Identifier
-
Mp00323:
Integer partitions
—Loehr-Warrington inverse⟶
Integer partitions
St000010: Integer partitions ⟶ ℤ
Values
[1] => [1] => 1
[2] => [1,1] => 2
[1,1] => [2] => 1
[3] => [2,1] => 2
[2,1] => [1,1,1] => 3
[1,1,1] => [3] => 1
[4] => [2,2] => 2
[3,1] => [1,1,1,1] => 4
[2,2] => [2,1,1] => 3
[2,1,1] => [3,1] => 2
[1,1,1,1] => [4] => 1
[5] => [3,2] => 2
[4,1] => [3,1,1] => 3
[3,2] => [1,1,1,1,1] => 5
[3,1,1] => [2,1,1,1] => 4
[2,2,1] => [2,2,1] => 3
[2,1,1,1] => [4,1] => 2
[1,1,1,1,1] => [5] => 1
[6] => [3,3] => 2
[5,1] => [3,2,1] => 3
[4,2] => [2,1,1,1,1] => 5
[4,1,1] => [2,2,1,1] => 4
[3,3] => [3,1,1,1] => 4
[3,2,1] => [1,1,1,1,1,1] => 6
[3,1,1,1] => [4,1,1] => 3
[2,2,2] => [2,2,2] => 3
[2,2,1,1] => [4,2] => 2
[2,1,1,1,1] => [5,1] => 2
[1,1,1,1,1,1] => [6] => 1
[7] => [4,3] => 2
[6,1] => [3,3,1] => 3
[5,2] => [3,2,1,1] => 4
[5,1,1] => [4,2,1] => 3
[4,3] => [2,2,1,1,1] => 5
[4,2,1] => [1,1,1,1,1,1,1] => 7
[4,1,1,1] => [2,2,2,1] => 4
[3,3,1] => [2,1,1,1,1,1] => 6
[3,2,2] => [3,1,1,1,1] => 5
[3,2,1,1] => [4,1,1,1] => 4
[3,1,1,1,1] => [5,1,1] => 3
[2,2,2,1] => [3,2,2] => 3
[2,2,1,1,1] => [5,2] => 2
[2,1,1,1,1,1] => [6,1] => 2
[1,1,1,1,1,1,1] => [7] => 1
[8] => [4,4] => 2
[7,1] => [4,3,1] => 3
[6,2] => [3,3,1,1] => 4
[6,1,1] => [4,2,2] => 3
[5,3] => [2,2,2,1,1] => 5
[5,2,1] => [4,1,1,1,1] => 5
[5,1,1,1] => [2,2,2,2] => 4
[4,4] => [4,2,1,1] => 4
[4,3,1] => [1,1,1,1,1,1,1,1] => 8
[4,2,2] => [2,1,1,1,1,1,1] => 7
[4,2,1,1] => [3,1,1,1,1,1] => 6
[4,1,1,1,1] => [5,2,1] => 3
[3,3,2] => [2,2,1,1,1,1] => 6
[3,3,1,1] => [3,2,1,1,1] => 5
[3,2,2,1] => [3,2,2,1] => 4
[3,2,1,1,1] => [5,1,1,1] => 4
[3,1,1,1,1,1] => [6,1,1] => 3
[2,2,2,2] => [3,3,2] => 3
[2,2,2,1,1] => [5,3] => 2
[2,2,1,1,1,1] => [6,2] => 2
[2,1,1,1,1,1,1] => [7,1] => 2
[1,1,1,1,1,1,1,1] => [8] => 1
[9] => [5,4] => 2
[8,1] => [4,4,1] => 3
[7,2] => [4,3,1,1] => 4
[7,1,1] => [4,3,2] => 3
[6,3] => [4,2,2,1] => 4
[6,2,1] => [4,2,1,1,1] => 5
[6,1,1,1] => [5,2,2] => 3
[5,4] => [2,2,2,2,1] => 5
[5,3,1] => [3,1,1,1,1,1,1] => 7
[5,2,2] => [3,2,1,1,1,1] => 6
[5,2,1,1] => [3,3,1,1,1] => 5
[5,1,1,1,1] => [3,2,2,2] => 4
[4,4,1] => [4,1,1,1,1,1] => 6
[4,3,2] => [1,1,1,1,1,1,1,1,1] => 9
[4,3,1,1] => [2,1,1,1,1,1,1,1] => 8
[4,2,2,1] => [2,2,1,1,1,1,1] => 7
[4,2,1,1,1] => [5,1,1,1,1] => 5
[4,1,1,1,1,1] => [6,2,1] => 3
[3,3,3] => [3,2,2,1,1] => 5
[3,3,2,1] => [2,2,2,1,1,1] => 6
[3,3,1,1,1] => [5,2,1,1] => 4
[3,2,2,2] => [3,3,2,1] => 4
[3,2,2,1,1] => [5,3,1] => 3
[3,2,1,1,1,1] => [6,1,1,1] => 4
[3,1,1,1,1,1,1] => [7,1,1] => 3
[2,2,2,2,1] => [3,3,3] => 3
[2,2,2,1,1,1] => [6,3] => 2
[2,2,1,1,1,1,1] => [7,2] => 2
[2,1,1,1,1,1,1,1] => [8,1] => 2
[1,1,1,1,1,1,1,1,1] => [9] => 1
[10] => [5,5] => 2
[9,1] => [5,4,1] => 3
[8,2] => [4,4,1,1] => 4
[8,1,1] => [4,4,2] => 3
[7,3] => [4,3,2,1] => 4
>>> Load all 1211 entries. <<<[7,2,1] => [4,3,1,1,1] => 5
[7,1,1,1] => [5,3,2] => 3
[6,4] => [2,2,2,2,2] => 5
[6,3,1] => [4,2,1,1,1,1] => 6
[6,2,2] => [3,3,2,1,1] => 5
[6,2,1,1] => [5,3,1,1] => 4
[6,1,1,1,1] => [3,3,2,2] => 4
[5,5] => [4,2,2,2] => 4
[5,4,1] => [3,3,1,1,1,1] => 6
[5,3,2] => [2,1,1,1,1,1,1,1,1] => 9
[5,3,1,1] => [2,2,1,1,1,1,1,1] => 8
[5,2,2,1] => [2,2,2,1,1,1,1] => 7
[5,2,1,1,1] => [3,2,2,2,1] => 5
[5,1,1,1,1,1] => [6,2,2] => 3
[4,4,2] => [3,1,1,1,1,1,1,1] => 8
[4,4,1,1] => [3,2,1,1,1,1,1] => 7
[4,3,3] => [4,1,1,1,1,1,1] => 7
[4,3,2,1] => [1,1,1,1,1,1,1,1,1,1] => 10
[4,3,1,1,1] => [5,1,1,1,1,1] => 6
[4,2,2,2] => [3,2,2,1,1,1] => 6
[4,2,2,1,1] => [5,2,1,1,1] => 5
[4,2,1,1,1,1] => [6,1,1,1,1] => 5
[4,1,1,1,1,1,1] => [7,2,1] => 3
[3,3,3,1] => [2,2,2,2,1,1] => 6
[3,3,2,2] => [4,2,2,1,1] => 5
[3,3,2,1,1] => [5,2,2,1] => 4
[3,3,1,1,1,1] => [6,2,1,1] => 4
[3,2,2,2,1] => [3,3,3,1] => 4
[3,2,2,1,1,1] => [6,3,1] => 3
[3,2,1,1,1,1,1] => [7,1,1,1] => 4
[3,1,1,1,1,1,1,1] => [8,1,1] => 3
[2,2,2,2,2] => [4,3,3] => 3
[2,2,2,2,1,1] => [6,4] => 2
[2,2,2,1,1,1,1] => [7,3] => 2
[2,2,1,1,1,1,1,1] => [8,2] => 2
[2,1,1,1,1,1,1,1,1] => [9,1] => 2
[1,1,1,1,1,1,1,1,1,1] => [10] => 1
[11] => [6,5] => 2
[10,1] => [5,5,1] => 3
[9,2] => [5,4,1,1] => 4
[9,1,1] => [5,4,2] => 3
[8,3] => [4,4,2,1] => 4
[8,2,1] => [4,4,1,1,1] => 5
[8,1,1,1] => [5,3,3] => 3
[7,4] => [5,2,2,2] => 4
[7,3,1] => [4,3,1,1,1,1] => 6
[7,2,2] => [4,3,2,1,1] => 5
[7,2,1,1] => [5,3,2,1] => 4
[7,1,1,1,1] => [6,3,2] => 3
[6,5] => [3,2,2,2,2] => 5
[6,4,1] => [3,2,2,2,1,1] => 6
[6,3,2] => [4,2,1,1,1,1,1] => 7
[6,3,1,1] => [5,2,1,1,1,1] => 6
[6,2,2,1] => [5,2,2,1,1] => 5
[6,2,1,1,1] => [3,3,2,2,1] => 5
[6,1,1,1,1,1] => [3,3,3,2] => 4
[5,5,1] => [5,3,1,1,1] => 5
[5,4,2] => [2,2,1,1,1,1,1,1,1] => 9
[5,4,1,1] => [2,2,2,1,1,1,1,1] => 8
[5,3,3] => [3,2,1,1,1,1,1,1] => 8
[5,3,2,1] => [1,1,1,1,1,1,1,1,1,1,1] => 11
[5,3,1,1,1] => [3,2,2,1,1,1,1] => 7
[5,2,2,2] => [2,2,2,2,1,1,1] => 7
[5,2,2,1,1] => [4,2,2,1,1,1] => 6
[5,2,1,1,1,1] => [6,3,1,1] => 4
[5,1,1,1,1,1,1] => [7,2,2] => 3
[4,4,3] => [3,3,1,1,1,1,1] => 7
[4,4,2,1] => [2,1,1,1,1,1,1,1,1,1] => 10
[4,4,1,1,1] => [3,3,2,1,1,1] => 6
[4,3,3,1] => [3,1,1,1,1,1,1,1,1] => 9
[4,3,2,2] => [4,1,1,1,1,1,1,1] => 8
[4,3,2,1,1] => [5,1,1,1,1,1,1] => 7
[4,3,1,1,1,1] => [6,1,1,1,1,1] => 6
[4,2,2,2,1] => [3,3,3,1,1] => 5
[4,2,2,1,1,1] => [6,2,1,1,1] => 5
[4,2,1,1,1,1,1] => [7,1,1,1,1] => 5
[4,1,1,1,1,1,1,1] => [8,2,1] => 3
[3,3,3,2] => [2,2,2,2,2,1] => 6
[3,3,3,1,1] => [4,2,2,2,1] => 5
[3,3,2,2,1] => [4,3,2,2] => 4
[3,3,2,1,1,1] => [6,2,2,1] => 4
[3,3,1,1,1,1,1] => [7,2,1,1] => 4
[3,2,2,2,2] => [4,3,3,1] => 4
[3,2,2,2,1,1] => [6,4,1] => 3
[3,2,2,1,1,1,1] => [7,3,1] => 3
[3,2,1,1,1,1,1,1] => [8,1,1,1] => 4
[3,1,1,1,1,1,1,1,1] => [9,1,1] => 3
[2,2,2,2,2,1] => [4,4,3] => 3
[2,2,2,2,1,1,1] => [7,4] => 2
[2,2,2,1,1,1,1,1] => [8,3] => 2
[2,2,1,1,1,1,1,1,1] => [9,2] => 2
[2,1,1,1,1,1,1,1,1,1] => [10,1] => 2
[1,1,1,1,1,1,1,1,1,1,1] => [11] => 1
[12] => [6,6] => 2
[11,1] => [6,5,1] => 3
[10,2] => [5,5,1,1] => 4
[10,1,1] => [5,5,2] => 3
[9,3] => [5,4,2,1] => 4
[9,2,1] => [5,4,1,1,1] => 5
[9,1,1,1] => [5,4,3] => 3
[8,4] => [4,4,2,2] => 4
[8,3,1] => [4,4,1,1,1,1] => 6
[8,2,2] => [4,4,2,1,1] => 5
[8,2,1,1] => [5,3,3,1] => 4
[8,1,1,1,1] => [6,3,3] => 3
[7,5] => [3,3,2,2,2] => 5
[7,4,1] => [5,3,2,1,1] => 5
[7,3,2] => [4,3,1,1,1,1,1] => 7
[7,3,1,1] => [3,3,2,2,1,1] => 6
[7,2,2,1] => [5,2,2,2,1] => 5
[7,2,1,1,1] => [6,3,2,1] => 4
[7,1,1,1,1,1] => [3,3,3,3] => 4
[6,6] => [5,3,2,2] => 4
[6,5,1] => [3,2,2,2,2,1] => 6
[6,4,2] => [3,2,2,1,1,1,1,1] => 8
[6,4,1,1] => [3,2,2,2,1,1,1] => 7
[6,3,3] => [3,3,2,1,1,1,1] => 7
[6,3,2,1] => [5,1,1,1,1,1,1,1] => 8
[6,3,1,1,1] => [4,3,2,1,1,1] => 6
[6,2,2,2] => [4,2,2,2,1,1] => 6
[6,2,2,1,1] => [4,3,2,2,1] => 5
[6,2,1,1,1,1] => [3,3,3,2,1] => 5
[6,1,1,1,1,1,1] => [7,3,2] => 3
[5,5,2] => [5,2,1,1,1,1,1] => 7
[5,5,1,1] => [5,2,2,1,1,1] => 6
[5,4,3] => [2,2,2,1,1,1,1,1,1] => 9
[5,4,2,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => 12
[5,4,1,1,1] => [2,2,2,2,1,1,1,1] => 8
[5,3,3,1] => [2,1,1,1,1,1,1,1,1,1,1] => 11
[5,3,2,2] => [3,1,1,1,1,1,1,1,1,1] => 10
[5,3,2,1,1] => [4,1,1,1,1,1,1,1,1] => 9
[5,3,1,1,1,1] => [6,2,1,1,1,1] => 6
[5,2,2,2,1] => [2,2,2,2,2,1,1] => 7
[5,2,2,1,1,1] => [6,2,2,1,1] => 5
[5,2,1,1,1,1,1] => [7,3,1,1] => 4
[5,1,1,1,1,1,1,1] => [8,2,2] => 3
[4,4,4] => [5,3,1,1,1,1] => 6
[4,4,3,1] => [2,2,1,1,1,1,1,1,1,1] => 10
[4,4,2,2] => [3,2,1,1,1,1,1,1,1] => 9
[4,4,2,1,1] => [4,2,1,1,1,1,1,1] => 8
[4,4,1,1,1,1] => [6,3,1,1,1] => 5
[4,3,3,2] => [3,3,1,1,1,1,1,1] => 8
[4,3,3,1,1] => [4,2,2,1,1,1,1] => 7
[4,3,2,2,1] => [3,3,3,1,1,1] => 6
[4,3,2,1,1,1] => [6,1,1,1,1,1,1] => 7
[4,3,1,1,1,1,1] => [7,1,1,1,1,1] => 6
[4,2,2,2,2] => [4,3,3,1,1] => 5
[4,2,2,2,1,1] => [6,4,1,1] => 4
[4,2,2,1,1,1,1] => [7,2,1,1,1] => 5
[4,2,1,1,1,1,1,1] => [8,1,1,1,1] => 5
[4,1,1,1,1,1,1,1,1] => [9,2,1] => 3
[3,3,3,3] => [4,2,2,2,2] => 5
[3,3,3,2,1] => [2,2,2,2,2,2] => 6
[3,3,3,1,1,1] => [6,2,2,2] => 4
[3,3,2,2,2] => [4,3,3,2] => 4
[3,3,2,2,1,1] => [6,4,2] => 3
[3,3,2,1,1,1,1] => [7,2,2,1] => 4
[3,3,1,1,1,1,1,1] => [8,2,1,1] => 4
[3,2,2,2,2,1] => [4,4,3,1] => 4
[3,2,2,2,1,1,1] => [7,4,1] => 3
[3,2,2,1,1,1,1,1] => [8,3,1] => 3
[3,2,1,1,1,1,1,1,1] => [9,1,1,1] => 4
[3,1,1,1,1,1,1,1,1,1] => [10,1,1] => 3
[2,2,2,2,2,2] => [4,4,4] => 3
[2,2,2,2,2,1,1] => [7,5] => 2
[2,2,2,2,1,1,1,1] => [8,4] => 2
[2,2,2,1,1,1,1,1,1] => [9,3] => 2
[2,2,1,1,1,1,1,1,1,1] => [10,2] => 2
[2,1,1,1,1,1,1,1,1,1,1] => [11,1] => 2
[1,1,1,1,1,1,1,1,1,1,1,1] => [12] => 1
[13] => [7,6] => 2
[12,1] => [6,6,1] => 3
[11,2] => [6,5,1,1] => 4
[11,1,1] => [6,5,2] => 3
[10,3] => [5,5,2,1] => 4
[10,2,1] => [5,5,1,1,1] => 5
[10,1,1,1] => [5,5,3] => 3
[9,4] => [5,4,2,2] => 4
[9,3,1] => [5,4,1,1,1,1] => 6
[9,2,2] => [5,4,2,1,1] => 5
[9,2,1,1] => [5,4,3,1] => 4
[9,1,1,1,1] => [6,4,3] => 3
[8,5] => [5,3,3,2] => 4
[8,4,1] => [5,3,3,1,1] => 5
[8,3,2] => [4,4,1,1,1,1,1] => 7
[8,3,1,1] => [4,4,2,1,1,1] => 6
[8,2,2,1] => [5,2,2,2,2] => 5
[8,2,1,1,1] => [6,3,3,1] => 4
[8,1,1,1,1,1] => [7,3,3] => 3
[7,6] => [3,3,3,2,2] => 5
[7,5,1] => [3,3,2,2,2,1] => 6
[7,4,2] => [5,3,1,1,1,1,1] => 7
[7,4,1,1] => [5,2,2,2,1,1] => 6
[7,3,3] => [5,3,2,1,1,1] => 6
[7,3,2,1] => [4,2,2,1,1,1,1,1] => 8
[7,3,1,1,1] => [6,3,2,1,1] => 5
[7,2,2,2] => [4,3,3,2,1] => 5
[7,2,2,1,1] => [6,4,2,1] => 4
[7,2,1,1,1,1] => [3,3,3,3,1] => 5
[7,1,1,1,1,1,1] => [4,3,3,3] => 4
[6,6,1] => [5,3,2,2,1] => 5
[6,5,2] => [3,3,2,2,1,1,1] => 7
[6,5,1,1] => [4,3,2,2,1,1] => 6
[6,4,3] => [2,2,2,2,1,1,1,1,1] => 9
[6,4,2,1] => [4,1,1,1,1,1,1,1,1,1] => 10
[6,4,1,1,1] => [2,2,2,2,2,1,1,1] => 8
[6,3,3,1] => [4,2,1,1,1,1,1,1,1] => 9
[6,3,2,2] => [4,3,1,1,1,1,1,1] => 8
[6,3,2,1,1] => [4,3,2,1,1,1,1] => 7
[6,3,1,1,1,1] => [3,3,3,2,1,1] => 6
[6,2,2,2,1] => [2,2,2,2,2,2,1] => 7
[6,2,2,1,1,1] => [4,4,2,2,1] => 5
[6,2,1,1,1,1,1] => [7,3,2,1] => 4
[6,1,1,1,1,1,1,1] => [8,3,2] => 3
[5,5,3] => [3,2,2,2,1,1,1,1] => 8
[5,5,2,1] => [5,1,1,1,1,1,1,1,1] => 9
[5,5,1,1,1] => [4,2,2,2,1,1,1] => 7
[5,4,4] => [5,2,2,1,1,1,1] => 7
[5,4,3,1] => [1,1,1,1,1,1,1,1,1,1,1,1,1] => 13
[5,4,2,2] => [2,1,1,1,1,1,1,1,1,1,1,1] => 12
[5,4,2,1,1] => [3,1,1,1,1,1,1,1,1,1,1] => 11
[5,4,1,1,1,1] => [6,2,2,1,1,1] => 6
[5,3,3,2] => [2,2,1,1,1,1,1,1,1,1,1] => 11
[5,3,3,1,1] => [3,2,1,1,1,1,1,1,1,1] => 10
[5,3,2,2,1] => [3,3,1,1,1,1,1,1,1] => 9
[5,3,2,1,1,1] => [6,1,1,1,1,1,1,1] => 8
[5,3,1,1,1,1,1] => [7,2,1,1,1,1] => 6
[5,2,2,2,2] => [4,2,2,2,2,1] => 6
[5,2,2,2,1,1] => [6,2,2,2,1] => 5
[5,2,2,1,1,1,1] => [7,2,2,1,1] => 5
[5,2,1,1,1,1,1,1] => [8,3,1,1] => 4
[5,1,1,1,1,1,1,1,1] => [9,2,2] => 3
[4,4,4,1] => [5,2,1,1,1,1,1,1] => 8
[4,4,3,2] => [2,2,2,1,1,1,1,1,1,1] => 10
[4,4,3,1,1] => [3,2,2,1,1,1,1,1,1] => 9
[4,4,2,2,1] => [3,3,2,1,1,1,1,1] => 8
[4,4,2,1,1,1] => [6,2,1,1,1,1,1] => 7
[4,4,1,1,1,1,1] => [7,3,1,1,1] => 5
[4,3,3,3] => [3,3,3,1,1,1,1] => 7
[4,3,3,2,1] => [3,2,2,2,2,1,1] => 7
[4,3,3,1,1,1] => [6,3,1,1,1,1] => 6
[4,3,2,2,2] => [4,3,3,1,1,1] => 6
[4,3,2,2,1,1] => [6,4,1,1,1] => 5
[4,3,2,1,1,1,1] => [7,1,1,1,1,1,1] => 7
[4,3,1,1,1,1,1,1] => [8,1,1,1,1,1] => 6
[4,2,2,2,2,1] => [4,4,3,1,1] => 5
[4,2,2,2,1,1,1] => [7,4,1,1] => 4
[4,2,2,1,1,1,1,1] => [8,2,1,1,1] => 5
[4,2,1,1,1,1,1,1,1] => [9,1,1,1,1] => 5
[4,1,1,1,1,1,1,1,1,1] => [10,2,1] => 3
[3,3,3,3,1] => [3,2,2,2,2,2] => 6
[3,3,3,2,2] => [4,3,2,2,2] => 5
[3,3,3,2,1,1] => [6,3,2,2] => 4
[3,3,3,1,1,1,1] => [7,2,2,2] => 4
[3,3,2,2,2,1] => [4,4,3,2] => 4
[3,3,2,2,1,1,1] => [7,4,2] => 3
[3,3,2,1,1,1,1,1] => [8,2,2,1] => 4
[3,3,1,1,1,1,1,1,1] => [9,2,1,1] => 4
[3,2,2,2,2,2] => [4,4,4,1] => 4
[3,2,2,2,2,1,1] => [7,5,1] => 3
[3,2,2,2,1,1,1,1] => [8,4,1] => 3
[3,2,2,1,1,1,1,1,1] => [9,3,1] => 3
[3,2,1,1,1,1,1,1,1,1] => [10,1,1,1] => 4
[3,1,1,1,1,1,1,1,1,1,1] => [11,1,1] => 3
[2,2,2,2,2,2,1] => [5,4,4] => 3
[2,2,2,2,2,1,1,1] => [8,5] => 2
[2,2,2,2,1,1,1,1,1] => [9,4] => 2
[2,2,2,1,1,1,1,1,1,1] => [10,3] => 2
[2,2,1,1,1,1,1,1,1,1,1] => [11,2] => 2
[2,1,1,1,1,1,1,1,1,1,1,1] => [12,1] => 2
[1,1,1,1,1,1,1,1,1,1,1,1,1] => [13] => 1
[14] => [7,7] => 2
[13,1] => [7,6,1] => 3
[12,2] => [6,6,1,1] => 4
[12,1,1] => [6,6,2] => 3
[11,3] => [6,5,2,1] => 4
[11,2,1] => [6,5,1,1,1] => 5
[11,1,1,1] => [6,5,3] => 3
[10,4] => [5,5,2,2] => 4
[10,3,1] => [5,5,1,1,1,1] => 6
[10,2,2] => [5,5,2,1,1] => 5
[10,2,1,1] => [5,5,3,1] => 4
[10,1,1,1,1] => [6,4,4] => 3
[9,5] => [5,4,3,2] => 4
[9,4,1] => [5,4,3,1,1] => 5
[9,3,2] => [5,4,1,1,1,1,1] => 7
[9,3,1,1] => [5,4,2,1,1,1] => 6
[9,2,2,1] => [5,4,2,2,1] => 5
[9,2,1,1,1] => [6,4,3,1] => 4
[9,1,1,1,1,1] => [7,4,3] => 3
[8,6] => [3,3,3,3,2] => 5
[8,5,1] => [5,3,3,2,1] => 5
[8,4,2] => [5,3,2,1,1,1,1] => 7
[8,4,1,1] => [5,3,2,2,1,1] => 6
[8,3,3] => [5,3,3,1,1,1] => 6
[8,3,2,1] => [4,4,1,1,1,1,1,1] => 8
[8,3,1,1,1] => [6,3,3,1,1] => 5
[8,2,2,2] => [4,4,2,2,2] => 5
[8,2,2,1,1] => [6,4,2,2] => 4
[8,2,1,1,1,1] => [7,3,3,1] => 4
[8,1,1,1,1,1,1] => [4,4,3,3] => 4
[7,7] => [6,3,3,2] => 4
[7,6,1] => [3,3,3,2,2,1] => 6
[7,5,2] => [3,3,2,2,2,1,1] => 7
[7,5,1,1] => [4,4,2,2,1,1] => 6
[7,4,3] => [5,2,2,2,1,1,1] => 7
[7,4,2,1] => [5,2,1,1,1,1,1,1,1] => 9
[7,4,1,1,1] => [6,2,2,2,1,1] => 6
[7,3,3,1] => [5,3,1,1,1,1,1,1] => 8
[7,3,2,2] => [4,3,3,1,1,1,1] => 7
[7,3,2,1,1] => [6,4,1,1,1,1] => 6
[7,3,1,1,1,1] => [3,3,3,3,1,1] => 6
[7,2,2,2,1] => [6,2,2,2,2] => 5
[7,2,2,1,1,1] => [4,3,3,2,2] => 5
[7,2,1,1,1,1,1] => [4,3,3,3,1] => 5
[7,1,1,1,1,1,1,1] => [8,3,3] => 3
[6,6,2] => [4,3,3,2,1,1] => 6
[6,6,1,1] => [6,4,2,1,1] => 5
[6,5,3] => [2,2,2,2,2,1,1,1,1] => 9
[6,5,2,1] => [4,3,2,1,1,1,1,1] => 8
[6,5,1,1,1] => [2,2,2,2,2,2,1,1] => 8
[6,4,4] => [3,2,2,2,2,1,1,1] => 8
[6,4,3,1] => [3,1,1,1,1,1,1,1,1,1,1,1] => 12
[6,4,2,2] => [3,2,1,1,1,1,1,1,1,1,1] => 11
[6,4,2,1,1] => [3,3,1,1,1,1,1,1,1,1] => 10
[6,4,1,1,1,1] => [4,2,2,2,2,1,1] => 7
[6,3,3,2] => [3,2,2,1,1,1,1,1,1,1] => 10
[6,3,3,1,1] => [3,3,2,1,1,1,1,1,1] => 9
[6,3,2,2,1] => [3,3,3,1,1,1,1,1] => 8
[6,3,2,1,1,1] => [4,4,2,1,1,1,1] => 7
[6,3,1,1,1,1,1] => [7,3,2,1,1] => 5
[6,2,2,2,2] => [2,2,2,2,2,2,2] => 7
[6,2,2,2,1,1] => [4,2,2,2,2,2] => 6
[6,2,2,1,1,1,1] => [7,4,2,1] => 4
[6,2,1,1,1,1,1,1] => [8,3,2,1] => 4
[6,1,1,1,1,1,1,1,1] => [9,3,2] => 3
[5,5,4] => [4,3,2,2,1,1,1] => 7
[5,5,3,1] => [4,1,1,1,1,1,1,1,1,1,1] => 11
[5,5,2,2] => [4,2,1,1,1,1,1,1,1,1] => 10
[5,5,2,1,1] => [4,3,1,1,1,1,1,1,1] => 9
[5,5,1,1,1,1] => [5,2,2,2,2,1] => 6
[5,4,4,1] => [5,1,1,1,1,1,1,1,1,1] => 10
[5,4,3,2] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 14
[5,4,3,1,1] => [2,1,1,1,1,1,1,1,1,1,1,1,1] => 13
[5,4,2,2,1] => [2,2,1,1,1,1,1,1,1,1,1,1] => 12
[5,4,2,1,1,1] => [6,1,1,1,1,1,1,1,1] => 9
[5,4,1,1,1,1,1] => [7,2,2,1,1,1] => 6
[5,3,3,3] => [3,2,2,2,1,1,1,1,1] => 9
[5,3,3,2,1] => [2,2,2,1,1,1,1,1,1,1,1] => 11
[5,3,3,1,1,1] => [6,2,1,1,1,1,1,1] => 8
[5,3,2,2,2] => [3,3,2,2,1,1,1,1] => 8
[5,3,2,2,1,1] => [6,3,1,1,1,1,1] => 7
[5,3,2,1,1,1,1] => [7,1,1,1,1,1,1,1] => 8
[5,3,1,1,1,1,1,1] => [8,2,1,1,1,1] => 6
[5,2,2,2,2,1] => [4,4,3,2,1] => 5
[5,2,2,2,1,1,1] => [7,2,2,2,1] => 5
[5,2,2,1,1,1,1,1] => [8,2,2,1,1] => 5
[5,2,1,1,1,1,1,1,1] => [9,3,1,1] => 4
[5,1,1,1,1,1,1,1,1,1] => [10,2,2] => 3
[4,4,4,2] => [4,2,2,1,1,1,1,1,1] => 9
[4,4,4,1,1] => [4,2,2,2,1,1,1,1] => 8
[4,4,3,3] => [5,2,2,1,1,1,1,1] => 8
[4,4,3,2,1] => [2,2,2,2,1,1,1,1,1,1] => 10
[4,4,3,1,1,1] => [6,2,2,1,1,1,1] => 7
[4,4,2,2,2] => [3,3,3,2,1,1,1] => 7
[4,4,2,2,1,1] => [6,3,2,1,1,1] => 6
[4,4,2,1,1,1,1] => [7,2,1,1,1,1,1] => 7
[4,4,1,1,1,1,1,1] => [8,3,1,1,1] => 5
[4,3,3,3,1] => [3,2,2,2,2,2,1] => 7
[4,3,3,2,2] => [4,3,2,2,2,1] => 6
[4,3,3,2,1,1] => [6,3,2,2,1] => 5
[4,3,3,1,1,1,1] => [7,3,1,1,1,1] => 6
[4,3,2,2,2,1] => [4,4,3,1,1,1] => 6
[4,3,2,2,1,1,1] => [7,4,1,1,1] => 5
[4,3,2,1,1,1,1,1] => [8,1,1,1,1,1,1] => 7
[4,3,1,1,1,1,1,1,1] => [9,1,1,1,1,1] => 6
[4,2,2,2,2,2] => [4,4,4,1,1] => 5
[4,2,2,2,2,1,1] => [7,5,1,1] => 4
[4,2,2,2,1,1,1,1] => [8,4,1,1] => 4
[4,2,2,1,1,1,1,1,1] => [9,2,1,1,1] => 5
[4,2,1,1,1,1,1,1,1,1] => [10,1,1,1,1] => 5
[4,1,1,1,1,1,1,1,1,1,1] => [11,2,1] => 3
[3,3,3,3,2] => [3,3,2,2,2,2] => 6
[3,3,3,3,1,1] => [5,3,2,2,2] => 5
[3,3,3,2,2,1] => [5,3,3,3] => 4
[3,3,3,2,1,1,1] => [7,3,2,2] => 4
[3,3,3,1,1,1,1,1] => [8,2,2,2] => 4
[3,3,2,2,2,2] => [4,4,4,2] => 4
[3,3,2,2,2,1,1] => [7,5,2] => 3
[3,3,2,2,1,1,1,1] => [8,4,2] => 3
[3,3,2,1,1,1,1,1,1] => [9,2,2,1] => 4
[3,3,1,1,1,1,1,1,1,1] => [10,2,1,1] => 4
[3,2,2,2,2,2,1] => [5,4,4,1] => 4
[3,2,2,2,2,1,1,1] => [8,5,1] => 3
[3,2,2,2,1,1,1,1,1] => [9,4,1] => 3
[3,2,2,1,1,1,1,1,1,1] => [10,3,1] => 3
[3,2,1,1,1,1,1,1,1,1,1] => [11,1,1,1] => 4
[3,1,1,1,1,1,1,1,1,1,1,1] => [12,1,1] => 3
[2,2,2,2,2,2,2] => [5,5,4] => 3
[2,2,2,2,2,2,1,1] => [8,6] => 2
[2,2,2,2,2,1,1,1,1] => [9,5] => 2
[2,2,2,2,1,1,1,1,1,1] => [10,4] => 2
[2,2,2,1,1,1,1,1,1,1,1] => [11,3] => 2
[2,2,1,1,1,1,1,1,1,1,1,1] => [12,2] => 2
[2,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1] => 2
[1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14] => 1
[15] => [8,7] => 2
[14,1] => [7,7,1] => 3
[13,2] => [7,6,1,1] => 4
[13,1,1] => [7,6,2] => 3
[12,3] => [6,6,2,1] => 4
[12,2,1] => [6,6,1,1,1] => 5
[12,1,1,1] => [6,6,3] => 3
[11,4] => [6,5,2,2] => 4
[11,3,1] => [6,5,1,1,1,1] => 6
[11,2,2] => [6,5,2,1,1] => 5
[11,2,1,1] => [6,5,3,1] => 4
[11,1,1,1,1] => [6,5,4] => 3
[10,5] => [5,5,3,2] => 4
[10,4,1] => [5,5,3,1,1] => 5
[10,3,2] => [5,5,1,1,1,1,1] => 7
[10,3,1,1] => [5,5,2,1,1,1] => 6
[10,2,2,1] => [5,5,2,2,1] => 5
[10,2,1,1,1] => [6,4,4,1] => 4
[10,1,1,1,1,1] => [7,4,4] => 3
[9,6] => [6,3,3,3] => 4
[9,5,1] => [5,4,3,2,1] => 5
[9,4,2] => [5,4,2,1,1,1,1] => 7
[9,4,1,1] => [5,4,2,2,1,1] => 6
[9,3,3] => [5,4,3,1,1,1] => 6
[9,3,2,1] => [5,4,1,1,1,1,1,1] => 8
[9,3,1,1,1] => [6,4,3,1,1] => 5
[9,2,2,2] => [5,4,2,2,2] => 5
[9,2,2,1,1] => [6,4,3,2] => 4
[9,2,1,1,1,1] => [7,4,3,1] => 4
[9,1,1,1,1,1,1] => [8,4,3] => 3
[8,7] => [3,3,3,3,3] => 5
[8,6,1] => [3,3,3,3,2,1] => 6
[8,5,2] => [5,3,3,2,1,1] => 6
[8,5,1,1] => [6,4,2,2,1] => 5
[8,4,3] => [5,3,2,2,1,1,1] => 7
[8,4,2,1] => [5,3,1,1,1,1,1,1,1] => 9
[8,4,1,1,1] => [4,4,2,2,2,1] => 6
[8,3,3,1] => [5,3,2,1,1,1,1,1] => 8
[8,3,2,2] => [4,3,3,2,1,1,1] => 7
[8,3,2,1,1] => [6,4,2,1,1,1] => 6
[8,3,1,1,1,1] => [7,3,3,1,1] => 5
[8,2,2,2,1] => [6,3,2,2,2] => 5
[8,2,2,1,1,1] => [7,4,2,2] => 4
[8,2,1,1,1,1,1] => [4,4,3,3,1] => 5
[8,1,1,1,1,1,1,1] => [4,4,4,3] => 4
[7,7,1] => [6,3,3,2,1] => 5
[7,6,2] => [3,3,3,2,2,1,1] => 7
[7,6,1,1] => [4,3,2,2,2,2] => 6
[7,5,3] => [4,2,2,2,2,1,1,1] => 8
[7,5,2,1] => [4,4,2,1,1,1,1,1] => 8
[7,5,1,1,1] => [4,2,2,2,2,2,1] => 7
[7,4,4] => [4,3,2,2,2,1,1] => 7
[7,4,3,1] => [5,2,1,1,1,1,1,1,1,1] => 10
[7,4,2,2] => [4,3,2,1,1,1,1,1,1] => 9
[7,4,2,1,1] => [6,3,1,1,1,1,1,1] => 8
[7,4,1,1,1,1] => [4,3,3,2,2,1] => 6
[7,3,3,2] => [3,3,3,2,1,1,1,1] => 8
[7,3,3,1,1] => [6,3,2,1,1,1,1] => 7
[7,3,2,2,1] => [6,3,3,1,1,1] => 6
[7,3,2,1,1,1] => [3,3,3,3,1,1,1] => 7
[7,3,1,1,1,1,1] => [4,3,3,3,1,1] => 6
[7,2,2,2,2] => [5,2,2,2,2,2] => 6
[7,2,2,2,1,1] => [4,4,3,2,2] => 5
[7,2,2,1,1,1,1] => [4,3,3,3,2] => 5
[7,2,1,1,1,1,1,1] => [8,3,3,1] => 4
[7,1,1,1,1,1,1,1,1] => [9,3,3] => 3
[6,6,3] => [5,2,2,2,2,1,1] => 7
[6,6,2,1] => [6,4,1,1,1,1,1] => 7
[6,6,1,1,1] => [6,2,2,2,2,1] => 6
[6,5,4] => [2,2,2,2,2,2,1,1,1] => 9
[6,5,3,1] => [3,3,1,1,1,1,1,1,1,1,1] => 11
[6,5,2,2] => [3,3,2,1,1,1,1,1,1,1] => 10
[6,5,2,1,1] => [3,3,3,1,1,1,1,1,1] => 9
[6,5,1,1,1,1] => [2,2,2,2,2,2,2,1] => 8
[6,4,4,1] => [4,3,1,1,1,1,1,1,1,1] => 10
[6,4,3,2] => [2,1,1,1,1,1,1,1,1,1,1,1,1,1] => 14
[6,4,3,1,1] => [2,2,1,1,1,1,1,1,1,1,1,1,1] => 13
[6,4,2,2,1] => [2,2,2,1,1,1,1,1,1,1,1,1] => 12
[6,4,2,1,1,1] => [4,2,2,2,1,1,1,1,1] => 9
[6,4,1,1,1,1,1] => [7,2,2,2,1,1] => 6
[6,3,3,3] => [3,3,2,2,1,1,1,1,1] => 9
[6,3,3,2,1] => [2,2,2,2,1,1,1,1,1,1,1] => 11
[6,3,3,1,1,1] => [3,3,2,2,2,1,1,1] => 8
[6,3,2,2,2] => [3,2,2,2,2,2,1,1] => 8
[6,3,2,2,1,1] => [5,3,3,1,1,1,1] => 7
[6,3,2,1,1,1,1] => [7,4,1,1,1,1] => 6
[6,3,1,1,1,1,1,1] => [8,3,2,1,1] => 5
[6,2,2,2,2,1] => [3,2,2,2,2,2,2] => 7
[6,2,2,2,1,1,1] => [7,2,2,2,2] => 5
[6,2,2,1,1,1,1,1] => [8,4,2,1] => 4
[6,2,1,1,1,1,1,1,1] => [9,3,2,1] => 4
[6,1,1,1,1,1,1,1,1,1] => [10,3,2] => 3
[5,5,5] => [6,3,2,2,1,1] => 6
[5,5,4,1] => [4,4,1,1,1,1,1,1,1] => 9
[5,5,3,2] => [3,1,1,1,1,1,1,1,1,1,1,1,1] => 13
[5,5,3,1,1] => [3,2,1,1,1,1,1,1,1,1,1,1] => 12
[5,5,2,2,1] => [3,2,2,1,1,1,1,1,1,1,1] => 11
[5,5,2,1,1,1] => [4,3,3,1,1,1,1,1] => 8
[5,5,1,1,1,1,1] => [7,4,2,1,1] => 5
[5,4,4,2] => [4,1,1,1,1,1,1,1,1,1,1,1] => 12
[5,4,4,1,1] => [4,2,1,1,1,1,1,1,1,1,1] => 11
[5,4,3,3] => [5,1,1,1,1,1,1,1,1,1,1] => 11
[5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 15
[5,4,3,1,1,1] => [6,1,1,1,1,1,1,1,1,1] => 10
[5,4,2,2,2] => [4,2,2,1,1,1,1,1,1,1] => 10
[5,4,2,2,1,1] => [6,2,1,1,1,1,1,1,1] => 9
[5,4,2,1,1,1,1] => [7,1,1,1,1,1,1,1,1] => 9
[5,4,1,1,1,1,1,1] => [8,2,2,1,1,1] => 6
[5,3,3,3,1] => [2,2,2,2,2,1,1,1,1,1] => 10
[5,3,3,2,2] => [5,2,2,1,1,1,1,1,1] => 9
[5,3,3,2,1,1] => [6,2,2,1,1,1,1,1] => 8
[5,3,3,1,1,1,1] => [7,2,1,1,1,1,1,1] => 8
[5,3,2,2,2,1] => [4,4,3,1,1,1,1] => 7
[5,3,2,2,1,1,1] => [7,3,1,1,1,1,1] => 7
[5,3,2,1,1,1,1,1] => [8,1,1,1,1,1,1,1] => 8
[5,3,1,1,1,1,1,1,1] => [9,2,1,1,1,1] => 6
[5,2,2,2,2,2] => [4,4,4,2,1] => 5
[5,2,2,2,2,1,1] => [7,5,2,1] => 4
[5,2,2,2,1,1,1,1] => [8,2,2,2,1] => 5
[5,2,2,1,1,1,1,1,1] => [9,2,2,1,1] => 5
[5,2,1,1,1,1,1,1,1,1] => [10,3,1,1] => 4
[5,1,1,1,1,1,1,1,1,1,1] => [11,2,2] => 3
[4,4,4,3] => [4,3,2,2,1,1,1,1] => 8
[4,4,4,2,1] => [3,2,2,2,1,1,1,1,1,1] => 10
[4,4,4,1,1,1] => [4,4,2,2,1,1,1] => 7
[4,4,3,3,1] => [3,2,2,2,2,1,1,1,1] => 9
[4,4,3,2,2] => [5,2,2,2,1,1,1,1] => 8
[4,4,3,2,1,1] => [6,2,2,2,1,1,1] => 7
[4,4,3,1,1,1,1] => [7,2,2,1,1,1,1] => 7
[4,4,2,2,2,1] => [4,4,3,2,1,1] => 6
[4,4,2,2,1,1,1] => [7,3,2,1,1,1] => 6
[4,4,2,1,1,1,1,1] => [8,2,1,1,1,1,1] => 7
[4,4,1,1,1,1,1,1,1] => [9,3,1,1,1] => 5
[4,3,3,3,2] => [3,3,2,2,2,2,1] => 7
[4,3,3,3,1,1] => [5,3,2,2,2,1] => 6
[4,3,3,2,2,1] => [5,3,3,3,1] => 5
[4,3,3,2,1,1,1] => [7,3,2,2,1] => 5
[4,3,3,1,1,1,1,1] => [8,3,1,1,1,1] => 6
[4,3,2,2,2,2] => [4,4,4,1,1,1] => 6
[4,3,2,2,2,1,1] => [7,5,1,1,1] => 5
[4,3,2,2,1,1,1,1] => [8,4,1,1,1] => 5
[4,3,2,1,1,1,1,1,1] => [9,1,1,1,1,1,1] => 7
[4,3,1,1,1,1,1,1,1,1] => [10,1,1,1,1,1] => 6
[4,2,2,2,2,2,1] => [5,4,4,1,1] => 5
[4,2,2,2,2,1,1,1] => [8,5,1,1] => 4
[4,2,2,2,1,1,1,1,1] => [9,4,1,1] => 4
[4,2,2,1,1,1,1,1,1,1] => [10,2,1,1,1] => 5
[4,2,1,1,1,1,1,1,1,1,1] => [11,1,1,1,1] => 5
[4,1,1,1,1,1,1,1,1,1,1,1] => [12,2,1] => 3
[3,3,3,3,3] => [5,3,3,2,2] => 5
[3,3,3,3,2,1] => [3,3,3,2,2,2] => 6
[3,3,3,3,1,1,1] => [7,3,3,2] => 4
[3,3,3,2,2,2] => [5,4,3,3] => 4
[3,3,3,2,2,1,1] => [7,5,3] => 3
[3,3,3,2,1,1,1,1] => [8,3,2,2] => 4
[3,3,3,1,1,1,1,1,1] => [9,2,2,2] => 4
[3,3,2,2,2,2,1] => [5,4,4,2] => 4
[3,3,2,2,2,1,1,1] => [8,5,2] => 3
[3,3,2,2,1,1,1,1,1] => [9,4,2] => 3
[3,3,2,1,1,1,1,1,1,1] => [10,2,2,1] => 4
[3,3,1,1,1,1,1,1,1,1,1] => [11,2,1,1] => 4
[3,2,2,2,2,2,2] => [5,5,4,1] => 4
[3,2,2,2,2,2,1,1] => [8,6,1] => 3
[3,2,2,2,2,1,1,1,1] => [9,5,1] => 3
[3,2,2,2,1,1,1,1,1,1] => [10,4,1] => 3
[3,2,2,1,1,1,1,1,1,1,1] => [11,3,1] => 3
[3,2,1,1,1,1,1,1,1,1,1,1] => [12,1,1,1] => 4
[3,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1] => 3
[2,2,2,2,2,2,2,1] => [5,5,5] => 3
[2,2,2,2,2,2,1,1,1] => [9,6] => 2
[2,2,2,2,2,1,1,1,1,1] => [10,5] => 2
[2,2,2,2,1,1,1,1,1,1,1] => [11,4] => 2
[2,2,2,1,1,1,1,1,1,1,1,1] => [12,3] => 2
[2,2,1,1,1,1,1,1,1,1,1,1,1] => [13,2] => 2
[2,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1] => 2
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15] => 1
[16] => [8,8] => 2
[15,1] => [8,7,1] => 3
[14,2] => [7,7,1,1] => 4
[14,1,1] => [7,7,2] => 3
[13,3] => [7,6,2,1] => 4
[13,2,1] => [7,6,1,1,1] => 5
[13,1,1,1] => [7,6,3] => 3
[12,4] => [6,6,2,2] => 4
[12,3,1] => [6,6,1,1,1,1] => 6
[12,2,2] => [6,6,2,1,1] => 5
[12,2,1,1] => [6,6,3,1] => 4
[12,1,1,1,1] => [6,6,4] => 3
[11,5] => [6,5,3,2] => 4
[11,4,1] => [6,5,3,1,1] => 5
[11,3,2] => [6,5,1,1,1,1,1] => 7
[11,3,1,1] => [6,5,2,1,1,1] => 6
[11,2,2,1] => [6,5,2,2,1] => 5
[11,2,1,1,1] => [6,5,4,1] => 4
[11,1,1,1,1,1] => [7,5,4] => 3
[10,6] => [5,5,3,3] => 4
[10,5,1] => [5,5,3,2,1] => 5
[10,4,2] => [5,5,2,1,1,1,1] => 7
[10,4,1,1] => [5,5,2,2,1,1] => 6
[10,3,3] => [5,5,3,1,1,1] => 6
[10,3,2,1] => [5,5,1,1,1,1,1,1] => 8
[10,3,1,1,1] => [6,4,4,1,1] => 5
[10,2,2,2] => [5,5,2,2,2] => 5
[10,2,2,1,1] => [6,4,4,2] => 4
[10,2,1,1,1,1] => [7,4,4,1] => 4
[10,1,1,1,1,1,1] => [8,4,4] => 3
[9,7] => [4,3,3,3,3] => 5
[9,6,1] => [6,3,3,3,1] => 5
[9,5,2] => [5,4,3,2,1,1] => 6
[9,5,1,1] => [6,4,3,2,1] => 5
[9,4,3] => [5,4,2,2,1,1,1] => 7
[9,4,2,1] => [5,4,1,1,1,1,1,1,1] => 9
[9,4,1,1,1] => [5,4,2,2,2,1] => 6
[9,3,3,1] => [5,4,2,1,1,1,1,1] => 8
[9,3,2,2] => [5,4,3,1,1,1,1] => 7
[9,3,2,1,1] => [6,4,3,1,1,1] => 6
[9,3,1,1,1,1] => [7,4,3,1,1] => 5
[9,2,2,2,1] => [6,3,3,2,2] => 5
[9,2,2,1,1,1] => [7,4,3,2] => 4
[9,2,1,1,1,1,1] => [8,4,3,1] => 4
[9,1,1,1,1,1,1,1] => [4,4,4,4] => 4
[8,8] => [6,4,3,3] => 4
[8,7,1] => [3,3,3,3,3,1] => 6
[8,6,2] => [3,3,3,3,2,1,1] => 7
[8,6,1,1] => [4,3,3,2,2,2] => 6
[8,5,3] => [4,3,3,2,2,1,1] => 7
[8,5,2,1] => [6,4,2,1,1,1,1] => 7
[8,5,1,1,1] => [6,2,2,2,2,2] => 6
[8,4,4] => [6,4,2,2,1,1] => 6
[8,4,3,1] => [5,3,1,1,1,1,1,1,1,1] => 10
[8,4,2,2] => [4,3,3,1,1,1,1,1,1] => 9
[8,4,2,1,1] => [4,4,2,2,1,1,1,1] => 8
[8,4,1,1,1,1] => [7,4,2,2,1] => 5
[8,3,3,2] => [5,2,2,2,2,1,1,1] => 8
[8,3,3,1,1] => [6,3,2,2,1,1,1] => 7
[8,3,2,2,1] => [6,3,2,2,2,1] => 6
[8,3,2,1,1,1] => [7,4,2,1,1,1] => 6
[8,3,1,1,1,1,1] => [4,4,3,3,1,1] => 6
[8,2,2,2,2] => [4,4,4,2,2] => 5
[8,2,2,2,1,1] => [7,5,2,2] => 4
[8,2,2,1,1,1,1] => [4,4,3,3,2] => 5
[8,2,1,1,1,1,1,1] => [4,4,4,3,1] => 5
[8,1,1,1,1,1,1,1,1] => [9,4,3] => 3
[7,7,2] => [6,3,3,2,1,1] => 6
[7,7,1,1] => [6,4,2,2,2] => 5
[7,6,3] => [4,4,2,2,2,1,1] => 7
[7,6,2,1] => [3,3,3,2,2,1,1,1] => 8
[7,6,1,1,1] => [4,4,2,2,2,2] => 6
[7,5,4] => [2,2,2,2,2,2,2,1,1] => 9
[7,5,3,1] => [4,3,2,1,1,1,1,1,1,1] => 10
[7,5,2,2] => [3,3,3,2,1,1,1,1,1] => 9
[7,5,2,1,1] => [5,3,3,1,1,1,1,1] => 8
[7,5,1,1,1,1] => [2,2,2,2,2,2,2,2] => 8
[7,4,4,1] => [4,3,2,2,1,1,1,1,1] => 9
[7,4,3,2] => [5,2,1,1,1,1,1,1,1,1,1] => 11
[7,4,3,1,1] => [6,2,1,1,1,1,1,1,1,1] => 10
[7,4,2,2,1] => [6,2,2,1,1,1,1,1,1] => 9
[7,4,2,1,1,1] => [3,3,3,3,1,1,1,1] => 8
[7,4,1,1,1,1,1] => [4,3,3,3,2,1] => 6
[7,3,3,3] => [3,3,2,2,2,2,1,1] => 8
[7,3,3,2,1] => [6,2,2,2,1,1,1,1] => 8
[7,3,3,1,1,1] => [5,3,3,2,1,1,1] => 7
[7,3,2,2,2] => [5,3,2,2,2,1,1] => 7
[7,3,2,2,1,1] => [5,3,3,3,1,1] => 6
[7,3,2,1,1,1,1] => [4,3,3,3,1,1,1] => 7
[7,3,1,1,1,1,1,1] => [8,3,3,1,1] => 5
[7,2,2,2,2,1] => [3,3,2,2,2,2,2] => 7
[7,2,2,2,1,1,1] => [5,4,3,2,2] => 5
[7,2,2,1,1,1,1,1] => [8,4,2,2] => 4
[7,2,1,1,1,1,1,1,1] => [9,3,3,1] => 4
[7,1,1,1,1,1,1,1,1,1] => [10,3,3] => 3
[6,6,4] => [4,2,2,2,2,2,1,1] => 8
[6,6,3,1] => [6,3,1,1,1,1,1,1,1] => 9
[6,6,2,2] => [6,3,2,1,1,1,1,1] => 8
[6,6,2,1,1] => [6,3,3,1,1,1,1] => 7
[6,6,1,1,1,1] => [4,2,2,2,2,2,2] => 7
[6,5,5] => [6,2,2,2,2,1,1] => 7
[6,5,4,1] => [3,3,3,1,1,1,1,1,1,1] => 10
[6,5,3,2] => [2,2,1,1,1,1,1,1,1,1,1,1,1,1] => 14
[6,5,3,1,1] => [2,2,2,1,1,1,1,1,1,1,1,1,1] => 13
[6,5,2,2,1] => [2,2,2,2,1,1,1,1,1,1,1,1] => 12
[6,5,2,1,1,1] => [3,2,2,2,2,2,1,1,1] => 9
[6,5,1,1,1,1,1] => [7,2,2,2,2,1] => 6
[6,4,4,2] => [3,2,1,1,1,1,1,1,1,1,1,1,1] => 13
[6,4,4,1,1] => [3,2,2,1,1,1,1,1,1,1,1,1] => 12
[6,4,3,3] => [4,2,1,1,1,1,1,1,1,1,1,1] => 12
[6,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 16
[6,4,3,1,1,1] => [4,2,2,1,1,1,1,1,1,1,1] => 11
[6,4,2,2,2] => [3,2,2,2,1,1,1,1,1,1,1] => 11
[6,4,2,2,1,1] => [4,2,2,2,1,1,1,1,1,1] => 10
[6,4,2,1,1,1,1] => [7,3,1,1,1,1,1,1] => 8
[6,4,1,1,1,1,1,1] => [8,2,2,2,1,1] => 6
[6,3,3,3,1] => [2,2,2,2,2,1,1,1,1,1,1] => 11
[6,3,3,2,2] => [3,2,2,2,2,1,1,1,1,1] => 10
[6,3,3,2,1,1] => [5,2,2,2,1,1,1,1,1] => 9
[6,3,3,1,1,1,1] => [7,3,2,1,1,1,1] => 7
[6,3,2,2,2,1] => [3,2,2,2,2,2,2,1] => 8
[6,3,2,2,1,1,1] => [7,3,3,1,1,1] => 6
[6,3,2,1,1,1,1,1] => [8,4,1,1,1,1] => 6
[6,3,1,1,1,1,1,1,1] => [9,3,2,1,1] => 5
[6,2,2,2,2,2] => [5,3,2,2,2,2] => 6
[6,2,2,2,2,1,1] => [7,3,2,2,2] => 5
[6,2,2,2,1,1,1,1] => [8,2,2,2,2] => 5
[6,2,2,1,1,1,1,1,1] => [9,4,2,1] => 4
[6,2,1,1,1,1,1,1,1,1] => [10,3,2,1] => 4
[6,1,1,1,1,1,1,1,1,1,1] => [11,3,2] => 3
[5,5,5,1] => [6,4,1,1,1,1,1,1] => 8
[5,5,4,2] => [3,3,1,1,1,1,1,1,1,1,1,1] => 12
[5,5,4,1,1] => [3,3,2,1,1,1,1,1,1,1,1] => 11
[5,5,3,3] => [4,3,1,1,1,1,1,1,1,1,1] => 11
[5,5,3,2,1] => [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 15
[5,5,3,1,1,1] => [5,2,2,1,1,1,1,1,1,1] => 10
[5,5,2,2,2] => [3,3,2,2,1,1,1,1,1,1] => 10
[5,5,2,2,1,1] => [5,3,2,1,1,1,1,1,1] => 9
[5,5,2,1,1,1,1] => [7,4,1,1,1,1,1] => 7
[5,5,1,1,1,1,1,1] => [8,4,2,1,1] => 5
[5,4,4,3] => [4,4,1,1,1,1,1,1,1,1] => 10
[5,4,4,2,1] => [3,1,1,1,1,1,1,1,1,1,1,1,1,1] => 14
[5,4,4,1,1,1] => [4,4,2,1,1,1,1,1,1] => 9
[5,4,3,3,1] => [4,1,1,1,1,1,1,1,1,1,1,1,1] => 13
[5,4,3,2,2] => [5,1,1,1,1,1,1,1,1,1,1,1] => 12
[5,4,3,2,1,1] => [6,1,1,1,1,1,1,1,1,1,1] => 11
[5,4,3,1,1,1,1] => [7,1,1,1,1,1,1,1,1,1] => 10
[5,4,2,2,2,1] => [4,4,3,1,1,1,1,1] => 8
[5,4,2,2,1,1,1] => [7,2,1,1,1,1,1,1,1] => 9
[5,4,2,1,1,1,1,1] => [8,1,1,1,1,1,1,1,1] => 9
[5,4,1,1,1,1,1,1,1] => [9,2,2,1,1,1] => 6
[5,3,3,3,2] => [4,2,2,2,2,1,1,1,1] => 9
[5,3,3,3,1,1] => [4,3,3,2,1,1,1,1] => 8
[5,3,3,2,2,1] => [4,4,3,2,1,1,1] => 7
[5,3,3,2,1,1,1] => [7,2,2,1,1,1,1,1] => 8
[5,3,3,1,1,1,1,1] => [8,2,1,1,1,1,1,1] => 8
[5,3,2,2,2,2] => [4,4,4,1,1,1,1] => 7
[5,3,2,2,2,1,1] => [7,5,1,1,1,1] => 6
[5,3,2,2,1,1,1,1] => [8,3,1,1,1,1,1] => 7
[5,3,2,1,1,1,1,1,1] => [9,1,1,1,1,1,1,1] => 8
[5,3,1,1,1,1,1,1,1,1] => [10,2,1,1,1,1] => 6
[5,2,2,2,2,2,1] => [5,4,4,2,1] => 5
[5,2,2,2,2,1,1,1] => [8,5,2,1] => 4
[5,2,2,2,1,1,1,1,1] => [9,2,2,2,1] => 5
[5,2,2,1,1,1,1,1,1,1] => [10,2,2,1,1] => 5
[5,2,1,1,1,1,1,1,1,1,1] => [11,3,1,1] => 4
[5,1,1,1,1,1,1,1,1,1,1,1] => [12,2,2] => 3
[4,4,4,4] => [4,3,2,2,2,2,1] => 7
[4,4,4,3,1] => [2,2,2,2,2,2,1,1,1,1] => 10
[4,4,4,2,2] => [3,3,2,2,2,1,1,1,1] => 9
[4,4,4,2,1,1] => [5,3,2,2,1,1,1,1] => 8
[4,4,4,1,1,1,1] => [7,3,2,2,1,1] => 6
[4,4,3,3,2] => [4,3,2,2,2,1,1,1] => 8
[4,4,3,3,1,1] => [5,2,2,2,2,2,1] => 7
[4,4,3,2,2,1] => [4,4,3,2,2,1] => 6
[4,4,3,2,1,1,1] => [7,2,2,2,1,1,1] => 7
[4,4,3,1,1,1,1,1] => [8,2,2,1,1,1,1] => 7
[4,4,2,2,2,2] => [4,4,4,2,1,1] => 6
[4,4,2,2,2,1,1] => [7,5,2,1,1] => 5
[4,4,2,2,1,1,1,1] => [8,3,2,1,1,1] => 6
[4,4,2,1,1,1,1,1,1] => [9,2,1,1,1,1,1] => 7
[4,4,1,1,1,1,1,1,1,1] => [10,3,1,1,1] => 5
[4,3,3,3,3] => [5,3,3,2,2,1] => 6
[4,3,3,3,2,1] => [3,3,3,2,2,2,1] => 7
[4,3,3,3,1,1,1] => [7,3,3,2,1] => 5
[4,3,3,2,2,2] => [5,4,3,3,1] => 5
[4,3,3,2,2,1,1] => [7,5,3,1] => 4
[4,3,3,2,1,1,1,1] => [8,3,2,2,1] => 5
[4,3,3,1,1,1,1,1,1] => [9,3,1,1,1,1] => 6
[4,3,2,2,2,2,1] => [5,4,4,1,1,1] => 6
[4,3,2,2,2,1,1,1] => [8,5,1,1,1] => 5
[4,3,2,2,1,1,1,1,1] => [9,4,1,1,1] => 5
[4,3,2,1,1,1,1,1,1,1] => [10,1,1,1,1,1,1] => 7
[4,3,1,1,1,1,1,1,1,1,1] => [11,1,1,1,1,1] => 6
[4,2,2,2,2,2,2] => [5,5,4,1,1] => 5
[4,2,2,2,2,2,1,1] => [8,6,1,1] => 4
[4,2,2,2,2,1,1,1,1] => [9,5,1,1] => 4
[4,2,2,2,1,1,1,1,1,1] => [10,4,1,1] => 4
[4,2,2,1,1,1,1,1,1,1,1] => [11,2,1,1,1] => 5
[4,2,1,1,1,1,1,1,1,1,1,1] => [12,1,1,1,1] => 5
[4,1,1,1,1,1,1,1,1,1,1,1,1] => [13,2,1] => 3
[3,3,3,3,3,1] => [3,3,3,3,2,2] => 6
[3,3,3,3,2,2] => [5,3,3,3,2] => 5
[3,3,3,3,2,1,1] => [7,3,3,3] => 4
[3,3,3,3,1,1,1,1] => [8,3,3,2] => 4
[3,3,3,2,2,2,1] => [5,4,4,3] => 4
[3,3,3,2,2,1,1,1] => [8,5,3] => 3
[3,3,3,2,1,1,1,1,1] => [9,3,2,2] => 4
[3,3,3,1,1,1,1,1,1,1] => [10,2,2,2] => 4
[3,3,2,2,2,2,2] => [5,5,4,2] => 4
[3,3,2,2,2,2,1,1] => [8,6,2] => 3
[3,3,2,2,2,1,1,1,1] => [9,5,2] => 3
[3,3,2,2,1,1,1,1,1,1] => [10,4,2] => 3
[3,3,2,1,1,1,1,1,1,1,1] => [11,2,2,1] => 4
[3,3,1,1,1,1,1,1,1,1,1,1] => [12,2,1,1] => 4
[3,2,2,2,2,2,2,1] => [5,5,5,1] => 4
[3,2,2,2,2,2,1,1,1] => [9,6,1] => 3
[3,2,2,2,2,1,1,1,1,1] => [10,5,1] => 3
[3,2,2,2,1,1,1,1,1,1,1] => [11,4,1] => 3
[3,2,2,1,1,1,1,1,1,1,1,1] => [12,3,1] => 3
[3,2,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1,1] => 4
[3,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1,1] => 3
[2,2,2,2,2,2,2,2] => [6,5,5] => 3
[2,2,2,2,2,2,2,1,1] => [9,7] => 2
[2,2,2,2,2,2,1,1,1,1] => [10,6] => 2
[2,2,2,2,2,1,1,1,1,1,1] => [11,5] => 2
[2,2,2,2,1,1,1,1,1,1,1,1] => [12,4] => 2
[2,2,2,1,1,1,1,1,1,1,1,1,1] => [13,3] => 2
[2,2,1,1,1,1,1,1,1,1,1,1,1,1] => [14,2] => 2
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,1] => 2
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [16] => 1
[17] => [9,8] => 2
[16,1] => [8,8,1] => 3
[15,2] => [8,7,1,1] => 4
[15,1,1] => [8,7,2] => 3
[14,3] => [7,7,2,1] => 4
[14,2,1] => [7,7,1,1,1] => 5
[14,1,1,1] => [7,7,3] => 3
[13,4] => [7,6,2,2] => 4
[13,3,1] => [7,6,1,1,1,1] => 6
[13,2,2] => [7,6,2,1,1] => 5
[13,2,1,1] => [7,6,3,1] => 4
[13,1,1,1,1] => [7,6,4] => 3
[12,5] => [6,6,3,2] => 4
[12,4,1] => [6,6,3,1,1] => 5
[12,3,2] => [6,6,1,1,1,1,1] => 7
[12,3,1,1] => [6,6,2,1,1,1] => 6
[12,2,2,1] => [6,6,2,2,1] => 5
[12,2,1,1,1] => [6,6,4,1] => 4
[12,1,1,1,1,1] => [7,5,5] => 3
[11,6] => [6,5,3,3] => 4
[11,5,1] => [6,5,3,2,1] => 5
[11,4,2] => [6,5,2,1,1,1,1] => 7
[11,4,1,1] => [6,5,2,2,1,1] => 6
[11,3,3] => [6,5,3,1,1,1] => 6
[11,3,2,1] => [6,5,1,1,1,1,1,1] => 8
[11,3,1,1,1] => [6,5,4,1,1] => 5
[11,2,2,2] => [6,5,2,2,2] => 5
[11,2,2,1,1] => [6,5,4,2] => 4
[11,2,1,1,1,1] => [7,5,4,1] => 4
[11,1,1,1,1,1,1] => [8,5,4] => 3
[10,7] => [7,4,3,3] => 4
[10,6,1] => [5,5,3,3,1] => 5
[10,5,2] => [5,5,3,2,1,1] => 6
[10,5,1,1] => [6,4,4,2,1] => 5
[10,4,3] => [5,5,2,2,1,1,1] => 7
[10,4,2,1] => [5,5,1,1,1,1,1,1,1] => 9
[10,4,1,1,1] => [5,5,2,2,2,1] => 6
[10,3,3,1] => [5,5,2,1,1,1,1,1] => 8
[10,3,2,2] => [5,5,3,1,1,1,1] => 7
[10,3,2,1,1] => [6,4,4,1,1,1] => 6
[10,3,1,1,1,1] => [7,4,4,1,1] => 5
[10,2,2,2,1] => [6,3,3,3,2] => 5
[10,2,2,1,1,1] => [7,4,4,2] => 4
[10,2,1,1,1,1,1] => [8,4,4,1] => 4
[10,1,1,1,1,1,1,1] => [9,4,4] => 3
[9,8] => [4,4,3,3,3] => 5
[9,7,1] => [4,3,3,3,3,1] => 6
[9,6,2] => [6,3,3,3,1,1] => 6
[9,6,1,1] => [6,4,3,2,2] => 5
[9,5,3] => [5,4,2,2,2,1,1] => 7
[9,5,2,1] => [6,4,3,1,1,1,1] => 7
[9,5,1,1,1] => [6,4,2,2,2,1] => 6
[9,4,4] => [6,4,3,2,1,1] => 6
[9,4,3,1] => [5,4,1,1,1,1,1,1,1,1] => 10
[9,4,2,2] => [5,4,2,1,1,1,1,1,1] => 9
[9,4,2,1,1] => [5,4,3,1,1,1,1,1] => 8
[9,4,1,1,1,1] => [7,4,3,2,1] => 5
[9,3,3,2] => [5,4,2,2,1,1,1,1] => 8
[9,3,3,1,1] => [5,4,3,2,1,1,1] => 7
[9,3,2,2,1] => [6,3,3,2,2,1] => 6
[9,3,2,1,1,1] => [7,4,3,1,1,1] => 6
[9,3,1,1,1,1,1] => [8,4,3,1,1] => 5
[9,2,2,2,2] => [5,4,3,3,2] => 5
[9,2,2,2,1,1] => [7,5,3,2] => 4
[9,2,2,1,1,1,1] => [8,4,3,2] => 4
[9,2,1,1,1,1,1,1] => [4,4,4,4,1] => 5
[9,1,1,1,1,1,1,1,1] => [5,4,4,4] => 4
[8,8,1] => [6,4,3,3,1] => 5
[8,7,2] => [3,3,3,3,3,1,1] => 7
[8,7,1,1] => [4,3,3,3,2,2] => 6
[8,6,3] => [4,3,3,2,2,2,1] => 7
[8,6,2,1] => [3,3,3,3,2,1,1,1] => 8
[8,6,1,1,1] => [5,4,2,2,2,2] => 6
[8,5,4] => [6,2,2,2,2,2,1] => 7
[8,5,3,1] => [6,4,1,1,1,1,1,1,1] => 9
[8,5,2,2] => [6,3,2,2,1,1,1,1] => 8
[8,5,2,1,1] => [6,3,2,2,2,1,1] => 7
[8,5,1,1,1,1] => [7,2,2,2,2,2] => 6
[8,4,4,1] => [6,4,2,1,1,1,1,1] => 8
[8,4,3,2] => [5,2,2,1,1,1,1,1,1,1,1] => 11
[8,4,3,1,1] => [4,4,2,1,1,1,1,1,1,1] => 10
[8,4,2,2,1] => [5,3,2,2,1,1,1,1,1] => 9
[8,4,2,1,1,1] => [7,4,2,1,1,1,1] => 7
[8,4,1,1,1,1,1] => [4,4,3,3,2,1] => 6
[8,3,3,3] => [5,3,3,2,2,1,1] => 7
[8,3,3,2,1] => [6,2,2,2,2,1,1,1] => 8
[8,3,3,1,1,1] => [7,3,3,2,1,1] => 6
[8,3,2,2,2] => [5,4,3,3,1,1] => 6
[8,3,2,2,1,1] => [7,5,3,1,1] => 5
[8,3,2,1,1,1,1] => [4,4,3,3,1,1,1] => 7
[8,3,1,1,1,1,1,1] => [4,4,4,3,1,1] => 6
[8,2,2,2,2,1] => [7,3,3,2,2] => 5
[8,2,2,2,1,1,1] => [5,5,3,2,2] => 5
[8,2,2,1,1,1,1,1] => [4,4,4,3,2] => 5
[8,2,1,1,1,1,1,1,1] => [9,4,3,1] => 4
[8,1,1,1,1,1,1,1,1,1] => [10,4,3] => 3
[7,7,3] => [4,4,3,2,2,2] => 6
[7,7,2,1] => [6,3,3,2,1,1,1] => 7
[7,7,1,1,1] => [7,4,2,2,2] => 5
[7,6,4] => [2,2,2,2,2,2,2,2,1] => 9
[7,6,3,1] => [4,4,2,2,1,1,1,1,1] => 9
[7,6,2,2] => [5,3,3,2,1,1,1,1] => 8
[7,6,2,1,1] => [5,3,3,3,1,1,1] => 7
[7,6,1,1,1,1] => [3,2,2,2,2,2,2,2] => 8
[7,5,5] => [4,2,2,2,2,2,2,1] => 8
[7,5,4,1] => [3,3,3,2,1,1,1,1,1,1] => 10
[7,5,3,2] => [4,2,2,1,1,1,1,1,1,1,1,1] => 12
[7,5,3,1,1] => [4,2,2,2,1,1,1,1,1,1,1] => 11
[7,5,2,2,1] => [5,2,2,2,1,1,1,1,1,1] => 10
[7,5,2,1,1,1] => [3,2,2,2,2,2,2,1,1] => 9
[7,5,1,1,1,1,1] => [4,3,2,2,2,2,2] => 7
[7,4,4,2] => [4,3,2,1,1,1,1,1,1,1,1] => 11
[7,4,4,1,1] => [5,3,2,1,1,1,1,1,1,1] => 10
[7,4,3,3] => [4,3,2,2,1,1,1,1,1,1] => 10
[7,4,3,2,1] => [6,1,1,1,1,1,1,1,1,1,1,1] => 12
[7,4,3,1,1,1] => [4,3,3,2,1,1,1,1,1] => 9
[7,4,2,2,2] => [3,3,3,3,1,1,1,1,1] => 9
[7,4,2,2,1,1] => [4,4,3,2,1,1,1,1] => 8
[7,4,2,1,1,1,1] => [4,3,3,3,1,1,1,1] => 8
[7,4,1,1,1,1,1,1] => [8,4,2,2,1] => 5
[7,3,3,3,1] => [5,2,2,2,2,1,1,1,1] => 9
[7,3,3,2,2] => [5,3,2,2,2,1,1,1] => 8
[7,3,3,2,1,1] => [4,4,3,2,2,1,1] => 7
[7,3,3,1,1,1,1] => [4,3,3,3,2,1,1] => 7
[7,3,2,2,2,1] => [3,3,2,2,2,2,2,1] => 8
[7,3,2,2,1,1,1] => [5,4,3,2,2,1] => 6
[7,3,2,1,1,1,1,1] => [8,4,2,1,1,1] => 6
[7,3,1,1,1,1,1,1,1] => [9,3,3,1,1] => 5
[7,2,2,2,2,2] => [3,3,3,2,2,2,2] => 7
[7,2,2,2,2,1,1] => [5,3,3,2,2,2] => 6
[7,2,2,2,1,1,1,1] => [8,5,2,2] => 4
[7,2,2,1,1,1,1,1,1] => [9,4,2,2] => 4
[7,2,1,1,1,1,1,1,1,1] => [10,3,3,1] => 4
[7,1,1,1,1,1,1,1,1,1,1] => [11,3,3] => 3
[6,6,5] => [5,2,2,2,2,2,2] => 7
[6,6,4,1] => [5,3,3,1,1,1,1,1,1] => 9
[6,6,3,2] => [6,2,1,1,1,1,1,1,1,1,1] => 11
[6,6,3,1,1] => [6,2,2,1,1,1,1,1,1,1] => 10
[6,6,2,2,1] => [6,2,2,2,1,1,1,1,1] => 9
[6,6,2,1,1,1] => [4,3,2,2,2,2,1,1] => 8
[6,6,1,1,1,1,1] => [6,3,2,2,2,2] => 6
[6,5,5,1] => [6,3,3,1,1,1,1,1] => 8
[6,5,4,2] => [2,2,2,1,1,1,1,1,1,1,1,1,1,1] => 14
[6,5,4,1,1] => [2,2,2,2,1,1,1,1,1,1,1,1,1] => 13
[6,5,3,3] => [3,2,2,1,1,1,1,1,1,1,1,1,1] => 13
[6,5,3,2,1] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 17
[6,5,3,1,1,1] => [3,2,2,2,1,1,1,1,1,1,1,1] => 12
[6,5,2,2,2] => [2,2,2,2,2,1,1,1,1,1,1,1] => 12
[6,5,2,2,1,1] => [3,2,2,2,2,1,1,1,1,1,1] => 11
[6,5,2,1,1,1,1] => [7,3,3,1,1,1,1] => 7
[6,5,1,1,1,1,1,1] => [8,2,2,2,2,1] => 6
[6,4,4,3] => [3,3,2,1,1,1,1,1,1,1,1,1] => 12
[6,4,4,2,1] => [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 16
[6,4,4,1,1,1] => [3,3,2,2,1,1,1,1,1,1,1] => 11
[6,4,3,3,1] => [3,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 15
[6,4,3,2,2] => [4,1,1,1,1,1,1,1,1,1,1,1,1,1] => 14
[6,4,3,2,1,1] => [5,1,1,1,1,1,1,1,1,1,1,1,1] => 13
[6,4,3,1,1,1,1] => [7,2,1,1,1,1,1,1,1,1] => 10
[6,4,2,2,2,1] => [3,3,2,2,2,1,1,1,1,1] => 10
[6,4,2,2,1,1,1] => [7,2,2,1,1,1,1,1,1] => 9
[6,4,2,1,1,1,1,1] => [8,3,1,1,1,1,1,1] => 8
[6,4,1,1,1,1,1,1,1] => [9,2,2,2,1,1] => 6
[6,3,3,3,2] => [2,2,2,2,2,2,1,1,1,1,1] => 11
[6,3,3,3,1,1] => [3,2,2,2,2,2,1,1,1,1] => 10
[6,3,3,2,2,1] => [4,3,2,2,2,1,1,1,1] => 9
[6,3,3,2,1,1,1] => [7,2,2,2,1,1,1,1] => 8
[6,3,3,1,1,1,1,1] => [8,3,2,1,1,1,1] => 7
[6,3,2,2,2,2] => [5,3,2,2,2,2,1] => 7
[6,3,2,2,2,1,1] => [7,3,2,2,2,1] => 6
[6,3,2,2,1,1,1,1] => [8,3,3,1,1,1] => 6
[6,3,2,1,1,1,1,1,1] => [9,4,1,1,1,1] => 6
[6,3,1,1,1,1,1,1,1,1] => [10,3,2,1,1] => 5
[6,2,2,2,2,2,1] => [5,4,4,2,2] => 5
[6,2,2,2,2,1,1,1] => [8,3,2,2,2] => 5
[6,2,2,2,1,1,1,1,1] => [9,2,2,2,2] => 5
[6,2,2,1,1,1,1,1,1,1] => [10,4,2,1] => 4
[6,2,1,1,1,1,1,1,1,1,1] => [11,3,2,1] => 4
[6,1,1,1,1,1,1,1,1,1,1,1] => [12,3,2] => 3
[5,5,5,2] => [6,3,1,1,1,1,1,1,1,1] => 10
[5,5,5,1,1] => [6,3,2,1,1,1,1,1,1] => 9
[5,5,4,3] => [3,3,3,1,1,1,1,1,1,1,1] => 11
[5,5,4,2,1] => [2,2,1,1,1,1,1,1,1,1,1,1,1,1,1] => 15
[5,5,4,1,1,1] => [4,2,2,2,2,1,1,1,1,1] => 10
[5,5,3,3,1] => [3,2,1,1,1,1,1,1,1,1,1,1,1,1] => 14
[5,5,3,2,2] => [4,2,1,1,1,1,1,1,1,1,1,1,1] => 13
[5,5,3,2,1,1] => [5,2,1,1,1,1,1,1,1,1,1,1] => 12
[5,5,3,1,1,1,1] => [7,3,1,1,1,1,1,1,1] => 9
[5,5,2,2,2,1] => [3,3,2,2,2,2,1,1,1] => 9
[5,5,2,2,1,1,1] => [7,3,2,1,1,1,1,1] => 8
[5,5,2,1,1,1,1,1] => [8,4,1,1,1,1,1] => 7
[5,5,1,1,1,1,1,1,1] => [9,4,2,1,1] => 5
[5,4,4,4] => [3,3,3,2,2,1,1,1,1] => 9
[5,4,4,3,1] => [3,3,1,1,1,1,1,1,1,1,1,1,1] => 13
[5,4,4,2,2] => [4,3,1,1,1,1,1,1,1,1,1,1] => 12
[5,4,4,2,1,1] => [5,3,1,1,1,1,1,1,1,1,1] => 11
[5,4,4,1,1,1,1] => [7,4,1,1,1,1,1,1] => 8
[5,4,3,3,2] => [4,4,1,1,1,1,1,1,1,1,1] => 11
[5,4,3,3,1,1] => [4,3,3,1,1,1,1,1,1,1] => 10
[5,4,3,2,2,1] => [4,4,3,1,1,1,1,1,1] => 9
[5,4,3,2,1,1,1] => [7,1,1,1,1,1,1,1,1,1,1] => 11
[5,4,3,1,1,1,1,1] => [8,1,1,1,1,1,1,1,1,1] => 10
[5,4,2,2,2,2] => [4,4,4,1,1,1,1,1] => 8
[5,4,2,2,2,1,1] => [7,5,1,1,1,1,1] => 7
[5,4,2,2,1,1,1,1] => [8,2,1,1,1,1,1,1,1] => 9
[5,4,2,1,1,1,1,1,1] => [9,1,1,1,1,1,1,1,1] => 9
[5,4,1,1,1,1,1,1,1,1] => [10,2,2,1,1,1] => 6
[5,3,3,3,3] => [4,4,2,2,2,1,1,1] => 8
[5,3,3,3,2,1] => [3,3,3,2,2,2,1,1] => 8
[5,3,3,3,1,1,1] => [7,3,2,2,1,1,1] => 7
[5,3,3,2,2,2] => [4,4,4,2,1,1,1] => 7
[5,3,3,2,2,1,1] => [7,5,2,1,1,1] => 6
[5,3,3,2,1,1,1,1] => [8,2,2,1,1,1,1,1] => 8
[5,3,3,1,1,1,1,1,1] => [9,2,1,1,1,1,1,1] => 8
[5,3,2,2,2,2,1] => [5,4,4,1,1,1,1] => 7
[5,3,2,2,2,1,1,1] => [8,5,1,1,1,1] => 6
[5,3,2,2,1,1,1,1,1] => [9,3,1,1,1,1,1] => 7
[5,3,2,1,1,1,1,1,1,1] => [10,1,1,1,1,1,1,1] => 8
[5,3,1,1,1,1,1,1,1,1,1] => [11,2,1,1,1,1] => 6
[5,2,2,2,2,2,2] => [5,5,4,2,1] => 5
[5,2,2,2,2,2,1,1] => [8,6,2,1] => 4
[5,2,2,2,2,1,1,1,1] => [9,5,2,1] => 4
[5,2,2,2,1,1,1,1,1,1] => [10,2,2,2,1] => 5
[5,2,2,1,1,1,1,1,1,1,1] => [11,2,2,1,1] => 5
[5,2,1,1,1,1,1,1,1,1,1,1] => [12,3,1,1] => 4
[5,1,1,1,1,1,1,1,1,1,1,1,1] => [13,2,2] => 3
[4,4,4,4,1] => [5,2,2,2,2,2,1,1] => 8
[4,4,4,3,2] => [2,2,2,2,2,2,2,1,1,1] => 10
[4,4,4,3,1,1] => [4,2,2,2,2,2,1,1,1] => 9
[4,4,4,2,2,1] => [4,3,3,2,2,1,1,1] => 8
[4,4,4,2,1,1,1] => [7,2,2,2,2,1,1] => 7
[4,4,4,1,1,1,1,1] => [8,3,2,2,1,1] => 6
[4,4,3,3,3] => [6,4,2,2,1,1,1] => 7
[4,4,3,3,2,1] => [4,4,2,2,2,2,1] => 7
[4,4,3,3,1,1,1] => [7,4,2,2,1,1] => 6
[4,4,3,2,2,2] => [4,4,4,2,2,1] => 6
[4,4,3,2,2,1,1] => [7,5,2,2,1] => 5
[4,4,3,2,1,1,1,1] => [8,2,2,2,1,1,1] => 7
[4,4,3,1,1,1,1,1,1] => [9,2,2,1,1,1,1] => 7
[4,4,2,2,2,2,1] => [5,4,4,2,1,1] => 6
[4,4,2,2,2,1,1,1] => [8,5,2,1,1] => 5
[4,4,2,2,1,1,1,1,1] => [9,3,2,1,1,1] => 6
[4,4,2,1,1,1,1,1,1,1] => [10,2,1,1,1,1,1] => 7
[4,4,1,1,1,1,1,1,1,1,1] => [11,3,1,1,1] => 5
[4,3,3,3,3,1] => [3,3,3,3,2,2,1] => 7
[4,3,3,3,2,2] => [5,3,3,3,2,1] => 6
[4,3,3,3,2,1,1] => [7,3,3,3,1] => 5
[4,3,3,3,1,1,1,1] => [8,3,3,2,1] => 5
[4,3,3,2,2,2,1] => [5,4,4,3,1] => 5
[4,3,3,2,2,1,1,1] => [8,5,3,1] => 4
[4,3,3,2,1,1,1,1,1] => [9,3,2,2,1] => 5
[4,3,3,1,1,1,1,1,1,1] => [10,3,1,1,1,1] => 6
[4,3,2,2,2,2,2] => [5,5,4,1,1,1] => 6
[4,3,2,2,2,2,1,1] => [8,6,1,1,1] => 5
[4,3,2,2,2,1,1,1,1] => [9,5,1,1,1] => 5
[4,3,2,2,1,1,1,1,1,1] => [10,4,1,1,1] => 5
[4,3,2,1,1,1,1,1,1,1,1] => [11,1,1,1,1,1,1] => 7
[4,3,1,1,1,1,1,1,1,1,1,1] => [12,1,1,1,1,1] => 6
[4,2,2,2,2,2,2,1] => [5,5,5,1,1] => 5
[4,2,2,2,2,2,1,1,1] => [9,6,1,1] => 4
[4,2,2,2,2,1,1,1,1,1] => [10,5,1,1] => 4
[4,2,2,2,1,1,1,1,1,1,1] => [11,4,1,1] => 4
[4,2,2,1,1,1,1,1,1,1,1,1] => [12,2,1,1,1] => 5
[4,2,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1,1,1] => 5
[4,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,2,1] => 3
[3,3,3,3,3,2] => [3,3,3,3,3,2] => 6
[3,3,3,3,3,1,1] => [5,3,3,3,3] => 5
[3,3,3,3,2,2,1] => [6,4,4,3] => 4
[3,3,3,3,2,1,1,1] => [8,3,3,3] => 4
[3,3,3,3,1,1,1,1,1] => [9,3,3,2] => 4
[3,3,3,2,2,2,2] => [5,5,4,3] => 4
[3,3,3,2,2,2,1,1] => [8,6,3] => 3
[3,3,3,2,2,1,1,1,1] => [9,5,3] => 3
[3,3,3,2,1,1,1,1,1,1] => [10,3,2,2] => 4
[3,3,3,1,1,1,1,1,1,1,1] => [11,2,2,2] => 4
[3,3,2,2,2,2,2,1] => [5,5,5,2] => 4
[3,3,2,2,2,2,1,1,1] => [9,6,2] => 3
[3,3,2,2,2,1,1,1,1,1] => [10,5,2] => 3
[3,3,2,2,1,1,1,1,1,1,1] => [11,4,2] => 3
[3,3,2,1,1,1,1,1,1,1,1,1] => [12,2,2,1] => 4
[3,3,1,1,1,1,1,1,1,1,1,1,1] => [13,2,1,1] => 4
[3,2,2,2,2,2,2,2] => [6,5,5,1] => 4
[3,2,2,2,2,2,2,1,1] => [9,7,1] => 3
[3,2,2,2,2,2,1,1,1,1] => [10,6,1] => 3
[3,2,2,2,2,1,1,1,1,1,1] => [11,5,1] => 3
[3,2,2,2,1,1,1,1,1,1,1,1] => [12,4,1] => 3
[3,2,2,1,1,1,1,1,1,1,1,1,1] => [13,3,1] => 3
[3,2,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1,1,1] => 4
[3,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,1,1] => 3
[2,2,2,2,2,2,2,2,1] => [6,6,5] => 3
[2,2,2,2,2,2,2,1,1,1] => [10,7] => 2
[2,2,2,2,2,2,1,1,1,1,1] => [11,6] => 2
[2,2,2,2,2,1,1,1,1,1,1,1] => [12,5] => 2
[2,2,2,2,1,1,1,1,1,1,1,1,1] => [13,4] => 2
[2,2,2,1,1,1,1,1,1,1,1,1,1,1] => [14,3] => 2
[2,2,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,2] => 2
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [16,1] => 2
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [17] => 1
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Generating function
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Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
1,1 1,1,1 1,2,1,1 1,2,2,1,1 1,3,3,2,1,1 1,3,4,3,2,1,1 1,4,5,5,3,2,1,1 1,4,7,6,5,3,2,1,1 1,5,8,9,7,5,3,2,1,1 1,5,10,11,10,7,5,3,2,1,1 1,6,12,15,13,11,7,5,3,2,1,1 1,6,14,18,18,14,11,7,5,3,2,1,1
$F_{1} = q$
$F_{2} = q + q^{2}$
$F_{3} = q + q^{2} + q^{3}$
$F_{4} = q + 2\ q^{2} + q^{3} + q^{4}$
$F_{5} = q + 2\ q^{2} + 2\ q^{3} + q^{4} + q^{5}$
$F_{6} = q + 3\ q^{2} + 3\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$
$F_{7} = q + 3\ q^{2} + 4\ q^{3} + 3\ q^{4} + 2\ q^{5} + q^{6} + q^{7}$
$F_{8} = q + 4\ q^{2} + 5\ q^{3} + 5\ q^{4} + 3\ q^{5} + 2\ q^{6} + q^{7} + q^{8}$
$F_{9} = q + 4\ q^{2} + 7\ q^{3} + 6\ q^{4} + 5\ q^{5} + 3\ q^{6} + 2\ q^{7} + q^{8} + q^{9}$
$F_{10} = q + 5\ q^{2} + 8\ q^{3} + 9\ q^{4} + 7\ q^{5} + 5\ q^{6} + 3\ q^{7} + 2\ q^{8} + q^{9} + q^{10}$
$F_{11} = q + 5\ q^{2} + 10\ q^{3} + 11\ q^{4} + 10\ q^{5} + 7\ q^{6} + 5\ q^{7} + 3\ q^{8} + 2\ q^{9} + q^{10} + q^{11}$
$F_{12} = q + 6\ q^{2} + 12\ q^{3} + 15\ q^{4} + 13\ q^{5} + 11\ q^{6} + 7\ q^{7} + 5\ q^{8} + 3\ q^{9} + 2\ q^{10} + q^{11} + q^{12}$
$F_{13} = q + 6\ q^{2} + 14\ q^{3} + 18\ q^{4} + 18\ q^{5} + 14\ q^{6} + 11\ q^{7} + 7\ q^{8} + 5\ q^{9} + 3\ q^{10} + 2\ q^{11} + q^{12} + q^{13}$
$F_{14} = q + 7\ q^{2} + 16\ q^{3} + 23\ q^{4} + 23\ q^{5} + 20\ q^{6} + 15\ q^{7} + 11\ q^{8} + 7\ q^{9} + 5\ q^{10} + 3\ q^{11} + 2\ q^{12} + q^{13} + q^{14}$
$F_{15} = q + 7\ q^{2} + 19\ q^{3} + 27\ q^{4} + 30\ q^{5} + 26\ q^{6} + 21\ q^{7} + 15\ q^{8} + 11\ q^{9} + 7\ q^{10} + 5\ q^{11} + 3\ q^{12} + 2\ q^{13} + q^{14} + q^{15}$
$F_{16} = q + 8\ q^{2} + 21\ q^{3} + 34\ q^{4} + 37\ q^{5} + 35\ q^{6} + 28\ q^{7} + 22\ q^{8} + 15\ q^{9} + 11\ q^{10} + 7\ q^{11} + 5\ q^{12} + 3\ q^{13} + 2\ q^{14} + q^{15} + q^{16}$
$F_{17} = q + 8\ q^{2} + 24\ q^{3} + 39\ q^{4} + 47\ q^{5} + 44\ q^{6} + 38\ q^{7} + 29\ q^{8} + 22\ q^{9} + 15\ q^{10} + 11\ q^{11} + 7\ q^{12} + 5\ q^{13} + 3\ q^{14} + 2\ q^{15} + q^{16} + q^{17}$
Description
The length of the partition.
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Loehr-Warrington inverse
Description
Return a partition whose length is the diagonal inversion number of the preimage.
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