[5,5],[1]] => [5,5] => 0
[[4,4,4],[1,1]] => [4,4,4] => 0
[[3,3,2,2],[1,1]] => [3,3,2,2] => 0
[[4,4,2],[2]] => [4,4,2] => 0
[[3,3,3,2],[2,1]] => [3,3,3,2] => 1
[[4,3,2],[1]] => [4,3,2] => 1
[[3,3,3,2],[1,1]] => [3,3,3,2] => 1
[[4,4,2],[1]] => [4,4,2] => 0
[[2,2,2,2,2],[1,1]] => [2,2,2,2,2] => 0
[[3,2,2,2],[1]] => [3,2,2,2] => 2
[[3,3,2,2],[2]] => [3,3,2,2] => 0
[[3,3,2,2],[1]] => [3,3,2,2] => 0
[[2,2,2,2,2],[1]] => [2,2,2,2,2] => 0
[[3,3,3,2],[2]] => [3,3,3,2] => 1
[[3,3,3,2],[1]] => [3,3,3,2] => 1
[[3,3,3,3],[2,2]] => [3,3,3,3] => 0
[[4,3,3],[2]] => [4,3,3] => 3
[[4,4,3],[3]] => [4,4,3] => 1
[[4,4,3],[2]] => [4,4,3] => 1
[[3,3,3,3],[2,1]] => [3,3,3,3] => 0
[[4,3,3],[1]] => [4,3,3] => 3
[[3,3,3,3],[1,1]] => [3,3,3,3] => 0
[[4,4,3],[1]] => [4,4,3] => 1
[[4,4,4],[3]] => [4,4,4] => 0
[[4,4,4],[2]] => [4,4,4] => 0
[[4,4,4],[1]] => [4,4,4] => 0
[[3,3,3,3],[2]] => [3,3,3,3] => 0
[[3,3,3,3],[1]] => [3,3,3,3] => 0
[[4,4,4,1],[1,1]] => [4,4,4,1] => 2
[[4,4,3,1],[1]] => [4,4,3,1] => 0
[[4,4,4,1],[2]] => [4,4,4,1] => 2
[[4,4,4,1],[1]] => [4,4,4,1] => 2
[[3,3,3,3,1],[1]] => [3,3,3,3,1] => 1
[[4,4,4,2],[1,1,1]] => [4,4,4,2] => 0
[[6,6],[4]] => [6,6] => 2
[[5,5,5],[4,3]] => [5,5,5] => 1
[[6,5],[3]] => [6,5] => 1
[[5,5,5],[3,3]] => [5,5,5] => 1
[[6,6],[3]] => [6,6] => 2
[[4,4,4,4],[3,3,2]] => [4,4,4,4] => 0
[[6,4],[2]] => [6,4] => 0
[[5,5,4],[3,2]] => [5,5,4] => 0
[[4,4,4,4],[3,2,2]] => [4,4,4,4] => 0
[[5,4,4],[2,2]] => [5,4,4] => 1
[[4,4,4,4],[2,2,2]] => [4,4,4,4] => 0
[[5,5,4],[2,2]] => [5,5,4] => 0
[[5,5,5],[4,2]] => [5,5,5] => 1
[[6,5],[2]] => [6,5] => 1
[[5,5,5],[3,2]] => [5,5,5] => 1
[[6,6],[2]] => [6,6] => 2
[[5,5,5],[2,2]] => [5,5,5] => 1
[[3,3,3,3,3],[2,2,2,1]] => [3,3,3,3,3] => 1
[[6,3],[1]] => [6,3] => 1
[[5,5,3],[3,1]] => [5,5,3] => 1
[[5,4,3],[2,1]] => [5,4,3] => 0
[[4,4,4,3],[2,2,1]] => [4,4,4,3] => 1
[[5,5,3],[2,1]] => [5,5,3] => 1
[[3,3,3,3,3],[2,2,1,1]] => [3,3,3,3,3] => 1
[[5,3,3],[1,1]] => [5,3,3] => 2
[[4,4,3,3],[2,1,1]] => [4,4,3,3] => 0
[[3,3,3,3,3],[2,1,1,1]] => [3,3,3,3,3] => 1
[[4,3,3,3],[1,1,1]] => [4,3,3,3] => 2
[[3,3,3,3,3],[1,1,1,1]] => [3,3,3,3,3] => 1
[[4,4,3,3],[1,1,1]] => [4,4,3,3] => 0
[[5,4,3],[1,1]] => [5,4,3] => 0
[[4,4,4,3],[2,1,1]] => [4,4,4,3] => 1
[[5,5,3],[1,1]] => [5,5,3] => 1
[[4,4,4,3],[1,1,1]] => [4,4,4,3] => 1
[[4,4,4,4],[3,3,1]] => [4,4,4,4] => 0
[[6,4],[1]] => [6,4] => 0
[[5,5,4],[3,1]] => [5,5,4] => 0
[[4,4,4,4],[3,2,1]] => [4,4,4,4] => 0
[[5,4,4],[2,1]] => [5,4,4] => 1
[[4,4,4,4],[2,2,1]] => [4,4,4,4] => 0
[[5,5,4],[2,1]] => [5,5,4] => 0
[[5,5,5],[4,1]] => [5,5,5] => 1
[[6,5],[1]] => [6,5] => 1
[[5,5,5],[3,1]] => [5,5,5] => 1
[[6,6],[1]] => [6,6] => 2
[[5,5,5],[2,1]] => [5,5,5] => 1
[[4,4,4,4],[3,1,1]] => [4,4,4,4] => 0
[[5,4,4],[1,1]] => [5,4,4] => 1
[[4,4,4,4],[2,1,1]] => [4,4,4,4] => 0
[[5,5,4],[1,1]] => [5,5,4] => 0
[[4,4,4,4],[1,1,1]] => [4,4,4,4] => 0
[[5,5,5],[1,1]] => [5,5,5] => 1
[[2,2,2,2,2,2],[1,1,1,1]] => [2,2,2,2,2,2] => 2
[[5,5,2],[3]] => [5,5,2] => 2
[[5,4,2],[2]] => [5,4,2] => 1
[[4,4,4,2],[2,2]] => [4,4,4,2] => 0
[[5,5,2],[2]] => [5,5,2] => 2
[[5,3,2],[1]] => [5,3,2] => 0
[[4,4,3,2],[2,1]] => [4,4,3,2] => 2
[[4,3,3,2],[1,1]] => [4,3,3,2] => 0
[[3,3,3,3,2],[1,1,1]] => [3,3,3,3,2] => 0
[[4,4,3,2],[1,1]] => [4,4,3,2] => 2
[[5,4,2],[1]] => [5,4,2] => 1
[[4,4,4,2],[2,1]] => [4,4,4,2] => 0
[[5,5,2],[1]] => [5,5,2] => 2
[[4,4,4,2],[1,1]] => [4,4,4,2] => 0
[[2,2,2,2,2,2],[1,1,1]] => [2,2,2,2,2,2] => 2
[[4,4,2,2],[2]] => [4,4,2,2] => 1
[[4,3,2,2],[1]] => [4,3,2,2] => 2
[[3,3,3,2,2],[1,1]] => [3,3,3,2,2] => 1
[[4,4,2,2],[1]] => [4,4,2,2] => 1
[[2,2,2,2,2,2],[1,1]] => [2,2,2,2,2,2] => 2
[[3,3,2,2,2],[1]] => [3,3,2,2,2] => 2
[[3,3,3,2,2],[1]] => [3,3,3,2,2] => 1
[[4,4,3,2],[2]] => [4,4,3,2] => 2
[[4,3,3,2],[1]] => [4,3,3,2] => 0
[[3,3,3,3,2],[1,1]] => [3,3,3,3,2] => 0
[[4,4,3,2],[1]] => [4,4,3,2] => 2
[[4,4,4,2],[3]] => [4,4,4,2] => 0
[[4,4,4,2],[2]] => [4,4,4,2] => 0
[[4,4,4,2],[1]] => [4,4,4,2] => 0
[[3,3,3,3,2],[2]] => [3,3,3,3,2] => 0
[[3,3,3,3,2],[1]] => [3,3,3,3,2] => 0
[[3,3,3,3,3],[2,2,2]] => [3,3,3,3,3] => 1
[[5,5,3],[3]] => [5,5,3] => 1
[[5,4,3],[2]] => [5,4,3] => 0
[[4,4,4,3],[2,2]] => [4,4,4,3] => 1
[[5,5,3],[2]] => [5,5,3] => 1
[[3,3,3,3,3],[2,2,1]] => [3,3,3,3,3] => 1
[[5,3,3],[1]] => [5,3,3] => 2
[[4,4,3,3],[2,1]] => [4,4,3,3] => 0
[[3,3,3,3,3],[2,1,1]] => [3,3,3,3,3] => 1
[[4,3,3,3],[1,1]] => [4,3,3,3] => 2
[[3,3,3,3,3],[1,1,1]] => [3,3,3,3,3] => 1
[[4,4,3,3],[1,1]] => [4,4,3,3] => 0
[[4,4,4,3],[3,1]] => [4,4,4,3] => 1
[[5,4,3],[1]] => [5,4,3] => 0
[[4,4,4,3],[2,1]] => [4,4,4,3] => 1
[[5,5,3],[1]] => [5,5,3] => 1
[[4,4,4,3],[1,1]] => [4,4,4,3] => 1
[[4,4,4,4],[3,3]] => [4,4,4,4] => 0
[[5,5,4],[3]] => [5,5,4] => 0
[[4,4,4,4],[3,2]] => [4,4,4,4] => 0
[[5,4,4],[2]] => [5,4,4] => 1
[[4,4,4,4],[2,2]] => [4,4,4,4] => 0
[[5,5,4],[2]] => [5,5,4] => 0
[[5,5,5],[4]] => [5,5,5] => 1
[[5,5,5],[3]] => [5,5,5] => 1
[[5,5,5],[2]] => [5,5,5] => 1
[[4,4,4,4],[3,1]] => [4,4,4,4] => 0
[[5,4,4],[1]] => [5,4,4] => 1
[[4,4,4,4],[2,1]] => [4,4,4,4] => 0
[[5,5,4],[1]] => [5,5,4] => 0
[[4,4,4,4],[1,1]] => [4,4,4,4] => 0
[[5,5,5],[1]] => [5,5,5] => 1
[[3,3,3,3,3],[2,2]] => [3,3,3,3,3] => 1
[[4,3,3,3],[2]] => [4,3,3,3] => 2
[[4,4,3,3],[3]] => [4,4,3,3] => 0
[[4,4,3,3],[2]] => [4,4,3,3] => 0
[[3,3,3,3,3],[2,1]] => [3,3,3,3,3] => 1
[[4,3,3,3],[1]] => [4,3,3,3] => 2
[[3,3,3,3,3],[1,1]] => [3,3,3,3,3] => 1
[[4,4,3,3],[1]] => [4,4,3,3] => 0
[[4,4,4,3],[3]] => [4,4,4,3] => 1
[[4,4,4,3],[2]] => [4,4,4,3] => 1
[[4,4,4,3],[1]] => [4,4,4,3] => 1
[[3,3,3,3,3],[2]] => [3,3,3,3,3] => 1
[[3,3,3,3,3],[1]] => [3,3,3,3,3] => 1
[[4,4,4,4],[3]] => [4,4,4,4] => 0
[[4,4,4,4],[2]] => [4,4,4,4] => 0
[[4,4,4,4],[1]] => [4,4,4,4] => 0
[[5,5,5,1],[1]] => [5,5,5,1] => 0
[[4,4,4,4,1],[1]] => [4,4,4,4,1] => 1
[[7,7],[5]] => [7,7] => 2
[[7,7],[4]] => [7,7] => 2
[[2,2,2,2,2,2,2],[1,1,1,1,1]] => [2,2,2,2,2,2,2] => 2
[[2,2,2,2,2,2,2],[1,1,1,1]] => [2,2,2,2,2,2,2] => 2
[[8,8],[6]] => [8,8] => 0
[[2,2,2,2,2,2,2,2],[1,1,1,1,1,1]] => [2,2,2,2,2,2,2,2] => 0
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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 3,5,1 12,11,3,2 36,34,11,6
$F_{1} = q$
$F_{2} = 2 + q$
$F_{3} = 3 + 5\ q + q^{2}$
$F_{4} = 12 + 11\ q + 3\ q^{2} + 2\ q^{3}$
$F_{5} = 36 + 34\ q + 11\ q^{2} + 6\ q^{3}$
Description
The Grundy value of an integer partition.
Consider the two-player game on an integer partition.
In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition.
The first player that cannot move lose. This happens exactly when the empty partition is reached.
The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1].
This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.
Consider the two-player game on an integer partition.
In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition.
The first player that cannot move lose. This happens exactly when the empty partition is reached.
The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1].
This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.
Map
outer shape
Description
The outer shape of the skew partition.
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