Identifier
-
Mp00040:
Integer compositions
—to partition⟶
Integer partitions
St000185: Integer partitions ⟶ ℤ
Values
[1] => [1] => 0
[1,1] => [1,1] => 1
[2] => [2] => 0
[1,1,1] => [1,1,1] => 3
[1,2] => [2,1] => 1
[2,1] => [2,1] => 1
[3] => [3] => 0
[1,1,1,1] => [1,1,1,1] => 6
[1,1,2] => [2,1,1] => 3
[1,2,1] => [2,1,1] => 3
[1,3] => [3,1] => 1
[2,1,1] => [2,1,1] => 3
[2,2] => [2,2] => 2
[3,1] => [3,1] => 1
[4] => [4] => 0
[1,1,1,1,1] => [1,1,1,1,1] => 10
[1,1,1,2] => [2,1,1,1] => 6
[1,1,2,1] => [2,1,1,1] => 6
[1,1,3] => [3,1,1] => 3
[1,2,1,1] => [2,1,1,1] => 6
[1,2,2] => [2,2,1] => 4
[1,3,1] => [3,1,1] => 3
[1,4] => [4,1] => 1
[2,1,1,1] => [2,1,1,1] => 6
[2,1,2] => [2,2,1] => 4
[2,2,1] => [2,2,1] => 4
[2,3] => [3,2] => 2
[3,1,1] => [3,1,1] => 3
[3,2] => [3,2] => 2
[4,1] => [4,1] => 1
[5] => [5] => 0
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 15
[1,1,1,1,2] => [2,1,1,1,1] => 10
[1,1,1,2,1] => [2,1,1,1,1] => 10
[1,1,1,3] => [3,1,1,1] => 6
[1,1,2,1,1] => [2,1,1,1,1] => 10
[1,1,2,2] => [2,2,1,1] => 7
[1,1,3,1] => [3,1,1,1] => 6
[1,1,4] => [4,1,1] => 3
[1,2,1,1,1] => [2,1,1,1,1] => 10
[1,2,1,2] => [2,2,1,1] => 7
[1,2,2,1] => [2,2,1,1] => 7
[1,2,3] => [3,2,1] => 4
[1,3,1,1] => [3,1,1,1] => 6
[1,3,2] => [3,2,1] => 4
[1,4,1] => [4,1,1] => 3
[1,5] => [5,1] => 1
[2,1,1,1,1] => [2,1,1,1,1] => 10
[2,1,1,2] => [2,2,1,1] => 7
[2,1,2,1] => [2,2,1,1] => 7
[2,1,3] => [3,2,1] => 4
[2,2,1,1] => [2,2,1,1] => 7
[2,2,2] => [2,2,2] => 6
[2,3,1] => [3,2,1] => 4
[2,4] => [4,2] => 2
[3,1,1,1] => [3,1,1,1] => 6
[3,1,2] => [3,2,1] => 4
[3,2,1] => [3,2,1] => 4
[3,3] => [3,3] => 3
[4,1,1] => [4,1,1] => 3
[4,2] => [4,2] => 2
[5,1] => [5,1] => 1
[6] => [6] => 0
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 21
[1,1,1,1,1,2] => [2,1,1,1,1,1] => 15
[1,1,1,1,2,1] => [2,1,1,1,1,1] => 15
[1,1,1,1,3] => [3,1,1,1,1] => 10
[1,1,1,2,1,1] => [2,1,1,1,1,1] => 15
[1,1,1,2,2] => [2,2,1,1,1] => 11
[1,1,1,3,1] => [3,1,1,1,1] => 10
[1,1,1,4] => [4,1,1,1] => 6
[1,1,2,1,1,1] => [2,1,1,1,1,1] => 15
[1,1,2,1,2] => [2,2,1,1,1] => 11
[1,1,2,2,1] => [2,2,1,1,1] => 11
[1,1,2,3] => [3,2,1,1] => 7
[1,1,3,1,1] => [3,1,1,1,1] => 10
[1,1,3,2] => [3,2,1,1] => 7
[1,1,4,1] => [4,1,1,1] => 6
[1,1,5] => [5,1,1] => 3
[1,2,1,1,1,1] => [2,1,1,1,1,1] => 15
[1,2,1,1,2] => [2,2,1,1,1] => 11
[1,2,1,2,1] => [2,2,1,1,1] => 11
[1,2,1,3] => [3,2,1,1] => 7
[1,2,2,1,1] => [2,2,1,1,1] => 11
[1,2,2,2] => [2,2,2,1] => 9
[1,2,3,1] => [3,2,1,1] => 7
[1,2,4] => [4,2,1] => 4
[1,3,1,1,1] => [3,1,1,1,1] => 10
[1,3,1,2] => [3,2,1,1] => 7
[1,3,2,1] => [3,2,1,1] => 7
[1,3,3] => [3,3,1] => 5
[1,4,1,1] => [4,1,1,1] => 6
[1,4,2] => [4,2,1] => 4
[1,5,1] => [5,1,1] => 3
[1,6] => [6,1] => 1
[2,1,1,1,1,1] => [2,1,1,1,1,1] => 15
[2,1,1,1,2] => [2,2,1,1,1] => 11
[2,1,1,2,1] => [2,2,1,1,1] => 11
[2,1,1,3] => [3,2,1,1] => 7
[2,1,2,1,1] => [2,2,1,1,1] => 11
[2,1,2,2] => [2,2,2,1] => 9
>>> Load all 1217 entries. <<<[2,1,3,1] => [3,2,1,1] => 7
[2,1,4] => [4,2,1] => 4
[2,2,1,1,1] => [2,2,1,1,1] => 11
[2,2,1,2] => [2,2,2,1] => 9
[2,2,2,1] => [2,2,2,1] => 9
[2,2,3] => [3,2,2] => 6
[2,3,1,1] => [3,2,1,1] => 7
[2,3,2] => [3,2,2] => 6
[2,4,1] => [4,2,1] => 4
[2,5] => [5,2] => 2
[3,1,1,1,1] => [3,1,1,1,1] => 10
[3,1,1,2] => [3,2,1,1] => 7
[3,1,2,1] => [3,2,1,1] => 7
[3,1,3] => [3,3,1] => 5
[3,2,1,1] => [3,2,1,1] => 7
[3,2,2] => [3,2,2] => 6
[3,3,1] => [3,3,1] => 5
[3,4] => [4,3] => 3
[4,1,1,1] => [4,1,1,1] => 6
[4,1,2] => [4,2,1] => 4
[4,2,1] => [4,2,1] => 4
[4,3] => [4,3] => 3
[5,1,1] => [5,1,1] => 3
[5,2] => [5,2] => 2
[6,1] => [6,1] => 1
[7] => [7] => 0
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 28
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => 21
[1,1,1,1,1,2,1] => [2,1,1,1,1,1,1] => 21
[1,1,1,1,1,3] => [3,1,1,1,1,1] => 15
[1,1,1,1,2,1,1] => [2,1,1,1,1,1,1] => 21
[1,1,1,1,2,2] => [2,2,1,1,1,1] => 16
[1,1,1,1,3,1] => [3,1,1,1,1,1] => 15
[1,1,1,1,4] => [4,1,1,1,1] => 10
[1,1,1,2,1,1,1] => [2,1,1,1,1,1,1] => 21
[1,1,1,2,1,2] => [2,2,1,1,1,1] => 16
[1,1,1,2,2,1] => [2,2,1,1,1,1] => 16
[1,1,1,2,3] => [3,2,1,1,1] => 11
[1,1,1,3,1,1] => [3,1,1,1,1,1] => 15
[1,1,1,3,2] => [3,2,1,1,1] => 11
[1,1,1,4,1] => [4,1,1,1,1] => 10
[1,1,1,5] => [5,1,1,1] => 6
[1,1,2,1,1,1,1] => [2,1,1,1,1,1,1] => 21
[1,1,2,1,1,2] => [2,2,1,1,1,1] => 16
[1,1,2,1,2,1] => [2,2,1,1,1,1] => 16
[1,1,2,1,3] => [3,2,1,1,1] => 11
[1,1,2,2,1,1] => [2,2,1,1,1,1] => 16
[1,1,2,2,2] => [2,2,2,1,1] => 13
[1,1,2,3,1] => [3,2,1,1,1] => 11
[1,1,2,4] => [4,2,1,1] => 7
[1,1,3,1,1,1] => [3,1,1,1,1,1] => 15
[1,1,3,1,2] => [3,2,1,1,1] => 11
[1,1,3,2,1] => [3,2,1,1,1] => 11
[1,1,3,3] => [3,3,1,1] => 8
[1,1,4,1,1] => [4,1,1,1,1] => 10
[1,1,4,2] => [4,2,1,1] => 7
[1,1,5,1] => [5,1,1,1] => 6
[1,1,6] => [6,1,1] => 3
[1,2,1,1,1,1,1] => [2,1,1,1,1,1,1] => 21
[1,2,1,1,1,2] => [2,2,1,1,1,1] => 16
[1,2,1,1,2,1] => [2,2,1,1,1,1] => 16
[1,2,1,1,3] => [3,2,1,1,1] => 11
[1,2,1,2,1,1] => [2,2,1,1,1,1] => 16
[1,2,1,2,2] => [2,2,2,1,1] => 13
[1,2,1,3,1] => [3,2,1,1,1] => 11
[1,2,1,4] => [4,2,1,1] => 7
[1,2,2,1,1,1] => [2,2,1,1,1,1] => 16
[1,2,2,1,2] => [2,2,2,1,1] => 13
[1,2,2,2,1] => [2,2,2,1,1] => 13
[1,2,2,3] => [3,2,2,1] => 9
[1,2,3,1,1] => [3,2,1,1,1] => 11
[1,2,3,2] => [3,2,2,1] => 9
[1,2,4,1] => [4,2,1,1] => 7
[1,2,5] => [5,2,1] => 4
[1,3,1,1,1,1] => [3,1,1,1,1,1] => 15
[1,3,1,1,2] => [3,2,1,1,1] => 11
[1,3,1,2,1] => [3,2,1,1,1] => 11
[1,3,1,3] => [3,3,1,1] => 8
[1,3,2,1,1] => [3,2,1,1,1] => 11
[1,3,2,2] => [3,2,2,1] => 9
[1,3,3,1] => [3,3,1,1] => 8
[1,3,4] => [4,3,1] => 5
[1,4,1,1,1] => [4,1,1,1,1] => 10
[1,4,1,2] => [4,2,1,1] => 7
[1,4,2,1] => [4,2,1,1] => 7
[1,4,3] => [4,3,1] => 5
[1,5,1,1] => [5,1,1,1] => 6
[1,5,2] => [5,2,1] => 4
[1,6,1] => [6,1,1] => 3
[1,7] => [7,1] => 1
[2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 21
[2,1,1,1,1,2] => [2,2,1,1,1,1] => 16
[2,1,1,1,2,1] => [2,2,1,1,1,1] => 16
[2,1,1,1,3] => [3,2,1,1,1] => 11
[2,1,1,2,1,1] => [2,2,1,1,1,1] => 16
[2,1,1,2,2] => [2,2,2,1,1] => 13
[2,1,1,3,1] => [3,2,1,1,1] => 11
[2,1,1,4] => [4,2,1,1] => 7
[2,1,2,1,1,1] => [2,2,1,1,1,1] => 16
[2,1,2,1,2] => [2,2,2,1,1] => 13
[2,1,2,2,1] => [2,2,2,1,1] => 13
[2,1,2,3] => [3,2,2,1] => 9
[2,1,3,1,1] => [3,2,1,1,1] => 11
[2,1,3,2] => [3,2,2,1] => 9
[2,1,4,1] => [4,2,1,1] => 7
[2,1,5] => [5,2,1] => 4
[2,2,1,1,1,1] => [2,2,1,1,1,1] => 16
[2,2,1,1,2] => [2,2,2,1,1] => 13
[2,2,1,2,1] => [2,2,2,1,1] => 13
[2,2,1,3] => [3,2,2,1] => 9
[2,2,2,1,1] => [2,2,2,1,1] => 13
[2,2,2,2] => [2,2,2,2] => 12
[2,2,3,1] => [3,2,2,1] => 9
[2,2,4] => [4,2,2] => 6
[2,3,1,1,1] => [3,2,1,1,1] => 11
[2,3,1,2] => [3,2,2,1] => 9
[2,3,2,1] => [3,2,2,1] => 9
[2,3,3] => [3,3,2] => 7
[2,4,1,1] => [4,2,1,1] => 7
[2,4,2] => [4,2,2] => 6
[2,5,1] => [5,2,1] => 4
[2,6] => [6,2] => 2
[3,1,1,1,1,1] => [3,1,1,1,1,1] => 15
[3,1,1,1,2] => [3,2,1,1,1] => 11
[3,1,1,2,1] => [3,2,1,1,1] => 11
[3,1,1,3] => [3,3,1,1] => 8
[3,1,2,1,1] => [3,2,1,1,1] => 11
[3,1,2,2] => [3,2,2,1] => 9
[3,1,3,1] => [3,3,1,1] => 8
[3,1,4] => [4,3,1] => 5
[3,2,1,1,1] => [3,2,1,1,1] => 11
[3,2,1,2] => [3,2,2,1] => 9
[3,2,2,1] => [3,2,2,1] => 9
[3,2,3] => [3,3,2] => 7
[3,3,1,1] => [3,3,1,1] => 8
[3,3,2] => [3,3,2] => 7
[3,4,1] => [4,3,1] => 5
[3,5] => [5,3] => 3
[4,1,1,1,1] => [4,1,1,1,1] => 10
[4,1,1,2] => [4,2,1,1] => 7
[4,1,2,1] => [4,2,1,1] => 7
[4,1,3] => [4,3,1] => 5
[4,2,1,1] => [4,2,1,1] => 7
[4,2,2] => [4,2,2] => 6
[4,3,1] => [4,3,1] => 5
[4,4] => [4,4] => 4
[5,1,1,1] => [5,1,1,1] => 6
[5,1,2] => [5,2,1] => 4
[5,2,1] => [5,2,1] => 4
[5,3] => [5,3] => 3
[6,1,1] => [6,1,1] => 3
[6,2] => [6,2] => 2
[7,1] => [7,1] => 1
[8] => [8] => 0
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 36
[1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1] => 28
[1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1] => 28
[1,1,1,1,1,1,3] => [3,1,1,1,1,1,1] => 21
[1,1,1,1,1,2,1,1] => [2,1,1,1,1,1,1,1] => 28
[1,1,1,1,1,2,2] => [2,2,1,1,1,1,1] => 22
[1,1,1,1,1,3,1] => [3,1,1,1,1,1,1] => 21
[1,1,1,1,1,4] => [4,1,1,1,1,1] => 15
[1,1,1,1,2,1,1,1] => [2,1,1,1,1,1,1,1] => 28
[1,1,1,1,2,1,2] => [2,2,1,1,1,1,1] => 22
[1,1,1,1,2,2,1] => [2,2,1,1,1,1,1] => 22
[1,1,1,1,2,3] => [3,2,1,1,1,1] => 16
[1,1,1,1,3,1,1] => [3,1,1,1,1,1,1] => 21
[1,1,1,1,3,2] => [3,2,1,1,1,1] => 16
[1,1,1,1,4,1] => [4,1,1,1,1,1] => 15
[1,1,1,1,5] => [5,1,1,1,1] => 10
[1,1,1,2,1,1,1,1] => [2,1,1,1,1,1,1,1] => 28
[1,1,1,2,1,1,2] => [2,2,1,1,1,1,1] => 22
[1,1,1,2,1,2,1] => [2,2,1,1,1,1,1] => 22
[1,1,1,2,1,3] => [3,2,1,1,1,1] => 16
[1,1,1,2,2,1,1] => [2,2,1,1,1,1,1] => 22
[1,1,1,2,2,2] => [2,2,2,1,1,1] => 18
[1,1,1,2,3,1] => [3,2,1,1,1,1] => 16
[1,1,1,2,4] => [4,2,1,1,1] => 11
[1,1,1,3,1,1,1] => [3,1,1,1,1,1,1] => 21
[1,1,1,3,1,2] => [3,2,1,1,1,1] => 16
[1,1,1,3,2,1] => [3,2,1,1,1,1] => 16
[1,1,1,3,3] => [3,3,1,1,1] => 12
[1,1,1,4,1,1] => [4,1,1,1,1,1] => 15
[1,1,1,4,2] => [4,2,1,1,1] => 11
[1,1,1,5,1] => [5,1,1,1,1] => 10
[1,1,1,6] => [6,1,1,1] => 6
[1,1,2,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 28
[1,1,2,1,1,1,2] => [2,2,1,1,1,1,1] => 22
[1,1,2,1,1,2,1] => [2,2,1,1,1,1,1] => 22
[1,1,2,1,1,3] => [3,2,1,1,1,1] => 16
[1,1,2,1,2,1,1] => [2,2,1,1,1,1,1] => 22
[1,1,2,1,2,2] => [2,2,2,1,1,1] => 18
[1,1,2,1,3,1] => [3,2,1,1,1,1] => 16
[1,1,2,1,4] => [4,2,1,1,1] => 11
[1,1,2,2,1,1,1] => [2,2,1,1,1,1,1] => 22
[1,1,2,2,1,2] => [2,2,2,1,1,1] => 18
[1,1,2,2,2,1] => [2,2,2,1,1,1] => 18
[1,1,2,2,3] => [3,2,2,1,1] => 13
[1,1,2,3,1,1] => [3,2,1,1,1,1] => 16
[1,1,2,3,2] => [3,2,2,1,1] => 13
[1,1,2,4,1] => [4,2,1,1,1] => 11
[1,1,2,5] => [5,2,1,1] => 7
[1,1,3,1,1,1,1] => [3,1,1,1,1,1,1] => 21
[1,1,3,1,1,2] => [3,2,1,1,1,1] => 16
[1,1,3,1,2,1] => [3,2,1,1,1,1] => 16
[1,1,3,1,3] => [3,3,1,1,1] => 12
[1,1,3,2,1,1] => [3,2,1,1,1,1] => 16
[1,1,3,2,2] => [3,2,2,1,1] => 13
[1,1,3,3,1] => [3,3,1,1,1] => 12
[1,1,3,4] => [4,3,1,1] => 8
[1,1,4,1,1,1] => [4,1,1,1,1,1] => 15
[1,1,4,1,2] => [4,2,1,1,1] => 11
[1,1,4,2,1] => [4,2,1,1,1] => 11
[1,1,4,3] => [4,3,1,1] => 8
[1,1,5,1,1] => [5,1,1,1,1] => 10
[1,1,5,2] => [5,2,1,1] => 7
[1,1,6,1] => [6,1,1,1] => 6
[1,1,7] => [7,1,1] => 3
[1,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 28
[1,2,1,1,1,1,2] => [2,2,1,1,1,1,1] => 22
[1,2,1,1,1,2,1] => [2,2,1,1,1,1,1] => 22
[1,2,1,1,1,3] => [3,2,1,1,1,1] => 16
[1,2,1,1,2,1,1] => [2,2,1,1,1,1,1] => 22
[1,2,1,1,2,2] => [2,2,2,1,1,1] => 18
[1,2,1,1,3,1] => [3,2,1,1,1,1] => 16
[1,2,1,1,4] => [4,2,1,1,1] => 11
[1,2,1,2,1,1,1] => [2,2,1,1,1,1,1] => 22
[1,2,1,2,1,2] => [2,2,2,1,1,1] => 18
[1,2,1,2,2,1] => [2,2,2,1,1,1] => 18
[1,2,1,2,3] => [3,2,2,1,1] => 13
[1,2,1,3,1,1] => [3,2,1,1,1,1] => 16
[1,2,1,3,2] => [3,2,2,1,1] => 13
[1,2,1,4,1] => [4,2,1,1,1] => 11
[1,2,1,5] => [5,2,1,1] => 7
[1,2,2,1,1,1,1] => [2,2,1,1,1,1,1] => 22
[1,2,2,1,1,2] => [2,2,2,1,1,1] => 18
[1,2,2,1,2,1] => [2,2,2,1,1,1] => 18
[1,2,2,1,3] => [3,2,2,1,1] => 13
[1,2,2,2,1,1] => [2,2,2,1,1,1] => 18
[1,2,2,2,2] => [2,2,2,2,1] => 16
[1,2,2,3,1] => [3,2,2,1,1] => 13
[1,2,2,4] => [4,2,2,1] => 9
[1,2,3,1,1,1] => [3,2,1,1,1,1] => 16
[1,2,3,1,2] => [3,2,2,1,1] => 13
[1,2,3,2,1] => [3,2,2,1,1] => 13
[1,2,3,3] => [3,3,2,1] => 10
[1,2,4,1,1] => [4,2,1,1,1] => 11
[1,2,4,2] => [4,2,2,1] => 9
[1,2,5,1] => [5,2,1,1] => 7
[1,2,6] => [6,2,1] => 4
[1,3,1,1,1,1,1] => [3,1,1,1,1,1,1] => 21
[1,3,1,1,1,2] => [3,2,1,1,1,1] => 16
[1,3,1,1,2,1] => [3,2,1,1,1,1] => 16
[1,3,1,1,3] => [3,3,1,1,1] => 12
[1,3,1,2,1,1] => [3,2,1,1,1,1] => 16
[1,3,1,2,2] => [3,2,2,1,1] => 13
[1,3,1,3,1] => [3,3,1,1,1] => 12
[1,3,1,4] => [4,3,1,1] => 8
[1,3,2,1,1,1] => [3,2,1,1,1,1] => 16
[1,3,2,1,2] => [3,2,2,1,1] => 13
[1,3,2,2,1] => [3,2,2,1,1] => 13
[1,3,2,3] => [3,3,2,1] => 10
[1,3,3,1,1] => [3,3,1,1,1] => 12
[1,3,3,2] => [3,3,2,1] => 10
[1,3,4,1] => [4,3,1,1] => 8
[1,3,5] => [5,3,1] => 5
[1,4,1,1,1,1] => [4,1,1,1,1,1] => 15
[1,4,1,1,2] => [4,2,1,1,1] => 11
[1,4,1,2,1] => [4,2,1,1,1] => 11
[1,4,1,3] => [4,3,1,1] => 8
[1,4,2,1,1] => [4,2,1,1,1] => 11
[1,4,2,2] => [4,2,2,1] => 9
[1,4,3,1] => [4,3,1,1] => 8
[1,4,4] => [4,4,1] => 6
[1,5,1,1,1] => [5,1,1,1,1] => 10
[1,5,1,2] => [5,2,1,1] => 7
[1,5,2,1] => [5,2,1,1] => 7
[1,5,3] => [5,3,1] => 5
[1,6,1,1] => [6,1,1,1] => 6
[1,6,2] => [6,2,1] => 4
[1,7,1] => [7,1,1] => 3
[1,8] => [8,1] => 1
[2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 28
[2,1,1,1,1,1,2] => [2,2,1,1,1,1,1] => 22
[2,1,1,1,1,2,1] => [2,2,1,1,1,1,1] => 22
[2,1,1,1,1,3] => [3,2,1,1,1,1] => 16
[2,1,1,1,2,1,1] => [2,2,1,1,1,1,1] => 22
[2,1,1,1,2,2] => [2,2,2,1,1,1] => 18
[2,1,1,1,3,1] => [3,2,1,1,1,1] => 16
[2,1,1,1,4] => [4,2,1,1,1] => 11
[2,1,1,2,1,1,1] => [2,2,1,1,1,1,1] => 22
[2,1,1,2,1,2] => [2,2,2,1,1,1] => 18
[2,1,1,2,2,1] => [2,2,2,1,1,1] => 18
[2,1,1,2,3] => [3,2,2,1,1] => 13
[2,1,1,3,1,1] => [3,2,1,1,1,1] => 16
[2,1,1,3,2] => [3,2,2,1,1] => 13
[2,1,1,4,1] => [4,2,1,1,1] => 11
[2,1,1,5] => [5,2,1,1] => 7
[2,1,2,1,1,1,1] => [2,2,1,1,1,1,1] => 22
[2,1,2,1,1,2] => [2,2,2,1,1,1] => 18
[2,1,2,1,2,1] => [2,2,2,1,1,1] => 18
[2,1,2,1,3] => [3,2,2,1,1] => 13
[2,1,2,2,1,1] => [2,2,2,1,1,1] => 18
[2,1,2,2,2] => [2,2,2,2,1] => 16
[2,1,2,3,1] => [3,2,2,1,1] => 13
[2,1,2,4] => [4,2,2,1] => 9
[2,1,3,1,1,1] => [3,2,1,1,1,1] => 16
[2,1,3,1,2] => [3,2,2,1,1] => 13
[2,1,3,2,1] => [3,2,2,1,1] => 13
[2,1,3,3] => [3,3,2,1] => 10
[2,1,4,1,1] => [4,2,1,1,1] => 11
[2,1,4,2] => [4,2,2,1] => 9
[2,1,5,1] => [5,2,1,1] => 7
[2,1,6] => [6,2,1] => 4
[2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 22
[2,2,1,1,1,2] => [2,2,2,1,1,1] => 18
[2,2,1,1,2,1] => [2,2,2,1,1,1] => 18
[2,2,1,1,3] => [3,2,2,1,1] => 13
[2,2,1,2,1,1] => [2,2,2,1,1,1] => 18
[2,2,1,2,2] => [2,2,2,2,1] => 16
[2,2,1,3,1] => [3,2,2,1,1] => 13
[2,2,1,4] => [4,2,2,1] => 9
[2,2,2,1,1,1] => [2,2,2,1,1,1] => 18
[2,2,2,1,2] => [2,2,2,2,1] => 16
[2,2,2,2,1] => [2,2,2,2,1] => 16
[2,2,2,3] => [3,2,2,2] => 12
[2,2,3,1,1] => [3,2,2,1,1] => 13
[2,2,3,2] => [3,2,2,2] => 12
[2,2,4,1] => [4,2,2,1] => 9
[2,2,5] => [5,2,2] => 6
[2,3,1,1,1,1] => [3,2,1,1,1,1] => 16
[2,3,1,1,2] => [3,2,2,1,1] => 13
[2,3,1,2,1] => [3,2,2,1,1] => 13
[2,3,1,3] => [3,3,2,1] => 10
[2,3,2,1,1] => [3,2,2,1,1] => 13
[2,3,2,2] => [3,2,2,2] => 12
[2,3,3,1] => [3,3,2,1] => 10
[2,3,4] => [4,3,2] => 7
[2,4,1,1,1] => [4,2,1,1,1] => 11
[2,4,1,2] => [4,2,2,1] => 9
[2,4,2,1] => [4,2,2,1] => 9
[2,4,3] => [4,3,2] => 7
[2,5,1,1] => [5,2,1,1] => 7
[2,5,2] => [5,2,2] => 6
[2,6,1] => [6,2,1] => 4
[2,7] => [7,2] => 2
[3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 21
[3,1,1,1,1,2] => [3,2,1,1,1,1] => 16
[3,1,1,1,2,1] => [3,2,1,1,1,1] => 16
[3,1,1,1,3] => [3,3,1,1,1] => 12
[3,1,1,2,1,1] => [3,2,1,1,1,1] => 16
[3,1,1,2,2] => [3,2,2,1,1] => 13
[3,1,1,3,1] => [3,3,1,1,1] => 12
[3,1,1,4] => [4,3,1,1] => 8
[3,1,2,1,1,1] => [3,2,1,1,1,1] => 16
[3,1,2,1,2] => [3,2,2,1,1] => 13
[3,1,2,2,1] => [3,2,2,1,1] => 13
[3,1,2,3] => [3,3,2,1] => 10
[3,1,3,1,1] => [3,3,1,1,1] => 12
[3,1,3,2] => [3,3,2,1] => 10
[3,1,4,1] => [4,3,1,1] => 8
[3,1,5] => [5,3,1] => 5
[3,2,1,1,1,1] => [3,2,1,1,1,1] => 16
[3,2,1,1,2] => [3,2,2,1,1] => 13
[3,2,1,2,1] => [3,2,2,1,1] => 13
[3,2,1,3] => [3,3,2,1] => 10
[3,2,2,1,1] => [3,2,2,1,1] => 13
[3,2,2,2] => [3,2,2,2] => 12
[3,2,3,1] => [3,3,2,1] => 10
[3,2,4] => [4,3,2] => 7
[3,3,1,1,1] => [3,3,1,1,1] => 12
[3,3,1,2] => [3,3,2,1] => 10
[3,3,2,1] => [3,3,2,1] => 10
[3,3,3] => [3,3,3] => 9
[3,4,1,1] => [4,3,1,1] => 8
[3,4,2] => [4,3,2] => 7
[3,5,1] => [5,3,1] => 5
[3,6] => [6,3] => 3
[4,1,1,1,1,1] => [4,1,1,1,1,1] => 15
[4,1,1,1,2] => [4,2,1,1,1] => 11
[4,1,1,2,1] => [4,2,1,1,1] => 11
[4,1,1,3] => [4,3,1,1] => 8
[4,1,2,1,1] => [4,2,1,1,1] => 11
[4,1,2,2] => [4,2,2,1] => 9
[4,1,3,1] => [4,3,1,1] => 8
[4,1,4] => [4,4,1] => 6
[4,2,1,1,1] => [4,2,1,1,1] => 11
[4,2,1,2] => [4,2,2,1] => 9
[4,2,2,1] => [4,2,2,1] => 9
[4,2,3] => [4,3,2] => 7
[4,3,1,1] => [4,3,1,1] => 8
[4,3,2] => [4,3,2] => 7
[4,4,1] => [4,4,1] => 6
[4,5] => [5,4] => 4
[5,1,1,1,1] => [5,1,1,1,1] => 10
[5,1,1,2] => [5,2,1,1] => 7
[5,1,2,1] => [5,2,1,1] => 7
[5,1,3] => [5,3,1] => 5
[5,2,1,1] => [5,2,1,1] => 7
[5,2,2] => [5,2,2] => 6
[5,3,1] => [5,3,1] => 5
[5,4] => [5,4] => 4
[6,1,1,1] => [6,1,1,1] => 6
[6,1,2] => [6,2,1] => 4
[6,2,1] => [6,2,1] => 4
[6,3] => [6,3] => 3
[7,1,1] => [7,1,1] => 3
[7,2] => [7,2] => 2
[8,1] => [8,1] => 1
[9] => [9] => 0
[1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 45
[1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1] => 36
[1,1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1,1] => 36
[1,1,1,1,1,1,1,3] => [3,1,1,1,1,1,1,1] => 28
[1,1,1,1,1,1,2,1,1] => [2,1,1,1,1,1,1,1,1] => 36
[1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1] => 29
[1,1,1,1,1,1,3,1] => [3,1,1,1,1,1,1,1] => 28
[1,1,1,1,1,1,4] => [4,1,1,1,1,1,1] => 21
[1,1,1,1,1,2,1,1,1] => [2,1,1,1,1,1,1,1,1] => 36
[1,1,1,1,1,2,1,2] => [2,2,1,1,1,1,1,1] => 29
[1,1,1,1,1,2,2,1] => [2,2,1,1,1,1,1,1] => 29
[1,1,1,1,1,2,3] => [3,2,1,1,1,1,1] => 22
[1,1,1,1,1,3,1,1] => [3,1,1,1,1,1,1,1] => 28
[1,1,1,1,1,3,2] => [3,2,1,1,1,1,1] => 22
[1,1,1,1,1,4,1] => [4,1,1,1,1,1,1] => 21
[1,1,1,1,1,5] => [5,1,1,1,1,1] => 15
[1,1,1,1,2,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 36
[1,1,1,1,2,1,1,2] => [2,2,1,1,1,1,1,1] => 29
[1,1,1,1,2,1,2,1] => [2,2,1,1,1,1,1,1] => 29
[1,1,1,1,2,1,3] => [3,2,1,1,1,1,1] => 22
[1,1,1,1,2,2,1,1] => [2,2,1,1,1,1,1,1] => 29
[1,1,1,1,2,2,2] => [2,2,2,1,1,1,1] => 24
[1,1,1,1,2,3,1] => [3,2,1,1,1,1,1] => 22
[1,1,1,1,2,4] => [4,2,1,1,1,1] => 16
[1,1,1,1,3,1,1,1] => [3,1,1,1,1,1,1,1] => 28
[1,1,1,1,3,1,2] => [3,2,1,1,1,1,1] => 22
[1,1,1,1,3,2,1] => [3,2,1,1,1,1,1] => 22
[1,1,1,1,3,3] => [3,3,1,1,1,1] => 17
[1,1,1,1,4,1,1] => [4,1,1,1,1,1,1] => 21
[1,1,1,1,4,2] => [4,2,1,1,1,1] => 16
[1,1,1,1,5,1] => [5,1,1,1,1,1] => 15
[1,1,1,1,6] => [6,1,1,1,1] => 10
[1,1,1,2,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 36
[1,1,1,2,1,1,1,2] => [2,2,1,1,1,1,1,1] => 29
[1,1,1,2,1,1,2,1] => [2,2,1,1,1,1,1,1] => 29
[1,1,1,2,1,1,3] => [3,2,1,1,1,1,1] => 22
[1,1,1,2,1,2,1,1] => [2,2,1,1,1,1,1,1] => 29
[1,1,1,2,1,2,2] => [2,2,2,1,1,1,1] => 24
[1,1,1,2,1,3,1] => [3,2,1,1,1,1,1] => 22
[1,1,1,2,1,4] => [4,2,1,1,1,1] => 16
[1,1,1,2,2,1,1,1] => [2,2,1,1,1,1,1,1] => 29
[1,1,1,2,2,1,2] => [2,2,2,1,1,1,1] => 24
[1,1,1,2,2,2,1] => [2,2,2,1,1,1,1] => 24
[1,1,1,2,2,3] => [3,2,2,1,1,1] => 18
[1,1,1,2,3,1,1] => [3,2,1,1,1,1,1] => 22
[1,1,1,2,3,2] => [3,2,2,1,1,1] => 18
[1,1,1,2,4,1] => [4,2,1,1,1,1] => 16
[1,1,1,2,5] => [5,2,1,1,1] => 11
[1,1,1,3,1,1,1,1] => [3,1,1,1,1,1,1,1] => 28
[1,1,1,3,1,1,2] => [3,2,1,1,1,1,1] => 22
[1,1,1,3,1,2,1] => [3,2,1,1,1,1,1] => 22
[1,1,1,3,1,3] => [3,3,1,1,1,1] => 17
[1,1,1,3,2,1,1] => [3,2,1,1,1,1,1] => 22
[1,1,1,3,2,2] => [3,2,2,1,1,1] => 18
[1,1,1,3,3,1] => [3,3,1,1,1,1] => 17
[1,1,1,3,4] => [4,3,1,1,1] => 12
[1,1,1,4,1,1,1] => [4,1,1,1,1,1,1] => 21
[1,1,1,4,1,2] => [4,2,1,1,1,1] => 16
[1,1,1,4,2,1] => [4,2,1,1,1,1] => 16
[1,1,1,4,3] => [4,3,1,1,1] => 12
[1,1,1,5,1,1] => [5,1,1,1,1,1] => 15
[1,1,1,5,2] => [5,2,1,1,1] => 11
[1,1,1,6,1] => [6,1,1,1,1] => 10
[1,1,1,7] => [7,1,1,1] => 6
[1,1,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 36
[1,1,2,1,1,1,1,2] => [2,2,1,1,1,1,1,1] => 29
[1,1,2,1,1,1,2,1] => [2,2,1,1,1,1,1,1] => 29
[1,1,2,1,1,1,3] => [3,2,1,1,1,1,1] => 22
[1,1,2,1,1,2,1,1] => [2,2,1,1,1,1,1,1] => 29
[1,1,2,1,1,2,2] => [2,2,2,1,1,1,1] => 24
[1,1,2,1,1,3,1] => [3,2,1,1,1,1,1] => 22
[1,1,2,1,1,4] => [4,2,1,1,1,1] => 16
[1,1,2,1,2,1,1,1] => [2,2,1,1,1,1,1,1] => 29
[1,1,2,1,2,1,2] => [2,2,2,1,1,1,1] => 24
[1,1,2,1,2,2,1] => [2,2,2,1,1,1,1] => 24
[1,1,2,1,2,3] => [3,2,2,1,1,1] => 18
[1,1,2,1,3,1,1] => [3,2,1,1,1,1,1] => 22
[1,1,2,1,3,2] => [3,2,2,1,1,1] => 18
[1,1,2,1,4,1] => [4,2,1,1,1,1] => 16
[1,1,2,1,5] => [5,2,1,1,1] => 11
[1,1,2,2,1,1,1,1] => [2,2,1,1,1,1,1,1] => 29
[1,1,2,2,1,1,2] => [2,2,2,1,1,1,1] => 24
[1,1,2,2,1,2,1] => [2,2,2,1,1,1,1] => 24
[1,1,2,2,1,3] => [3,2,2,1,1,1] => 18
[1,1,2,2,2,1,1] => [2,2,2,1,1,1,1] => 24
[1,1,2,2,2,2] => [2,2,2,2,1,1] => 21
[1,1,2,2,3,1] => [3,2,2,1,1,1] => 18
[1,1,2,2,4] => [4,2,2,1,1] => 13
[1,1,2,3,1,1,1] => [3,2,1,1,1,1,1] => 22
[1,1,2,3,1,2] => [3,2,2,1,1,1] => 18
[1,1,2,3,2,1] => [3,2,2,1,1,1] => 18
[1,1,2,3,3] => [3,3,2,1,1] => 14
[1,1,2,4,1,1] => [4,2,1,1,1,1] => 16
[1,1,2,4,2] => [4,2,2,1,1] => 13
[1,1,2,5,1] => [5,2,1,1,1] => 11
[1,1,2,6] => [6,2,1,1] => 7
[1,1,3,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 28
[1,1,3,1,1,1,2] => [3,2,1,1,1,1,1] => 22
[1,1,3,1,1,2,1] => [3,2,1,1,1,1,1] => 22
[1,1,3,1,1,3] => [3,3,1,1,1,1] => 17
[1,1,3,1,2,1,1] => [3,2,1,1,1,1,1] => 22
[1,1,3,1,2,2] => [3,2,2,1,1,1] => 18
[1,1,3,1,3,1] => [3,3,1,1,1,1] => 17
[1,1,3,1,4] => [4,3,1,1,1] => 12
[1,1,3,2,1,1,1] => [3,2,1,1,1,1,1] => 22
[1,1,3,2,1,2] => [3,2,2,1,1,1] => 18
[1,1,3,2,2,1] => [3,2,2,1,1,1] => 18
[1,1,3,2,3] => [3,3,2,1,1] => 14
[1,1,3,3,1,1] => [3,3,1,1,1,1] => 17
[1,1,3,3,2] => [3,3,2,1,1] => 14
[1,1,3,4,1] => [4,3,1,1,1] => 12
[1,1,3,5] => [5,3,1,1] => 8
[1,1,4,1,1,1,1] => [4,1,1,1,1,1,1] => 21
[1,1,4,1,1,2] => [4,2,1,1,1,1] => 16
[1,1,4,1,2,1] => [4,2,1,1,1,1] => 16
[1,1,4,1,3] => [4,3,1,1,1] => 12
[1,1,4,2,1,1] => [4,2,1,1,1,1] => 16
[1,1,4,2,2] => [4,2,2,1,1] => 13
[1,1,4,3,1] => [4,3,1,1,1] => 12
[1,1,4,4] => [4,4,1,1] => 9
[1,1,5,1,1,1] => [5,1,1,1,1,1] => 15
[1,1,5,1,2] => [5,2,1,1,1] => 11
[1,1,5,2,1] => [5,2,1,1,1] => 11
[1,1,5,3] => [5,3,1,1] => 8
[1,1,6,1,1] => [6,1,1,1,1] => 10
[1,1,6,2] => [6,2,1,1] => 7
[1,1,7,1] => [7,1,1,1] => 6
[1,1,8] => [8,1,1] => 3
[1,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 36
[1,2,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1] => 29
[1,2,1,1,1,1,2,1] => [2,2,1,1,1,1,1,1] => 29
[1,2,1,1,1,1,3] => [3,2,1,1,1,1,1] => 22
[1,2,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1] => 29
[1,2,1,1,1,2,2] => [2,2,2,1,1,1,1] => 24
[1,2,1,1,1,3,1] => [3,2,1,1,1,1,1] => 22
[1,2,1,1,1,4] => [4,2,1,1,1,1] => 16
[1,2,1,1,2,1,1,1] => [2,2,1,1,1,1,1,1] => 29
[1,2,1,1,2,1,2] => [2,2,2,1,1,1,1] => 24
[1,2,1,1,2,2,1] => [2,2,2,1,1,1,1] => 24
[1,2,1,1,2,3] => [3,2,2,1,1,1] => 18
[1,2,1,1,3,1,1] => [3,2,1,1,1,1,1] => 22
[1,2,1,1,3,2] => [3,2,2,1,1,1] => 18
[1,2,1,1,4,1] => [4,2,1,1,1,1] => 16
[1,2,1,1,5] => [5,2,1,1,1] => 11
[1,2,1,2,1,1,1,1] => [2,2,1,1,1,1,1,1] => 29
[1,2,1,2,1,1,2] => [2,2,2,1,1,1,1] => 24
[1,2,1,2,1,2,1] => [2,2,2,1,1,1,1] => 24
[1,2,1,2,1,3] => [3,2,2,1,1,1] => 18
[1,2,1,2,2,1,1] => [2,2,2,1,1,1,1] => 24
[1,2,1,2,2,2] => [2,2,2,2,1,1] => 21
[1,2,1,2,3,1] => [3,2,2,1,1,1] => 18
[1,2,1,2,4] => [4,2,2,1,1] => 13
[1,2,1,3,1,1,1] => [3,2,1,1,1,1,1] => 22
[1,2,1,3,1,2] => [3,2,2,1,1,1] => 18
[1,2,1,3,2,1] => [3,2,2,1,1,1] => 18
[1,2,1,3,3] => [3,3,2,1,1] => 14
[1,2,1,4,1,1] => [4,2,1,1,1,1] => 16
[1,2,1,4,2] => [4,2,2,1,1] => 13
[1,2,1,5,1] => [5,2,1,1,1] => 11
[1,2,1,6] => [6,2,1,1] => 7
[1,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 29
[1,2,2,1,1,1,2] => [2,2,2,1,1,1,1] => 24
[1,2,2,1,1,2,1] => [2,2,2,1,1,1,1] => 24
[1,2,2,1,1,3] => [3,2,2,1,1,1] => 18
[1,2,2,1,2,1,1] => [2,2,2,1,1,1,1] => 24
[1,2,2,1,2,2] => [2,2,2,2,1,1] => 21
[1,2,2,1,3,1] => [3,2,2,1,1,1] => 18
[1,2,2,1,4] => [4,2,2,1,1] => 13
[1,2,2,2,1,1,1] => [2,2,2,1,1,1,1] => 24
[1,2,2,2,1,2] => [2,2,2,2,1,1] => 21
[1,2,2,2,2,1] => [2,2,2,2,1,1] => 21
[1,2,2,2,3] => [3,2,2,2,1] => 16
[1,2,2,3,1,1] => [3,2,2,1,1,1] => 18
[1,2,2,3,2] => [3,2,2,2,1] => 16
[1,2,2,4,1] => [4,2,2,1,1] => 13
[1,2,2,5] => [5,2,2,1] => 9
[1,2,3,1,1,1,1] => [3,2,1,1,1,1,1] => 22
[1,2,3,1,1,2] => [3,2,2,1,1,1] => 18
[1,2,3,1,2,1] => [3,2,2,1,1,1] => 18
[1,2,3,1,3] => [3,3,2,1,1] => 14
[1,2,3,2,1,1] => [3,2,2,1,1,1] => 18
[1,2,3,2,2] => [3,2,2,2,1] => 16
[1,2,3,3,1] => [3,3,2,1,1] => 14
[1,2,3,4] => [4,3,2,1] => 10
[1,2,4,1,1,1] => [4,2,1,1,1,1] => 16
[1,2,4,1,2] => [4,2,2,1,1] => 13
[1,2,4,2,1] => [4,2,2,1,1] => 13
[1,2,4,3] => [4,3,2,1] => 10
[1,2,5,1,1] => [5,2,1,1,1] => 11
[1,2,5,2] => [5,2,2,1] => 9
[1,2,6,1] => [6,2,1,1] => 7
[1,2,7] => [7,2,1] => 4
[1,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 28
[1,3,1,1,1,1,2] => [3,2,1,1,1,1,1] => 22
[1,3,1,1,1,2,1] => [3,2,1,1,1,1,1] => 22
[1,3,1,1,1,3] => [3,3,1,1,1,1] => 17
[1,3,1,1,2,1,1] => [3,2,1,1,1,1,1] => 22
[1,3,1,1,2,2] => [3,2,2,1,1,1] => 18
[1,3,1,1,3,1] => [3,3,1,1,1,1] => 17
[1,3,1,1,4] => [4,3,1,1,1] => 12
[1,3,1,2,1,1,1] => [3,2,1,1,1,1,1] => 22
[1,3,1,2,1,2] => [3,2,2,1,1,1] => 18
[1,3,1,2,2,1] => [3,2,2,1,1,1] => 18
[1,3,1,2,3] => [3,3,2,1,1] => 14
[1,3,1,3,1,1] => [3,3,1,1,1,1] => 17
[1,3,1,3,2] => [3,3,2,1,1] => 14
[1,3,1,4,1] => [4,3,1,1,1] => 12
[1,3,1,5] => [5,3,1,1] => 8
[1,3,2,1,1,1,1] => [3,2,1,1,1,1,1] => 22
[1,3,2,1,1,2] => [3,2,2,1,1,1] => 18
[1,3,2,1,2,1] => [3,2,2,1,1,1] => 18
[1,3,2,1,3] => [3,3,2,1,1] => 14
[1,3,2,2,1,1] => [3,2,2,1,1,1] => 18
[1,3,2,2,2] => [3,2,2,2,1] => 16
[1,3,2,3,1] => [3,3,2,1,1] => 14
[1,3,2,4] => [4,3,2,1] => 10
[1,3,3,1,1,1] => [3,3,1,1,1,1] => 17
[1,3,3,1,2] => [3,3,2,1,1] => 14
[1,3,3,2,1] => [3,3,2,1,1] => 14
[1,3,3,3] => [3,3,3,1] => 12
[1,3,4,1,1] => [4,3,1,1,1] => 12
[1,3,4,2] => [4,3,2,1] => 10
[1,3,5,1] => [5,3,1,1] => 8
[1,3,6] => [6,3,1] => 5
[1,4,1,1,1,1,1] => [4,1,1,1,1,1,1] => 21
[1,4,1,1,1,2] => [4,2,1,1,1,1] => 16
[1,4,1,1,2,1] => [4,2,1,1,1,1] => 16
[1,4,1,1,3] => [4,3,1,1,1] => 12
[1,4,1,2,1,1] => [4,2,1,1,1,1] => 16
[1,4,1,2,2] => [4,2,2,1,1] => 13
[1,4,1,3,1] => [4,3,1,1,1] => 12
[1,4,1,4] => [4,4,1,1] => 9
[1,4,2,1,1,1] => [4,2,1,1,1,1] => 16
[1,4,2,1,2] => [4,2,2,1,1] => 13
[1,4,2,2,1] => [4,2,2,1,1] => 13
[1,4,2,3] => [4,3,2,1] => 10
[1,4,3,1,1] => [4,3,1,1,1] => 12
[1,4,3,2] => [4,3,2,1] => 10
[1,4,4,1] => [4,4,1,1] => 9
[1,4,5] => [5,4,1] => 6
[1,5,1,1,1,1] => [5,1,1,1,1,1] => 15
[1,5,1,1,2] => [5,2,1,1,1] => 11
[1,5,1,2,1] => [5,2,1,1,1] => 11
[1,5,1,3] => [5,3,1,1] => 8
[1,5,2,1,1] => [5,2,1,1,1] => 11
[1,5,2,2] => [5,2,2,1] => 9
[1,5,3,1] => [5,3,1,1] => 8
[1,5,4] => [5,4,1] => 6
[1,6,1,1,1] => [6,1,1,1,1] => 10
[1,6,1,2] => [6,2,1,1] => 7
[1,6,2,1] => [6,2,1,1] => 7
[1,6,3] => [6,3,1] => 5
[1,7,1,1] => [7,1,1,1] => 6
[1,7,2] => [7,2,1] => 4
[1,8,1] => [8,1,1] => 3
[1,9] => [9,1] => 1
[2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 36
[2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1] => 29
[2,1,1,1,1,1,2,1] => [2,2,1,1,1,1,1,1] => 29
[2,1,1,1,1,1,3] => [3,2,1,1,1,1,1] => 22
[2,1,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1] => 29
[2,1,1,1,1,2,2] => [2,2,2,1,1,1,1] => 24
[2,1,1,1,1,3,1] => [3,2,1,1,1,1,1] => 22
[2,1,1,1,1,4] => [4,2,1,1,1,1] => 16
[2,1,1,1,2,1,1,1] => [2,2,1,1,1,1,1,1] => 29
[2,1,1,1,2,1,2] => [2,2,2,1,1,1,1] => 24
[2,1,1,1,2,2,1] => [2,2,2,1,1,1,1] => 24
[2,1,1,1,2,3] => [3,2,2,1,1,1] => 18
[2,1,1,1,3,1,1] => [3,2,1,1,1,1,1] => 22
[2,1,1,1,3,2] => [3,2,2,1,1,1] => 18
[2,1,1,1,4,1] => [4,2,1,1,1,1] => 16
[2,1,1,1,5] => [5,2,1,1,1] => 11
[2,1,1,2,1,1,1,1] => [2,2,1,1,1,1,1,1] => 29
[2,1,1,2,1,1,2] => [2,2,2,1,1,1,1] => 24
[2,1,1,2,1,2,1] => [2,2,2,1,1,1,1] => 24
[2,1,1,2,1,3] => [3,2,2,1,1,1] => 18
[2,1,1,2,2,1,1] => [2,2,2,1,1,1,1] => 24
[2,1,1,2,2,2] => [2,2,2,2,1,1] => 21
[2,1,1,2,3,1] => [3,2,2,1,1,1] => 18
[2,1,1,2,4] => [4,2,2,1,1] => 13
[2,1,1,3,1,1,1] => [3,2,1,1,1,1,1] => 22
[2,1,1,3,1,2] => [3,2,2,1,1,1] => 18
[2,1,1,3,2,1] => [3,2,2,1,1,1] => 18
[2,1,1,3,3] => [3,3,2,1,1] => 14
[2,1,1,4,1,1] => [4,2,1,1,1,1] => 16
[2,1,1,4,2] => [4,2,2,1,1] => 13
[2,1,1,5,1] => [5,2,1,1,1] => 11
[2,1,1,6] => [6,2,1,1] => 7
[2,1,2,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 29
[2,1,2,1,1,1,2] => [2,2,2,1,1,1,1] => 24
[2,1,2,1,1,2,1] => [2,2,2,1,1,1,1] => 24
[2,1,2,1,1,3] => [3,2,2,1,1,1] => 18
[2,1,2,1,2,1,1] => [2,2,2,1,1,1,1] => 24
[2,1,2,1,2,2] => [2,2,2,2,1,1] => 21
[2,1,2,1,3,1] => [3,2,2,1,1,1] => 18
[2,1,2,1,4] => [4,2,2,1,1] => 13
[2,1,2,2,1,1,1] => [2,2,2,1,1,1,1] => 24
[2,1,2,2,1,2] => [2,2,2,2,1,1] => 21
[2,1,2,2,2,1] => [2,2,2,2,1,1] => 21
[2,1,2,2,3] => [3,2,2,2,1] => 16
[2,1,2,3,1,1] => [3,2,2,1,1,1] => 18
[2,1,2,3,2] => [3,2,2,2,1] => 16
[2,1,2,4,1] => [4,2,2,1,1] => 13
[2,1,2,5] => [5,2,2,1] => 9
[2,1,3,1,1,1,1] => [3,2,1,1,1,1,1] => 22
[2,1,3,1,1,2] => [3,2,2,1,1,1] => 18
[2,1,3,1,2,1] => [3,2,2,1,1,1] => 18
[2,1,3,1,3] => [3,3,2,1,1] => 14
[2,1,3,2,1,1] => [3,2,2,1,1,1] => 18
[2,1,3,2,2] => [3,2,2,2,1] => 16
[2,1,3,3,1] => [3,3,2,1,1] => 14
[2,1,3,4] => [4,3,2,1] => 10
[2,1,4,1,1,1] => [4,2,1,1,1,1] => 16
[2,1,4,1,2] => [4,2,2,1,1] => 13
[2,1,4,2,1] => [4,2,2,1,1] => 13
[2,1,4,3] => [4,3,2,1] => 10
[2,1,5,1,1] => [5,2,1,1,1] => 11
[2,1,5,2] => [5,2,2,1] => 9
[2,1,6,1] => [6,2,1,1] => 7
[2,1,7] => [7,2,1] => 4
[2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 29
[2,2,1,1,1,1,2] => [2,2,2,1,1,1,1] => 24
[2,2,1,1,1,2,1] => [2,2,2,1,1,1,1] => 24
[2,2,1,1,1,3] => [3,2,2,1,1,1] => 18
[2,2,1,1,2,1,1] => [2,2,2,1,1,1,1] => 24
[2,2,1,1,2,2] => [2,2,2,2,1,1] => 21
[2,2,1,1,3,1] => [3,2,2,1,1,1] => 18
[2,2,1,1,4] => [4,2,2,1,1] => 13
[2,2,1,2,1,1,1] => [2,2,2,1,1,1,1] => 24
[2,2,1,2,1,2] => [2,2,2,2,1,1] => 21
[2,2,1,2,2,1] => [2,2,2,2,1,1] => 21
[2,2,1,2,3] => [3,2,2,2,1] => 16
[2,2,1,3,1,1] => [3,2,2,1,1,1] => 18
[2,2,1,3,2] => [3,2,2,2,1] => 16
[2,2,1,4,1] => [4,2,2,1,1] => 13
[2,2,1,5] => [5,2,2,1] => 9
[2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => 24
[2,2,2,1,1,2] => [2,2,2,2,1,1] => 21
[2,2,2,1,2,1] => [2,2,2,2,1,1] => 21
[2,2,2,1,3] => [3,2,2,2,1] => 16
[2,2,2,2,1,1] => [2,2,2,2,1,1] => 21
[2,2,2,2,2] => [2,2,2,2,2] => 20
[2,2,2,3,1] => [3,2,2,2,1] => 16
[2,2,2,4] => [4,2,2,2] => 12
[2,2,3,1,1,1] => [3,2,2,1,1,1] => 18
[2,2,3,1,2] => [3,2,2,2,1] => 16
[2,2,3,2,1] => [3,2,2,2,1] => 16
[2,2,3,3] => [3,3,2,2] => 13
[2,2,4,1,1] => [4,2,2,1,1] => 13
[2,2,4,2] => [4,2,2,2] => 12
[2,2,5,1] => [5,2,2,1] => 9
[2,2,6] => [6,2,2] => 6
[2,3,1,1,1,1,1] => [3,2,1,1,1,1,1] => 22
[2,3,1,1,1,2] => [3,2,2,1,1,1] => 18
[2,3,1,1,2,1] => [3,2,2,1,1,1] => 18
[2,3,1,1,3] => [3,3,2,1,1] => 14
[2,3,1,2,1,1] => [3,2,2,1,1,1] => 18
[2,3,1,2,2] => [3,2,2,2,1] => 16
[2,3,1,3,1] => [3,3,2,1,1] => 14
[2,3,1,4] => [4,3,2,1] => 10
[2,3,2,1,1,1] => [3,2,2,1,1,1] => 18
[2,3,2,1,2] => [3,2,2,2,1] => 16
[2,3,2,2,1] => [3,2,2,2,1] => 16
[2,3,2,3] => [3,3,2,2] => 13
[2,3,3,1,1] => [3,3,2,1,1] => 14
[2,3,3,2] => [3,3,2,2] => 13
[2,3,4,1] => [4,3,2,1] => 10
[2,3,5] => [5,3,2] => 7
[2,4,1,1,1,1] => [4,2,1,1,1,1] => 16
[2,4,1,1,2] => [4,2,2,1,1] => 13
[2,4,1,2,1] => [4,2,2,1,1] => 13
[2,4,1,3] => [4,3,2,1] => 10
[2,4,2,1,1] => [4,2,2,1,1] => 13
[2,4,2,2] => [4,2,2,2] => 12
[2,4,3,1] => [4,3,2,1] => 10
[2,4,4] => [4,4,2] => 8
[2,5,1,1,1] => [5,2,1,1,1] => 11
[2,5,1,2] => [5,2,2,1] => 9
[2,5,2,1] => [5,2,2,1] => 9
[2,5,3] => [5,3,2] => 7
[2,6,1,1] => [6,2,1,1] => 7
[2,6,2] => [6,2,2] => 6
[2,7,1] => [7,2,1] => 4
[2,8] => [8,2] => 2
[3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 28
[3,1,1,1,1,1,2] => [3,2,1,1,1,1,1] => 22
[3,1,1,1,1,2,1] => [3,2,1,1,1,1,1] => 22
[3,1,1,1,1,3] => [3,3,1,1,1,1] => 17
[3,1,1,1,2,1,1] => [3,2,1,1,1,1,1] => 22
[3,1,1,1,2,2] => [3,2,2,1,1,1] => 18
[3,1,1,1,3,1] => [3,3,1,1,1,1] => 17
[3,1,1,1,4] => [4,3,1,1,1] => 12
[3,1,1,2,1,1,1] => [3,2,1,1,1,1,1] => 22
[3,1,1,2,1,2] => [3,2,2,1,1,1] => 18
[3,1,1,2,2,1] => [3,2,2,1,1,1] => 18
[3,1,1,2,3] => [3,3,2,1,1] => 14
[3,1,1,3,1,1] => [3,3,1,1,1,1] => 17
[3,1,1,3,2] => [3,3,2,1,1] => 14
[3,1,1,4,1] => [4,3,1,1,1] => 12
[3,1,1,5] => [5,3,1,1] => 8
[3,1,2,1,1,1,1] => [3,2,1,1,1,1,1] => 22
[3,1,2,1,1,2] => [3,2,2,1,1,1] => 18
[3,1,2,1,2,1] => [3,2,2,1,1,1] => 18
[3,1,2,1,3] => [3,3,2,1,1] => 14
[3,1,2,2,1,1] => [3,2,2,1,1,1] => 18
[3,1,2,2,2] => [3,2,2,2,1] => 16
[3,1,2,3,1] => [3,3,2,1,1] => 14
[3,1,2,4] => [4,3,2,1] => 10
[3,1,3,1,1,1] => [3,3,1,1,1,1] => 17
[3,1,3,1,2] => [3,3,2,1,1] => 14
[3,1,3,2,1] => [3,3,2,1,1] => 14
[3,1,3,3] => [3,3,3,1] => 12
[3,1,4,1,1] => [4,3,1,1,1] => 12
[3,1,4,2] => [4,3,2,1] => 10
[3,1,5,1] => [5,3,1,1] => 8
[3,1,6] => [6,3,1] => 5
[3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => 22
[3,2,1,1,1,2] => [3,2,2,1,1,1] => 18
[3,2,1,1,2,1] => [3,2,2,1,1,1] => 18
[3,2,1,1,3] => [3,3,2,1,1] => 14
[3,2,1,2,1,1] => [3,2,2,1,1,1] => 18
[3,2,1,2,2] => [3,2,2,2,1] => 16
[3,2,1,3,1] => [3,3,2,1,1] => 14
[3,2,1,4] => [4,3,2,1] => 10
[3,2,2,1,1,1] => [3,2,2,1,1,1] => 18
[3,2,2,1,2] => [3,2,2,2,1] => 16
[3,2,2,2,1] => [3,2,2,2,1] => 16
[3,2,2,3] => [3,3,2,2] => 13
[3,2,3,1,1] => [3,3,2,1,1] => 14
[3,2,3,2] => [3,3,2,2] => 13
[3,2,4,1] => [4,3,2,1] => 10
[3,2,5] => [5,3,2] => 7
[3,3,1,1,1,1] => [3,3,1,1,1,1] => 17
[3,3,1,1,2] => [3,3,2,1,1] => 14
[3,3,1,2,1] => [3,3,2,1,1] => 14
[3,3,1,3] => [3,3,3,1] => 12
[3,3,2,1,1] => [3,3,2,1,1] => 14
[3,3,2,2] => [3,3,2,2] => 13
[3,3,3,1] => [3,3,3,1] => 12
[3,3,4] => [4,3,3] => 9
[3,4,1,1,1] => [4,3,1,1,1] => 12
[3,4,1,2] => [4,3,2,1] => 10
[3,4,2,1] => [4,3,2,1] => 10
[3,4,3] => [4,3,3] => 9
[3,5,1,1] => [5,3,1,1] => 8
[3,5,2] => [5,3,2] => 7
[3,6,1] => [6,3,1] => 5
[3,7] => [7,3] => 3
[4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => 21
[4,1,1,1,1,2] => [4,2,1,1,1,1] => 16
[4,1,1,1,2,1] => [4,2,1,1,1,1] => 16
[4,1,1,1,3] => [4,3,1,1,1] => 12
[4,1,1,2,1,1] => [4,2,1,1,1,1] => 16
[4,1,1,2,2] => [4,2,2,1,1] => 13
[4,1,1,3,1] => [4,3,1,1,1] => 12
[4,1,1,4] => [4,4,1,1] => 9
[4,1,2,1,1,1] => [4,2,1,1,1,1] => 16
[4,1,2,1,2] => [4,2,2,1,1] => 13
[4,1,2,2,1] => [4,2,2,1,1] => 13
[4,1,2,3] => [4,3,2,1] => 10
[4,1,3,1,1] => [4,3,1,1,1] => 12
[4,1,3,2] => [4,3,2,1] => 10
[4,1,4,1] => [4,4,1,1] => 9
[4,1,5] => [5,4,1] => 6
[4,2,1,1,1,1] => [4,2,1,1,1,1] => 16
[4,2,1,1,2] => [4,2,2,1,1] => 13
[4,2,1,2,1] => [4,2,2,1,1] => 13
[4,2,1,3] => [4,3,2,1] => 10
[4,2,2,1,1] => [4,2,2,1,1] => 13
[4,2,2,2] => [4,2,2,2] => 12
[4,2,3,1] => [4,3,2,1] => 10
[4,2,4] => [4,4,2] => 8
[4,3,1,1,1] => [4,3,1,1,1] => 12
[4,3,1,2] => [4,3,2,1] => 10
[4,3,2,1] => [4,3,2,1] => 10
[4,3,3] => [4,3,3] => 9
[4,4,1,1] => [4,4,1,1] => 9
[4,4,2] => [4,4,2] => 8
[4,5,1] => [5,4,1] => 6
[4,6] => [6,4] => 4
[5,1,1,1,1,1] => [5,1,1,1,1,1] => 15
[5,1,1,1,2] => [5,2,1,1,1] => 11
[5,1,1,2,1] => [5,2,1,1,1] => 11
[5,1,1,3] => [5,3,1,1] => 8
[5,1,2,1,1] => [5,2,1,1,1] => 11
[5,1,2,2] => [5,2,2,1] => 9
[5,1,3,1] => [5,3,1,1] => 8
[5,1,4] => [5,4,1] => 6
[5,2,1,1,1] => [5,2,1,1,1] => 11
[5,2,1,2] => [5,2,2,1] => 9
[5,2,2,1] => [5,2,2,1] => 9
[5,2,3] => [5,3,2] => 7
[5,3,1,1] => [5,3,1,1] => 8
[5,3,2] => [5,3,2] => 7
[5,4,1] => [5,4,1] => 6
[5,5] => [5,5] => 5
[6,1,1,1,1] => [6,1,1,1,1] => 10
[6,1,1,2] => [6,2,1,1] => 7
[6,1,2,1] => [6,2,1,1] => 7
[6,1,3] => [6,3,1] => 5
[6,2,1,1] => [6,2,1,1] => 7
[6,2,2] => [6,2,2] => 6
[6,3,1] => [6,3,1] => 5
[6,4] => [6,4] => 4
[7,1,1,1] => [7,1,1,1] => 6
[7,1,2] => [7,2,1] => 4
[7,2,1] => [7,2,1] => 4
[7,3] => [7,3] => 3
[8,1,1] => [8,1,1] => 3
[8,2] => [8,2] => 2
[9,1] => [9,1] => 1
[10] => [10] => 0
[1,5,4,1] => [5,4,1,1] => 9
[1,4,5,1] => [5,4,1,1] => 9
[1,1,2,2,3,3] => [3,3,2,2,1,1] => 22
[1,1,3,3,2,2] => [3,3,2,2,1,1] => 22
[1,1,3,2,2,3] => [3,3,2,2,1,1] => 22
[2,2,1,1,3,3] => [3,3,2,2,1,1] => 22
[2,2,2,2,2,2] => [2,2,2,2,2,2] => 30
[2,2,3,3,1,1] => [3,3,2,2,1,1] => 22
[2,2,3,1,1,3] => [3,3,2,2,1,1] => 22
[2,2,4,4] => [4,4,2,2] => 14
[2,1,1,2,3,3] => [3,3,2,2,1,1] => 22
[2,1,3,3,1,2] => [3,3,2,2,1,1] => 22
[2,1,3,2,1,3] => [3,3,2,2,1,1] => 22
[2,1,4,5] => [5,4,2,1] => 11
[3,3,1,1,2,2] => [3,3,2,2,1,1] => 22
[3,3,2,2,1,1] => [3,3,2,2,1,1] => 22
[3,3,2,1,1,2] => [3,3,2,2,1,1] => 22
[3,3,3,3] => [3,3,3,3] => 18
[3,2,1,1,2,3] => [3,3,2,2,1,1] => 22
[3,2,2,3,1,1] => [3,3,2,2,1,1] => 22
[3,2,2,1,1,3] => [3,3,2,2,1,1] => 22
[3,2,3,4] => [4,3,3,2] => 15
[3,1,1,3,2,2] => [3,3,2,2,1,1] => 22
[3,1,1,2,2,3] => [3,3,2,2,1,1] => 22
[3,1,2,3,1,2] => [3,3,2,2,1,1] => 22
[3,1,2,2,1,3] => [3,3,2,2,1,1] => 22
[3,1,3,5] => [5,3,3,1] => 12
[4,4,2,2] => [4,4,2,2] => 14
[4,3,2,3] => [4,3,3,2] => 15
[4,2,2,4] => [4,4,2,2] => 14
[4,1,2,5] => [5,4,2,1] => 11
[5,4,1,2] => [5,4,2,1] => 11
[5,3,1,3] => [5,3,3,1] => 12
[5,2,1,4] => [5,4,2,1] => 11
[6,6] => [6,6] => 6
[1,2,2,3,3,1] => [3,3,2,2,1,1] => 22
[1,2,2,4,3] => [4,3,2,2,1] => 17
[1,2,3,3,2,1] => [3,3,2,2,1,1] => 22
[1,2,3,4,2] => [4,3,2,2,1] => 17
[1,2,3,2,3,1] => [3,3,2,2,1,1] => 22
[1,2,3,3,3] => [3,3,3,2,1] => 19
[1,2,4,4,1] => [4,4,2,1,1] => 15
[1,2,4,3,2] => [4,3,2,2,1] => 17
[1,2,4,2,3] => [4,3,2,2,1] => 17
[1,2,5,4] => [5,4,2,1] => 11
[1,3,3,2,2,1] => [3,3,2,2,1,1] => 22
[1,3,3,3,2] => [3,3,3,2,1] => 19
[1,3,4,3,1] => [4,3,3,1,1] => 16
[1,3,4,2,2] => [4,3,2,2,1] => 17
[1,3,5,3] => [5,3,3,1] => 12
[1,3,2,3,2,1] => [3,3,2,2,1,1] => 22
[1,3,2,4,2] => [4,3,2,2,1] => 17
[1,3,2,2,3,1] => [3,3,2,2,1,1] => 22
[1,3,2,3,3] => [3,3,3,2,1] => 19
[1,3,3,4,1] => [4,3,3,1,1] => 16
[1,3,3,2,3] => [3,3,3,2,1] => 19
[1,3,4,4] => [4,4,3,1] => 13
[1,4,4,2,1] => [4,4,2,1,1] => 15
[1,4,5,2] => [5,4,2,1] => 11
[1,4,3,3,1] => [4,3,3,1,1] => 16
[1,4,3,2,2] => [4,3,2,2,1] => 17
[1,4,4,3] => [4,4,3,1] => 13
[1,4,2,4,1] => [4,4,2,1,1] => 15
[1,4,2,3,2] => [4,3,2,2,1] => 17
[1,4,2,2,3] => [4,3,2,2,1] => 17
[1,4,3,4] => [4,4,3,1] => 13
[1,5,4,2] => [5,4,2,1] => 11
[1,5,3,3] => [5,3,3,1] => 12
[1,5,2,4] => [5,4,2,1] => 11
[2,3,2,3,2] => [3,3,2,2,2] => 21
[2,3,3,3,1] => [3,3,3,2,1] => 19
[2,3,3,2,2] => [3,3,2,2,2] => 21
[2,3,4,3] => [4,3,3,2] => 15
[2,4,3,2,1] => [4,3,2,2,1] => 17
[2,4,4,2] => [4,4,2,2] => 14
[2,4,2,3,1] => [4,3,2,2,1] => 17
[2,4,3,3] => [4,3,3,2] => 15
[2,5,4,1] => [5,4,2,1] => 11
[2,5,3,2] => [5,3,2,2] => 13
[2,5,2,3] => [5,3,2,2] => 13
[2,6,4] => [6,4,2] => 8
[2,2,3,3,2] => [3,3,2,2,2] => 21
[2,2,4,3,1] => [4,3,2,2,1] => 17
[2,2,5,3] => [5,3,2,2] => 13
[2,2,2,3,3] => [3,3,2,2,2] => 21
[2,2,3,4,1] => [4,3,2,2,1] => 17
[2,2,3,2,3] => [3,3,2,2,2] => 21
[2,3,4,2,1] => [4,3,2,2,1] => 17
[2,3,5,2] => [5,3,2,2] => 13
[2,3,2,4,1] => [4,3,2,2,1] => 17
[2,3,2,2,3] => [3,3,2,2,2] => 21
[2,3,3,4] => [4,3,3,2] => 15
[2,4,5,1] => [5,4,2,1] => 11
[2,4,2,4] => [4,4,2,2] => 14
[2,5,5] => [5,5,2] => 9
[3,4,2,2,1] => [4,3,2,2,1] => 17
[3,4,3,2] => [4,3,3,2] => 15
[3,5,3,1] => [5,3,3,1] => 12
[3,5,2,2] => [5,3,2,2] => 13
[3,3,3,2,1] => [3,3,3,2,1] => 19
[3,3,4,2] => [4,3,3,2] => 15
[3,3,2,3,1] => [3,3,3,2,1] => 19
[3,3,2,2,2] => [3,3,2,2,2] => 21
[3,4,4,1] => [4,4,3,1] => 13
[3,4,2,3] => [4,3,3,2] => 15
[3,5,4] => [5,4,3] => 10
[3,2,4,2,1] => [4,3,2,2,1] => 17
[3,2,5,2] => [5,3,2,2] => 13
[3,2,3,3,1] => [3,3,3,2,1] => 19
[3,2,3,2,2] => [3,3,2,2,2] => 21
[3,2,4,3] => [4,3,3,2] => 15
[3,2,2,4,1] => [4,3,2,2,1] => 17
[3,2,2,3,2] => [3,3,2,2,2] => 21
[3,2,2,2,3] => [3,3,2,2,2] => 21
[3,3,5,1] => [5,3,3,1] => 12
[3,3,2,4] => [4,3,3,2] => 15
[3,4,5] => [5,4,3] => 10
[4,5,2,1] => [5,4,2,1] => 11
[4,6,2] => [6,4,2] => 8
[4,4,3,1] => [4,4,3,1] => 13
[4,5,3] => [5,4,3] => 10
[4,3,4,1] => [4,4,3,1] => 13
[4,3,3,2] => [4,3,3,2] => 15
[4,4,4] => [4,4,4] => 12
[4,2,5,1] => [5,4,2,1] => 11
[4,2,4,2] => [4,4,2,2] => 14
[4,2,3,3] => [4,3,3,2] => 15
[4,3,5] => [5,4,3] => 10
[5,5,2] => [5,5,2] => 9
[5,4,3] => [5,4,3] => 10
[5,3,4] => [5,4,3] => 10
[5,2,5] => [5,5,2] => 9
[1,5,2,1,2] => [5,2,2,1,1] => 13
[1,2,2,2,2,2] => [2,2,2,2,2,1] => 25
[1,4,7] => [7,4,1] => 6
[1,2,2,2,5] => [5,2,2,2,1] => 16
[1,1,2,3,5] => [5,3,2,1,1] => 14
[3,1,2,1,5] => [5,3,2,1,1] => 14
[1,3,2,1,5] => [5,3,2,1,1] => 14
[1,1,4,1,5] => [5,4,1,1,1] => 13
[1,1,2,5,3] => [5,3,2,1,1] => 14
[3,1,2,3,3] => [3,3,3,2,1] => 19
[1,1,4,3,3] => [4,3,3,1,1] => 16
[5,1,2,1,3] => [5,3,2,1,1] => 14
[3,3,2,1,3] => [3,3,3,2,1] => 19
[1,5,2,1,3] => [5,3,2,1,1] => 14
[3,1,4,1,3] => [4,3,3,1,1] => 16
[1,3,4,1,3] => [4,3,3,1,1] => 16
[5,2,2,2,1] => [5,2,2,2,1] => 16
[7,4,1] => [7,4,1] => 6
[3,1,2,5,1] => [5,3,2,1,1] => 14
[1,3,2,5,1] => [5,3,2,1,1] => 14
[1,1,4,5,1] => [5,4,1,1,1] => 13
[5,1,2,3,1] => [5,3,2,1,1] => 14
[1,5,2,3,1] => [5,3,2,1,1] => 14
[3,1,4,3,1] => [4,3,3,1,1] => 16
[5,3,2,1,1] => [5,3,2,1,1] => 14
[3,5,2,1,1] => [5,3,2,1,1] => 14
[5,1,4,1,1] => [5,4,1,1,1] => 13
[3,3,4,1,1] => [4,3,3,1,1] => 16
[1,5,4,1,1] => [5,4,1,1,1] => 13
[4,3,2,2,1] => [4,3,2,2,1] => 17
[4,2,3,2,1] => [4,3,2,2,1] => 17
[4,2,2,3,1] => [4,3,2,2,1] => 17
[5,4,2,1] => [5,4,2,1] => 11
[5,3,3,1] => [5,3,3,1] => 12
[5,3,2,2] => [5,3,2,2] => 13
[5,2,4,1] => [5,4,2,1] => 11
[5,2,3,2] => [5,3,2,2] => 13
[5,2,2,3] => [5,3,2,2] => 13
[6,4,2] => [6,4,2] => 8
[6,2,4] => [6,4,2] => 8
[7,5] => [7,5] => 5
[2,1,2,5,1] => [5,2,2,1,1] => 13
[2,2,2,2,2,1] => [2,2,2,2,2,1] => 25
[1,2,5,2,2] => [5,2,2,2,1] => 16
[1,3,5,2,1] => [5,3,2,1,1] => 14
[1,5,2,2,2] => [5,2,2,2,1] => 16
[1,5,3,2,1] => [5,3,2,1,1] => 14
[2,5,2,2,1] => [5,2,2,2,1] => 16
[2,2,5,2,1] => [5,2,2,2,1] => 16
[1,1,2,3,2,3] => [3,3,2,2,1,1] => 22
[1,3,1,5,1] => [5,3,1,1,1] => 12
[1,5,1,3,1] => [5,3,1,1,1] => 12
[1,4,1,4,1] => [4,4,1,1,1] => 13
[1,4,3,2,1] => [4,3,2,1,1] => 14
[1,2,3,4,1] => [4,3,2,1,1] => 14
[1,2,5,2,1] => [5,2,2,1,1] => 13
[1,4,3,4,1] => [4,4,3,1,1] => 17
[1,4,5,2,1] => [5,4,2,1,1] => 15
[1,2,5,4,1] => [5,4,2,1,1] => 15
[1,6,4] => [6,4,1] => 6
[1,5,5] => [5,5,1] => 7
[1,4,6] => [6,4,1] => 6
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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 1,2,0,1 1,2,1,3,0,0,1 1,2,2,3,3,0,4,0,0,0,1 1,2,2,4,6,0,5,6,0,0,5,0,0,0,0,1 1,2,2,5,6,3,7,12,0,4,5,10,0,0,0,6,0,0,0,0,0,1 1,2,2,5,7,6,7,15,6,12,5,20,1,10,0,6,15,0,0,0,0,7,0,0,0,0,0,0,1
$F_{1} = 1$
$F_{2} = 1 + q$
$F_{3} = 1 + 2\ q + q^{3}$
$F_{4} = 1 + 2\ q + q^{2} + 3\ q^{3} + q^{6}$
$F_{5} = 1 + 2\ q + 2\ q^{2} + 3\ q^{3} + 3\ q^{4} + 4\ q^{6} + q^{10}$
$F_{6} = 1 + 2\ q + 2\ q^{2} + 4\ q^{3} + 6\ q^{4} + 5\ q^{6} + 6\ q^{7} + 5\ q^{10} + q^{15}$
$F_{7} = 1 + 2\ q + 2\ q^{2} + 5\ q^{3} + 6\ q^{4} + 3\ q^{5} + 7\ q^{6} + 12\ q^{7} + 4\ q^{9} + 5\ q^{10} + 10\ q^{11} + 6\ q^{15} + q^{21}$
$F_{8} = 1 + 2\ q + 2\ q^{2} + 5\ q^{3} + 7\ q^{4} + 6\ q^{5} + 7\ q^{6} + 15\ q^{7} + 6\ q^{8} + 12\ q^{9} + 5\ q^{10} + 20\ q^{11} + q^{12} + 10\ q^{13} + 6\ q^{15} + 15\ q^{16} + 7\ q^{21} + q^{28}$
$F_{9} = 1 + 2\ q + 2\ q^{2} + 5\ q^{3} + 8\ q^{4} + 6\ q^{5} + 10\ q^{6} + 18\ q^{7} + 12\ q^{8} + 13\ q^{9} + 17\ q^{10} + 20\ q^{11} + 14\ q^{12} + 30\ q^{13} + 6\ q^{15} + 35\ q^{16} + 20\ q^{18} + 7\ q^{21} + 21\ q^{22} + 8\ q^{28} + q^{36}$
$F_{10} = 1 + 2\ q + 2\ q^{2} + 5\ q^{3} + 8\ q^{4} + 7\ q^{5} + 13\ q^{6} + 18\ q^{7} + 15\ q^{8} + 21\ q^{9} + 29\ q^{10} + 20\ q^{11} + 28\ q^{12} + 36\ q^{13} + 30\ q^{14} + 6\ q^{15} + 50\ q^{16} + 15\ q^{17} + 60\ q^{18} + q^{20} + 22\ q^{21} + 42\ q^{22} + 35\ q^{24} + 8\ q^{28} + 28\ q^{29} + 9\ q^{36} + q^{45}$
Description
The weighted size of a partition.
Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is
$$\sum_{i=0}^m i \cdot \lambda_i.$$
This is also the sum of the leg lengths of the cells in $\lambda$, or
$$ \sum_i \binom{\lambda^{\prime}_i}{2} $$
where $\lambda^{\prime}$ is the conjugate partition of $\lambda$.
This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2].
This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].
Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is
$$\sum_{i=0}^m i \cdot \lambda_i.$$
This is also the sum of the leg lengths of the cells in $\lambda$, or
$$ \sum_i \binom{\lambda^{\prime}_i}{2} $$
where $\lambda^{\prime}$ is the conjugate partition of $\lambda$.
This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2].
This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].
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Description
Sends a composition to the partition obtained by sorting the entries.
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