Identifier
-
Mp00202:
Integer partitions
—first row removal⟶
Integer partitions
St000992: Integer partitions ⟶ ℤ
Values
[1] => [] => 0
[2] => [] => 0
[1,1] => [1] => 1
[3] => [] => 0
[2,1] => [1] => 1
[1,1,1] => [1,1] => 0
[4] => [] => 0
[3,1] => [1] => 1
[2,2] => [2] => 2
[2,1,1] => [1,1] => 0
[1,1,1,1] => [1,1,1] => 1
[5] => [] => 0
[4,1] => [1] => 1
[3,2] => [2] => 2
[3,1,1] => [1,1] => 0
[2,2,1] => [2,1] => 1
[2,1,1,1] => [1,1,1] => 1
[1,1,1,1,1] => [1,1,1,1] => 0
[6] => [] => 0
[5,1] => [1] => 1
[4,2] => [2] => 2
[4,1,1] => [1,1] => 0
[3,3] => [3] => 3
[3,2,1] => [2,1] => 1
[3,1,1,1] => [1,1,1] => 1
[2,2,2] => [2,2] => 0
[2,2,1,1] => [2,1,1] => 2
[2,1,1,1,1] => [1,1,1,1] => 0
[1,1,1,1,1,1] => [1,1,1,1,1] => 1
[7] => [] => 0
[6,1] => [1] => 1
[5,2] => [2] => 2
[5,1,1] => [1,1] => 0
[4,3] => [3] => 3
[4,2,1] => [2,1] => 1
[4,1,1,1] => [1,1,1] => 1
[3,3,1] => [3,1] => 2
[3,2,2] => [2,2] => 0
[3,2,1,1] => [2,1,1] => 2
[3,1,1,1,1] => [1,1,1,1] => 0
[2,2,2,1] => [2,2,1] => 1
[2,2,1,1,1] => [2,1,1,1] => 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] => 0
[8] => [] => 0
[7,1] => [1] => 1
[6,2] => [2] => 2
[6,1,1] => [1,1] => 0
[5,3] => [3] => 3
[5,2,1] => [2,1] => 1
[5,1,1,1] => [1,1,1] => 1
[4,4] => [4] => 4
[4,3,1] => [3,1] => 2
[4,2,2] => [2,2] => 0
[4,2,1,1] => [2,1,1] => 2
[4,1,1,1,1] => [1,1,1,1] => 0
[3,3,2] => [3,2] => 1
[3,3,1,1] => [3,1,1] => 3
[3,2,2,1] => [2,2,1] => 1
[3,2,1,1,1] => [2,1,1,1] => 1
[3,1,1,1,1,1] => [1,1,1,1,1] => 1
[2,2,2,2] => [2,2,2] => 2
[2,2,2,1,1] => [2,2,1,1] => 0
[2,2,1,1,1,1] => [2,1,1,1,1] => 2
[2,1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[9] => [] => 0
[8,1] => [1] => 1
[7,2] => [2] => 2
[7,1,1] => [1,1] => 0
[6,3] => [3] => 3
[6,2,1] => [2,1] => 1
[6,1,1,1] => [1,1,1] => 1
[5,4] => [4] => 4
[5,3,1] => [3,1] => 2
[5,2,2] => [2,2] => 0
[5,2,1,1] => [2,1,1] => 2
[5,1,1,1,1] => [1,1,1,1] => 0
[4,4,1] => [4,1] => 3
[4,3,2] => [3,2] => 1
[4,3,1,1] => [3,1,1] => 3
[4,2,2,1] => [2,2,1] => 1
[4,2,1,1,1] => [2,1,1,1] => 1
[4,1,1,1,1,1] => [1,1,1,1,1] => 1
[3,3,3] => [3,3] => 0
[3,3,2,1] => [3,2,1] => 2
[3,3,1,1,1] => [3,1,1,1] => 2
[3,2,2,2] => [2,2,2] => 2
[3,2,2,1,1] => [2,2,1,1] => 0
[3,2,1,1,1,1] => [2,1,1,1,1] => 2
[3,1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[2,2,2,2,1] => [2,2,2,1] => 1
[2,2,2,1,1,1] => [2,2,1,1,1] => 1
[2,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 0
[10] => [] => 0
[9,1] => [1] => 1
[8,2] => [2] => 2
[8,1,1] => [1,1] => 0
[7,3] => [3] => 3
>>> Load all 1201 entries. <<<[7,2,1] => [2,1] => 1
[7,1,1,1] => [1,1,1] => 1
[6,4] => [4] => 4
[6,3,1] => [3,1] => 2
[6,2,2] => [2,2] => 0
[6,2,1,1] => [2,1,1] => 2
[6,1,1,1,1] => [1,1,1,1] => 0
[5,5] => [5] => 5
[5,4,1] => [4,1] => 3
[5,3,2] => [3,2] => 1
[5,3,1,1] => [3,1,1] => 3
[5,2,2,1] => [2,2,1] => 1
[5,2,1,1,1] => [2,1,1,1] => 1
[5,1,1,1,1,1] => [1,1,1,1,1] => 1
[4,4,2] => [4,2] => 2
[4,4,1,1] => [4,1,1] => 4
[4,3,3] => [3,3] => 0
[4,3,2,1] => [3,2,1] => 2
[4,3,1,1,1] => [3,1,1,1] => 2
[4,2,2,2] => [2,2,2] => 2
[4,2,2,1,1] => [2,2,1,1] => 0
[4,2,1,1,1,1] => [2,1,1,1,1] => 2
[4,1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[3,3,3,1] => [3,3,1] => 1
[3,3,2,2] => [3,2,2] => 3
[3,3,2,1,1] => [3,2,1,1] => 1
[3,3,1,1,1,1] => [3,1,1,1,1] => 3
[3,2,2,2,1] => [2,2,2,1] => 1
[3,2,2,1,1,1] => [2,2,1,1,1] => 1
[3,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[3,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[2,2,2,2,2] => [2,2,2,2] => 0
[2,2,2,2,1,1] => [2,2,2,1,1] => 2
[2,2,2,1,1,1,1] => [2,2,1,1,1,1] => 0
[2,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1] => [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
[11] => [] => 0
[10,1] => [1] => 1
[9,2] => [2] => 2
[9,1,1] => [1,1] => 0
[8,3] => [3] => 3
[8,2,1] => [2,1] => 1
[8,1,1,1] => [1,1,1] => 1
[7,4] => [4] => 4
[7,3,1] => [3,1] => 2
[7,2,2] => [2,2] => 0
[7,2,1,1] => [2,1,1] => 2
[7,1,1,1,1] => [1,1,1,1] => 0
[6,5] => [5] => 5
[6,4,1] => [4,1] => 3
[6,3,2] => [3,2] => 1
[6,3,1,1] => [3,1,1] => 3
[6,2,2,1] => [2,2,1] => 1
[6,2,1,1,1] => [2,1,1,1] => 1
[6,1,1,1,1,1] => [1,1,1,1,1] => 1
[5,5,1] => [5,1] => 4
[5,4,2] => [4,2] => 2
[5,4,1,1] => [4,1,1] => 4
[5,3,3] => [3,3] => 0
[5,3,2,1] => [3,2,1] => 2
[5,3,1,1,1] => [3,1,1,1] => 2
[5,2,2,2] => [2,2,2] => 2
[5,2,2,1,1] => [2,2,1,1] => 0
[5,2,1,1,1,1] => [2,1,1,1,1] => 2
[5,1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[4,4,3] => [4,3] => 1
[4,4,2,1] => [4,2,1] => 3
[4,4,1,1,1] => [4,1,1,1] => 3
[4,3,3,1] => [3,3,1] => 1
[4,3,2,2] => [3,2,2] => 3
[4,3,2,1,1] => [3,2,1,1] => 1
[4,3,1,1,1,1] => [3,1,1,1,1] => 3
[4,2,2,2,1] => [2,2,2,1] => 1
[4,2,2,1,1,1] => [2,2,1,1,1] => 1
[4,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[4,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[3,3,3,2] => [3,3,2] => 2
[3,3,3,1,1] => [3,3,1,1] => 0
[3,3,2,2,1] => [3,2,2,1] => 2
[3,3,2,1,1,1] => [3,2,1,1,1] => 2
[3,3,1,1,1,1,1] => [3,1,1,1,1,1] => 2
[3,2,2,2,2] => [2,2,2,2] => 0
[3,2,2,2,1,1] => [2,2,2,1,1] => 2
[3,2,2,1,1,1,1] => [2,2,1,1,1,1] => 0
[3,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,1] => [2,2,2,2,1] => 1
[2,2,2,2,1,1,1] => [2,2,2,1,1,1] => 1
[2,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 1
[2,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 0
[12] => [] => 0
[11,1] => [1] => 1
[10,2] => [2] => 2
[10,1,1] => [1,1] => 0
[9,3] => [3] => 3
[9,2,1] => [2,1] => 1
[9,1,1,1] => [1,1,1] => 1
[8,4] => [4] => 4
[8,3,1] => [3,1] => 2
[8,2,2] => [2,2] => 0
[8,2,1,1] => [2,1,1] => 2
[8,1,1,1,1] => [1,1,1,1] => 0
[7,5] => [5] => 5
[7,4,1] => [4,1] => 3
[7,3,2] => [3,2] => 1
[7,3,1,1] => [3,1,1] => 3
[7,2,2,1] => [2,2,1] => 1
[7,2,1,1,1] => [2,1,1,1] => 1
[7,1,1,1,1,1] => [1,1,1,1,1] => 1
[6,6] => [6] => 6
[6,5,1] => [5,1] => 4
[6,4,2] => [4,2] => 2
[6,4,1,1] => [4,1,1] => 4
[6,3,3] => [3,3] => 0
[6,3,2,1] => [3,2,1] => 2
[6,3,1,1,1] => [3,1,1,1] => 2
[6,2,2,2] => [2,2,2] => 2
[6,2,2,1,1] => [2,2,1,1] => 0
[6,2,1,1,1,1] => [2,1,1,1,1] => 2
[6,1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[5,5,2] => [5,2] => 3
[5,5,1,1] => [5,1,1] => 5
[5,4,3] => [4,3] => 1
[5,4,2,1] => [4,2,1] => 3
[5,4,1,1,1] => [4,1,1,1] => 3
[5,3,3,1] => [3,3,1] => 1
[5,3,2,2] => [3,2,2] => 3
[5,3,2,1,1] => [3,2,1,1] => 1
[5,3,1,1,1,1] => [3,1,1,1,1] => 3
[5,2,2,2,1] => [2,2,2,1] => 1
[5,2,2,1,1,1] => [2,2,1,1,1] => 1
[5,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[5,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[4,4,4] => [4,4] => 0
[4,4,3,1] => [4,3,1] => 2
[4,4,2,2] => [4,2,2] => 4
[4,4,2,1,1] => [4,2,1,1] => 2
[4,4,1,1,1,1] => [4,1,1,1,1] => 4
[4,3,3,2] => [3,3,2] => 2
[4,3,3,1,1] => [3,3,1,1] => 0
[4,3,2,2,1] => [3,2,2,1] => 2
[4,3,2,1,1,1] => [3,2,1,1,1] => 2
[4,3,1,1,1,1,1] => [3,1,1,1,1,1] => 2
[4,2,2,2,2] => [2,2,2,2] => 0
[4,2,2,2,1,1] => [2,2,2,1,1] => 2
[4,2,2,1,1,1,1] => [2,2,1,1,1,1] => 0
[4,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 2
[4,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 0
[3,3,3,3] => [3,3,3] => 3
[3,3,3,2,1] => [3,3,2,1] => 1
[3,3,3,1,1,1] => [3,3,1,1,1] => 1
[3,3,2,2,2] => [3,2,2,2] => 1
[3,3,2,2,1,1] => [3,2,2,1,1] => 3
[3,3,2,1,1,1,1] => [3,2,1,1,1,1] => 1
[3,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 3
[3,2,2,2,2,1] => [2,2,2,2,1] => 1
[3,2,2,2,1,1,1] => [2,2,2,1,1,1] => 1
[3,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 1
[3,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 1
[3,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2] => [2,2,2,2,2] => 2
[2,2,2,2,2,1,1] => [2,2,2,2,1,1] => 0
[2,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => 2
[2,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[13] => [] => 0
[12,1] => [1] => 1
[11,2] => [2] => 2
[11,1,1] => [1,1] => 0
[10,3] => [3] => 3
[10,2,1] => [2,1] => 1
[10,1,1,1] => [1,1,1] => 1
[9,4] => [4] => 4
[9,3,1] => [3,1] => 2
[9,2,2] => [2,2] => 0
[9,2,1,1] => [2,1,1] => 2
[9,1,1,1,1] => [1,1,1,1] => 0
[8,5] => [5] => 5
[8,4,1] => [4,1] => 3
[8,3,2] => [3,2] => 1
[8,3,1,1] => [3,1,1] => 3
[8,2,2,1] => [2,2,1] => 1
[8,2,1,1,1] => [2,1,1,1] => 1
[8,1,1,1,1,1] => [1,1,1,1,1] => 1
[7,6] => [6] => 6
[7,5,1] => [5,1] => 4
[7,4,2] => [4,2] => 2
[7,4,1,1] => [4,1,1] => 4
[7,3,3] => [3,3] => 0
[7,3,2,1] => [3,2,1] => 2
[7,3,1,1,1] => [3,1,1,1] => 2
[7,2,2,2] => [2,2,2] => 2
[7,2,2,1,1] => [2,2,1,1] => 0
[7,2,1,1,1,1] => [2,1,1,1,1] => 2
[7,1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[6,6,1] => [6,1] => 5
[6,5,2] => [5,2] => 3
[6,5,1,1] => [5,1,1] => 5
[6,4,3] => [4,3] => 1
[6,4,2,1] => [4,2,1] => 3
[6,4,1,1,1] => [4,1,1,1] => 3
[6,3,3,1] => [3,3,1] => 1
[6,3,2,2] => [3,2,2] => 3
[6,3,2,1,1] => [3,2,1,1] => 1
[6,3,1,1,1,1] => [3,1,1,1,1] => 3
[6,2,2,2,1] => [2,2,2,1] => 1
[6,2,2,1,1,1] => [2,2,1,1,1] => 1
[6,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[6,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[5,5,3] => [5,3] => 2
[5,5,2,1] => [5,2,1] => 4
[5,5,1,1,1] => [5,1,1,1] => 4
[5,4,4] => [4,4] => 0
[5,4,3,1] => [4,3,1] => 2
[5,4,2,2] => [4,2,2] => 4
[5,4,2,1,1] => [4,2,1,1] => 2
[5,4,1,1,1,1] => [4,1,1,1,1] => 4
[5,3,3,2] => [3,3,2] => 2
[5,3,3,1,1] => [3,3,1,1] => 0
[5,3,2,2,1] => [3,2,2,1] => 2
[5,3,2,1,1,1] => [3,2,1,1,1] => 2
[5,3,1,1,1,1,1] => [3,1,1,1,1,1] => 2
[5,2,2,2,2] => [2,2,2,2] => 0
[5,2,2,2,1,1] => [2,2,2,1,1] => 2
[5,2,2,1,1,1,1] => [2,2,1,1,1,1] => 0
[5,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 2
[5,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 0
[4,4,4,1] => [4,4,1] => 1
[4,4,3,2] => [4,3,2] => 3
[4,4,3,1,1] => [4,3,1,1] => 1
[4,4,2,2,1] => [4,2,2,1] => 3
[4,4,2,1,1,1] => [4,2,1,1,1] => 3
[4,4,1,1,1,1,1] => [4,1,1,1,1,1] => 3
[4,3,3,3] => [3,3,3] => 3
[4,3,3,2,1] => [3,3,2,1] => 1
[4,3,3,1,1,1] => [3,3,1,1,1] => 1
[4,3,2,2,2] => [3,2,2,2] => 1
[4,3,2,2,1,1] => [3,2,2,1,1] => 3
[4,3,2,1,1,1,1] => [3,2,1,1,1,1] => 1
[4,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 3
[4,2,2,2,2,1] => [2,2,2,2,1] => 1
[4,2,2,2,1,1,1] => [2,2,2,1,1,1] => 1
[4,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 1
[4,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 1
[4,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[3,3,3,3,1] => [3,3,3,1] => 2
[3,3,3,2,2] => [3,3,2,2] => 0
[3,3,3,2,1,1] => [3,3,2,1,1] => 2
[3,3,3,1,1,1,1] => [3,3,1,1,1,1] => 0
[3,3,2,2,2,1] => [3,2,2,2,1] => 2
[3,3,2,2,1,1,1] => [3,2,2,1,1,1] => 2
[3,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => 2
[3,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 2
[3,2,2,2,2,2] => [2,2,2,2,2] => 2
[3,2,2,2,2,1,1] => [2,2,2,2,1,1] => 0
[3,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => 2
[3,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 0
[3,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,2,1] => [2,2,2,2,2,1] => 1
[2,2,2,2,2,1,1,1] => [2,2,2,2,1,1,1] => 1
[2,2,2,2,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => 1
[2,2,2,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => 1
[2,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => 0
[14] => [] => 0
[13,1] => [1] => 1
[12,2] => [2] => 2
[12,1,1] => [1,1] => 0
[11,3] => [3] => 3
[11,2,1] => [2,1] => 1
[11,1,1,1] => [1,1,1] => 1
[10,4] => [4] => 4
[10,3,1] => [3,1] => 2
[10,2,2] => [2,2] => 0
[10,2,1,1] => [2,1,1] => 2
[10,1,1,1,1] => [1,1,1,1] => 0
[9,5] => [5] => 5
[9,4,1] => [4,1] => 3
[9,3,2] => [3,2] => 1
[9,3,1,1] => [3,1,1] => 3
[9,2,2,1] => [2,2,1] => 1
[9,2,1,1,1] => [2,1,1,1] => 1
[9,1,1,1,1,1] => [1,1,1,1,1] => 1
[8,6] => [6] => 6
[8,5,1] => [5,1] => 4
[8,4,2] => [4,2] => 2
[8,4,1,1] => [4,1,1] => 4
[8,3,3] => [3,3] => 0
[8,3,2,1] => [3,2,1] => 2
[8,3,1,1,1] => [3,1,1,1] => 2
[8,2,2,2] => [2,2,2] => 2
[8,2,2,1,1] => [2,2,1,1] => 0
[8,2,1,1,1,1] => [2,1,1,1,1] => 2
[8,1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[7,7] => [7] => 7
[7,6,1] => [6,1] => 5
[7,5,2] => [5,2] => 3
[7,5,1,1] => [5,1,1] => 5
[7,4,3] => [4,3] => 1
[7,4,2,1] => [4,2,1] => 3
[7,4,1,1,1] => [4,1,1,1] => 3
[7,3,3,1] => [3,3,1] => 1
[7,3,2,2] => [3,2,2] => 3
[7,3,2,1,1] => [3,2,1,1] => 1
[7,3,1,1,1,1] => [3,1,1,1,1] => 3
[7,2,2,2,1] => [2,2,2,1] => 1
[7,2,2,1,1,1] => [2,2,1,1,1] => 1
[7,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[7,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[6,6,2] => [6,2] => 4
[6,6,1,1] => [6,1,1] => 6
[6,5,3] => [5,3] => 2
[6,5,2,1] => [5,2,1] => 4
[6,5,1,1,1] => [5,1,1,1] => 4
[6,4,4] => [4,4] => 0
[6,4,3,1] => [4,3,1] => 2
[6,4,2,2] => [4,2,2] => 4
[6,4,2,1,1] => [4,2,1,1] => 2
[6,4,1,1,1,1] => [4,1,1,1,1] => 4
[6,3,3,2] => [3,3,2] => 2
[6,3,3,1,1] => [3,3,1,1] => 0
[6,3,2,2,1] => [3,2,2,1] => 2
[6,3,2,1,1,1] => [3,2,1,1,1] => 2
[6,3,1,1,1,1,1] => [3,1,1,1,1,1] => 2
[6,2,2,2,2] => [2,2,2,2] => 0
[6,2,2,2,1,1] => [2,2,2,1,1] => 2
[6,2,2,1,1,1,1] => [2,2,1,1,1,1] => 0
[6,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 2
[6,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 0
[5,5,4] => [5,4] => 1
[5,5,3,1] => [5,3,1] => 3
[5,5,2,2] => [5,2,2] => 5
[5,5,2,1,1] => [5,2,1,1] => 3
[5,5,1,1,1,1] => [5,1,1,1,1] => 5
[5,4,4,1] => [4,4,1] => 1
[5,4,3,2] => [4,3,2] => 3
[5,4,3,1,1] => [4,3,1,1] => 1
[5,4,2,2,1] => [4,2,2,1] => 3
[5,4,2,1,1,1] => [4,2,1,1,1] => 3
[5,4,1,1,1,1,1] => [4,1,1,1,1,1] => 3
[5,3,3,3] => [3,3,3] => 3
[5,3,3,2,1] => [3,3,2,1] => 1
[5,3,3,1,1,1] => [3,3,1,1,1] => 1
[5,3,2,2,2] => [3,2,2,2] => 1
[5,3,2,2,1,1] => [3,2,2,1,1] => 3
[5,3,2,1,1,1,1] => [3,2,1,1,1,1] => 1
[5,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 3
[5,2,2,2,2,1] => [2,2,2,2,1] => 1
[5,2,2,2,1,1,1] => [2,2,2,1,1,1] => 1
[5,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 1
[5,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 1
[5,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[4,4,4,2] => [4,4,2] => 2
[4,4,4,1,1] => [4,4,1,1] => 0
[4,4,3,3] => [4,3,3] => 4
[4,4,3,2,1] => [4,3,2,1] => 2
[4,4,3,1,1,1] => [4,3,1,1,1] => 2
[4,4,2,2,2] => [4,2,2,2] => 2
[4,4,2,2,1,1] => [4,2,2,1,1] => 4
[4,4,2,1,1,1,1] => [4,2,1,1,1,1] => 2
[4,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => 4
[4,3,3,3,1] => [3,3,3,1] => 2
[4,3,3,2,2] => [3,3,2,2] => 0
[4,3,3,2,1,1] => [3,3,2,1,1] => 2
[4,3,3,1,1,1,1] => [3,3,1,1,1,1] => 0
[4,3,2,2,2,1] => [3,2,2,2,1] => 2
[4,3,2,2,1,1,1] => [3,2,2,1,1,1] => 2
[4,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => 2
[4,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 2
[4,2,2,2,2,2] => [2,2,2,2,2] => 2
[4,2,2,2,2,1,1] => [2,2,2,2,1,1] => 0
[4,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => 2
[4,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 0
[4,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 2
[4,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 0
[3,3,3,3,2] => [3,3,3,2] => 1
[3,3,3,3,1,1] => [3,3,3,1,1] => 3
[3,3,3,2,2,1] => [3,3,2,2,1] => 1
[3,3,3,2,1,1,1] => [3,3,2,1,1,1] => 1
[3,3,3,1,1,1,1,1] => [3,3,1,1,1,1,1] => 1
[3,3,2,2,2,2] => [3,2,2,2,2] => 3
[3,3,2,2,2,1,1] => [3,2,2,2,1,1] => 1
[3,3,2,2,1,1,1,1] => [3,2,2,1,1,1,1] => 3
[3,3,2,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1] => 1
[3,3,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1] => 3
[3,2,2,2,2,2,1] => [2,2,2,2,2,1] => 1
[3,2,2,2,2,1,1,1] => [2,2,2,2,1,1,1] => 1
[3,2,2,2,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => 1
[3,2,2,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => 1
[3,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => 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
[2,2,2,2,2,2,2] => [2,2,2,2,2,2] => 0
[2,2,2,2,2,2,1,1] => [2,2,2,2,2,1,1] => 2
[2,2,2,2,2,1,1,1,1] => [2,2,2,2,1,1,1,1] => 0
[2,2,2,2,1,1,1,1,1,1] => [2,2,2,1,1,1,1,1,1] => 2
[2,2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => 0
[15] => [] => 0
[14,1] => [1] => 1
[13,2] => [2] => 2
[13,1,1] => [1,1] => 0
[12,3] => [3] => 3
[12,2,1] => [2,1] => 1
[12,1,1,1] => [1,1,1] => 1
[11,4] => [4] => 4
[11,3,1] => [3,1] => 2
[11,2,2] => [2,2] => 0
[11,2,1,1] => [2,1,1] => 2
[11,1,1,1,1] => [1,1,1,1] => 0
[10,5] => [5] => 5
[10,4,1] => [4,1] => 3
[10,3,2] => [3,2] => 1
[10,3,1,1] => [3,1,1] => 3
[10,2,2,1] => [2,2,1] => 1
[10,2,1,1,1] => [2,1,1,1] => 1
[10,1,1,1,1,1] => [1,1,1,1,1] => 1
[9,6] => [6] => 6
[9,5,1] => [5,1] => 4
[9,4,2] => [4,2] => 2
[9,4,1,1] => [4,1,1] => 4
[9,3,3] => [3,3] => 0
[9,3,2,1] => [3,2,1] => 2
[9,3,1,1,1] => [3,1,1,1] => 2
[9,2,2,2] => [2,2,2] => 2
[9,2,2,1,1] => [2,2,1,1] => 0
[9,2,1,1,1,1] => [2,1,1,1,1] => 2
[9,1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[8,7] => [7] => 7
[8,6,1] => [6,1] => 5
[8,5,2] => [5,2] => 3
[8,5,1,1] => [5,1,1] => 5
[8,4,3] => [4,3] => 1
[8,4,2,1] => [4,2,1] => 3
[8,4,1,1,1] => [4,1,1,1] => 3
[8,3,3,1] => [3,3,1] => 1
[8,3,2,2] => [3,2,2] => 3
[8,3,2,1,1] => [3,2,1,1] => 1
[8,3,1,1,1,1] => [3,1,1,1,1] => 3
[8,2,2,2,1] => [2,2,2,1] => 1
[8,2,2,1,1,1] => [2,2,1,1,1] => 1
[8,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[8,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[7,7,1] => [7,1] => 6
[7,6,2] => [6,2] => 4
[7,6,1,1] => [6,1,1] => 6
[7,5,3] => [5,3] => 2
[7,5,2,1] => [5,2,1] => 4
[7,5,1,1,1] => [5,1,1,1] => 4
[7,4,4] => [4,4] => 0
[7,4,3,1] => [4,3,1] => 2
[7,4,2,2] => [4,2,2] => 4
[7,4,2,1,1] => [4,2,1,1] => 2
[7,4,1,1,1,1] => [4,1,1,1,1] => 4
[7,3,3,2] => [3,3,2] => 2
[7,3,3,1,1] => [3,3,1,1] => 0
[7,3,2,2,1] => [3,2,2,1] => 2
[7,3,2,1,1,1] => [3,2,1,1,1] => 2
[7,3,1,1,1,1,1] => [3,1,1,1,1,1] => 2
[7,2,2,2,2] => [2,2,2,2] => 0
[7,2,2,2,1,1] => [2,2,2,1,1] => 2
[7,2,2,1,1,1,1] => [2,2,1,1,1,1] => 0
[7,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 2
[7,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 0
[6,6,3] => [6,3] => 3
[6,6,2,1] => [6,2,1] => 5
[6,6,1,1,1] => [6,1,1,1] => 5
[6,5,4] => [5,4] => 1
[6,5,3,1] => [5,3,1] => 3
[6,5,2,2] => [5,2,2] => 5
[6,5,2,1,1] => [5,2,1,1] => 3
[6,5,1,1,1,1] => [5,1,1,1,1] => 5
[6,4,4,1] => [4,4,1] => 1
[6,4,3,2] => [4,3,2] => 3
[6,4,3,1,1] => [4,3,1,1] => 1
[6,4,2,2,1] => [4,2,2,1] => 3
[6,4,2,1,1,1] => [4,2,1,1,1] => 3
[6,4,1,1,1,1,1] => [4,1,1,1,1,1] => 3
[6,3,3,3] => [3,3,3] => 3
[6,3,3,2,1] => [3,3,2,1] => 1
[6,3,3,1,1,1] => [3,3,1,1,1] => 1
[6,3,2,2,2] => [3,2,2,2] => 1
[6,3,2,2,1,1] => [3,2,2,1,1] => 3
[6,3,2,1,1,1,1] => [3,2,1,1,1,1] => 1
[6,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 3
[6,2,2,2,2,1] => [2,2,2,2,1] => 1
[6,2,2,2,1,1,1] => [2,2,2,1,1,1] => 1
[6,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 1
[6,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 1
[6,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[5,5,5] => [5,5] => 0
[5,5,4,1] => [5,4,1] => 2
[5,5,3,2] => [5,3,2] => 4
[5,5,3,1,1] => [5,3,1,1] => 2
[5,5,2,2,1] => [5,2,2,1] => 4
[5,5,2,1,1,1] => [5,2,1,1,1] => 4
[5,5,1,1,1,1,1] => [5,1,1,1,1,1] => 4
[5,4,4,2] => [4,4,2] => 2
[5,4,4,1,1] => [4,4,1,1] => 0
[5,4,3,3] => [4,3,3] => 4
[5,4,3,2,1] => [4,3,2,1] => 2
[5,4,3,1,1,1] => [4,3,1,1,1] => 2
[5,4,2,2,2] => [4,2,2,2] => 2
[5,4,2,2,1,1] => [4,2,2,1,1] => 4
[5,4,2,1,1,1,1] => [4,2,1,1,1,1] => 2
[5,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => 4
[5,3,3,3,1] => [3,3,3,1] => 2
[5,3,3,2,2] => [3,3,2,2] => 0
[5,3,3,2,1,1] => [3,3,2,1,1] => 2
[5,3,3,1,1,1,1] => [3,3,1,1,1,1] => 0
[5,3,2,2,2,1] => [3,2,2,2,1] => 2
[5,3,2,2,1,1,1] => [3,2,2,1,1,1] => 2
[5,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => 2
[5,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 2
[5,2,2,2,2,2] => [2,2,2,2,2] => 2
[5,2,2,2,2,1,1] => [2,2,2,2,1,1] => 0
[5,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => 2
[5,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 0
[5,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 2
[5,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 0
[4,4,4,3] => [4,4,3] => 3
[4,4,4,2,1] => [4,4,2,1] => 1
[4,4,4,1,1,1] => [4,4,1,1,1] => 1
[4,4,3,3,1] => [4,3,3,1] => 3
[4,4,3,2,2] => [4,3,2,2] => 1
[4,4,3,2,1,1] => [4,3,2,1,1] => 3
[4,4,3,1,1,1,1] => [4,3,1,1,1,1] => 1
[4,4,2,2,2,1] => [4,2,2,2,1] => 3
[4,4,2,2,1,1,1] => [4,2,2,1,1,1] => 3
[4,4,2,1,1,1,1,1] => [4,2,1,1,1,1,1] => 3
[4,4,1,1,1,1,1,1,1] => [4,1,1,1,1,1,1,1] => 3
[4,3,3,3,2] => [3,3,3,2] => 1
[4,3,3,3,1,1] => [3,3,3,1,1] => 3
[4,3,3,2,2,1] => [3,3,2,2,1] => 1
[4,3,3,2,1,1,1] => [3,3,2,1,1,1] => 1
[4,3,3,1,1,1,1,1] => [3,3,1,1,1,1,1] => 1
[4,3,2,2,2,2] => [3,2,2,2,2] => 3
[4,3,2,2,2,1,1] => [3,2,2,2,1,1] => 1
[4,3,2,2,1,1,1,1] => [3,2,2,1,1,1,1] => 3
[4,3,2,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1] => 1
[4,3,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1] => 3
[4,2,2,2,2,2,1] => [2,2,2,2,2,1] => 1
[4,2,2,2,2,1,1,1] => [2,2,2,2,1,1,1] => 1
[4,2,2,2,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => 1
[4,2,2,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => 1
[4,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => 1
[4,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[3,3,3,3,3] => [3,3,3,3] => 0
[3,3,3,3,2,1] => [3,3,3,2,1] => 2
[3,3,3,3,1,1,1] => [3,3,3,1,1,1] => 2
[3,3,3,2,2,2] => [3,3,2,2,2] => 2
[3,3,3,2,2,1,1] => [3,3,2,2,1,1] => 0
[3,3,3,2,1,1,1,1] => [3,3,2,1,1,1,1] => 2
[3,3,3,1,1,1,1,1,1] => [3,3,1,1,1,1,1,1] => 0
[3,3,2,2,2,2,1] => [3,2,2,2,2,1] => 2
[3,3,2,2,2,1,1,1] => [3,2,2,2,1,1,1] => 2
[3,3,2,2,1,1,1,1,1] => [3,2,2,1,1,1,1,1] => 2
[3,3,2,1,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1,1] => 2
[3,3,1,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1,1] => 2
[3,2,2,2,2,2,2] => [2,2,2,2,2,2] => 0
[3,2,2,2,2,2,1,1] => [2,2,2,2,2,1,1] => 2
[3,2,2,2,2,1,1,1,1] => [2,2,2,2,1,1,1,1] => 0
[3,2,2,2,1,1,1,1,1,1] => [2,2,2,1,1,1,1,1,1] => 2
[3,2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => 0
[3,2,1,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1] => 2
[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] => 0
[16] => [] => 0
[15,1] => [1] => 1
[14,2] => [2] => 2
[14,1,1] => [1,1] => 0
[13,3] => [3] => 3
[13,2,1] => [2,1] => 1
[13,1,1,1] => [1,1,1] => 1
[12,4] => [4] => 4
[12,3,1] => [3,1] => 2
[12,2,2] => [2,2] => 0
[12,2,1,1] => [2,1,1] => 2
[12,1,1,1,1] => [1,1,1,1] => 0
[11,5] => [5] => 5
[11,4,1] => [4,1] => 3
[11,3,2] => [3,2] => 1
[11,3,1,1] => [3,1,1] => 3
[11,2,2,1] => [2,2,1] => 1
[11,2,1,1,1] => [2,1,1,1] => 1
[11,1,1,1,1,1] => [1,1,1,1,1] => 1
[10,6] => [6] => 6
[10,5,1] => [5,1] => 4
[10,4,2] => [4,2] => 2
[10,4,1,1] => [4,1,1] => 4
[10,3,3] => [3,3] => 0
[10,3,2,1] => [3,2,1] => 2
[10,3,1,1,1] => [3,1,1,1] => 2
[10,2,2,2] => [2,2,2] => 2
[10,2,2,1,1] => [2,2,1,1] => 0
[10,2,1,1,1,1] => [2,1,1,1,1] => 2
[10,1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[9,7] => [7] => 7
[9,6,1] => [6,1] => 5
[9,5,2] => [5,2] => 3
[9,5,1,1] => [5,1,1] => 5
[9,4,3] => [4,3] => 1
[9,4,2,1] => [4,2,1] => 3
[9,4,1,1,1] => [4,1,1,1] => 3
[9,3,3,1] => [3,3,1] => 1
[9,3,2,2] => [3,2,2] => 3
[9,3,2,1,1] => [3,2,1,1] => 1
[9,3,1,1,1,1] => [3,1,1,1,1] => 3
[9,2,2,2,1] => [2,2,2,1] => 1
[9,2,2,1,1,1] => [2,2,1,1,1] => 1
[9,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[9,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[8,8] => [8] => 8
[8,7,1] => [7,1] => 6
[8,6,2] => [6,2] => 4
[8,6,1,1] => [6,1,1] => 6
[8,5,3] => [5,3] => 2
[8,5,2,1] => [5,2,1] => 4
[8,5,1,1,1] => [5,1,1,1] => 4
[8,4,4] => [4,4] => 0
[8,4,3,1] => [4,3,1] => 2
[8,4,2,2] => [4,2,2] => 4
[8,4,2,1,1] => [4,2,1,1] => 2
[8,4,1,1,1,1] => [4,1,1,1,1] => 4
[8,3,3,2] => [3,3,2] => 2
[8,3,3,1,1] => [3,3,1,1] => 0
[8,3,2,2,1] => [3,2,2,1] => 2
[8,3,2,1,1,1] => [3,2,1,1,1] => 2
[8,3,1,1,1,1,1] => [3,1,1,1,1,1] => 2
[8,2,2,2,2] => [2,2,2,2] => 0
[8,2,2,2,1,1] => [2,2,2,1,1] => 2
[8,2,2,1,1,1,1] => [2,2,1,1,1,1] => 0
[8,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 2
[8,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 0
[7,7,2] => [7,2] => 5
[7,7,1,1] => [7,1,1] => 7
[7,6,3] => [6,3] => 3
[7,6,2,1] => [6,2,1] => 5
[7,6,1,1,1] => [6,1,1,1] => 5
[7,5,4] => [5,4] => 1
[7,5,3,1] => [5,3,1] => 3
[7,5,2,2] => [5,2,2] => 5
[7,5,2,1,1] => [5,2,1,1] => 3
[7,5,1,1,1,1] => [5,1,1,1,1] => 5
[7,4,4,1] => [4,4,1] => 1
[7,4,3,2] => [4,3,2] => 3
[7,4,3,1,1] => [4,3,1,1] => 1
[7,4,2,2,1] => [4,2,2,1] => 3
[7,4,2,1,1,1] => [4,2,1,1,1] => 3
[7,4,1,1,1,1,1] => [4,1,1,1,1,1] => 3
[7,3,3,3] => [3,3,3] => 3
[7,3,3,2,1] => [3,3,2,1] => 1
[7,3,3,1,1,1] => [3,3,1,1,1] => 1
[7,3,2,2,2] => [3,2,2,2] => 1
[7,3,2,2,1,1] => [3,2,2,1,1] => 3
[7,3,2,1,1,1,1] => [3,2,1,1,1,1] => 1
[7,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 3
[7,2,2,2,2,1] => [2,2,2,2,1] => 1
[7,2,2,2,1,1,1] => [2,2,2,1,1,1] => 1
[7,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 1
[7,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 1
[7,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[6,6,4] => [6,4] => 2
[6,6,3,1] => [6,3,1] => 4
[6,6,2,2] => [6,2,2] => 6
[6,6,2,1,1] => [6,2,1,1] => 4
[6,6,1,1,1,1] => [6,1,1,1,1] => 6
[6,5,5] => [5,5] => 0
[6,5,4,1] => [5,4,1] => 2
[6,5,3,2] => [5,3,2] => 4
[6,5,3,1,1] => [5,3,1,1] => 2
[6,5,2,2,1] => [5,2,2,1] => 4
[6,5,2,1,1,1] => [5,2,1,1,1] => 4
[6,5,1,1,1,1,1] => [5,1,1,1,1,1] => 4
[6,4,4,2] => [4,4,2] => 2
[6,4,4,1,1] => [4,4,1,1] => 0
[6,4,3,3] => [4,3,3] => 4
[6,4,3,2,1] => [4,3,2,1] => 2
[6,4,3,1,1,1] => [4,3,1,1,1] => 2
[6,4,2,2,2] => [4,2,2,2] => 2
[6,4,2,2,1,1] => [4,2,2,1,1] => 4
[6,4,2,1,1,1,1] => [4,2,1,1,1,1] => 2
[6,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => 4
[6,3,3,3,1] => [3,3,3,1] => 2
[6,3,3,2,2] => [3,3,2,2] => 0
[6,3,3,2,1,1] => [3,3,2,1,1] => 2
[6,3,3,1,1,1,1] => [3,3,1,1,1,1] => 0
[6,3,2,2,2,1] => [3,2,2,2,1] => 2
[6,3,2,2,1,1,1] => [3,2,2,1,1,1] => 2
[6,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => 2
[6,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 2
[6,2,2,2,2,2] => [2,2,2,2,2] => 2
[6,2,2,2,2,1,1] => [2,2,2,2,1,1] => 0
[6,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => 2
[6,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 0
[6,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 2
[6,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 0
[5,5,5,1] => [5,5,1] => 1
[5,5,4,2] => [5,4,2] => 3
[5,5,4,1,1] => [5,4,1,1] => 1
[5,5,3,3] => [5,3,3] => 5
[5,5,3,2,1] => [5,3,2,1] => 3
[5,5,3,1,1,1] => [5,3,1,1,1] => 3
[5,5,2,2,2] => [5,2,2,2] => 3
[5,5,2,2,1,1] => [5,2,2,1,1] => 5
[5,5,2,1,1,1,1] => [5,2,1,1,1,1] => 3
[5,5,1,1,1,1,1,1] => [5,1,1,1,1,1,1] => 5
[5,4,4,3] => [4,4,3] => 3
[5,4,4,2,1] => [4,4,2,1] => 1
[5,4,4,1,1,1] => [4,4,1,1,1] => 1
[5,4,3,3,1] => [4,3,3,1] => 3
[5,4,3,2,2] => [4,3,2,2] => 1
[5,4,3,2,1,1] => [4,3,2,1,1] => 3
[5,4,3,1,1,1,1] => [4,3,1,1,1,1] => 1
[5,4,2,2,2,1] => [4,2,2,2,1] => 3
[5,4,2,2,1,1,1] => [4,2,2,1,1,1] => 3
[5,4,2,1,1,1,1,1] => [4,2,1,1,1,1,1] => 3
[5,4,1,1,1,1,1,1,1] => [4,1,1,1,1,1,1,1] => 3
[5,3,3,3,2] => [3,3,3,2] => 1
[5,3,3,3,1,1] => [3,3,3,1,1] => 3
[5,3,3,2,2,1] => [3,3,2,2,1] => 1
[5,3,3,2,1,1,1] => [3,3,2,1,1,1] => 1
[5,3,3,1,1,1,1,1] => [3,3,1,1,1,1,1] => 1
[5,3,2,2,2,2] => [3,2,2,2,2] => 3
[5,3,2,2,2,1,1] => [3,2,2,2,1,1] => 1
[5,3,2,2,1,1,1,1] => [3,2,2,1,1,1,1] => 3
[5,3,2,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1] => 1
[5,3,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1] => 3
[5,2,2,2,2,2,1] => [2,2,2,2,2,1] => 1
[5,2,2,2,2,1,1,1] => [2,2,2,2,1,1,1] => 1
[5,2,2,2,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => 1
[5,2,2,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => 1
[5,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => 1
[5,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[4,4,4,4] => [4,4,4] => 4
[4,4,4,3,1] => [4,4,3,1] => 2
[4,4,4,2,2] => [4,4,2,2] => 0
[4,4,4,2,1,1] => [4,4,2,1,1] => 2
[4,4,4,1,1,1,1] => [4,4,1,1,1,1] => 0
[4,4,3,3,2] => [4,3,3,2] => 2
[4,4,3,3,1,1] => [4,3,3,1,1] => 4
[4,4,3,2,2,1] => [4,3,2,2,1] => 2
[4,4,3,2,1,1,1] => [4,3,2,1,1,1] => 2
[4,4,3,1,1,1,1,1] => [4,3,1,1,1,1,1] => 2
[4,4,2,2,2,2] => [4,2,2,2,2] => 4
[4,4,2,2,2,1,1] => [4,2,2,2,1,1] => 2
[4,4,2,2,1,1,1,1] => [4,2,2,1,1,1,1] => 4
[4,4,2,1,1,1,1,1,1] => [4,2,1,1,1,1,1,1] => 2
[4,4,1,1,1,1,1,1,1,1] => [4,1,1,1,1,1,1,1,1] => 4
[4,3,3,3,3] => [3,3,3,3] => 0
[4,3,3,3,2,1] => [3,3,3,2,1] => 2
[4,3,3,3,1,1,1] => [3,3,3,1,1,1] => 2
[4,3,3,2,2,2] => [3,3,2,2,2] => 2
[4,3,3,2,2,1,1] => [3,3,2,2,1,1] => 0
[4,3,3,2,1,1,1,1] => [3,3,2,1,1,1,1] => 2
[4,3,3,1,1,1,1,1,1] => [3,3,1,1,1,1,1,1] => 0
[4,3,2,2,2,2,1] => [3,2,2,2,2,1] => 2
[4,3,2,2,2,1,1,1] => [3,2,2,2,1,1,1] => 2
[4,3,2,2,1,1,1,1,1] => [3,2,2,1,1,1,1,1] => 2
[4,3,2,1,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1,1] => 2
[4,3,1,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1,1] => 2
[4,2,2,2,2,2,2] => [2,2,2,2,2,2] => 0
[4,2,2,2,2,2,1,1] => [2,2,2,2,2,1,1] => 2
[4,2,2,2,2,1,1,1,1] => [2,2,2,2,1,1,1,1] => 0
[4,2,2,2,1,1,1,1,1,1] => [2,2,2,1,1,1,1,1,1] => 2
[4,2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => 0
[4,2,1,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1] => 2
[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] => 0
[17] => [] => 0
[16,1] => [1] => 1
[15,2] => [2] => 2
[15,1,1] => [1,1] => 0
[14,3] => [3] => 3
[14,2,1] => [2,1] => 1
[14,1,1,1] => [1,1,1] => 1
[13,4] => [4] => 4
[13,3,1] => [3,1] => 2
[13,2,2] => [2,2] => 0
[13,2,1,1] => [2,1,1] => 2
[13,1,1,1,1] => [1,1,1,1] => 0
[12,5] => [5] => 5
[12,4,1] => [4,1] => 3
[12,3,2] => [3,2] => 1
[12,3,1,1] => [3,1,1] => 3
[12,2,2,1] => [2,2,1] => 1
[12,2,1,1,1] => [2,1,1,1] => 1
[12,1,1,1,1,1] => [1,1,1,1,1] => 1
[11,6] => [6] => 6
[11,5,1] => [5,1] => 4
[11,4,2] => [4,2] => 2
[11,4,1,1] => [4,1,1] => 4
[11,3,3] => [3,3] => 0
[11,3,2,1] => [3,2,1] => 2
[11,3,1,1,1] => [3,1,1,1] => 2
[11,2,2,2] => [2,2,2] => 2
[11,2,2,1,1] => [2,2,1,1] => 0
[11,2,1,1,1,1] => [2,1,1,1,1] => 2
[11,1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[10,7] => [7] => 7
[10,6,1] => [6,1] => 5
[10,5,2] => [5,2] => 3
[10,5,1,1] => [5,1,1] => 5
[10,4,3] => [4,3] => 1
[10,4,2,1] => [4,2,1] => 3
[10,4,1,1,1] => [4,1,1,1] => 3
[10,3,3,1] => [3,3,1] => 1
[10,3,2,2] => [3,2,2] => 3
[10,3,2,1,1] => [3,2,1,1] => 1
[10,3,1,1,1,1] => [3,1,1,1,1] => 3
[10,2,2,2,1] => [2,2,2,1] => 1
[10,2,2,1,1,1] => [2,2,1,1,1] => 1
[10,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[10,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[9,8] => [8] => 8
[9,7,1] => [7,1] => 6
[9,6,2] => [6,2] => 4
[9,6,1,1] => [6,1,1] => 6
[9,5,3] => [5,3] => 2
[9,5,2,1] => [5,2,1] => 4
[9,5,1,1,1] => [5,1,1,1] => 4
[9,4,4] => [4,4] => 0
[9,4,3,1] => [4,3,1] => 2
[9,4,2,2] => [4,2,2] => 4
[9,4,2,1,1] => [4,2,1,1] => 2
[9,4,1,1,1,1] => [4,1,1,1,1] => 4
[9,3,3,2] => [3,3,2] => 2
[9,3,3,1,1] => [3,3,1,1] => 0
[9,3,2,2,1] => [3,2,2,1] => 2
[9,3,2,1,1,1] => [3,2,1,1,1] => 2
[9,3,1,1,1,1,1] => [3,1,1,1,1,1] => 2
[9,2,2,2,2] => [2,2,2,2] => 0
[9,2,2,2,1,1] => [2,2,2,1,1] => 2
[9,2,2,1,1,1,1] => [2,2,1,1,1,1] => 0
[9,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 2
[9,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 0
[8,8,1] => [8,1] => 7
[8,7,2] => [7,2] => 5
[8,7,1,1] => [7,1,1] => 7
[8,6,3] => [6,3] => 3
[8,6,2,1] => [6,2,1] => 5
[8,6,1,1,1] => [6,1,1,1] => 5
[8,5,4] => [5,4] => 1
[8,5,3,1] => [5,3,1] => 3
[8,5,2,2] => [5,2,2] => 5
[8,5,2,1,1] => [5,2,1,1] => 3
[8,5,1,1,1,1] => [5,1,1,1,1] => 5
[8,4,4,1] => [4,4,1] => 1
[8,4,3,2] => [4,3,2] => 3
[8,4,3,1,1] => [4,3,1,1] => 1
[8,4,2,2,1] => [4,2,2,1] => 3
[8,4,2,1,1,1] => [4,2,1,1,1] => 3
[8,4,1,1,1,1,1] => [4,1,1,1,1,1] => 3
[8,3,3,3] => [3,3,3] => 3
[8,3,3,2,1] => [3,3,2,1] => 1
[8,3,3,1,1,1] => [3,3,1,1,1] => 1
[8,3,2,2,2] => [3,2,2,2] => 1
[8,3,2,2,1,1] => [3,2,2,1,1] => 3
[8,3,2,1,1,1,1] => [3,2,1,1,1,1] => 1
[8,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 3
[8,2,2,2,2,1] => [2,2,2,2,1] => 1
[8,2,2,2,1,1,1] => [2,2,2,1,1,1] => 1
[8,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 1
[8,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 1
[8,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[7,7,3] => [7,3] => 4
[7,7,2,1] => [7,2,1] => 6
[7,7,1,1,1] => [7,1,1,1] => 6
[7,6,4] => [6,4] => 2
[7,6,3,1] => [6,3,1] => 4
[7,6,2,2] => [6,2,2] => 6
[7,6,2,1,1] => [6,2,1,1] => 4
[7,6,1,1,1,1] => [6,1,1,1,1] => 6
[7,5,5] => [5,5] => 0
[7,5,4,1] => [5,4,1] => 2
[7,5,3,2] => [5,3,2] => 4
[7,5,3,1,1] => [5,3,1,1] => 2
[7,5,2,2,1] => [5,2,2,1] => 4
[7,5,2,1,1,1] => [5,2,1,1,1] => 4
[7,5,1,1,1,1,1] => [5,1,1,1,1,1] => 4
[7,4,4,2] => [4,4,2] => 2
[7,4,4,1,1] => [4,4,1,1] => 0
[7,4,3,3] => [4,3,3] => 4
[7,4,3,2,1] => [4,3,2,1] => 2
[7,4,3,1,1,1] => [4,3,1,1,1] => 2
[7,4,2,2,2] => [4,2,2,2] => 2
[7,4,2,2,1,1] => [4,2,2,1,1] => 4
[7,4,2,1,1,1,1] => [4,2,1,1,1,1] => 2
[7,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => 4
[7,3,3,3,1] => [3,3,3,1] => 2
[7,3,3,2,2] => [3,3,2,2] => 0
[7,3,3,2,1,1] => [3,3,2,1,1] => 2
[7,3,3,1,1,1,1] => [3,3,1,1,1,1] => 0
[7,3,2,2,2,1] => [3,2,2,2,1] => 2
[7,3,2,2,1,1,1] => [3,2,2,1,1,1] => 2
[7,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => 2
[7,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 2
[7,2,2,2,2,2] => [2,2,2,2,2] => 2
[7,2,2,2,2,1,1] => [2,2,2,2,1,1] => 0
[7,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => 2
[7,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 0
[7,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 2
[7,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 0
[6,6,5] => [6,5] => 1
[6,6,4,1] => [6,4,1] => 3
[6,6,3,2] => [6,3,2] => 5
[6,6,3,1,1] => [6,3,1,1] => 3
[6,6,2,2,1] => [6,2,2,1] => 5
[6,6,2,1,1,1] => [6,2,1,1,1] => 5
[6,6,1,1,1,1,1] => [6,1,1,1,1,1] => 5
[6,5,5,1] => [5,5,1] => 1
[6,5,4,2] => [5,4,2] => 3
[6,5,4,1,1] => [5,4,1,1] => 1
[6,5,3,3] => [5,3,3] => 5
[6,5,3,2,1] => [5,3,2,1] => 3
[6,5,3,1,1,1] => [5,3,1,1,1] => 3
[6,5,2,2,2] => [5,2,2,2] => 3
[6,5,2,2,1,1] => [5,2,2,1,1] => 5
[6,5,2,1,1,1,1] => [5,2,1,1,1,1] => 3
[6,5,1,1,1,1,1,1] => [5,1,1,1,1,1,1] => 5
[6,4,4,3] => [4,4,3] => 3
[6,4,4,2,1] => [4,4,2,1] => 1
[6,4,4,1,1,1] => [4,4,1,1,1] => 1
[6,4,3,3,1] => [4,3,3,1] => 3
[6,4,3,2,2] => [4,3,2,2] => 1
[6,4,3,2,1,1] => [4,3,2,1,1] => 3
[6,4,3,1,1,1,1] => [4,3,1,1,1,1] => 1
[6,4,2,2,2,1] => [4,2,2,2,1] => 3
[6,4,2,2,1,1,1] => [4,2,2,1,1,1] => 3
[6,4,2,1,1,1,1,1] => [4,2,1,1,1,1,1] => 3
[6,4,1,1,1,1,1,1,1] => [4,1,1,1,1,1,1,1] => 3
[6,3,3,3,2] => [3,3,3,2] => 1
[6,3,3,3,1,1] => [3,3,3,1,1] => 3
[6,3,3,2,2,1] => [3,3,2,2,1] => 1
[6,3,3,2,1,1,1] => [3,3,2,1,1,1] => 1
[6,3,3,1,1,1,1,1] => [3,3,1,1,1,1,1] => 1
[6,3,2,2,2,2] => [3,2,2,2,2] => 3
[6,3,2,2,2,1,1] => [3,2,2,2,1,1] => 1
[6,3,2,2,1,1,1,1] => [3,2,2,1,1,1,1] => 3
[6,3,2,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1] => 1
[6,3,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1] => 3
[6,2,2,2,2,2,1] => [2,2,2,2,2,1] => 1
[6,2,2,2,2,1,1,1] => [2,2,2,2,1,1,1] => 1
[6,2,2,2,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => 1
[6,2,2,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => 1
[6,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => 1
[6,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[5,5,5,2] => [5,5,2] => 2
[5,5,5,1,1] => [5,5,1,1] => 0
[5,5,4,3] => [5,4,3] => 4
[5,5,4,2,1] => [5,4,2,1] => 2
[5,5,4,1,1,1] => [5,4,1,1,1] => 2
[5,5,3,3,1] => [5,3,3,1] => 4
[5,5,3,2,2] => [5,3,2,2] => 2
[5,5,3,2,1,1] => [5,3,2,1,1] => 4
[5,5,3,1,1,1,1] => [5,3,1,1,1,1] => 2
[5,5,2,2,2,1] => [5,2,2,2,1] => 4
[5,5,2,2,1,1,1] => [5,2,2,1,1,1] => 4
[5,5,2,1,1,1,1,1] => [5,2,1,1,1,1,1] => 4
[5,5,1,1,1,1,1,1,1] => [5,1,1,1,1,1,1,1] => 4
[5,4,4,4] => [4,4,4] => 4
[5,4,4,3,1] => [4,4,3,1] => 2
[5,4,4,2,2] => [4,4,2,2] => 0
[5,4,4,2,1,1] => [4,4,2,1,1] => 2
[5,4,4,1,1,1,1] => [4,4,1,1,1,1] => 0
[5,4,3,3,2] => [4,3,3,2] => 2
[5,4,3,3,1,1] => [4,3,3,1,1] => 4
[5,4,3,2,2,1] => [4,3,2,2,1] => 2
[5,4,3,2,1,1,1] => [4,3,2,1,1,1] => 2
[5,4,3,1,1,1,1,1] => [4,3,1,1,1,1,1] => 2
[5,4,2,2,2,2] => [4,2,2,2,2] => 4
[5,4,2,2,2,1,1] => [4,2,2,2,1,1] => 2
[5,4,2,2,1,1,1,1] => [4,2,2,1,1,1,1] => 4
[5,4,2,1,1,1,1,1,1] => [4,2,1,1,1,1,1,1] => 2
[5,4,1,1,1,1,1,1,1,1] => [4,1,1,1,1,1,1,1,1] => 4
[5,3,3,3,3] => [3,3,3,3] => 0
[5,3,3,3,2,1] => [3,3,3,2,1] => 2
[5,3,3,3,1,1,1] => [3,3,3,1,1,1] => 2
[5,3,3,2,2,2] => [3,3,2,2,2] => 2
[5,3,3,2,2,1,1] => [3,3,2,2,1,1] => 0
[5,3,3,2,1,1,1,1] => [3,3,2,1,1,1,1] => 2
[5,3,3,1,1,1,1,1,1] => [3,3,1,1,1,1,1,1] => 0
[5,3,2,2,2,2,1] => [3,2,2,2,2,1] => 2
[5,3,2,2,2,1,1,1] => [3,2,2,2,1,1,1] => 2
[5,3,2,2,1,1,1,1,1] => [3,2,2,1,1,1,1,1] => 2
[5,3,2,1,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1,1] => 2
[5,3,1,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1,1] => 2
[5,2,2,2,2,2,2] => [2,2,2,2,2,2] => 0
[5,2,2,2,2,2,1,1] => [2,2,2,2,2,1,1] => 2
[5,2,2,2,2,1,1,1,1] => [2,2,2,2,1,1,1,1] => 0
[5,2,2,2,1,1,1,1,1,1] => [2,2,2,1,1,1,1,1,1] => 2
[5,2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => 0
[5,2,1,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1] => 2
[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] => 0
[4,4,4,3,2] => [4,4,3,2] => 1
[4,4,4,3,1,1] => [4,4,3,1,1] => 3
[4,4,4,2,2,1] => [4,4,2,2,1] => 1
[4,4,3,3,2,1] => [4,3,3,2,1] => 3
[6,5,4,3,2,1] => [5,4,3,2,1] => 3
[5,5,4,3,2,1] => [5,4,3,2,1] => 3
[6,4,4,3,2,1] => [4,4,3,2,1] => 2
[5,4,4,3,2,1] => [4,4,3,2,1] => 2
[4,4,4,3,2,1] => [4,4,3,2,1] => 2
[6,5,3,3,2,1] => [5,3,3,2,1] => 4
[5,5,3,3,2,1] => [5,3,3,2,1] => 4
[6,4,3,3,2,1] => [4,3,3,2,1] => 3
[5,4,3,3,2,1] => [4,3,3,2,1] => 3
[6,3,3,3,2,1] => [3,3,3,2,1] => 2
[6,5,4,2,2,1] => [5,4,2,2,1] => 2
[5,5,4,2,2,1] => [5,4,2,2,1] => 2
[6,4,4,2,2,1] => [4,4,2,2,1] => 1
[5,4,4,2,2,1] => [4,4,2,2,1] => 1
[6,5,3,2,2,1] => [5,3,2,2,1] => 3
[5,5,3,2,2,1] => [5,3,2,2,1] => 3
[6,4,3,2,2,1] => [4,3,2,2,1] => 2
[6,5,2,2,2,1] => [5,2,2,2,1] => 4
[6,5,4,3,1,1] => [5,4,3,1,1] => 4
[5,5,4,3,1,1] => [5,4,3,1,1] => 4
[6,4,4,3,1,1] => [4,4,3,1,1] => 3
[5,4,4,3,1,1] => [4,4,3,1,1] => 3
[6,5,3,3,1,1] => [5,3,3,1,1] => 5
[5,5,3,3,1,1] => [5,3,3,1,1] => 5
[6,4,3,3,1,1] => [4,3,3,1,1] => 4
[6,5,4,2,1,1] => [5,4,2,1,1] => 3
[5,5,4,2,1,1] => [5,4,2,1,1] => 3
[6,4,4,2,1,1] => [4,4,2,1,1] => 2
[6,5,3,2,1,1] => [5,3,2,1,1] => 4
[6,5,4,1,1,1] => [5,4,1,1,1] => 2
[6,5,4,3,2] => [5,4,3,2] => 2
[5,5,4,3,2] => [5,4,3,2] => 2
[6,4,4,3,2] => [4,4,3,2] => 1
[5,4,4,3,2] => [4,4,3,2] => 1
[6,5,3,3,2] => [5,3,3,2] => 3
[5,5,3,3,2] => [5,3,3,2] => 3
[6,4,3,3,2] => [4,3,3,2] => 2
[6,5,4,2,2] => [5,4,2,2] => 1
[5,5,4,2,2] => [5,4,2,2] => 1
[6,4,4,2,2] => [4,4,2,2] => 0
[6,5,3,2,2] => [5,3,2,2] => 2
[6,5,4,3,1] => [5,4,3,1] => 3
[5,5,4,3,1] => [5,4,3,1] => 3
[6,4,4,3,1] => [4,4,3,1] => 2
[6,5,3,3,1] => [5,3,3,1] => 4
[6,5,4,2,1] => [5,4,2,1] => 2
[6,5,4,3] => [5,4,3] => 4
[7,6,5,4,3,2,1] => [6,5,4,3,2,1] => 3
[6,6,5,4,3,2,1] => [6,5,4,3,2,1] => 3
[7,2,2,2,2,2,1] => [2,2,2,2,2,1] => 1
[7,4,3,2,1,1,1] => [4,3,2,1,1,1] => 2
[6,4,3,2,1,1,1] => [4,3,2,1,1,1] => 2
[6,6,2,1,1,1,1] => [6,2,1,1,1,1] => 4
[7,6,1,1,1,1,1] => [6,1,1,1,1,1] => 5
[7,5,4,3,2,1] => [5,4,3,2,1] => 3
[7,4,4,3,2,1] => [4,4,3,2,1] => 2
[7,4,3,3,2,1] => [4,3,3,2,1] => 3
[7,3,3,3,2,1] => [3,3,3,2,1] => 2
[7,4,3,2,1,1] => [4,3,2,1,1] => 3
[7,3,3,3,3] => [3,3,3,3] => 0
[7,5,4,3,2] => [5,4,3,2] => 2
[7,5,4,3,1] => [5,4,3,1] => 3
[7,5,4,2,1] => [5,4,2,1] => 2
[7,6,3,2,1] => [6,3,2,1] => 4
[6,6,3,2,1] => [6,3,2,1] => 4
[7,5,4,3] => [5,4,3] => 4
[7,6,5] => [6,5] => 1
[6,2,2,2,2,2,2] => [2,2,2,2,2,2] => 0
[6,6,6] => [6,6] => 0
[7,6,6] => [6,6] => 0
[10,4,4] => [4,4] => 0
[10,10] => [10] => 10
[11,7,3] => [7,3] => 4
[8,2,2,2,2,2,2] => [2,2,2,2,2,2] => 0
[9,6,3] => [6,3] => 3
[8,6,4,2] => [6,4,2] => 4
[8,6,4] => [6,4] => 2
[10,6,4] => [6,4] => 2
[15,5,5] => [5,5] => 0
[9,7,5,3,1] => [7,5,3,1] => 4
[11,9,7,5,3,1] => [9,7,5,3,1] => 5
[8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 4
[7,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 4
[9,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 4
[8,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 4
[8,6,5,4,3,2,1] => [6,5,4,3,2,1] => 3
[9,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 4
[9,6,4] => [6,4] => 2
[10,7,3] => [7,3] => 4
[8,5,4,2] => [5,4,2] => 3
[9,7,4] => [7,4] => 3
[8,5,5,1] => [5,5,1] => 1
[9,5,5,1] => [5,5,1] => 1
[9,9] => [9] => 9
[7,7,1,1,1,1] => [7,1,1,1,1] => 7
[8,5,4,3,2,1] => [5,4,3,2,1] => 3
[8,7,5] => [7,5] => 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 2,2,1 3,3,1 4,4,2,1 5,6,3,1 6,8,5,2,1 8,11,7,3,1 10,14,10,5,2,1 12,19,14,7,3,1 15,24,19,11,5,2,1 18,31,26,15,7,3,1
$F_{1} = 1$
$F_{2} = 1 + q$
$F_{3} = 2 + q$
$F_{4} = 2 + 2\ q + q^{2}$
$F_{5} = 3 + 3\ q + q^{2}$
$F_{6} = 4 + 4\ q + 2\ q^{2} + q^{3}$
$F_{7} = 5 + 6\ q + 3\ q^{2} + q^{3}$
$F_{8} = 6 + 8\ q + 5\ q^{2} + 2\ q^{3} + q^{4}$
$F_{9} = 8 + 11\ q + 7\ q^{2} + 3\ q^{3} + q^{4}$
$F_{10} = 10 + 14\ q + 10\ q^{2} + 5\ q^{3} + 2\ q^{4} + q^{5}$
$F_{11} = 12 + 19\ q + 14\ q^{2} + 7\ q^{3} + 3\ q^{4} + q^{5}$
$F_{12} = 15 + 24\ q + 19\ q^{2} + 11\ q^{3} + 5\ q^{4} + 2\ q^{5} + q^{6}$
$F_{13} = 18 + 31\ q + 26\ q^{2} + 15\ q^{3} + 7\ q^{4} + 3\ q^{5} + q^{6}$
Description
The alternating sum of the parts of an integer partition.
For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Map
first row removal
Description
Removes the first entry of an integer partition
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