Identifier
-
Mp00044:
Integer partitions
—conjugate⟶
Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000475: Integer partitions ⟶ ℤ
Values
[1] => [1] => [] => [] => 0
[2] => [1,1] => [1] => [1] => 1
[1,1] => [2] => [] => [] => 0
[3] => [1,1,1] => [1,1] => [2] => 0
[2,1] => [2,1] => [1] => [1] => 1
[1,1,1] => [3] => [] => [] => 0
[4] => [1,1,1,1] => [1,1,1] => [3] => 0
[3,1] => [2,1,1] => [1,1] => [2] => 0
[2,2] => [2,2] => [2] => [1,1] => 2
[2,1,1] => [3,1] => [1] => [1] => 1
[1,1,1,1] => [4] => [] => [] => 0
[5] => [1,1,1,1,1] => [1,1,1,1] => [4] => 0
[4,1] => [2,1,1,1] => [1,1,1] => [3] => 0
[3,2] => [2,2,1] => [2,1] => [2,1] => 1
[3,1,1] => [3,1,1] => [1,1] => [2] => 0
[2,2,1] => [3,2] => [2] => [1,1] => 2
[2,1,1,1] => [4,1] => [1] => [1] => 1
[1,1,1,1,1] => [5] => [] => [] => 0
[6] => [1,1,1,1,1,1] => [1,1,1,1,1] => [5] => 0
[5,1] => [2,1,1,1,1] => [1,1,1,1] => [4] => 0
[4,2] => [2,2,1,1] => [2,1,1] => [3,1] => 1
[4,1,1] => [3,1,1,1] => [1,1,1] => [3] => 0
[3,3] => [2,2,2] => [2,2] => [2,2] => 0
[3,2,1] => [3,2,1] => [2,1] => [2,1] => 1
[3,1,1,1] => [4,1,1] => [1,1] => [2] => 0
[2,2,2] => [3,3] => [3] => [1,1,1] => 3
[2,2,1,1] => [4,2] => [2] => [1,1] => 2
[2,1,1,1,1] => [5,1] => [1] => [1] => 1
[1,1,1,1,1,1] => [6] => [] => [] => 0
[7] => [1,1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => 0
[6,1] => [2,1,1,1,1,1] => [1,1,1,1,1] => [5] => 0
[5,2] => [2,2,1,1,1] => [2,1,1,1] => [4,1] => 1
[5,1,1] => [3,1,1,1,1] => [1,1,1,1] => [4] => 0
[4,3] => [2,2,2,1] => [2,2,1] => [3,2] => 0
[4,2,1] => [3,2,1,1] => [2,1,1] => [3,1] => 1
[4,1,1,1] => [4,1,1,1] => [1,1,1] => [3] => 0
[3,3,1] => [3,2,2] => [2,2] => [2,2] => 0
[3,2,2] => [3,3,1] => [3,1] => [2,1,1] => 2
[3,2,1,1] => [4,2,1] => [2,1] => [2,1] => 1
[3,1,1,1,1] => [5,1,1] => [1,1] => [2] => 0
[2,2,2,1] => [4,3] => [3] => [1,1,1] => 3
[2,2,1,1,1] => [5,2] => [2] => [1,1] => 2
[2,1,1,1,1,1] => [6,1] => [1] => [1] => 1
[1,1,1,1,1,1,1] => [7] => [] => [] => 0
[8] => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [7] => 0
[7,1] => [2,1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => 0
[6,2] => [2,2,1,1,1,1] => [2,1,1,1,1] => [5,1] => 1
[6,1,1] => [3,1,1,1,1,1] => [1,1,1,1,1] => [5] => 0
[5,3] => [2,2,2,1,1] => [2,2,1,1] => [4,2] => 0
[5,2,1] => [3,2,1,1,1] => [2,1,1,1] => [4,1] => 1
[5,1,1,1] => [4,1,1,1,1] => [1,1,1,1] => [4] => 0
[4,4] => [2,2,2,2] => [2,2,2] => [3,3] => 0
[4,3,1] => [3,2,2,1] => [2,2,1] => [3,2] => 0
[4,2,2] => [3,3,1,1] => [3,1,1] => [3,1,1] => 2
[4,2,1,1] => [4,2,1,1] => [2,1,1] => [3,1] => 1
[4,1,1,1,1] => [5,1,1,1] => [1,1,1] => [3] => 0
[3,3,2] => [3,3,2] => [3,2] => [2,2,1] => 1
[3,3,1,1] => [4,2,2] => [2,2] => [2,2] => 0
[3,2,2,1] => [4,3,1] => [3,1] => [2,1,1] => 2
[3,2,1,1,1] => [5,2,1] => [2,1] => [2,1] => 1
[3,1,1,1,1,1] => [6,1,1] => [1,1] => [2] => 0
[2,2,2,2] => [4,4] => [4] => [1,1,1,1] => 4
[2,2,2,1,1] => [5,3] => [3] => [1,1,1] => 3
[2,2,1,1,1,1] => [6,2] => [2] => [1,1] => 2
[2,1,1,1,1,1,1] => [7,1] => [1] => [1] => 1
[1,1,1,1,1,1,1,1] => [8] => [] => [] => 0
[9] => [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [8] => 0
[8,1] => [2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [7] => 0
[7,2] => [2,2,1,1,1,1,1] => [2,1,1,1,1,1] => [6,1] => 1
[7,1,1] => [3,1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => 0
[6,3] => [2,2,2,1,1,1] => [2,2,1,1,1] => [5,2] => 0
[6,2,1] => [3,2,1,1,1,1] => [2,1,1,1,1] => [5,1] => 1
[6,1,1,1] => [4,1,1,1,1,1] => [1,1,1,1,1] => [5] => 0
[5,4] => [2,2,2,2,1] => [2,2,2,1] => [4,3] => 0
[5,3,1] => [3,2,2,1,1] => [2,2,1,1] => [4,2] => 0
[5,2,2] => [3,3,1,1,1] => [3,1,1,1] => [4,1,1] => 2
[5,2,1,1] => [4,2,1,1,1] => [2,1,1,1] => [4,1] => 1
[5,1,1,1,1] => [5,1,1,1,1] => [1,1,1,1] => [4] => 0
[4,4,1] => [3,2,2,2] => [2,2,2] => [3,3] => 0
[4,3,2] => [3,3,2,1] => [3,2,1] => [3,2,1] => 1
[4,3,1,1] => [4,2,2,1] => [2,2,1] => [3,2] => 0
[4,2,2,1] => [4,3,1,1] => [3,1,1] => [3,1,1] => 2
[4,2,1,1,1] => [5,2,1,1] => [2,1,1] => [3,1] => 1
[4,1,1,1,1,1] => [6,1,1,1] => [1,1,1] => [3] => 0
[3,3,3] => [3,3,3] => [3,3] => [2,2,2] => 0
[3,3,2,1] => [4,3,2] => [3,2] => [2,2,1] => 1
[3,3,1,1,1] => [5,2,2] => [2,2] => [2,2] => 0
[3,2,2,2] => [4,4,1] => [4,1] => [2,1,1,1] => 3
[3,2,2,1,1] => [5,3,1] => [3,1] => [2,1,1] => 2
[3,2,1,1,1,1] => [6,2,1] => [2,1] => [2,1] => 1
[3,1,1,1,1,1,1] => [7,1,1] => [1,1] => [2] => 0
[2,2,2,2,1] => [5,4] => [4] => [1,1,1,1] => 4
[2,2,2,1,1,1] => [6,3] => [3] => [1,1,1] => 3
[2,2,1,1,1,1,1] => [7,2] => [2] => [1,1] => 2
[2,1,1,1,1,1,1,1] => [8,1] => [1] => [1] => 1
[1,1,1,1,1,1,1,1,1] => [9] => [] => [] => 0
[10] => [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [9] => 0
[9,1] => [2,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [8] => 0
[8,2] => [2,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => [7,1] => 1
[8,1,1] => [3,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [7] => 0
[7,3] => [2,2,2,1,1,1,1] => [2,2,1,1,1,1] => [6,2] => 0
>>> Load all 1128 entries. <<<[7,2,1] => [3,2,1,1,1,1,1] => [2,1,1,1,1,1] => [6,1] => 1
[7,1,1,1] => [4,1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => 0
[6,4] => [2,2,2,2,1,1] => [2,2,2,1,1] => [5,3] => 0
[6,3,1] => [3,2,2,1,1,1] => [2,2,1,1,1] => [5,2] => 0
[6,2,2] => [3,3,1,1,1,1] => [3,1,1,1,1] => [5,1,1] => 2
[6,2,1,1] => [4,2,1,1,1,1] => [2,1,1,1,1] => [5,1] => 1
[6,1,1,1,1] => [5,1,1,1,1,1] => [1,1,1,1,1] => [5] => 0
[5,5] => [2,2,2,2,2] => [2,2,2,2] => [4,4] => 0
[5,4,1] => [3,2,2,2,1] => [2,2,2,1] => [4,3] => 0
[5,3,2] => [3,3,2,1,1] => [3,2,1,1] => [4,2,1] => 1
[5,3,1,1] => [4,2,2,1,1] => [2,2,1,1] => [4,2] => 0
[5,2,2,1] => [4,3,1,1,1] => [3,1,1,1] => [4,1,1] => 2
[5,2,1,1,1] => [5,2,1,1,1] => [2,1,1,1] => [4,1] => 1
[5,1,1,1,1,1] => [6,1,1,1,1] => [1,1,1,1] => [4] => 0
[4,4,2] => [3,3,2,2] => [3,2,2] => [3,3,1] => 1
[4,4,1,1] => [4,2,2,2] => [2,2,2] => [3,3] => 0
[4,3,3] => [3,3,3,1] => [3,3,1] => [3,2,2] => 0
[4,3,2,1] => [4,3,2,1] => [3,2,1] => [3,2,1] => 1
[4,3,1,1,1] => [5,2,2,1] => [2,2,1] => [3,2] => 0
[4,2,2,2] => [4,4,1,1] => [4,1,1] => [3,1,1,1] => 3
[4,2,2,1,1] => [5,3,1,1] => [3,1,1] => [3,1,1] => 2
[4,2,1,1,1,1] => [6,2,1,1] => [2,1,1] => [3,1] => 1
[4,1,1,1,1,1,1] => [7,1,1,1] => [1,1,1] => [3] => 0
[3,3,3,1] => [4,3,3] => [3,3] => [2,2,2] => 0
[3,3,2,2] => [4,4,2] => [4,2] => [2,2,1,1] => 2
[3,3,2,1,1] => [5,3,2] => [3,2] => [2,2,1] => 1
[3,3,1,1,1,1] => [6,2,2] => [2,2] => [2,2] => 0
[3,2,2,2,1] => [5,4,1] => [4,1] => [2,1,1,1] => 3
[3,2,2,1,1,1] => [6,3,1] => [3,1] => [2,1,1] => 2
[3,2,1,1,1,1,1] => [7,2,1] => [2,1] => [2,1] => 1
[3,1,1,1,1,1,1,1] => [8,1,1] => [1,1] => [2] => 0
[2,2,2,2,2] => [5,5] => [5] => [1,1,1,1,1] => 5
[2,2,2,2,1,1] => [6,4] => [4] => [1,1,1,1] => 4
[2,2,2,1,1,1,1] => [7,3] => [3] => [1,1,1] => 3
[2,2,1,1,1,1,1,1] => [8,2] => [2] => [1,1] => 2
[2,1,1,1,1,1,1,1,1] => [9,1] => [1] => [1] => 1
[1,1,1,1,1,1,1,1,1,1] => [10] => [] => [] => 0
[11] => [1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [10] => 0
[10,1] => [2,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [9] => 0
[9,2] => [2,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => [8,1] => 1
[9,1,1] => [3,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [8] => 0
[8,3] => [2,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => [7,2] => 0
[8,2,1] => [3,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => [7,1] => 1
[8,1,1,1] => [4,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [7] => 0
[7,4] => [2,2,2,2,1,1,1] => [2,2,2,1,1,1] => [6,3] => 0
[7,3,1] => [3,2,2,1,1,1,1] => [2,2,1,1,1,1] => [6,2] => 0
[7,2,2] => [3,3,1,1,1,1,1] => [3,1,1,1,1,1] => [6,1,1] => 2
[7,2,1,1] => [4,2,1,1,1,1,1] => [2,1,1,1,1,1] => [6,1] => 1
[7,1,1,1,1] => [5,1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => 0
[6,5] => [2,2,2,2,2,1] => [2,2,2,2,1] => [5,4] => 0
[6,4,1] => [3,2,2,2,1,1] => [2,2,2,1,1] => [5,3] => 0
[6,3,2] => [3,3,2,1,1,1] => [3,2,1,1,1] => [5,2,1] => 1
[6,3,1,1] => [4,2,2,1,1,1] => [2,2,1,1,1] => [5,2] => 0
[6,2,2,1] => [4,3,1,1,1,1] => [3,1,1,1,1] => [5,1,1] => 2
[6,2,1,1,1] => [5,2,1,1,1,1] => [2,1,1,1,1] => [5,1] => 1
[6,1,1,1,1,1] => [6,1,1,1,1,1] => [1,1,1,1,1] => [5] => 0
[5,5,1] => [3,2,2,2,2] => [2,2,2,2] => [4,4] => 0
[5,4,2] => [3,3,2,2,1] => [3,2,2,1] => [4,3,1] => 1
[5,4,1,1] => [4,2,2,2,1] => [2,2,2,1] => [4,3] => 0
[5,3,3] => [3,3,3,1,1] => [3,3,1,1] => [4,2,2] => 0
[5,3,2,1] => [4,3,2,1,1] => [3,2,1,1] => [4,2,1] => 1
[5,3,1,1,1] => [5,2,2,1,1] => [2,2,1,1] => [4,2] => 0
[5,2,2,2] => [4,4,1,1,1] => [4,1,1,1] => [4,1,1,1] => 3
[5,2,2,1,1] => [5,3,1,1,1] => [3,1,1,1] => [4,1,1] => 2
[5,2,1,1,1,1] => [6,2,1,1,1] => [2,1,1,1] => [4,1] => 1
[5,1,1,1,1,1,1] => [7,1,1,1,1] => [1,1,1,1] => [4] => 0
[4,4,3] => [3,3,3,2] => [3,3,2] => [3,3,2] => 0
[4,4,2,1] => [4,3,2,2] => [3,2,2] => [3,3,1] => 1
[4,4,1,1,1] => [5,2,2,2] => [2,2,2] => [3,3] => 0
[4,3,3,1] => [4,3,3,1] => [3,3,1] => [3,2,2] => 0
[4,3,2,2] => [4,4,2,1] => [4,2,1] => [3,2,1,1] => 2
[4,3,2,1,1] => [5,3,2,1] => [3,2,1] => [3,2,1] => 1
[4,3,1,1,1,1] => [6,2,2,1] => [2,2,1] => [3,2] => 0
[4,2,2,2,1] => [5,4,1,1] => [4,1,1] => [3,1,1,1] => 3
[4,2,2,1,1,1] => [6,3,1,1] => [3,1,1] => [3,1,1] => 2
[4,2,1,1,1,1,1] => [7,2,1,1] => [2,1,1] => [3,1] => 1
[4,1,1,1,1,1,1,1] => [8,1,1,1] => [1,1,1] => [3] => 0
[3,3,3,2] => [4,4,3] => [4,3] => [2,2,2,1] => 1
[3,3,3,1,1] => [5,3,3] => [3,3] => [2,2,2] => 0
[3,3,2,2,1] => [5,4,2] => [4,2] => [2,2,1,1] => 2
[3,3,2,1,1,1] => [6,3,2] => [3,2] => [2,2,1] => 1
[3,3,1,1,1,1,1] => [7,2,2] => [2,2] => [2,2] => 0
[3,2,2,2,2] => [5,5,1] => [5,1] => [2,1,1,1,1] => 4
[3,2,2,2,1,1] => [6,4,1] => [4,1] => [2,1,1,1] => 3
[3,2,2,1,1,1,1] => [7,3,1] => [3,1] => [2,1,1] => 2
[3,2,1,1,1,1,1,1] => [8,2,1] => [2,1] => [2,1] => 1
[3,1,1,1,1,1,1,1,1] => [9,1,1] => [1,1] => [2] => 0
[2,2,2,2,2,1] => [6,5] => [5] => [1,1,1,1,1] => 5
[2,2,2,2,1,1,1] => [7,4] => [4] => [1,1,1,1] => 4
[2,2,2,1,1,1,1,1] => [8,3] => [3] => [1,1,1] => 3
[2,2,1,1,1,1,1,1,1] => [9,2] => [2] => [1,1] => 2
[2,1,1,1,1,1,1,1,1,1] => [10,1] => [1] => [1] => 1
[1,1,1,1,1,1,1,1,1,1,1] => [11] => [] => [] => 0
[12] => [1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => [11] => 0
[11,1] => [2,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [10] => 0
[10,2] => [2,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => [9,1] => 1
[10,1,1] => [3,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [9] => 0
[9,3] => [2,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => [8,2] => 0
[9,2,1] => [3,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => [8,1] => 1
[9,1,1,1] => [4,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [8] => 0
[8,4] => [2,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => [7,3] => 0
[8,3,1] => [3,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => [7,2] => 0
[8,2,2] => [3,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => [7,1,1] => 2
[8,2,1,1] => [4,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => [7,1] => 1
[8,1,1,1,1] => [5,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [7] => 0
[7,5] => [2,2,2,2,2,1,1] => [2,2,2,2,1,1] => [6,4] => 0
[7,4,1] => [3,2,2,2,1,1,1] => [2,2,2,1,1,1] => [6,3] => 0
[7,3,2] => [3,3,2,1,1,1,1] => [3,2,1,1,1,1] => [6,2,1] => 1
[7,3,1,1] => [4,2,2,1,1,1,1] => [2,2,1,1,1,1] => [6,2] => 0
[7,2,2,1] => [4,3,1,1,1,1,1] => [3,1,1,1,1,1] => [6,1,1] => 2
[7,2,1,1,1] => [5,2,1,1,1,1,1] => [2,1,1,1,1,1] => [6,1] => 1
[7,1,1,1,1,1] => [6,1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => 0
[6,6] => [2,2,2,2,2,2] => [2,2,2,2,2] => [5,5] => 0
[6,5,1] => [3,2,2,2,2,1] => [2,2,2,2,1] => [5,4] => 0
[6,4,2] => [3,3,2,2,1,1] => [3,2,2,1,1] => [5,3,1] => 1
[6,4,1,1] => [4,2,2,2,1,1] => [2,2,2,1,1] => [5,3] => 0
[6,3,3] => [3,3,3,1,1,1] => [3,3,1,1,1] => [5,2,2] => 0
[6,3,2,1] => [4,3,2,1,1,1] => [3,2,1,1,1] => [5,2,1] => 1
[6,3,1,1,1] => [5,2,2,1,1,1] => [2,2,1,1,1] => [5,2] => 0
[6,2,2,2] => [4,4,1,1,1,1] => [4,1,1,1,1] => [5,1,1,1] => 3
[6,2,2,1,1] => [5,3,1,1,1,1] => [3,1,1,1,1] => [5,1,1] => 2
[6,2,1,1,1,1] => [6,2,1,1,1,1] => [2,1,1,1,1] => [5,1] => 1
[6,1,1,1,1,1,1] => [7,1,1,1,1,1] => [1,1,1,1,1] => [5] => 0
[5,5,2] => [3,3,2,2,2] => [3,2,2,2] => [4,4,1] => 1
[5,5,1,1] => [4,2,2,2,2] => [2,2,2,2] => [4,4] => 0
[5,4,3] => [3,3,3,2,1] => [3,3,2,1] => [4,3,2] => 0
[5,4,2,1] => [4,3,2,2,1] => [3,2,2,1] => [4,3,1] => 1
[5,4,1,1,1] => [5,2,2,2,1] => [2,2,2,1] => [4,3] => 0
[5,3,3,1] => [4,3,3,1,1] => [3,3,1,1] => [4,2,2] => 0
[5,3,2,2] => [4,4,2,1,1] => [4,2,1,1] => [4,2,1,1] => 2
[5,3,2,1,1] => [5,3,2,1,1] => [3,2,1,1] => [4,2,1] => 1
[5,3,1,1,1,1] => [6,2,2,1,1] => [2,2,1,1] => [4,2] => 0
[5,2,2,2,1] => [5,4,1,1,1] => [4,1,1,1] => [4,1,1,1] => 3
[5,2,2,1,1,1] => [6,3,1,1,1] => [3,1,1,1] => [4,1,1] => 2
[5,2,1,1,1,1,1] => [7,2,1,1,1] => [2,1,1,1] => [4,1] => 1
[5,1,1,1,1,1,1,1] => [8,1,1,1,1] => [1,1,1,1] => [4] => 0
[4,4,4] => [3,3,3,3] => [3,3,3] => [3,3,3] => 0
[4,4,3,1] => [4,3,3,2] => [3,3,2] => [3,3,2] => 0
[4,4,2,2] => [4,4,2,2] => [4,2,2] => [3,3,1,1] => 2
[4,4,2,1,1] => [5,3,2,2] => [3,2,2] => [3,3,1] => 1
[4,4,1,1,1,1] => [6,2,2,2] => [2,2,2] => [3,3] => 0
[4,3,3,2] => [4,4,3,1] => [4,3,1] => [3,2,2,1] => 1
[4,3,3,1,1] => [5,3,3,1] => [3,3,1] => [3,2,2] => 0
[4,3,2,2,1] => [5,4,2,1] => [4,2,1] => [3,2,1,1] => 2
[4,3,2,1,1,1] => [6,3,2,1] => [3,2,1] => [3,2,1] => 1
[4,3,1,1,1,1,1] => [7,2,2,1] => [2,2,1] => [3,2] => 0
[4,2,2,2,2] => [5,5,1,1] => [5,1,1] => [3,1,1,1,1] => 4
[4,2,2,2,1,1] => [6,4,1,1] => [4,1,1] => [3,1,1,1] => 3
[4,2,2,1,1,1,1] => [7,3,1,1] => [3,1,1] => [3,1,1] => 2
[4,2,1,1,1,1,1,1] => [8,2,1,1] => [2,1,1] => [3,1] => 1
[4,1,1,1,1,1,1,1,1] => [9,1,1,1] => [1,1,1] => [3] => 0
[3,3,3,3] => [4,4,4] => [4,4] => [2,2,2,2] => 0
[3,3,3,2,1] => [5,4,3] => [4,3] => [2,2,2,1] => 1
[3,3,3,1,1,1] => [6,3,3] => [3,3] => [2,2,2] => 0
[3,3,2,2,2] => [5,5,2] => [5,2] => [2,2,1,1,1] => 3
[3,3,2,2,1,1] => [6,4,2] => [4,2] => [2,2,1,1] => 2
[3,3,2,1,1,1,1] => [7,3,2] => [3,2] => [2,2,1] => 1
[3,3,1,1,1,1,1,1] => [8,2,2] => [2,2] => [2,2] => 0
[3,2,2,2,2,1] => [6,5,1] => [5,1] => [2,1,1,1,1] => 4
[3,2,2,2,1,1,1] => [7,4,1] => [4,1] => [2,1,1,1] => 3
[3,2,2,1,1,1,1,1] => [8,3,1] => [3,1] => [2,1,1] => 2
[3,2,1,1,1,1,1,1,1] => [9,2,1] => [2,1] => [2,1] => 1
[3,1,1,1,1,1,1,1,1,1] => [10,1,1] => [1,1] => [2] => 0
[2,2,2,2,2,2] => [6,6] => [6] => [1,1,1,1,1,1] => 6
[2,2,2,2,2,1,1] => [7,5] => [5] => [1,1,1,1,1] => 5
[2,2,2,2,1,1,1,1] => [8,4] => [4] => [1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1] => [9,3] => [3] => [1,1,1] => 3
[2,2,1,1,1,1,1,1,1,1] => [10,2] => [2] => [1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1] => [11,1] => [1] => [1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1] => [12] => [] => [] => 0
[13] => [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] => [12] => 0
[12,1] => [2,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => [11] => 0
[11,2] => [2,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => [10,1] => 1
[11,1,1] => [3,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [10] => 0
[10,3] => [2,2,2,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => [9,2] => 0
[10,2,1] => [3,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => [9,1] => 1
[10,1,1,1] => [4,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [9] => 0
[9,4] => [2,2,2,2,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => [8,3] => 0
[9,3,1] => [3,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => [8,2] => 0
[9,2,2] => [3,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => [8,1,1] => 2
[9,2,1,1] => [4,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => [8,1] => 1
[9,1,1,1,1] => [5,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [8] => 0
[8,5] => [2,2,2,2,2,1,1,1] => [2,2,2,2,1,1,1] => [7,4] => 0
[8,4,1] => [3,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => [7,3] => 0
[8,3,2] => [3,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => [7,2,1] => 1
[8,3,1,1] => [4,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => [7,2] => 0
[8,2,2,1] => [4,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => [7,1,1] => 2
[8,2,1,1,1] => [5,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => [7,1] => 1
[8,1,1,1,1,1] => [6,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [7] => 0
[7,6] => [2,2,2,2,2,2,1] => [2,2,2,2,2,1] => [6,5] => 0
[7,5,1] => [3,2,2,2,2,1,1] => [2,2,2,2,1,1] => [6,4] => 0
[7,4,2] => [3,3,2,2,1,1,1] => [3,2,2,1,1,1] => [6,3,1] => 1
[7,4,1,1] => [4,2,2,2,1,1,1] => [2,2,2,1,1,1] => [6,3] => 0
[7,3,3] => [3,3,3,1,1,1,1] => [3,3,1,1,1,1] => [6,2,2] => 0
[7,3,2,1] => [4,3,2,1,1,1,1] => [3,2,1,1,1,1] => [6,2,1] => 1
[7,3,1,1,1] => [5,2,2,1,1,1,1] => [2,2,1,1,1,1] => [6,2] => 0
[7,2,2,2] => [4,4,1,1,1,1,1] => [4,1,1,1,1,1] => [6,1,1,1] => 3
[7,2,2,1,1] => [5,3,1,1,1,1,1] => [3,1,1,1,1,1] => [6,1,1] => 2
[7,2,1,1,1,1] => [6,2,1,1,1,1,1] => [2,1,1,1,1,1] => [6,1] => 1
[7,1,1,1,1,1,1] => [7,1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => 0
[6,6,1] => [3,2,2,2,2,2] => [2,2,2,2,2] => [5,5] => 0
[6,5,2] => [3,3,2,2,2,1] => [3,2,2,2,1] => [5,4,1] => 1
[6,5,1,1] => [4,2,2,2,2,1] => [2,2,2,2,1] => [5,4] => 0
[6,4,3] => [3,3,3,2,1,1] => [3,3,2,1,1] => [5,3,2] => 0
[6,4,2,1] => [4,3,2,2,1,1] => [3,2,2,1,1] => [5,3,1] => 1
[6,4,1,1,1] => [5,2,2,2,1,1] => [2,2,2,1,1] => [5,3] => 0
[6,3,3,1] => [4,3,3,1,1,1] => [3,3,1,1,1] => [5,2,2] => 0
[6,3,2,2] => [4,4,2,1,1,1] => [4,2,1,1,1] => [5,2,1,1] => 2
[6,3,2,1,1] => [5,3,2,1,1,1] => [3,2,1,1,1] => [5,2,1] => 1
[6,3,1,1,1,1] => [6,2,2,1,1,1] => [2,2,1,1,1] => [5,2] => 0
[6,2,2,2,1] => [5,4,1,1,1,1] => [4,1,1,1,1] => [5,1,1,1] => 3
[6,2,2,1,1,1] => [6,3,1,1,1,1] => [3,1,1,1,1] => [5,1,1] => 2
[6,2,1,1,1,1,1] => [7,2,1,1,1,1] => [2,1,1,1,1] => [5,1] => 1
[6,1,1,1,1,1,1,1] => [8,1,1,1,1,1] => [1,1,1,1,1] => [5] => 0
[5,5,3] => [3,3,3,2,2] => [3,3,2,2] => [4,4,2] => 0
[5,5,2,1] => [4,3,2,2,2] => [3,2,2,2] => [4,4,1] => 1
[5,5,1,1,1] => [5,2,2,2,2] => [2,2,2,2] => [4,4] => 0
[5,4,4] => [3,3,3,3,1] => [3,3,3,1] => [4,3,3] => 0
[5,4,3,1] => [4,3,3,2,1] => [3,3,2,1] => [4,3,2] => 0
[5,4,2,2] => [4,4,2,2,1] => [4,2,2,1] => [4,3,1,1] => 2
[5,4,2,1,1] => [5,3,2,2,1] => [3,2,2,1] => [4,3,1] => 1
[5,4,1,1,1,1] => [6,2,2,2,1] => [2,2,2,1] => [4,3] => 0
[5,3,3,2] => [4,4,3,1,1] => [4,3,1,1] => [4,2,2,1] => 1
[5,3,3,1,1] => [5,3,3,1,1] => [3,3,1,1] => [4,2,2] => 0
[5,3,2,2,1] => [5,4,2,1,1] => [4,2,1,1] => [4,2,1,1] => 2
[5,3,2,1,1,1] => [6,3,2,1,1] => [3,2,1,1] => [4,2,1] => 1
[5,3,1,1,1,1,1] => [7,2,2,1,1] => [2,2,1,1] => [4,2] => 0
[5,2,2,2,2] => [5,5,1,1,1] => [5,1,1,1] => [4,1,1,1,1] => 4
[5,2,2,2,1,1] => [6,4,1,1,1] => [4,1,1,1] => [4,1,1,1] => 3
[5,2,2,1,1,1,1] => [7,3,1,1,1] => [3,1,1,1] => [4,1,1] => 2
[5,2,1,1,1,1,1,1] => [8,2,1,1,1] => [2,1,1,1] => [4,1] => 1
[5,1,1,1,1,1,1,1,1] => [9,1,1,1,1] => [1,1,1,1] => [4] => 0
[4,4,4,1] => [4,3,3,3] => [3,3,3] => [3,3,3] => 0
[4,4,3,2] => [4,4,3,2] => [4,3,2] => [3,3,2,1] => 1
[4,4,3,1,1] => [5,3,3,2] => [3,3,2] => [3,3,2] => 0
[4,4,2,2,1] => [5,4,2,2] => [4,2,2] => [3,3,1,1] => 2
[4,4,2,1,1,1] => [6,3,2,2] => [3,2,2] => [3,3,1] => 1
[4,4,1,1,1,1,1] => [7,2,2,2] => [2,2,2] => [3,3] => 0
[4,3,3,3] => [4,4,4,1] => [4,4,1] => [3,2,2,2] => 0
[4,3,3,2,1] => [5,4,3,1] => [4,3,1] => [3,2,2,1] => 1
[4,3,3,1,1,1] => [6,3,3,1] => [3,3,1] => [3,2,2] => 0
[4,3,2,2,2] => [5,5,2,1] => [5,2,1] => [3,2,1,1,1] => 3
[4,3,2,2,1,1] => [6,4,2,1] => [4,2,1] => [3,2,1,1] => 2
[4,3,2,1,1,1,1] => [7,3,2,1] => [3,2,1] => [3,2,1] => 1
[4,3,1,1,1,1,1,1] => [8,2,2,1] => [2,2,1] => [3,2] => 0
[4,2,2,2,2,1] => [6,5,1,1] => [5,1,1] => [3,1,1,1,1] => 4
[4,2,2,2,1,1,1] => [7,4,1,1] => [4,1,1] => [3,1,1,1] => 3
[4,2,2,1,1,1,1,1] => [8,3,1,1] => [3,1,1] => [3,1,1] => 2
[4,2,1,1,1,1,1,1,1] => [9,2,1,1] => [2,1,1] => [3,1] => 1
[4,1,1,1,1,1,1,1,1,1] => [10,1,1,1] => [1,1,1] => [3] => 0
[3,3,3,3,1] => [5,4,4] => [4,4] => [2,2,2,2] => 0
[3,3,3,2,2] => [5,5,3] => [5,3] => [2,2,2,1,1] => 2
[3,3,3,2,1,1] => [6,4,3] => [4,3] => [2,2,2,1] => 1
[3,3,3,1,1,1,1] => [7,3,3] => [3,3] => [2,2,2] => 0
[3,3,2,2,2,1] => [6,5,2] => [5,2] => [2,2,1,1,1] => 3
[3,3,2,2,1,1,1] => [7,4,2] => [4,2] => [2,2,1,1] => 2
[3,3,2,1,1,1,1,1] => [8,3,2] => [3,2] => [2,2,1] => 1
[3,3,1,1,1,1,1,1,1] => [9,2,2] => [2,2] => [2,2] => 0
[3,2,2,2,2,2] => [6,6,1] => [6,1] => [2,1,1,1,1,1] => 5
[3,2,2,2,2,1,1] => [7,5,1] => [5,1] => [2,1,1,1,1] => 4
[3,2,2,2,1,1,1,1] => [8,4,1] => [4,1] => [2,1,1,1] => 3
[3,2,2,1,1,1,1,1,1] => [9,3,1] => [3,1] => [2,1,1] => 2
[3,2,1,1,1,1,1,1,1,1] => [10,2,1] => [2,1] => [2,1] => 1
[3,1,1,1,1,1,1,1,1,1,1] => [11,1,1] => [1,1] => [2] => 0
[2,2,2,2,2,2,1] => [7,6] => [6] => [1,1,1,1,1,1] => 6
[2,2,2,2,2,1,1,1] => [8,5] => [5] => [1,1,1,1,1] => 5
[2,2,2,2,1,1,1,1,1] => [9,4] => [4] => [1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1,1] => [10,3] => [3] => [1,1,1] => 3
[2,2,1,1,1,1,1,1,1,1,1] => [11,2] => [2] => [1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1,1] => [12,1] => [1] => [1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1] => [13] => [] => [] => 0
[13,1] => [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] => [12] => 0
[12,2] => [2,2,1,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1] => [11,1] => 1
[12,1,1] => [3,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => [11] => 0
[11,3] => [2,2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => [10,2] => 0
[11,2,1] => [3,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => [10,1] => 1
[11,1,1,1] => [4,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [10] => 0
[10,4] => [2,2,2,2,1,1,1,1,1,1] => [2,2,2,1,1,1,1,1,1] => [9,3] => 0
[10,3,1] => [3,2,2,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => [9,2] => 0
[10,2,2] => [3,3,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1] => [9,1,1] => 2
[10,2,1,1] => [4,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => [9,1] => 1
[10,1,1,1,1] => [5,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [9] => 0
[9,5] => [2,2,2,2,2,1,1,1,1] => [2,2,2,2,1,1,1,1] => [8,4] => 0
[9,4,1] => [3,2,2,2,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => [8,3] => 0
[9,3,2] => [3,3,2,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1] => [8,2,1] => 1
[9,3,1,1] => [4,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => [8,2] => 0
[9,2,2,1] => [4,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => [8,1,1] => 2
[9,2,1,1,1] => [5,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => [8,1] => 1
[9,1,1,1,1,1] => [6,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [8] => 0
[8,6] => [2,2,2,2,2,2,1,1] => [2,2,2,2,2,1,1] => [7,5] => 0
[8,5,1] => [3,2,2,2,2,1,1,1] => [2,2,2,2,1,1,1] => [7,4] => 0
[8,4,2] => [3,3,2,2,1,1,1,1] => [3,2,2,1,1,1,1] => [7,3,1] => 1
[8,4,1,1] => [4,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => [7,3] => 0
[8,3,3] => [3,3,3,1,1,1,1,1] => [3,3,1,1,1,1,1] => [7,2,2] => 0
[8,3,2,1] => [4,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => [7,2,1] => 1
[8,3,1,1,1] => [5,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => [7,2] => 0
[8,2,2,2] => [4,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => [7,1,1,1] => 3
[8,2,2,1,1] => [5,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => [7,1,1] => 2
[8,2,1,1,1,1] => [6,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => [7,1] => 1
[8,1,1,1,1,1,1] => [7,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [7] => 0
[7,7] => [2,2,2,2,2,2,2] => [2,2,2,2,2,2] => [6,6] => 0
[7,6,1] => [3,2,2,2,2,2,1] => [2,2,2,2,2,1] => [6,5] => 0
[7,5,2] => [3,3,2,2,2,1,1] => [3,2,2,2,1,1] => [6,4,1] => 1
[7,5,1,1] => [4,2,2,2,2,1,1] => [2,2,2,2,1,1] => [6,4] => 0
[7,4,3] => [3,3,3,2,1,1,1] => [3,3,2,1,1,1] => [6,3,2] => 0
[7,4,2,1] => [4,3,2,2,1,1,1] => [3,2,2,1,1,1] => [6,3,1] => 1
[7,4,1,1,1] => [5,2,2,2,1,1,1] => [2,2,2,1,1,1] => [6,3] => 0
[7,3,3,1] => [4,3,3,1,1,1,1] => [3,3,1,1,1,1] => [6,2,2] => 0
[7,3,2,2] => [4,4,2,1,1,1,1] => [4,2,1,1,1,1] => [6,2,1,1] => 2
[7,3,2,1,1] => [5,3,2,1,1,1,1] => [3,2,1,1,1,1] => [6,2,1] => 1
[7,3,1,1,1,1] => [6,2,2,1,1,1,1] => [2,2,1,1,1,1] => [6,2] => 0
[7,2,2,2,1] => [5,4,1,1,1,1,1] => [4,1,1,1,1,1] => [6,1,1,1] => 3
[7,2,2,1,1,1] => [6,3,1,1,1,1,1] => [3,1,1,1,1,1] => [6,1,1] => 2
[7,2,1,1,1,1,1] => [7,2,1,1,1,1,1] => [2,1,1,1,1,1] => [6,1] => 1
[7,1,1,1,1,1,1,1] => [8,1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => 0
[6,6,2] => [3,3,2,2,2,2] => [3,2,2,2,2] => [5,5,1] => 1
[6,6,1,1] => [4,2,2,2,2,2] => [2,2,2,2,2] => [5,5] => 0
[6,5,3] => [3,3,3,2,2,1] => [3,3,2,2,1] => [5,4,2] => 0
[6,5,2,1] => [4,3,2,2,2,1] => [3,2,2,2,1] => [5,4,1] => 1
[6,5,1,1,1] => [5,2,2,2,2,1] => [2,2,2,2,1] => [5,4] => 0
[6,4,4] => [3,3,3,3,1,1] => [3,3,3,1,1] => [5,3,3] => 0
[6,4,3,1] => [4,3,3,2,1,1] => [3,3,2,1,1] => [5,3,2] => 0
[6,4,2,2] => [4,4,2,2,1,1] => [4,2,2,1,1] => [5,3,1,1] => 2
[6,4,2,1,1] => [5,3,2,2,1,1] => [3,2,2,1,1] => [5,3,1] => 1
[6,4,1,1,1,1] => [6,2,2,2,1,1] => [2,2,2,1,1] => [5,3] => 0
[6,3,3,2] => [4,4,3,1,1,1] => [4,3,1,1,1] => [5,2,2,1] => 1
[6,3,3,1,1] => [5,3,3,1,1,1] => [3,3,1,1,1] => [5,2,2] => 0
[6,3,2,2,1] => [5,4,2,1,1,1] => [4,2,1,1,1] => [5,2,1,1] => 2
[6,3,2,1,1,1] => [6,3,2,1,1,1] => [3,2,1,1,1] => [5,2,1] => 1
[6,3,1,1,1,1,1] => [7,2,2,1,1,1] => [2,2,1,1,1] => [5,2] => 0
[6,2,2,2,2] => [5,5,1,1,1,1] => [5,1,1,1,1] => [5,1,1,1,1] => 4
[6,2,2,2,1,1] => [6,4,1,1,1,1] => [4,1,1,1,1] => [5,1,1,1] => 3
[6,2,2,1,1,1,1] => [7,3,1,1,1,1] => [3,1,1,1,1] => [5,1,1] => 2
[6,2,1,1,1,1,1,1] => [8,2,1,1,1,1] => [2,1,1,1,1] => [5,1] => 1
[6,1,1,1,1,1,1,1,1] => [9,1,1,1,1,1] => [1,1,1,1,1] => [5] => 0
[5,5,4] => [3,3,3,3,2] => [3,3,3,2] => [4,4,3] => 0
[5,5,3,1] => [4,3,3,2,2] => [3,3,2,2] => [4,4,2] => 0
[5,5,2,2] => [4,4,2,2,2] => [4,2,2,2] => [4,4,1,1] => 2
[5,5,2,1,1] => [5,3,2,2,2] => [3,2,2,2] => [4,4,1] => 1
[5,5,1,1,1,1] => [6,2,2,2,2] => [2,2,2,2] => [4,4] => 0
[5,4,4,1] => [4,3,3,3,1] => [3,3,3,1] => [4,3,3] => 0
[5,4,3,2] => [4,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[5,4,3,1,1] => [5,3,3,2,1] => [3,3,2,1] => [4,3,2] => 0
[5,4,2,2,1] => [5,4,2,2,1] => [4,2,2,1] => [4,3,1,1] => 2
[5,4,2,1,1,1] => [6,3,2,2,1] => [3,2,2,1] => [4,3,1] => 1
[5,4,1,1,1,1,1] => [7,2,2,2,1] => [2,2,2,1] => [4,3] => 0
[5,3,3,3] => [4,4,4,1,1] => [4,4,1,1] => [4,2,2,2] => 0
[5,3,3,2,1] => [5,4,3,1,1] => [4,3,1,1] => [4,2,2,1] => 1
[5,3,3,1,1,1] => [6,3,3,1,1] => [3,3,1,1] => [4,2,2] => 0
[5,3,2,2,2] => [5,5,2,1,1] => [5,2,1,1] => [4,2,1,1,1] => 3
[5,3,2,2,1,1] => [6,4,2,1,1] => [4,2,1,1] => [4,2,1,1] => 2
[5,3,2,1,1,1,1] => [7,3,2,1,1] => [3,2,1,1] => [4,2,1] => 1
[5,3,1,1,1,1,1,1] => [8,2,2,1,1] => [2,2,1,1] => [4,2] => 0
[5,2,2,2,2,1] => [6,5,1,1,1] => [5,1,1,1] => [4,1,1,1,1] => 4
[5,2,2,2,1,1,1] => [7,4,1,1,1] => [4,1,1,1] => [4,1,1,1] => 3
[5,2,2,1,1,1,1,1] => [8,3,1,1,1] => [3,1,1,1] => [4,1,1] => 2
[5,2,1,1,1,1,1,1,1] => [9,2,1,1,1] => [2,1,1,1] => [4,1] => 1
[5,1,1,1,1,1,1,1,1,1] => [10,1,1,1,1] => [1,1,1,1] => [4] => 0
[4,4,4,2] => [4,4,3,3] => [4,3,3] => [3,3,3,1] => 1
[4,4,4,1,1] => [5,3,3,3] => [3,3,3] => [3,3,3] => 0
[4,4,3,3] => [4,4,4,2] => [4,4,2] => [3,3,2,2] => 0
[4,4,3,2,1] => [5,4,3,2] => [4,3,2] => [3,3,2,1] => 1
[4,4,3,1,1,1] => [6,3,3,2] => [3,3,2] => [3,3,2] => 0
[4,4,2,2,2] => [5,5,2,2] => [5,2,2] => [3,3,1,1,1] => 3
[4,4,2,2,1,1] => [6,4,2,2] => [4,2,2] => [3,3,1,1] => 2
[4,4,2,1,1,1,1] => [7,3,2,2] => [3,2,2] => [3,3,1] => 1
[4,4,1,1,1,1,1,1] => [8,2,2,2] => [2,2,2] => [3,3] => 0
[4,3,3,3,1] => [5,4,4,1] => [4,4,1] => [3,2,2,2] => 0
[4,3,3,2,2] => [5,5,3,1] => [5,3,1] => [3,2,2,1,1] => 2
[4,3,3,2,1,1] => [6,4,3,1] => [4,3,1] => [3,2,2,1] => 1
[4,3,3,1,1,1,1] => [7,3,3,1] => [3,3,1] => [3,2,2] => 0
[4,3,2,2,2,1] => [6,5,2,1] => [5,2,1] => [3,2,1,1,1] => 3
[4,3,2,2,1,1,1] => [7,4,2,1] => [4,2,1] => [3,2,1,1] => 2
[4,3,2,1,1,1,1,1] => [8,3,2,1] => [3,2,1] => [3,2,1] => 1
[4,3,1,1,1,1,1,1,1] => [9,2,2,1] => [2,2,1] => [3,2] => 0
[4,2,2,2,2,2] => [6,6,1,1] => [6,1,1] => [3,1,1,1,1,1] => 5
[4,2,2,2,2,1,1] => [7,5,1,1] => [5,1,1] => [3,1,1,1,1] => 4
[4,2,2,2,1,1,1,1] => [8,4,1,1] => [4,1,1] => [3,1,1,1] => 3
[4,2,2,1,1,1,1,1,1] => [9,3,1,1] => [3,1,1] => [3,1,1] => 2
[4,2,1,1,1,1,1,1,1,1] => [10,2,1,1] => [2,1,1] => [3,1] => 1
[4,1,1,1,1,1,1,1,1,1,1] => [11,1,1,1] => [1,1,1] => [3] => 0
[3,3,3,3,2] => [5,5,4] => [5,4] => [2,2,2,2,1] => 1
[3,3,3,3,1,1] => [6,4,4] => [4,4] => [2,2,2,2] => 0
[3,3,3,2,2,1] => [6,5,3] => [5,3] => [2,2,2,1,1] => 2
[3,3,3,2,1,1,1] => [7,4,3] => [4,3] => [2,2,2,1] => 1
[3,3,3,1,1,1,1,1] => [8,3,3] => [3,3] => [2,2,2] => 0
[3,3,2,2,2,2] => [6,6,2] => [6,2] => [2,2,1,1,1,1] => 4
[3,3,2,2,2,1,1] => [7,5,2] => [5,2] => [2,2,1,1,1] => 3
[3,3,2,2,1,1,1,1] => [8,4,2] => [4,2] => [2,2,1,1] => 2
[3,3,2,1,1,1,1,1,1] => [9,3,2] => [3,2] => [2,2,1] => 1
[3,3,1,1,1,1,1,1,1,1] => [10,2,2] => [2,2] => [2,2] => 0
[3,2,2,2,2,2,1] => [7,6,1] => [6,1] => [2,1,1,1,1,1] => 5
[3,2,2,2,2,1,1,1] => [8,5,1] => [5,1] => [2,1,1,1,1] => 4
[3,2,2,2,1,1,1,1,1] => [9,4,1] => [4,1] => [2,1,1,1] => 3
[3,2,2,1,1,1,1,1,1,1] => [10,3,1] => [3,1] => [2,1,1] => 2
[3,2,1,1,1,1,1,1,1,1,1] => [11,2,1] => [2,1] => [2,1] => 1
[3,1,1,1,1,1,1,1,1,1,1,1] => [12,1,1] => [1,1] => [2] => 0
[2,2,2,2,2,2,2] => [7,7] => [7] => [1,1,1,1,1,1,1] => 7
[2,2,2,2,2,2,1,1] => [8,6] => [6] => [1,1,1,1,1,1] => 6
[2,2,2,2,2,1,1,1,1] => [9,5] => [5] => [1,1,1,1,1] => 5
[2,2,2,2,1,1,1,1,1,1] => [10,4] => [4] => [1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1,1,1] => [11,3] => [3] => [1,1,1] => 3
[2,2,1,1,1,1,1,1,1,1,1,1] => [12,2] => [2] => [1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1] => [1] => [1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14] => [] => [] => 0
[13,1,1] => [3,1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => [12] => 0
[12,2,1] => [3,2,1,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1] => [11,1] => 1
[12,1,1,1] => [4,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => [11] => 0
[11,3,1] => [3,2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => [10,2] => 0
[11,2,2] => [3,3,1,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1,1] => [10,1,1] => 2
[11,2,1,1] => [4,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => [10,1] => 1
[11,1,1,1,1] => [5,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [10] => 0
[10,4,1] => [3,2,2,2,1,1,1,1,1,1] => [2,2,2,1,1,1,1,1,1] => [9,3] => 0
[10,3,2] => [3,3,2,1,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1,1] => [9,2,1] => 1
[10,3,1,1] => [4,2,2,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => [9,2] => 0
[10,2,2,1] => [4,3,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1] => [9,1,1] => 2
[10,2,1,1,1] => [5,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => [9,1] => 1
[10,1,1,1,1,1] => [6,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [9] => 0
[9,5,1] => [3,2,2,2,2,1,1,1,1] => [2,2,2,2,1,1,1,1] => [8,4] => 0
[9,4,2] => [3,3,2,2,1,1,1,1,1] => [3,2,2,1,1,1,1,1] => [8,3,1] => 1
[9,4,1,1] => [4,2,2,2,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => [8,3] => 0
[9,3,3] => [3,3,3,1,1,1,1,1,1] => [3,3,1,1,1,1,1,1] => [8,2,2] => 0
[9,3,2,1] => [4,3,2,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1] => [8,2,1] => 1
[9,3,1,1,1] => [5,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => [8,2] => 0
[9,2,2,2] => [4,4,1,1,1,1,1,1,1] => [4,1,1,1,1,1,1,1] => [8,1,1,1] => 3
[9,2,2,1,1] => [5,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => [8,1,1] => 2
[9,2,1,1,1,1] => [6,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => [8,1] => 1
[9,1,1,1,1,1,1] => [7,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [8] => 0
[8,6,1] => [3,2,2,2,2,2,1,1] => [2,2,2,2,2,1,1] => [7,5] => 0
[8,5,2] => [3,3,2,2,2,1,1,1] => [3,2,2,2,1,1,1] => [7,4,1] => 1
[8,5,1,1] => [4,2,2,2,2,1,1,1] => [2,2,2,2,1,1,1] => [7,4] => 0
[8,4,3] => [3,3,3,2,1,1,1,1] => [3,3,2,1,1,1,1] => [7,3,2] => 0
[8,4,2,1] => [4,3,2,2,1,1,1,1] => [3,2,2,1,1,1,1] => [7,3,1] => 1
[8,4,1,1,1] => [5,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => [7,3] => 0
[8,3,3,1] => [4,3,3,1,1,1,1,1] => [3,3,1,1,1,1,1] => [7,2,2] => 0
[8,3,2,2] => [4,4,2,1,1,1,1,1] => [4,2,1,1,1,1,1] => [7,2,1,1] => 2
[8,3,2,1,1] => [5,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => [7,2,1] => 1
[8,3,1,1,1,1] => [6,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => [7,2] => 0
[8,2,2,2,1] => [5,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => [7,1,1,1] => 3
[8,2,2,1,1,1] => [6,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => [7,1,1] => 2
[8,2,1,1,1,1,1] => [7,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => [7,1] => 1
[8,1,1,1,1,1,1,1] => [8,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [7] => 0
[7,7,1] => [3,2,2,2,2,2,2] => [2,2,2,2,2,2] => [6,6] => 0
[7,6,2] => [3,3,2,2,2,2,1] => [3,2,2,2,2,1] => [6,5,1] => 1
[7,6,1,1] => [4,2,2,2,2,2,1] => [2,2,2,2,2,1] => [6,5] => 0
[7,5,3] => [3,3,3,2,2,1,1] => [3,3,2,2,1,1] => [6,4,2] => 0
[7,5,2,1] => [4,3,2,2,2,1,1] => [3,2,2,2,1,1] => [6,4,1] => 1
[7,5,1,1,1] => [5,2,2,2,2,1,1] => [2,2,2,2,1,1] => [6,4] => 0
[7,4,4] => [3,3,3,3,1,1,1] => [3,3,3,1,1,1] => [6,3,3] => 0
[7,4,3,1] => [4,3,3,2,1,1,1] => [3,3,2,1,1,1] => [6,3,2] => 0
[7,4,2,2] => [4,4,2,2,1,1,1] => [4,2,2,1,1,1] => [6,3,1,1] => 2
[7,4,2,1,1] => [5,3,2,2,1,1,1] => [3,2,2,1,1,1] => [6,3,1] => 1
[7,4,1,1,1,1] => [6,2,2,2,1,1,1] => [2,2,2,1,1,1] => [6,3] => 0
[7,3,3,2] => [4,4,3,1,1,1,1] => [4,3,1,1,1,1] => [6,2,2,1] => 1
[7,3,3,1,1] => [5,3,3,1,1,1,1] => [3,3,1,1,1,1] => [6,2,2] => 0
[7,3,2,2,1] => [5,4,2,1,1,1,1] => [4,2,1,1,1,1] => [6,2,1,1] => 2
[7,3,2,1,1,1] => [6,3,2,1,1,1,1] => [3,2,1,1,1,1] => [6,2,1] => 1
[7,3,1,1,1,1,1] => [7,2,2,1,1,1,1] => [2,2,1,1,1,1] => [6,2] => 0
[7,2,2,2,2] => [5,5,1,1,1,1,1] => [5,1,1,1,1,1] => [6,1,1,1,1] => 4
[7,2,2,2,1,1] => [6,4,1,1,1,1,1] => [4,1,1,1,1,1] => [6,1,1,1] => 3
[7,2,2,1,1,1,1] => [7,3,1,1,1,1,1] => [3,1,1,1,1,1] => [6,1,1] => 2
[7,2,1,1,1,1,1,1] => [8,2,1,1,1,1,1] => [2,1,1,1,1,1] => [6,1] => 1
[7,1,1,1,1,1,1,1,1] => [9,1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => 0
[6,6,3] => [3,3,3,2,2,2] => [3,3,2,2,2] => [5,5,2] => 0
[6,6,2,1] => [4,3,2,2,2,2] => [3,2,2,2,2] => [5,5,1] => 1
[6,6,1,1,1] => [5,2,2,2,2,2] => [2,2,2,2,2] => [5,5] => 0
[6,5,4] => [3,3,3,3,2,1] => [3,3,3,2,1] => [5,4,3] => 0
[6,5,3,1] => [4,3,3,2,2,1] => [3,3,2,2,1] => [5,4,2] => 0
[6,5,2,2] => [4,4,2,2,2,1] => [4,2,2,2,1] => [5,4,1,1] => 2
[6,5,2,1,1] => [5,3,2,2,2,1] => [3,2,2,2,1] => [5,4,1] => 1
[6,5,1,1,1,1] => [6,2,2,2,2,1] => [2,2,2,2,1] => [5,4] => 0
[6,4,4,1] => [4,3,3,3,1,1] => [3,3,3,1,1] => [5,3,3] => 0
[6,4,3,2] => [4,4,3,2,1,1] => [4,3,2,1,1] => [5,3,2,1] => 1
[6,4,3,1,1] => [5,3,3,2,1,1] => [3,3,2,1,1] => [5,3,2] => 0
[6,4,2,2,1] => [5,4,2,2,1,1] => [4,2,2,1,1] => [5,3,1,1] => 2
[6,4,2,1,1,1] => [6,3,2,2,1,1] => [3,2,2,1,1] => [5,3,1] => 1
[6,4,1,1,1,1,1] => [7,2,2,2,1,1] => [2,2,2,1,1] => [5,3] => 0
[6,3,3,3] => [4,4,4,1,1,1] => [4,4,1,1,1] => [5,2,2,2] => 0
[6,3,3,2,1] => [5,4,3,1,1,1] => [4,3,1,1,1] => [5,2,2,1] => 1
[6,3,3,1,1,1] => [6,3,3,1,1,1] => [3,3,1,1,1] => [5,2,2] => 0
[6,3,2,2,2] => [5,5,2,1,1,1] => [5,2,1,1,1] => [5,2,1,1,1] => 3
[6,3,2,2,1,1] => [6,4,2,1,1,1] => [4,2,1,1,1] => [5,2,1,1] => 2
[6,3,2,1,1,1,1] => [7,3,2,1,1,1] => [3,2,1,1,1] => [5,2,1] => 1
[6,3,1,1,1,1,1,1] => [8,2,2,1,1,1] => [2,2,1,1,1] => [5,2] => 0
[6,2,2,2,2,1] => [6,5,1,1,1,1] => [5,1,1,1,1] => [5,1,1,1,1] => 4
[6,2,2,2,1,1,1] => [7,4,1,1,1,1] => [4,1,1,1,1] => [5,1,1,1] => 3
[6,2,2,1,1,1,1,1] => [8,3,1,1,1,1] => [3,1,1,1,1] => [5,1,1] => 2
[6,2,1,1,1,1,1,1,1] => [9,2,1,1,1,1] => [2,1,1,1,1] => [5,1] => 1
[6,1,1,1,1,1,1,1,1,1] => [10,1,1,1,1,1] => [1,1,1,1,1] => [5] => 0
[5,5,5] => [3,3,3,3,3] => [3,3,3,3] => [4,4,4] => 0
[5,5,4,1] => [4,3,3,3,2] => [3,3,3,2] => [4,4,3] => 0
[5,5,3,2] => [4,4,3,2,2] => [4,3,2,2] => [4,4,2,1] => 1
[5,5,3,1,1] => [5,3,3,2,2] => [3,3,2,2] => [4,4,2] => 0
[5,5,2,2,1] => [5,4,2,2,2] => [4,2,2,2] => [4,4,1,1] => 2
[5,5,2,1,1,1] => [6,3,2,2,2] => [3,2,2,2] => [4,4,1] => 1
[5,5,1,1,1,1,1] => [7,2,2,2,2] => [2,2,2,2] => [4,4] => 0
[5,4,4,2] => [4,4,3,3,1] => [4,3,3,1] => [4,3,3,1] => 1
[5,4,4,1,1] => [5,3,3,3,1] => [3,3,3,1] => [4,3,3] => 0
[5,4,3,3] => [4,4,4,2,1] => [4,4,2,1] => [4,3,2,2] => 0
[5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[5,4,3,1,1,1] => [6,3,3,2,1] => [3,3,2,1] => [4,3,2] => 0
[5,4,2,2,2] => [5,5,2,2,1] => [5,2,2,1] => [4,3,1,1,1] => 3
[5,4,2,2,1,1] => [6,4,2,2,1] => [4,2,2,1] => [4,3,1,1] => 2
[5,4,2,1,1,1,1] => [7,3,2,2,1] => [3,2,2,1] => [4,3,1] => 1
[5,4,1,1,1,1,1,1] => [8,2,2,2,1] => [2,2,2,1] => [4,3] => 0
[5,3,3,3,1] => [5,4,4,1,1] => [4,4,1,1] => [4,2,2,2] => 0
[5,3,3,2,2] => [5,5,3,1,1] => [5,3,1,1] => [4,2,2,1,1] => 2
[5,3,3,2,1,1] => [6,4,3,1,1] => [4,3,1,1] => [4,2,2,1] => 1
[5,3,3,1,1,1,1] => [7,3,3,1,1] => [3,3,1,1] => [4,2,2] => 0
[5,3,2,2,2,1] => [6,5,2,1,1] => [5,2,1,1] => [4,2,1,1,1] => 3
[5,3,2,2,1,1,1] => [7,4,2,1,1] => [4,2,1,1] => [4,2,1,1] => 2
[5,3,2,1,1,1,1,1] => [8,3,2,1,1] => [3,2,1,1] => [4,2,1] => 1
[5,3,1,1,1,1,1,1,1] => [9,2,2,1,1] => [2,2,1,1] => [4,2] => 0
[5,2,2,2,2,2] => [6,6,1,1,1] => [6,1,1,1] => [4,1,1,1,1,1] => 5
[5,2,2,2,2,1,1] => [7,5,1,1,1] => [5,1,1,1] => [4,1,1,1,1] => 4
[5,2,2,2,1,1,1,1] => [8,4,1,1,1] => [4,1,1,1] => [4,1,1,1] => 3
[5,2,2,1,1,1,1,1,1] => [9,3,1,1,1] => [3,1,1,1] => [4,1,1] => 2
[5,2,1,1,1,1,1,1,1,1] => [10,2,1,1,1] => [2,1,1,1] => [4,1] => 1
[5,1,1,1,1,1,1,1,1,1,1] => [11,1,1,1,1] => [1,1,1,1] => [4] => 0
[4,4,4,3] => [4,4,4,3] => [4,4,3] => [3,3,3,2] => 0
[4,4,4,2,1] => [5,4,3,3] => [4,3,3] => [3,3,3,1] => 1
[4,4,4,1,1,1] => [6,3,3,3] => [3,3,3] => [3,3,3] => 0
[4,4,3,3,1] => [5,4,4,2] => [4,4,2] => [3,3,2,2] => 0
[4,4,3,2,2] => [5,5,3,2] => [5,3,2] => [3,3,2,1,1] => 2
[4,4,3,2,1,1] => [6,4,3,2] => [4,3,2] => [3,3,2,1] => 1
[4,4,3,1,1,1,1] => [7,3,3,2] => [3,3,2] => [3,3,2] => 0
[4,4,2,2,2,1] => [6,5,2,2] => [5,2,2] => [3,3,1,1,1] => 3
[4,4,2,2,1,1,1] => [7,4,2,2] => [4,2,2] => [3,3,1,1] => 2
[4,4,2,1,1,1,1,1] => [8,3,2,2] => [3,2,2] => [3,3,1] => 1
[4,4,1,1,1,1,1,1,1] => [9,2,2,2] => [2,2,2] => [3,3] => 0
[4,3,3,3,2] => [5,5,4,1] => [5,4,1] => [3,2,2,2,1] => 1
[4,3,3,3,1,1] => [6,4,4,1] => [4,4,1] => [3,2,2,2] => 0
[4,3,3,2,2,1] => [6,5,3,1] => [5,3,1] => [3,2,2,1,1] => 2
[4,3,3,2,1,1,1] => [7,4,3,1] => [4,3,1] => [3,2,2,1] => 1
[4,3,3,1,1,1,1,1] => [8,3,3,1] => [3,3,1] => [3,2,2] => 0
[4,3,2,2,2,2] => [6,6,2,1] => [6,2,1] => [3,2,1,1,1,1] => 4
[4,3,2,2,2,1,1] => [7,5,2,1] => [5,2,1] => [3,2,1,1,1] => 3
[4,3,2,2,1,1,1,1] => [8,4,2,1] => [4,2,1] => [3,2,1,1] => 2
[4,3,2,1,1,1,1,1,1] => [9,3,2,1] => [3,2,1] => [3,2,1] => 1
[4,3,1,1,1,1,1,1,1,1] => [10,2,2,1] => [2,2,1] => [3,2] => 0
[4,2,2,2,2,2,1] => [7,6,1,1] => [6,1,1] => [3,1,1,1,1,1] => 5
[4,2,2,2,2,1,1,1] => [8,5,1,1] => [5,1,1] => [3,1,1,1,1] => 4
[4,2,2,2,1,1,1,1,1] => [9,4,1,1] => [4,1,1] => [3,1,1,1] => 3
[4,2,2,1,1,1,1,1,1,1] => [10,3,1,1] => [3,1,1] => [3,1,1] => 2
[4,2,1,1,1,1,1,1,1,1,1] => [11,2,1,1] => [2,1,1] => [3,1] => 1
[4,1,1,1,1,1,1,1,1,1,1,1] => [12,1,1,1] => [1,1,1] => [3] => 0
[3,3,3,3,3] => [5,5,5] => [5,5] => [2,2,2,2,2] => 0
[3,3,3,3,2,1] => [6,5,4] => [5,4] => [2,2,2,2,1] => 1
[3,3,3,3,1,1,1] => [7,4,4] => [4,4] => [2,2,2,2] => 0
[3,3,3,2,2,2] => [6,6,3] => [6,3] => [2,2,2,1,1,1] => 3
[3,3,3,2,2,1,1] => [7,5,3] => [5,3] => [2,2,2,1,1] => 2
[3,3,3,2,1,1,1,1] => [8,4,3] => [4,3] => [2,2,2,1] => 1
[3,3,3,1,1,1,1,1,1] => [9,3,3] => [3,3] => [2,2,2] => 0
[3,3,2,2,2,2,1] => [7,6,2] => [6,2] => [2,2,1,1,1,1] => 4
[3,3,2,2,2,1,1,1] => [8,5,2] => [5,2] => [2,2,1,1,1] => 3
[3,3,2,2,1,1,1,1,1] => [9,4,2] => [4,2] => [2,2,1,1] => 2
[3,3,2,1,1,1,1,1,1,1] => [10,3,2] => [3,2] => [2,2,1] => 1
[3,3,1,1,1,1,1,1,1,1,1] => [11,2,2] => [2,2] => [2,2] => 0
[3,2,2,2,2,2,2] => [7,7,1] => [7,1] => [2,1,1,1,1,1,1] => 6
[3,2,2,2,2,2,1,1] => [8,6,1] => [6,1] => [2,1,1,1,1,1] => 5
[3,2,2,2,2,1,1,1,1] => [9,5,1] => [5,1] => [2,1,1,1,1] => 4
[3,2,2,2,1,1,1,1,1,1] => [10,4,1] => [4,1] => [2,1,1,1] => 3
[3,2,2,1,1,1,1,1,1,1,1] => [11,3,1] => [3,1] => [2,1,1] => 2
[3,2,1,1,1,1,1,1,1,1,1,1] => [12,2,1] => [2,1] => [2,1] => 1
[3,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1] => [1,1] => [2] => 0
[2,2,2,2,2,2,2,1] => [8,7] => [7] => [1,1,1,1,1,1,1] => 7
[2,2,2,2,2,2,1,1,1] => [9,6] => [6] => [1,1,1,1,1,1] => 6
[2,2,2,2,2,1,1,1,1,1] => [10,5] => [5] => [1,1,1,1,1] => 5
[2,2,2,2,1,1,1,1,1,1,1] => [11,4] => [4] => [1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1,1,1,1] => [12,3] => [3] => [1,1,1] => 3
[2,2,1,1,1,1,1,1,1,1,1,1,1] => [13,2] => [2] => [1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1] => [1] => [1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15] => [] => [] => 0
[13,1,1,1] => [4,1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => [12] => 0
[12,2,1,1] => [4,2,1,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1] => [11,1] => 1
[12,1,1,1,1] => [5,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => [11] => 0
[11,3,1,1] => [4,2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => [10,2] => 0
[11,2,2,1] => [4,3,1,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1,1] => [10,1,1] => 2
[11,2,1,1,1] => [5,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => [10,1] => 1
[11,1,1,1,1,1] => [6,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [10] => 0
[10,4,1,1] => [4,2,2,2,1,1,1,1,1,1] => [2,2,2,1,1,1,1,1,1] => [9,3] => 0
[10,3,2,1] => [4,3,2,1,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1,1] => [9,2,1] => 1
[10,3,1,1,1] => [5,2,2,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => [9,2] => 0
[10,2,2,2] => [4,4,1,1,1,1,1,1,1,1] => [4,1,1,1,1,1,1,1,1] => [9,1,1,1] => 3
[10,2,2,1,1] => [5,3,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1] => [9,1,1] => 2
[10,2,1,1,1,1] => [6,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => [9,1] => 1
[10,1,1,1,1,1,1] => [7,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [9] => 0
[9,5,1,1] => [4,2,2,2,2,1,1,1,1] => [2,2,2,2,1,1,1,1] => [8,4] => 0
[9,4,2,1] => [4,3,2,2,1,1,1,1,1] => [3,2,2,1,1,1,1,1] => [8,3,1] => 1
[9,4,1,1,1] => [5,2,2,2,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => [8,3] => 0
[9,3,3,1] => [4,3,3,1,1,1,1,1,1] => [3,3,1,1,1,1,1,1] => [8,2,2] => 0
[9,3,2,2] => [4,4,2,1,1,1,1,1,1] => [4,2,1,1,1,1,1,1] => [8,2,1,1] => 2
[9,3,2,1,1] => [5,3,2,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1] => [8,2,1] => 1
[9,3,1,1,1,1] => [6,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => [8,2] => 0
[9,2,2,2,1] => [5,4,1,1,1,1,1,1,1] => [4,1,1,1,1,1,1,1] => [8,1,1,1] => 3
[9,2,2,1,1,1] => [6,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => [8,1,1] => 2
[9,2,1,1,1,1,1] => [7,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => [8,1] => 1
[9,1,1,1,1,1,1,1] => [8,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [8] => 0
[8,6,1,1] => [4,2,2,2,2,2,1,1] => [2,2,2,2,2,1,1] => [7,5] => 0
[8,5,2,1] => [4,3,2,2,2,1,1,1] => [3,2,2,2,1,1,1] => [7,4,1] => 1
[8,5,1,1,1] => [5,2,2,2,2,1,1,1] => [2,2,2,2,1,1,1] => [7,4] => 0
[8,4,3,1] => [4,3,3,2,1,1,1,1] => [3,3,2,1,1,1,1] => [7,3,2] => 0
[8,4,2,2] => [4,4,2,2,1,1,1,1] => [4,2,2,1,1,1,1] => [7,3,1,1] => 2
[8,4,2,1,1] => [5,3,2,2,1,1,1,1] => [3,2,2,1,1,1,1] => [7,3,1] => 1
[8,4,1,1,1,1] => [6,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => [7,3] => 0
[8,3,3,2] => [4,4,3,1,1,1,1,1] => [4,3,1,1,1,1,1] => [7,2,2,1] => 1
[8,3,3,1,1] => [5,3,3,1,1,1,1,1] => [3,3,1,1,1,1,1] => [7,2,2] => 0
[8,3,2,2,1] => [5,4,2,1,1,1,1,1] => [4,2,1,1,1,1,1] => [7,2,1,1] => 2
[8,3,2,1,1,1] => [6,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => [7,2,1] => 1
[8,3,1,1,1,1,1] => [7,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => [7,2] => 0
[8,2,2,2,2] => [5,5,1,1,1,1,1,1] => [5,1,1,1,1,1,1] => [7,1,1,1,1] => 4
[8,2,2,2,1,1] => [6,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => [7,1,1,1] => 3
[8,2,2,1,1,1,1] => [7,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => [7,1,1] => 2
[8,2,1,1,1,1,1,1] => [8,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => [7,1] => 1
[8,1,1,1,1,1,1,1,1] => [9,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [7] => 0
[7,7,1,1] => [4,2,2,2,2,2,2] => [2,2,2,2,2,2] => [6,6] => 0
[7,6,2,1] => [4,3,2,2,2,2,1] => [3,2,2,2,2,1] => [6,5,1] => 1
[7,6,1,1,1] => [5,2,2,2,2,2,1] => [2,2,2,2,2,1] => [6,5] => 0
[7,5,3,1] => [4,3,3,2,2,1,1] => [3,3,2,2,1,1] => [6,4,2] => 0
[7,5,2,2] => [4,4,2,2,2,1,1] => [4,2,2,2,1,1] => [6,4,1,1] => 2
[7,5,2,1,1] => [5,3,2,2,2,1,1] => [3,2,2,2,1,1] => [6,4,1] => 1
[7,5,1,1,1,1] => [6,2,2,2,2,1,1] => [2,2,2,2,1,1] => [6,4] => 0
[7,4,4,1] => [4,3,3,3,1,1,1] => [3,3,3,1,1,1] => [6,3,3] => 0
[7,4,3,2] => [4,4,3,2,1,1,1] => [4,3,2,1,1,1] => [6,3,2,1] => 1
[7,4,3,1,1] => [5,3,3,2,1,1,1] => [3,3,2,1,1,1] => [6,3,2] => 0
[7,4,2,2,1] => [5,4,2,2,1,1,1] => [4,2,2,1,1,1] => [6,3,1,1] => 2
[7,4,2,1,1,1] => [6,3,2,2,1,1,1] => [3,2,2,1,1,1] => [6,3,1] => 1
[7,4,1,1,1,1,1] => [7,2,2,2,1,1,1] => [2,2,2,1,1,1] => [6,3] => 0
[7,3,3,3] => [4,4,4,1,1,1,1] => [4,4,1,1,1,1] => [6,2,2,2] => 0
[7,3,3,2,1] => [5,4,3,1,1,1,1] => [4,3,1,1,1,1] => [6,2,2,1] => 1
[7,3,3,1,1,1] => [6,3,3,1,1,1,1] => [3,3,1,1,1,1] => [6,2,2] => 0
[7,3,2,2,2] => [5,5,2,1,1,1,1] => [5,2,1,1,1,1] => [6,2,1,1,1] => 3
[7,3,2,2,1,1] => [6,4,2,1,1,1,1] => [4,2,1,1,1,1] => [6,2,1,1] => 2
[7,3,2,1,1,1,1] => [7,3,2,1,1,1,1] => [3,2,1,1,1,1] => [6,2,1] => 1
[7,3,1,1,1,1,1,1] => [8,2,2,1,1,1,1] => [2,2,1,1,1,1] => [6,2] => 0
[7,2,2,2,2,1] => [6,5,1,1,1,1,1] => [5,1,1,1,1,1] => [6,1,1,1,1] => 4
[7,2,2,2,1,1,1] => [7,4,1,1,1,1,1] => [4,1,1,1,1,1] => [6,1,1,1] => 3
[7,2,2,1,1,1,1,1] => [8,3,1,1,1,1,1] => [3,1,1,1,1,1] => [6,1,1] => 2
[7,2,1,1,1,1,1,1,1] => [9,2,1,1,1,1,1] => [2,1,1,1,1,1] => [6,1] => 1
[7,1,1,1,1,1,1,1,1,1] => [10,1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => 0
[6,6,3,1] => [4,3,3,2,2,2] => [3,3,2,2,2] => [5,5,2] => 0
[6,6,2,2] => [4,4,2,2,2,2] => [4,2,2,2,2] => [5,5,1,1] => 2
[6,6,2,1,1] => [5,3,2,2,2,2] => [3,2,2,2,2] => [5,5,1] => 1
[6,6,1,1,1,1] => [6,2,2,2,2,2] => [2,2,2,2,2] => [5,5] => 0
[6,5,4,1] => [4,3,3,3,2,1] => [3,3,3,2,1] => [5,4,3] => 0
[6,5,3,2] => [4,4,3,2,2,1] => [4,3,2,2,1] => [5,4,2,1] => 1
[6,5,3,1,1] => [5,3,3,2,2,1] => [3,3,2,2,1] => [5,4,2] => 0
[6,5,2,2,1] => [5,4,2,2,2,1] => [4,2,2,2,1] => [5,4,1,1] => 2
[6,5,2,1,1,1] => [6,3,2,2,2,1] => [3,2,2,2,1] => [5,4,1] => 1
[6,5,1,1,1,1,1] => [7,2,2,2,2,1] => [2,2,2,2,1] => [5,4] => 0
[6,4,4,2] => [4,4,3,3,1,1] => [4,3,3,1,1] => [5,3,3,1] => 1
[6,4,4,1,1] => [5,3,3,3,1,1] => [3,3,3,1,1] => [5,3,3] => 0
[6,4,3,3] => [4,4,4,2,1,1] => [4,4,2,1,1] => [5,3,2,2] => 0
[6,4,3,2,1] => [5,4,3,2,1,1] => [4,3,2,1,1] => [5,3,2,1] => 1
[6,4,3,1,1,1] => [6,3,3,2,1,1] => [3,3,2,1,1] => [5,3,2] => 0
[6,4,2,2,2] => [5,5,2,2,1,1] => [5,2,2,1,1] => [5,3,1,1,1] => 3
[6,4,2,2,1,1] => [6,4,2,2,1,1] => [4,2,2,1,1] => [5,3,1,1] => 2
[6,4,2,1,1,1,1] => [7,3,2,2,1,1] => [3,2,2,1,1] => [5,3,1] => 1
[6,4,1,1,1,1,1,1] => [8,2,2,2,1,1] => [2,2,2,1,1] => [5,3] => 0
[6,3,3,3,1] => [5,4,4,1,1,1] => [4,4,1,1,1] => [5,2,2,2] => 0
[6,3,3,2,2] => [5,5,3,1,1,1] => [5,3,1,1,1] => [5,2,2,1,1] => 2
[6,3,3,2,1,1] => [6,4,3,1,1,1] => [4,3,1,1,1] => [5,2,2,1] => 1
[6,3,3,1,1,1,1] => [7,3,3,1,1,1] => [3,3,1,1,1] => [5,2,2] => 0
[6,3,2,2,2,1] => [6,5,2,1,1,1] => [5,2,1,1,1] => [5,2,1,1,1] => 3
[6,3,2,2,1,1,1] => [7,4,2,1,1,1] => [4,2,1,1,1] => [5,2,1,1] => 2
[6,3,2,1,1,1,1,1] => [8,3,2,1,1,1] => [3,2,1,1,1] => [5,2,1] => 1
[6,3,1,1,1,1,1,1,1] => [9,2,2,1,1,1] => [2,2,1,1,1] => [5,2] => 0
[6,2,2,2,2,2] => [6,6,1,1,1,1] => [6,1,1,1,1] => [5,1,1,1,1,1] => 5
[6,2,2,2,2,1,1] => [7,5,1,1,1,1] => [5,1,1,1,1] => [5,1,1,1,1] => 4
[6,2,2,2,1,1,1,1] => [8,4,1,1,1,1] => [4,1,1,1,1] => [5,1,1,1] => 3
[6,2,2,1,1,1,1,1,1] => [9,3,1,1,1,1] => [3,1,1,1,1] => [5,1,1] => 2
[6,2,1,1,1,1,1,1,1,1] => [10,2,1,1,1,1] => [2,1,1,1,1] => [5,1] => 1
[6,1,1,1,1,1,1,1,1,1,1] => [11,1,1,1,1,1] => [1,1,1,1,1] => [5] => 0
[5,5,5,1] => [4,3,3,3,3] => [3,3,3,3] => [4,4,4] => 0
[5,5,4,2] => [4,4,3,3,2] => [4,3,3,2] => [4,4,3,1] => 1
[5,5,4,1,1] => [5,3,3,3,2] => [3,3,3,2] => [4,4,3] => 0
[5,5,3,3] => [4,4,4,2,2] => [4,4,2,2] => [4,4,2,2] => 0
[5,5,3,2,1] => [5,4,3,2,2] => [4,3,2,2] => [4,4,2,1] => 1
[5,5,3,1,1,1] => [6,3,3,2,2] => [3,3,2,2] => [4,4,2] => 0
[5,5,2,2,2] => [5,5,2,2,2] => [5,2,2,2] => [4,4,1,1,1] => 3
[5,5,2,2,1,1] => [6,4,2,2,2] => [4,2,2,2] => [4,4,1,1] => 2
[5,5,2,1,1,1,1] => [7,3,2,2,2] => [3,2,2,2] => [4,4,1] => 1
[5,5,1,1,1,1,1,1] => [8,2,2,2,2] => [2,2,2,2] => [4,4] => 0
[5,4,4,3] => [4,4,4,3,1] => [4,4,3,1] => [4,3,3,2] => 0
[5,4,4,2,1] => [5,4,3,3,1] => [4,3,3,1] => [4,3,3,1] => 1
[5,4,4,1,1,1] => [6,3,3,3,1] => [3,3,3,1] => [4,3,3] => 0
[5,4,3,3,1] => [5,4,4,2,1] => [4,4,2,1] => [4,3,2,2] => 0
[5,4,3,2,2] => [5,5,3,2,1] => [5,3,2,1] => [4,3,2,1,1] => 2
[5,4,3,2,1,1] => [6,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[5,4,3,1,1,1,1] => [7,3,3,2,1] => [3,3,2,1] => [4,3,2] => 0
[5,4,2,2,2,1] => [6,5,2,2,1] => [5,2,2,1] => [4,3,1,1,1] => 3
[5,4,2,2,1,1,1] => [7,4,2,2,1] => [4,2,2,1] => [4,3,1,1] => 2
[5,4,2,1,1,1,1,1] => [8,3,2,2,1] => [3,2,2,1] => [4,3,1] => 1
[5,4,1,1,1,1,1,1,1] => [9,2,2,2,1] => [2,2,2,1] => [4,3] => 0
[5,3,3,3,2] => [5,5,4,1,1] => [5,4,1,1] => [4,2,2,2,1] => 1
[5,3,3,3,1,1] => [6,4,4,1,1] => [4,4,1,1] => [4,2,2,2] => 0
[5,3,3,2,2,1] => [6,5,3,1,1] => [5,3,1,1] => [4,2,2,1,1] => 2
[5,3,3,2,1,1,1] => [7,4,3,1,1] => [4,3,1,1] => [4,2,2,1] => 1
[5,3,3,1,1,1,1,1] => [8,3,3,1,1] => [3,3,1,1] => [4,2,2] => 0
[5,3,2,2,2,2] => [6,6,2,1,1] => [6,2,1,1] => [4,2,1,1,1,1] => 4
[5,3,2,2,2,1,1] => [7,5,2,1,1] => [5,2,1,1] => [4,2,1,1,1] => 3
[5,3,2,2,1,1,1,1] => [8,4,2,1,1] => [4,2,1,1] => [4,2,1,1] => 2
[5,3,2,1,1,1,1,1,1] => [9,3,2,1,1] => [3,2,1,1] => [4,2,1] => 1
[5,3,1,1,1,1,1,1,1,1] => [10,2,2,1,1] => [2,2,1,1] => [4,2] => 0
[5,2,2,2,2,2,1] => [7,6,1,1,1] => [6,1,1,1] => [4,1,1,1,1,1] => 5
[5,2,2,2,2,1,1,1] => [8,5,1,1,1] => [5,1,1,1] => [4,1,1,1,1] => 4
[5,2,2,2,1,1,1,1,1] => [9,4,1,1,1] => [4,1,1,1] => [4,1,1,1] => 3
[5,2,2,1,1,1,1,1,1,1] => [10,3,1,1,1] => [3,1,1,1] => [4,1,1] => 2
[5,2,1,1,1,1,1,1,1,1,1] => [11,2,1,1,1] => [2,1,1,1] => [4,1] => 1
[5,1,1,1,1,1,1,1,1,1,1,1] => [12,1,1,1,1] => [1,1,1,1] => [4] => 0
[4,4,4,4] => [4,4,4,4] => [4,4,4] => [3,3,3,3] => 0
[4,4,4,3,1] => [5,4,4,3] => [4,4,3] => [3,3,3,2] => 0
[4,4,4,2,2] => [5,5,3,3] => [5,3,3] => [3,3,3,1,1] => 2
[4,4,4,2,1,1] => [6,4,3,3] => [4,3,3] => [3,3,3,1] => 1
[4,4,4,1,1,1,1] => [7,3,3,3] => [3,3,3] => [3,3,3] => 0
[4,4,3,3,2] => [5,5,4,2] => [5,4,2] => [3,3,2,2,1] => 1
[4,4,3,3,1,1] => [6,4,4,2] => [4,4,2] => [3,3,2,2] => 0
[4,4,3,2,2,1] => [6,5,3,2] => [5,3,2] => [3,3,2,1,1] => 2
[4,4,3,2,1,1,1] => [7,4,3,2] => [4,3,2] => [3,3,2,1] => 1
[4,4,3,1,1,1,1,1] => [8,3,3,2] => [3,3,2] => [3,3,2] => 0
[4,4,2,2,2,2] => [6,6,2,2] => [6,2,2] => [3,3,1,1,1,1] => 4
[4,4,2,2,2,1,1] => [7,5,2,2] => [5,2,2] => [3,3,1,1,1] => 3
[4,4,2,2,1,1,1,1] => [8,4,2,2] => [4,2,2] => [3,3,1,1] => 2
[4,4,2,1,1,1,1,1,1] => [9,3,2,2] => [3,2,2] => [3,3,1] => 1
[4,4,1,1,1,1,1,1,1,1] => [10,2,2,2] => [2,2,2] => [3,3] => 0
[4,3,3,3,3] => [5,5,5,1] => [5,5,1] => [3,2,2,2,2] => 0
[4,3,3,3,2,1] => [6,5,4,1] => [5,4,1] => [3,2,2,2,1] => 1
[4,3,3,3,1,1,1] => [7,4,4,1] => [4,4,1] => [3,2,2,2] => 0
[4,3,3,2,2,2] => [6,6,3,1] => [6,3,1] => [3,2,2,1,1,1] => 3
[4,3,3,2,2,1,1] => [7,5,3,1] => [5,3,1] => [3,2,2,1,1] => 2
[4,3,3,2,1,1,1,1] => [8,4,3,1] => [4,3,1] => [3,2,2,1] => 1
[4,3,3,1,1,1,1,1,1] => [9,3,3,1] => [3,3,1] => [3,2,2] => 0
[4,3,2,2,2,2,1] => [7,6,2,1] => [6,2,1] => [3,2,1,1,1,1] => 4
[4,3,2,2,2,1,1,1] => [8,5,2,1] => [5,2,1] => [3,2,1,1,1] => 3
[4,3,2,2,1,1,1,1,1] => [9,4,2,1] => [4,2,1] => [3,2,1,1] => 2
[4,3,2,1,1,1,1,1,1,1] => [10,3,2,1] => [3,2,1] => [3,2,1] => 1
[4,3,1,1,1,1,1,1,1,1,1] => [11,2,2,1] => [2,2,1] => [3,2] => 0
[4,2,2,2,2,2,2] => [7,7,1,1] => [7,1,1] => [3,1,1,1,1,1,1] => 6
[4,2,2,2,2,2,1,1] => [8,6,1,1] => [6,1,1] => [3,1,1,1,1,1] => 5
[4,2,2,2,2,1,1,1,1] => [9,5,1,1] => [5,1,1] => [3,1,1,1,1] => 4
[4,2,2,2,1,1,1,1,1,1] => [10,4,1,1] => [4,1,1] => [3,1,1,1] => 3
[4,2,2,1,1,1,1,1,1,1,1] => [11,3,1,1] => [3,1,1] => [3,1,1] => 2
[4,2,1,1,1,1,1,1,1,1,1,1] => [12,2,1,1] => [2,1,1] => [3,1] => 1
[4,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1,1] => [1,1,1] => [3] => 0
[3,3,3,3,3,1] => [6,5,5] => [5,5] => [2,2,2,2,2] => 0
[3,3,3,3,2,2] => [6,6,4] => [6,4] => [2,2,2,2,1,1] => 2
[3,3,3,3,2,1,1] => [7,5,4] => [5,4] => [2,2,2,2,1] => 1
[3,3,3,3,1,1,1,1] => [8,4,4] => [4,4] => [2,2,2,2] => 0
[3,3,3,2,2,2,1] => [7,6,3] => [6,3] => [2,2,2,1,1,1] => 3
[3,3,3,2,2,1,1,1] => [8,5,3] => [5,3] => [2,2,2,1,1] => 2
[3,3,3,2,1,1,1,1,1] => [9,4,3] => [4,3] => [2,2,2,1] => 1
[3,3,3,1,1,1,1,1,1,1] => [10,3,3] => [3,3] => [2,2,2] => 0
[3,3,2,2,2,2,2] => [7,7,2] => [7,2] => [2,2,1,1,1,1,1] => 5
[3,3,2,2,2,2,1,1] => [8,6,2] => [6,2] => [2,2,1,1,1,1] => 4
[3,3,2,2,2,1,1,1,1] => [9,5,2] => [5,2] => [2,2,1,1,1] => 3
[3,3,2,2,1,1,1,1,1,1] => [10,4,2] => [4,2] => [2,2,1,1] => 2
[3,3,2,1,1,1,1,1,1,1,1] => [11,3,2] => [3,2] => [2,2,1] => 1
[3,3,1,1,1,1,1,1,1,1,1,1] => [12,2,2] => [2,2] => [2,2] => 0
[3,2,2,2,2,2,2,1] => [8,7,1] => [7,1] => [2,1,1,1,1,1,1] => 6
[3,2,2,2,2,2,1,1,1] => [9,6,1] => [6,1] => [2,1,1,1,1,1] => 5
[3,2,2,2,2,1,1,1,1,1] => [10,5,1] => [5,1] => [2,1,1,1,1] => 4
[3,2,2,2,1,1,1,1,1,1,1] => [11,4,1] => [4,1] => [2,1,1,1] => 3
[3,2,2,1,1,1,1,1,1,1,1,1] => [12,3,1] => [3,1] => [2,1,1] => 2
[3,2,1,1,1,1,1,1,1,1,1,1,1] => [13,2,1] => [2,1] => [2,1] => 1
[3,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1,1] => [1,1] => [2] => 0
[2,2,2,2,2,2,2,2] => [8,8] => [8] => [1,1,1,1,1,1,1,1] => 8
[2,2,2,2,2,2,2,1,1] => [9,7] => [7] => [1,1,1,1,1,1,1] => 7
[2,2,2,2,2,2,1,1,1,1] => [10,6] => [6] => [1,1,1,1,1,1] => 6
[2,2,2,2,2,1,1,1,1,1,1] => [11,5] => [5] => [1,1,1,1,1] => 5
[2,2,2,2,1,1,1,1,1,1,1,1] => [12,4] => [4] => [1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1,1,1,1,1] => [13,3] => [3] => [1,1,1] => 3
[2,2,1,1,1,1,1,1,1,1,1,1,1,1] => [14,2] => [2] => [1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,1] => [1] => [1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [16] => [] => [] => 0
[13,1,1,1,1] => [5,1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => [12] => 0
[12,2,1,1,1] => [5,2,1,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1] => [11,1] => 1
[12,1,1,1,1,1] => [6,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => [11] => 0
[11,3,1,1,1] => [5,2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => [10,2] => 0
[11,2,2,1,1] => [5,3,1,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1,1] => [10,1,1] => 2
[11,2,1,1,1,1] => [6,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => [10,1] => 1
[11,1,1,1,1,1,1] => [7,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [10] => 0
[10,4,1,1,1] => [5,2,2,2,1,1,1,1,1,1] => [2,2,2,1,1,1,1,1,1] => [9,3] => 0
[10,3,2,1,1] => [5,3,2,1,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1,1] => [9,2,1] => 1
[10,3,1,1,1,1] => [6,2,2,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => [9,2] => 0
[10,2,2,2,1] => [5,4,1,1,1,1,1,1,1,1] => [4,1,1,1,1,1,1,1,1] => [9,1,1,1] => 3
[10,2,2,1,1,1] => [6,3,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1] => [9,1,1] => 2
[10,2,1,1,1,1,1] => [7,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => [9,1] => 1
[10,1,1,1,1,1,1,1] => [8,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [9] => 0
[9,5,1,1,1] => [5,2,2,2,2,1,1,1,1] => [2,2,2,2,1,1,1,1] => [8,4] => 0
[9,4,2,1,1] => [5,3,2,2,1,1,1,1,1] => [3,2,2,1,1,1,1,1] => [8,3,1] => 1
[9,4,1,1,1,1] => [6,2,2,2,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => [8,3] => 0
[9,3,3,1,1] => [5,3,3,1,1,1,1,1,1] => [3,3,1,1,1,1,1,1] => [8,2,2] => 0
[9,3,2,2,1] => [5,4,2,1,1,1,1,1,1] => [4,2,1,1,1,1,1,1] => [8,2,1,1] => 2
[9,3,2,1,1,1] => [6,3,2,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1] => [8,2,1] => 1
[9,3,1,1,1,1,1] => [7,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => [8,2] => 0
[9,2,2,2,2] => [5,5,1,1,1,1,1,1,1] => [5,1,1,1,1,1,1,1] => [8,1,1,1,1] => 4
[9,2,2,2,1,1] => [6,4,1,1,1,1,1,1,1] => [4,1,1,1,1,1,1,1] => [8,1,1,1] => 3
[9,2,2,1,1,1,1] => [7,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => [8,1,1] => 2
[9,2,1,1,1,1,1,1] => [8,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => [8,1] => 1
[9,1,1,1,1,1,1,1,1] => [9,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [8] => 0
[8,6,1,1,1] => [5,2,2,2,2,2,1,1] => [2,2,2,2,2,1,1] => [7,5] => 0
[8,5,2,1,1] => [5,3,2,2,2,1,1,1] => [3,2,2,2,1,1,1] => [7,4,1] => 1
[8,5,1,1,1,1] => [6,2,2,2,2,1,1,1] => [2,2,2,2,1,1,1] => [7,4] => 0
[8,4,3,1,1] => [5,3,3,2,1,1,1,1] => [3,3,2,1,1,1,1] => [7,3,2] => 0
[8,4,2,2,1] => [5,4,2,2,1,1,1,1] => [4,2,2,1,1,1,1] => [7,3,1,1] => 2
[8,4,2,1,1,1] => [6,3,2,2,1,1,1,1] => [3,2,2,1,1,1,1] => [7,3,1] => 1
[8,4,1,1,1,1,1] => [7,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => [7,3] => 0
[8,3,3,2,1] => [5,4,3,1,1,1,1,1] => [4,3,1,1,1,1,1] => [7,2,2,1] => 1
[8,3,3,1,1,1] => [6,3,3,1,1,1,1,1] => [3,3,1,1,1,1,1] => [7,2,2] => 0
[8,3,2,2,2] => [5,5,2,1,1,1,1,1] => [5,2,1,1,1,1,1] => [7,2,1,1,1] => 3
[8,3,2,2,1,1] => [6,4,2,1,1,1,1,1] => [4,2,1,1,1,1,1] => [7,2,1,1] => 2
[8,3,2,1,1,1,1] => [7,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => [7,2,1] => 1
[8,3,1,1,1,1,1,1] => [8,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => [7,2] => 0
[8,2,2,2,2,1] => [6,5,1,1,1,1,1,1] => [5,1,1,1,1,1,1] => [7,1,1,1,1] => 4
[8,2,2,2,1,1,1] => [7,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => [7,1,1,1] => 3
[8,2,2,1,1,1,1,1] => [8,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => [7,1,1] => 2
[8,2,1,1,1,1,1,1,1] => [9,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => [7,1] => 1
[8,1,1,1,1,1,1,1,1,1] => [10,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [7] => 0
[7,7,1,1,1] => [5,2,2,2,2,2,2] => [2,2,2,2,2,2] => [6,6] => 0
[7,6,2,1,1] => [5,3,2,2,2,2,1] => [3,2,2,2,2,1] => [6,5,1] => 1
[7,6,1,1,1,1] => [6,2,2,2,2,2,1] => [2,2,2,2,2,1] => [6,5] => 0
[7,5,3,1,1] => [5,3,3,2,2,1,1] => [3,3,2,2,1,1] => [6,4,2] => 0
[7,5,2,2,1] => [5,4,2,2,2,1,1] => [4,2,2,2,1,1] => [6,4,1,1] => 2
[7,5,2,1,1,1] => [6,3,2,2,2,1,1] => [3,2,2,2,1,1] => [6,4,1] => 1
[7,5,1,1,1,1,1] => [7,2,2,2,2,1,1] => [2,2,2,2,1,1] => [6,4] => 0
[7,4,4,1,1] => [5,3,3,3,1,1,1] => [3,3,3,1,1,1] => [6,3,3] => 0
[7,4,3,2,1] => [5,4,3,2,1,1,1] => [4,3,2,1,1,1] => [6,3,2,1] => 1
[7,4,3,1,1,1] => [6,3,3,2,1,1,1] => [3,3,2,1,1,1] => [6,3,2] => 0
[7,4,2,2,2] => [5,5,2,2,1,1,1] => [5,2,2,1,1,1] => [6,3,1,1,1] => 3
[7,4,2,2,1,1] => [6,4,2,2,1,1,1] => [4,2,2,1,1,1] => [6,3,1,1] => 2
[7,4,2,1,1,1,1] => [7,3,2,2,1,1,1] => [3,2,2,1,1,1] => [6,3,1] => 1
[7,4,1,1,1,1,1,1] => [8,2,2,2,1,1,1] => [2,2,2,1,1,1] => [6,3] => 0
[7,3,3,3,1] => [5,4,4,1,1,1,1] => [4,4,1,1,1,1] => [6,2,2,2] => 0
[7,3,3,2,2] => [5,5,3,1,1,1,1] => [5,3,1,1,1,1] => [6,2,2,1,1] => 2
[7,3,3,2,1,1] => [6,4,3,1,1,1,1] => [4,3,1,1,1,1] => [6,2,2,1] => 1
[7,3,3,1,1,1,1] => [7,3,3,1,1,1,1] => [3,3,1,1,1,1] => [6,2,2] => 0
[7,3,2,2,2,1] => [6,5,2,1,1,1,1] => [5,2,1,1,1,1] => [6,2,1,1,1] => 3
[7,3,2,2,1,1,1] => [7,4,2,1,1,1,1] => [4,2,1,1,1,1] => [6,2,1,1] => 2
[7,3,2,1,1,1,1,1] => [8,3,2,1,1,1,1] => [3,2,1,1,1,1] => [6,2,1] => 1
[7,3,1,1,1,1,1,1,1] => [9,2,2,1,1,1,1] => [2,2,1,1,1,1] => [6,2] => 0
[7,2,2,2,2,2] => [6,6,1,1,1,1,1] => [6,1,1,1,1,1] => [6,1,1,1,1,1] => 5
[7,2,2,2,2,1,1] => [7,5,1,1,1,1,1] => [5,1,1,1,1,1] => [6,1,1,1,1] => 4
[7,2,2,2,1,1,1,1] => [8,4,1,1,1,1,1] => [4,1,1,1,1,1] => [6,1,1,1] => 3
[7,2,2,1,1,1,1,1,1] => [9,3,1,1,1,1,1] => [3,1,1,1,1,1] => [6,1,1] => 2
[7,2,1,1,1,1,1,1,1,1] => [10,2,1,1,1,1,1] => [2,1,1,1,1,1] => [6,1] => 1
[7,1,1,1,1,1,1,1,1,1,1] => [11,1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => 0
[6,6,3,1,1] => [5,3,3,2,2,2] => [3,3,2,2,2] => [5,5,2] => 0
[6,6,2,2,1] => [5,4,2,2,2,2] => [4,2,2,2,2] => [5,5,1,1] => 2
[6,6,2,1,1,1] => [6,3,2,2,2,2] => [3,2,2,2,2] => [5,5,1] => 1
[6,6,1,1,1,1,1] => [7,2,2,2,2,2] => [2,2,2,2,2] => [5,5] => 0
[6,5,4,1,1] => [5,3,3,3,2,1] => [3,3,3,2,1] => [5,4,3] => 0
[6,5,3,2,1] => [5,4,3,2,2,1] => [4,3,2,2,1] => [5,4,2,1] => 1
[6,5,3,1,1,1] => [6,3,3,2,2,1] => [3,3,2,2,1] => [5,4,2] => 0
[6,5,2,2,2] => [5,5,2,2,2,1] => [5,2,2,2,1] => [5,4,1,1,1] => 3
[6,5,2,2,1,1] => [6,4,2,2,2,1] => [4,2,2,2,1] => [5,4,1,1] => 2
[6,5,2,1,1,1,1] => [7,3,2,2,2,1] => [3,2,2,2,1] => [5,4,1] => 1
[6,5,1,1,1,1,1,1] => [8,2,2,2,2,1] => [2,2,2,2,1] => [5,4] => 0
[6,4,4,2,1] => [5,4,3,3,1,1] => [4,3,3,1,1] => [5,3,3,1] => 1
[6,4,4,1,1,1] => [6,3,3,3,1,1] => [3,3,3,1,1] => [5,3,3] => 0
[6,4,3,3,1] => [5,4,4,2,1,1] => [4,4,2,1,1] => [5,3,2,2] => 0
[6,4,3,2,2] => [5,5,3,2,1,1] => [5,3,2,1,1] => [5,3,2,1,1] => 2
[6,4,3,2,1,1] => [6,4,3,2,1,1] => [4,3,2,1,1] => [5,3,2,1] => 1
[6,4,3,1,1,1,1] => [7,3,3,2,1,1] => [3,3,2,1,1] => [5,3,2] => 0
[6,4,2,2,2,1] => [6,5,2,2,1,1] => [5,2,2,1,1] => [5,3,1,1,1] => 3
[6,4,2,2,1,1,1] => [7,4,2,2,1,1] => [4,2,2,1,1] => [5,3,1,1] => 2
[6,4,2,1,1,1,1,1] => [8,3,2,2,1,1] => [3,2,2,1,1] => [5,3,1] => 1
[6,4,1,1,1,1,1,1,1] => [9,2,2,2,1,1] => [2,2,2,1,1] => [5,3] => 0
[6,3,3,3,2] => [5,5,4,1,1,1] => [5,4,1,1,1] => [5,2,2,2,1] => 1
[6,3,3,3,1,1] => [6,4,4,1,1,1] => [4,4,1,1,1] => [5,2,2,2] => 0
[6,3,3,2,2,1] => [6,5,3,1,1,1] => [5,3,1,1,1] => [5,2,2,1,1] => 2
[6,3,3,2,1,1,1] => [7,4,3,1,1,1] => [4,3,1,1,1] => [5,2,2,1] => 1
[6,3,3,1,1,1,1,1] => [8,3,3,1,1,1] => [3,3,1,1,1] => [5,2,2] => 0
[6,3,2,2,2,2] => [6,6,2,1,1,1] => [6,2,1,1,1] => [5,2,1,1,1,1] => 4
[6,3,2,2,2,1,1] => [7,5,2,1,1,1] => [5,2,1,1,1] => [5,2,1,1,1] => 3
[6,3,2,2,1,1,1,1] => [8,4,2,1,1,1] => [4,2,1,1,1] => [5,2,1,1] => 2
[6,3,2,1,1,1,1,1,1] => [9,3,2,1,1,1] => [3,2,1,1,1] => [5,2,1] => 1
[6,3,1,1,1,1,1,1,1,1] => [10,2,2,1,1,1] => [2,2,1,1,1] => [5,2] => 0
[6,2,2,2,2,2,1] => [7,6,1,1,1,1] => [6,1,1,1,1] => [5,1,1,1,1,1] => 5
[6,2,2,2,2,1,1,1] => [8,5,1,1,1,1] => [5,1,1,1,1] => [5,1,1,1,1] => 4
[6,2,2,2,1,1,1,1,1] => [9,4,1,1,1,1] => [4,1,1,1,1] => [5,1,1,1] => 3
[6,2,2,1,1,1,1,1,1,1] => [10,3,1,1,1,1] => [3,1,1,1,1] => [5,1,1] => 2
[6,2,1,1,1,1,1,1,1,1,1] => [11,2,1,1,1,1] => [2,1,1,1,1] => [5,1] => 1
[6,1,1,1,1,1,1,1,1,1,1,1] => [12,1,1,1,1,1] => [1,1,1,1,1] => [5] => 0
[5,5,5,1,1] => [5,3,3,3,3] => [3,3,3,3] => [4,4,4] => 0
[5,5,4,2,1] => [5,4,3,3,2] => [4,3,3,2] => [4,4,3,1] => 1
[5,5,4,1,1,1] => [6,3,3,3,2] => [3,3,3,2] => [4,4,3] => 0
[5,5,3,3,1] => [5,4,4,2,2] => [4,4,2,2] => [4,4,2,2] => 0
[5,5,3,2,2] => [5,5,3,2,2] => [5,3,2,2] => [4,4,2,1,1] => 2
[5,5,3,2,1,1] => [6,4,3,2,2] => [4,3,2,2] => [4,4,2,1] => 1
[5,5,3,1,1,1,1] => [7,3,3,2,2] => [3,3,2,2] => [4,4,2] => 0
[5,5,2,2,2,1] => [6,5,2,2,2] => [5,2,2,2] => [4,4,1,1,1] => 3
[5,5,2,2,1,1,1] => [7,4,2,2,2] => [4,2,2,2] => [4,4,1,1] => 2
[5,5,2,1,1,1,1,1] => [8,3,2,2,2] => [3,2,2,2] => [4,4,1] => 1
[5,5,1,1,1,1,1,1,1] => [9,2,2,2,2] => [2,2,2,2] => [4,4] => 0
[5,4,4,3,1] => [5,4,4,3,1] => [4,4,3,1] => [4,3,3,2] => 0
[5,4,4,2,2] => [5,5,3,3,1] => [5,3,3,1] => [4,3,3,1,1] => 2
[5,4,4,2,1,1] => [6,4,3,3,1] => [4,3,3,1] => [4,3,3,1] => 1
[5,4,4,1,1,1,1] => [7,3,3,3,1] => [3,3,3,1] => [4,3,3] => 0
[5,4,3,3,2] => [5,5,4,2,1] => [5,4,2,1] => [4,3,2,2,1] => 1
[5,4,3,3,1,1] => [6,4,4,2,1] => [4,4,2,1] => [4,3,2,2] => 0
[5,4,3,2,2,1] => [6,5,3,2,1] => [5,3,2,1] => [4,3,2,1,1] => 2
[5,4,3,2,1,1,1] => [7,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[5,4,3,1,1,1,1,1] => [8,3,3,2,1] => [3,3,2,1] => [4,3,2] => 0
[5,4,2,2,2,2] => [6,6,2,2,1] => [6,2,2,1] => [4,3,1,1,1,1] => 4
[5,4,2,2,2,1,1] => [7,5,2,2,1] => [5,2,2,1] => [4,3,1,1,1] => 3
[5,4,2,2,1,1,1,1] => [8,4,2,2,1] => [4,2,2,1] => [4,3,1,1] => 2
[5,4,2,1,1,1,1,1,1] => [9,3,2,2,1] => [3,2,2,1] => [4,3,1] => 1
[5,4,1,1,1,1,1,1,1,1] => [10,2,2,2,1] => [2,2,2,1] => [4,3] => 0
[5,3,3,3,3] => [5,5,5,1,1] => [5,5,1,1] => [4,2,2,2,2] => 0
[5,3,3,3,2,1] => [6,5,4,1,1] => [5,4,1,1] => [4,2,2,2,1] => 1
[5,3,3,3,1,1,1] => [7,4,4,1,1] => [4,4,1,1] => [4,2,2,2] => 0
[5,3,3,2,2,2] => [6,6,3,1,1] => [6,3,1,1] => [4,2,2,1,1,1] => 3
[5,3,3,2,2,1,1] => [7,5,3,1,1] => [5,3,1,1] => [4,2,2,1,1] => 2
[5,3,3,2,1,1,1,1] => [8,4,3,1,1] => [4,3,1,1] => [4,2,2,1] => 1
[5,3,3,1,1,1,1,1,1] => [9,3,3,1,1] => [3,3,1,1] => [4,2,2] => 0
[5,3,2,2,2,2,1] => [7,6,2,1,1] => [6,2,1,1] => [4,2,1,1,1,1] => 4
[5,3,2,2,2,1,1,1] => [8,5,2,1,1] => [5,2,1,1] => [4,2,1,1,1] => 3
[5,3,2,2,1,1,1,1,1] => [9,4,2,1,1] => [4,2,1,1] => [4,2,1,1] => 2
[5,3,2,1,1,1,1,1,1,1] => [10,3,2,1,1] => [3,2,1,1] => [4,2,1] => 1
[5,3,1,1,1,1,1,1,1,1,1] => [11,2,2,1,1] => [2,2,1,1] => [4,2] => 0
[5,2,2,2,2,2,2] => [7,7,1,1,1] => [7,1,1,1] => [4,1,1,1,1,1,1] => 6
[5,2,2,2,2,2,1,1] => [8,6,1,1,1] => [6,1,1,1] => [4,1,1,1,1,1] => 5
[5,2,2,2,2,1,1,1,1] => [9,5,1,1,1] => [5,1,1,1] => [4,1,1,1,1] => 4
[5,2,2,2,1,1,1,1,1,1] => [10,4,1,1,1] => [4,1,1,1] => [4,1,1,1] => 3
[5,2,2,1,1,1,1,1,1,1,1] => [11,3,1,1,1] => [3,1,1,1] => [4,1,1] => 2
[5,2,1,1,1,1,1,1,1,1,1,1] => [12,2,1,1,1] => [2,1,1,1] => [4,1] => 1
[5,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1,1,1] => [1,1,1,1] => [4] => 0
[4,4,4,4,1] => [5,4,4,4] => [4,4,4] => [3,3,3,3] => 0
[4,4,4,3,2] => [5,5,4,3] => [5,4,3] => [3,3,3,2,1] => 1
[4,4,4,3,1,1] => [6,4,4,3] => [4,4,3] => [3,3,3,2] => 0
[4,4,4,2,2,1] => [6,5,3,3] => [5,3,3] => [3,3,3,1,1] => 2
[4,4,4,2,1,1,1] => [7,4,3,3] => [4,3,3] => [3,3,3,1] => 1
[4,4,4,1,1,1,1,1] => [8,3,3,3] => [3,3,3] => [3,3,3] => 0
[4,4,3,3,3] => [5,5,5,2] => [5,5,2] => [3,3,2,2,2] => 0
[4,4,3,3,2,1] => [6,5,4,2] => [5,4,2] => [3,3,2,2,1] => 1
[4,4,3,3,1,1,1] => [7,4,4,2] => [4,4,2] => [3,3,2,2] => 0
[4,4,3,2,2,2] => [6,6,3,2] => [6,3,2] => [3,3,2,1,1,1] => 3
[4,4,3,2,2,1,1] => [7,5,3,2] => [5,3,2] => [3,3,2,1,1] => 2
[4,4,3,2,1,1,1,1] => [8,4,3,2] => [4,3,2] => [3,3,2,1] => 1
[4,4,3,1,1,1,1,1,1] => [9,3,3,2] => [3,3,2] => [3,3,2] => 0
[4,4,2,2,2,2,1] => [7,6,2,2] => [6,2,2] => [3,3,1,1,1,1] => 4
[4,4,2,2,2,1,1,1] => [8,5,2,2] => [5,2,2] => [3,3,1,1,1] => 3
[4,4,2,2,1,1,1,1,1] => [9,4,2,2] => [4,2,2] => [3,3,1,1] => 2
[4,4,2,1,1,1,1,1,1,1] => [10,3,2,2] => [3,2,2] => [3,3,1] => 1
[4,4,1,1,1,1,1,1,1,1,1] => [11,2,2,2] => [2,2,2] => [3,3] => 0
[4,3,3,3,3,1] => [6,5,5,1] => [5,5,1] => [3,2,2,2,2] => 0
[4,3,3,3,2,2] => [6,6,4,1] => [6,4,1] => [3,2,2,2,1,1] => 2
[4,3,3,3,2,1,1] => [7,5,4,1] => [5,4,1] => [3,2,2,2,1] => 1
[4,3,3,3,1,1,1,1] => [8,4,4,1] => [4,4,1] => [3,2,2,2] => 0
[4,3,3,2,2,2,1] => [7,6,3,1] => [6,3,1] => [3,2,2,1,1,1] => 3
[4,3,3,2,2,1,1,1] => [8,5,3,1] => [5,3,1] => [3,2,2,1,1] => 2
[4,3,3,2,1,1,1,1,1] => [9,4,3,1] => [4,3,1] => [3,2,2,1] => 1
[4,3,3,1,1,1,1,1,1,1] => [10,3,3,1] => [3,3,1] => [3,2,2] => 0
[4,3,2,2,2,2,2] => [7,7,2,1] => [7,2,1] => [3,2,1,1,1,1,1] => 5
[4,3,2,2,2,2,1,1] => [8,6,2,1] => [6,2,1] => [3,2,1,1,1,1] => 4
[4,3,2,2,2,1,1,1,1] => [9,5,2,1] => [5,2,1] => [3,2,1,1,1] => 3
[4,3,2,2,1,1,1,1,1,1] => [10,4,2,1] => [4,2,1] => [3,2,1,1] => 2
[4,3,2,1,1,1,1,1,1,1,1] => [11,3,2,1] => [3,2,1] => [3,2,1] => 1
[4,3,1,1,1,1,1,1,1,1,1,1] => [12,2,2,1] => [2,2,1] => [3,2] => 0
[4,2,2,2,2,2,2,1] => [8,7,1,1] => [7,1,1] => [3,1,1,1,1,1,1] => 6
[4,2,2,2,2,2,1,1,1] => [9,6,1,1] => [6,1,1] => [3,1,1,1,1,1] => 5
[4,2,2,2,2,1,1,1,1,1] => [10,5,1,1] => [5,1,1] => [3,1,1,1,1] => 4
[4,2,2,2,1,1,1,1,1,1,1] => [11,4,1,1] => [4,1,1] => [3,1,1,1] => 3
[4,2,2,1,1,1,1,1,1,1,1,1] => [12,3,1,1] => [3,1,1] => [3,1,1] => 2
[4,2,1,1,1,1,1,1,1,1,1,1,1] => [13,2,1,1] => [2,1,1] => [3,1] => 1
[4,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1,1,1] => [1,1,1] => [3] => 0
[3,3,3,3,3,2] => [6,6,5] => [6,5] => [2,2,2,2,2,1] => 1
[3,3,3,3,3,1,1] => [7,5,5] => [5,5] => [2,2,2,2,2] => 0
[3,3,3,3,2,2,1] => [7,6,4] => [6,4] => [2,2,2,2,1,1] => 2
[3,3,3,3,2,1,1,1] => [8,5,4] => [5,4] => [2,2,2,2,1] => 1
[3,3,3,3,1,1,1,1,1] => [9,4,4] => [4,4] => [2,2,2,2] => 0
[3,3,3,2,2,2,2] => [7,7,3] => [7,3] => [2,2,2,1,1,1,1] => 4
[3,3,3,2,2,2,1,1] => [8,6,3] => [6,3] => [2,2,2,1,1,1] => 3
[3,3,3,2,2,1,1,1,1] => [9,5,3] => [5,3] => [2,2,2,1,1] => 2
[3,3,3,2,1,1,1,1,1,1] => [10,4,3] => [4,3] => [2,2,2,1] => 1
[3,3,3,1,1,1,1,1,1,1,1] => [11,3,3] => [3,3] => [2,2,2] => 0
[3,3,2,2,2,2,2,1] => [8,7,2] => [7,2] => [2,2,1,1,1,1,1] => 5
[3,3,2,2,2,2,1,1,1] => [9,6,2] => [6,2] => [2,2,1,1,1,1] => 4
[3,3,2,2,2,1,1,1,1,1] => [10,5,2] => [5,2] => [2,2,1,1,1] => 3
[3,3,2,2,1,1,1,1,1,1,1] => [11,4,2] => [4,2] => [2,2,1,1] => 2
[3,3,2,1,1,1,1,1,1,1,1,1] => [12,3,2] => [3,2] => [2,2,1] => 1
[3,3,1,1,1,1,1,1,1,1,1,1,1] => [13,2,2] => [2,2] => [2,2] => 0
[3,2,2,2,2,2,2,2] => [8,8,1] => [8,1] => [2,1,1,1,1,1,1,1] => 7
[3,2,2,2,2,2,2,1,1] => [9,7,1] => [7,1] => [2,1,1,1,1,1,1] => 6
[3,2,2,2,2,2,1,1,1,1] => [10,6,1] => [6,1] => [2,1,1,1,1,1] => 5
[3,2,2,2,2,1,1,1,1,1,1] => [11,5,1] => [5,1] => [2,1,1,1,1] => 4
[3,2,2,2,1,1,1,1,1,1,1,1] => [12,4,1] => [4,1] => [2,1,1,1] => 3
[3,2,2,1,1,1,1,1,1,1,1,1,1] => [13,3,1] => [3,1] => [2,1,1] => 2
[3,2,1,1,1,1,1,1,1,1,1,1,1,1] => [14,2,1] => [2,1] => [2,1] => 1
[3,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,1,1] => [1,1] => [2] => 0
[2,2,2,2,2,2,2,2,1] => [9,8] => [8] => [1,1,1,1,1,1,1,1] => 8
[2,2,2,2,2,2,2,1,1,1] => [10,7] => [7] => [1,1,1,1,1,1,1] => 7
[2,2,2,2,2,2,1,1,1,1,1] => [11,6] => [6] => [1,1,1,1,1,1] => 6
[2,2,2,2,2,1,1,1,1,1,1,1] => [12,5] => [5] => [1,1,1,1,1] => 5
[2,2,2,2,1,1,1,1,1,1,1,1,1] => [13,4] => [4] => [1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1,1,1,1,1,1] => [14,3] => [3] => [1,1,1] => 3
[2,2,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,2] => [2] => [1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [16,1] => [1] => [1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [17] => [] => [] => 0
[4,4,4,3,2,1] => [6,5,4,3] => [5,4,3] => [3,3,3,2,1] => 1
[5,4,3,3,2,1] => [6,5,4,2,1] => [5,4,2,1] => [4,3,2,2,1] => 1
[6,3,3,3,2,1] => [6,5,4,1,1,1] => [5,4,1,1,1] => [5,2,2,2,1] => 1
[5,4,4,2,2,1] => [6,5,3,3,1] => [5,3,3,1] => [4,3,3,1,1] => 2
[5,5,3,2,2,1] => [6,5,3,2,2] => [5,3,2,2] => [4,4,2,1,1] => 2
[6,4,3,2,2,1] => [6,5,3,2,1,1] => [5,3,2,1,1] => [5,3,2,1,1] => 2
[6,5,2,2,2,1] => [6,5,2,2,2,1] => [5,2,2,2,1] => [5,4,1,1,1] => 3
[5,4,4,3,1,1] => [6,4,4,3,1] => [4,4,3,1] => [4,3,3,2] => 0
[5,5,3,3,1,1] => [6,4,4,2,2] => [4,4,2,2] => [4,4,2,2] => 0
[6,4,3,3,1,1] => [6,4,4,2,1,1] => [4,4,2,1,1] => [5,3,2,2] => 0
[5,5,4,2,1,1] => [6,4,3,3,2] => [4,3,3,2] => [4,4,3,1] => 1
[6,4,4,2,1,1] => [6,4,3,3,1,1] => [4,3,3,1,1] => [5,3,3,1] => 1
[6,5,3,2,1,1] => [6,4,3,2,2,1] => [4,3,2,2,1] => [5,4,2,1] => 1
[6,5,4,1,1,1] => [6,3,3,3,2,1] => [3,3,3,2,1] => [5,4,3] => 0
[3,3,3,3,3,2,1] => [7,6,5] => [6,5] => [2,2,2,2,2,1] => 1
[7,2,2,2,2,2,1] => [7,6,1,1,1,1,1] => [6,1,1,1,1,1] => [6,1,1,1,1,1] => 5
[7,4,3,2,1,1,1] => [7,4,3,2,1,1,1] => [4,3,2,1,1,1] => [6,3,2,1] => 1
[6,4,3,2,1,1,1] => [7,4,3,2,1,1] => [4,3,2,1,1] => [5,3,2,1] => 1
[5,5,5,1,1,1,1] => [7,3,3,3,3] => [3,3,3,3] => [4,4,4] => 0
[6,5,4,1,1,1,1] => [7,3,3,3,2,1] => [3,3,3,2,1] => [5,4,3] => 0
[7,6,1,1,1,1,1] => [7,2,2,2,2,2,1] => [2,2,2,2,2,1] => [6,5] => 0
[2,2,2,2,2,2,2,2,2] => [9,9] => [9] => [1,1,1,1,1,1,1,1,1] => 9
[3,3,3,3,3,3] => [6,6,6] => [6,6] => [2,2,2,2,2,2] => 0
[3,3,3,3,3,3,1] => [7,6,6] => [6,6] => [2,2,2,2,2,2] => 0
[6,2,2,2,2,2,2] => [7,7,1,1,1,1] => [7,1,1,1,1] => [5,1,1,1,1,1,1] => 6
[7,7,1,1,1,1] => [6,2,2,2,2,2,2] => [2,2,2,2,2,2] => [6,6] => 0
[3,3,3,3,2,2,1,1,1,1] => [10,6,4] => [6,4] => [2,2,2,2,1,1] => 2
[4,4,3,3,2,2,1,1] => [8,6,4,2] => [6,4,2] => [3,3,2,2,1,1] => 2
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Generating function
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Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
1,1 2,1 3,1,1 4,2,1 6,3,1,1 8,4,2,1 11,6,3,1,1 15,8,4,2,1 20,11,6,3,1,1 26,15,8,4,2,1 35,20,11,6,3,1,1 45,26,15,8,4,2,1
$F_{1} = 1$
$F_{2} = 1 + q$
$F_{3} = 2 + q$
$F_{4} = 3 + q + q^{2}$
$F_{5} = 4 + 2\ q + q^{2}$
$F_{6} = 6 + 3\ q + q^{2} + q^{3}$
$F_{7} = 8 + 4\ q + 2\ q^{2} + q^{3}$
$F_{8} = 11 + 6\ q + 3\ q^{2} + q^{3} + q^{4}$
$F_{9} = 15 + 8\ q + 4\ q^{2} + 2\ q^{3} + q^{4}$
$F_{10} = 20 + 11\ q + 6\ q^{2} + 3\ q^{3} + q^{4} + q^{5}$
$F_{11} = 26 + 15\ q + 8\ q^{2} + 4\ q^{3} + 2\ q^{4} + q^{5}$
$F_{12} = 35 + 20\ q + 11\ q^{2} + 6\ q^{3} + 3\ q^{4} + q^{5} + q^{6}$
$F_{13} = 45 + 26\ q + 15\ q^{2} + 8\ q^{3} + 4\ q^{4} + 2\ q^{5} + q^{6}$
Description
The number of parts equal to 1 in a partition.
Map
first row removal
Description
Removes the first entry of an integer partition
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
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