Identifier
-
Mp00044:
Integer partitions
—conjugate⟶
Integer partitions
St001541: Integer partitions ⟶ ℤ
Values
[1] => [1] => 0
[2] => [1,1] => 0
[1,1] => [2] => 1
[3] => [1,1,1] => 0
[2,1] => [2,1] => 2
[1,1,1] => [3] => 3
[4] => [1,1,1,1] => 0
[3,1] => [2,1,1] => 3
[2,2] => [2,2] => 4
[2,1,1] => [3,1] => 5
[1,1,1,1] => [4] => 6
[5] => [1,1,1,1,1] => 0
[4,1] => [2,1,1,1] => 4
[3,2] => [2,2,1] => 6
[3,1,1] => [3,1,1] => 7
[2,2,1] => [3,2] => 8
[2,1,1,1] => [4,1] => 9
[1,1,1,1,1] => [5] => 10
[6] => [1,1,1,1,1,1] => 0
[5,1] => [2,1,1,1,1] => 5
[4,2] => [2,2,1,1] => 8
[4,1,1] => [3,1,1,1] => 9
[3,3] => [2,2,2] => 9
[3,2,1] => [3,2,1] => 11
[3,1,1,1] => [4,1,1] => 12
[2,2,2] => [3,3] => 12
[2,2,1,1] => [4,2] => 13
[2,1,1,1,1] => [5,1] => 14
[1,1,1,1,1,1] => [6] => 15
[7] => [1,1,1,1,1,1,1] => 0
[6,1] => [2,1,1,1,1,1] => 6
[5,2] => [2,2,1,1,1] => 10
[5,1,1] => [3,1,1,1,1] => 11
[4,3] => [2,2,2,1] => 12
[4,2,1] => [3,2,1,1] => 14
[4,1,1,1] => [4,1,1,1] => 15
[3,3,1] => [3,2,2] => 15
[3,2,2] => [3,3,1] => 16
[3,2,1,1] => [4,2,1] => 17
[3,1,1,1,1] => [5,1,1] => 18
[2,2,2,1] => [4,3] => 18
[2,2,1,1,1] => [5,2] => 19
[2,1,1,1,1,1] => [6,1] => 20
[1,1,1,1,1,1,1] => [7] => 21
[8] => [1,1,1,1,1,1,1,1] => 0
[7,1] => [2,1,1,1,1,1,1] => 7
[6,2] => [2,2,1,1,1,1] => 12
[6,1,1] => [3,1,1,1,1,1] => 13
[5,3] => [2,2,2,1,1] => 15
[5,2,1] => [3,2,1,1,1] => 17
[5,1,1,1] => [4,1,1,1,1] => 18
[4,4] => [2,2,2,2] => 16
[4,3,1] => [3,2,2,1] => 19
[4,2,2] => [3,3,1,1] => 20
[4,2,1,1] => [4,2,1,1] => 21
[4,1,1,1,1] => [5,1,1,1] => 22
[3,3,2] => [3,3,2] => 21
[3,3,1,1] => [4,2,2] => 22
[3,2,2,1] => [4,3,1] => 23
[3,2,1,1,1] => [5,2,1] => 24
[3,1,1,1,1,1] => [6,1,1] => 25
[2,2,2,2] => [4,4] => 24
[2,2,2,1,1] => [5,3] => 25
[2,2,1,1,1,1] => [6,2] => 26
[2,1,1,1,1,1,1] => [7,1] => 27
[1,1,1,1,1,1,1,1] => [8] => 28
[9] => [1,1,1,1,1,1,1,1,1] => 0
[8,1] => [2,1,1,1,1,1,1,1] => 8
[7,2] => [2,2,1,1,1,1,1] => 14
[7,1,1] => [3,1,1,1,1,1,1] => 15
[6,3] => [2,2,2,1,1,1] => 18
[6,2,1] => [3,2,1,1,1,1] => 20
[6,1,1,1] => [4,1,1,1,1,1] => 21
[5,4] => [2,2,2,2,1] => 20
[5,3,1] => [3,2,2,1,1] => 23
[5,2,2] => [3,3,1,1,1] => 24
[5,2,1,1] => [4,2,1,1,1] => 25
[5,1,1,1,1] => [5,1,1,1,1] => 26
[4,4,1] => [3,2,2,2] => 24
[4,3,2] => [3,3,2,1] => 26
[4,3,1,1] => [4,2,2,1] => 27
[4,2,2,1] => [4,3,1,1] => 28
[4,2,1,1,1] => [5,2,1,1] => 29
[4,1,1,1,1,1] => [6,1,1,1] => 30
[3,3,3] => [3,3,3] => 27
[3,3,2,1] => [4,3,2] => 29
[3,3,1,1,1] => [5,2,2] => 30
[3,2,2,2] => [4,4,1] => 30
[3,2,2,1,1] => [5,3,1] => 31
[3,2,1,1,1,1] => [6,2,1] => 32
[3,1,1,1,1,1,1] => [7,1,1] => 33
[2,2,2,2,1] => [5,4] => 32
[2,2,2,1,1,1] => [6,3] => 33
[2,2,1,1,1,1,1] => [7,2] => 34
[2,1,1,1,1,1,1,1] => [8,1] => 35
[1,1,1,1,1,1,1,1,1] => [9] => 36
[10] => [1,1,1,1,1,1,1,1,1,1] => 0
[9,1] => [2,1,1,1,1,1,1,1,1] => 9
[8,2] => [2,2,1,1,1,1,1,1] => 16
[8,1,1] => [3,1,1,1,1,1,1,1] => 17
[7,3] => [2,2,2,1,1,1,1] => 21
>>> Load all 1200 entries. <<<[7,2,1] => [3,2,1,1,1,1,1] => 23
[7,1,1,1] => [4,1,1,1,1,1,1] => 24
[6,4] => [2,2,2,2,1,1] => 24
[6,3,1] => [3,2,2,1,1,1] => 27
[6,2,2] => [3,3,1,1,1,1] => 28
[6,2,1,1] => [4,2,1,1,1,1] => 29
[6,1,1,1,1] => [5,1,1,1,1,1] => 30
[5,5] => [2,2,2,2,2] => 25
[5,4,1] => [3,2,2,2,1] => 29
[5,3,2] => [3,3,2,1,1] => 31
[5,3,1,1] => [4,2,2,1,1] => 32
[5,2,2,1] => [4,3,1,1,1] => 33
[5,2,1,1,1] => [5,2,1,1,1] => 34
[5,1,1,1,1,1] => [6,1,1,1,1] => 35
[4,4,2] => [3,3,2,2] => 32
[4,4,1,1] => [4,2,2,2] => 33
[4,3,3] => [3,3,3,1] => 33
[4,3,2,1] => [4,3,2,1] => 35
[4,3,1,1,1] => [5,2,2,1] => 36
[4,2,2,2] => [4,4,1,1] => 36
[4,2,2,1,1] => [5,3,1,1] => 37
[4,2,1,1,1,1] => [6,2,1,1] => 38
[4,1,1,1,1,1,1] => [7,1,1,1] => 39
[3,3,3,1] => [4,3,3] => 36
[3,3,2,2] => [4,4,2] => 37
[3,3,2,1,1] => [5,3,2] => 38
[3,3,1,1,1,1] => [6,2,2] => 39
[3,2,2,2,1] => [5,4,1] => 39
[3,2,2,1,1,1] => [6,3,1] => 40
[3,2,1,1,1,1,1] => [7,2,1] => 41
[3,1,1,1,1,1,1,1] => [8,1,1] => 42
[2,2,2,2,2] => [5,5] => 40
[2,2,2,2,1,1] => [6,4] => 41
[2,2,2,1,1,1,1] => [7,3] => 42
[2,2,1,1,1,1,1,1] => [8,2] => 43
[2,1,1,1,1,1,1,1,1] => [9,1] => 44
[1,1,1,1,1,1,1,1,1,1] => [10] => 45
[11] => [1,1,1,1,1,1,1,1,1,1,1] => 0
[10,1] => [2,1,1,1,1,1,1,1,1,1] => 10
[9,2] => [2,2,1,1,1,1,1,1,1] => 18
[9,1,1] => [3,1,1,1,1,1,1,1,1] => 19
[8,3] => [2,2,2,1,1,1,1,1] => 24
[8,2,1] => [3,2,1,1,1,1,1,1] => 26
[8,1,1,1] => [4,1,1,1,1,1,1,1] => 27
[7,4] => [2,2,2,2,1,1,1] => 28
[7,3,1] => [3,2,2,1,1,1,1] => 31
[7,2,2] => [3,3,1,1,1,1,1] => 32
[7,2,1,1] => [4,2,1,1,1,1,1] => 33
[7,1,1,1,1] => [5,1,1,1,1,1,1] => 34
[6,5] => [2,2,2,2,2,1] => 30
[6,4,1] => [3,2,2,2,1,1] => 34
[6,3,2] => [3,3,2,1,1,1] => 36
[6,3,1,1] => [4,2,2,1,1,1] => 37
[6,2,2,1] => [4,3,1,1,1,1] => 38
[6,2,1,1,1] => [5,2,1,1,1,1] => 39
[6,1,1,1,1,1] => [6,1,1,1,1,1] => 40
[5,5,1] => [3,2,2,2,2] => 35
[5,4,2] => [3,3,2,2,1] => 38
[5,4,1,1] => [4,2,2,2,1] => 39
[5,3,3] => [3,3,3,1,1] => 39
[5,3,2,1] => [4,3,2,1,1] => 41
[5,3,1,1,1] => [5,2,2,1,1] => 42
[5,2,2,2] => [4,4,1,1,1] => 42
[5,2,2,1,1] => [5,3,1,1,1] => 43
[5,2,1,1,1,1] => [6,2,1,1,1] => 44
[5,1,1,1,1,1,1] => [7,1,1,1,1] => 45
[4,4,3] => [3,3,3,2] => 40
[4,4,2,1] => [4,3,2,2] => 42
[4,4,1,1,1] => [5,2,2,2] => 43
[4,3,3,1] => [4,3,3,1] => 43
[4,3,2,2] => [4,4,2,1] => 44
[4,3,2,1,1] => [5,3,2,1] => 45
[4,3,1,1,1,1] => [6,2,2,1] => 46
[4,2,2,2,1] => [5,4,1,1] => 46
[4,2,2,1,1,1] => [6,3,1,1] => 47
[4,2,1,1,1,1,1] => [7,2,1,1] => 48
[4,1,1,1,1,1,1,1] => [8,1,1,1] => 49
[3,3,3,2] => [4,4,3] => 45
[3,3,3,1,1] => [5,3,3] => 46
[3,3,2,2,1] => [5,4,2] => 47
[3,3,2,1,1,1] => [6,3,2] => 48
[3,3,1,1,1,1,1] => [7,2,2] => 49
[3,2,2,2,2] => [5,5,1] => 48
[3,2,2,2,1,1] => [6,4,1] => 49
[3,2,2,1,1,1,1] => [7,3,1] => 50
[3,2,1,1,1,1,1,1] => [8,2,1] => 51
[3,1,1,1,1,1,1,1,1] => [9,1,1] => 52
[2,2,2,2,2,1] => [6,5] => 50
[2,2,2,2,1,1,1] => [7,4] => 51
[2,2,2,1,1,1,1,1] => [8,3] => 52
[2,2,1,1,1,1,1,1,1] => [9,2] => 53
[2,1,1,1,1,1,1,1,1,1] => [10,1] => 54
[1,1,1,1,1,1,1,1,1,1,1] => [11] => 55
[12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 0
[11,1] => [2,1,1,1,1,1,1,1,1,1,1] => 11
[10,2] => [2,2,1,1,1,1,1,1,1,1] => 20
[10,1,1] => [3,1,1,1,1,1,1,1,1,1] => 21
[9,3] => [2,2,2,1,1,1,1,1,1] => 27
[9,2,1] => [3,2,1,1,1,1,1,1,1] => 29
[9,1,1,1] => [4,1,1,1,1,1,1,1,1] => 30
[8,4] => [2,2,2,2,1,1,1,1] => 32
[8,3,1] => [3,2,2,1,1,1,1,1] => 35
[8,2,2] => [3,3,1,1,1,1,1,1] => 36
[8,2,1,1] => [4,2,1,1,1,1,1,1] => 37
[8,1,1,1,1] => [5,1,1,1,1,1,1,1] => 38
[7,5] => [2,2,2,2,2,1,1] => 35
[7,4,1] => [3,2,2,2,1,1,1] => 39
[7,3,2] => [3,3,2,1,1,1,1] => 41
[7,3,1,1] => [4,2,2,1,1,1,1] => 42
[7,2,2,1] => [4,3,1,1,1,1,1] => 43
[7,2,1,1,1] => [5,2,1,1,1,1,1] => 44
[7,1,1,1,1,1] => [6,1,1,1,1,1,1] => 45
[6,6] => [2,2,2,2,2,2] => 36
[6,5,1] => [3,2,2,2,2,1] => 41
[6,4,2] => [3,3,2,2,1,1] => 44
[6,4,1,1] => [4,2,2,2,1,1] => 45
[6,3,3] => [3,3,3,1,1,1] => 45
[6,3,2,1] => [4,3,2,1,1,1] => 47
[6,3,1,1,1] => [5,2,2,1,1,1] => 48
[6,2,2,2] => [4,4,1,1,1,1] => 48
[6,2,2,1,1] => [5,3,1,1,1,1] => 49
[6,2,1,1,1,1] => [6,2,1,1,1,1] => 50
[6,1,1,1,1,1,1] => [7,1,1,1,1,1] => 51
[5,5,2] => [3,3,2,2,2] => 45
[5,5,1,1] => [4,2,2,2,2] => 46
[5,4,3] => [3,3,3,2,1] => 47
[5,4,2,1] => [4,3,2,2,1] => 49
[5,4,1,1,1] => [5,2,2,2,1] => 50
[5,3,3,1] => [4,3,3,1,1] => 50
[5,3,2,2] => [4,4,2,1,1] => 51
[5,3,2,1,1] => [5,3,2,1,1] => 52
[5,3,1,1,1,1] => [6,2,2,1,1] => 53
[5,2,2,2,1] => [5,4,1,1,1] => 53
[5,2,2,1,1,1] => [6,3,1,1,1] => 54
[5,2,1,1,1,1,1] => [7,2,1,1,1] => 55
[5,1,1,1,1,1,1,1] => [8,1,1,1,1] => 56
[4,4,4] => [3,3,3,3] => 48
[4,4,3,1] => [4,3,3,2] => 51
[4,4,2,2] => [4,4,2,2] => 52
[4,4,2,1,1] => [5,3,2,2] => 53
[4,4,1,1,1,1] => [6,2,2,2] => 54
[4,3,3,2] => [4,4,3,1] => 53
[4,3,3,1,1] => [5,3,3,1] => 54
[4,3,2,2,1] => [5,4,2,1] => 55
[4,3,2,1,1,1] => [6,3,2,1] => 56
[4,3,1,1,1,1,1] => [7,2,2,1] => 57
[4,2,2,2,2] => [5,5,1,1] => 56
[4,2,2,2,1,1] => [6,4,1,1] => 57
[4,2,2,1,1,1,1] => [7,3,1,1] => 58
[4,2,1,1,1,1,1,1] => [8,2,1,1] => 59
[4,1,1,1,1,1,1,1,1] => [9,1,1,1] => 60
[3,3,3,3] => [4,4,4] => 54
[3,3,3,2,1] => [5,4,3] => 56
[3,3,3,1,1,1] => [6,3,3] => 57
[3,3,2,2,2] => [5,5,2] => 57
[3,3,2,2,1,1] => [6,4,2] => 58
[3,3,2,1,1,1,1] => [7,3,2] => 59
[3,3,1,1,1,1,1,1] => [8,2,2] => 60
[3,2,2,2,2,1] => [6,5,1] => 59
[3,2,2,2,1,1,1] => [7,4,1] => 60
[3,2,2,1,1,1,1,1] => [8,3,1] => 61
[3,2,1,1,1,1,1,1,1] => [9,2,1] => 62
[3,1,1,1,1,1,1,1,1,1] => [10,1,1] => 63
[2,2,2,2,2,2] => [6,6] => 60
[2,2,2,2,2,1,1] => [7,5] => 61
[2,2,2,2,1,1,1,1] => [8,4] => 62
[2,2,2,1,1,1,1,1,1] => [9,3] => 63
[2,2,1,1,1,1,1,1,1,1] => [10,2] => 64
[2,1,1,1,1,1,1,1,1,1,1] => [11,1] => 65
[1,1,1,1,1,1,1,1,1,1,1,1] => [12] => 66
[13] => [1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[12,1] => [2,1,1,1,1,1,1,1,1,1,1,1] => 12
[11,2] => [2,2,1,1,1,1,1,1,1,1,1] => 22
[11,1,1] => [3,1,1,1,1,1,1,1,1,1,1] => 23
[10,3] => [2,2,2,1,1,1,1,1,1,1] => 30
[10,2,1] => [3,2,1,1,1,1,1,1,1,1] => 32
[10,1,1,1] => [4,1,1,1,1,1,1,1,1,1] => 33
[9,4] => [2,2,2,2,1,1,1,1,1] => 36
[9,3,1] => [3,2,2,1,1,1,1,1,1] => 39
[9,2,2] => [3,3,1,1,1,1,1,1,1] => 40
[9,2,1,1] => [4,2,1,1,1,1,1,1,1] => 41
[9,1,1,1,1] => [5,1,1,1,1,1,1,1,1] => 42
[8,5] => [2,2,2,2,2,1,1,1] => 40
[8,4,1] => [3,2,2,2,1,1,1,1] => 44
[8,3,2] => [3,3,2,1,1,1,1,1] => 46
[8,3,1,1] => [4,2,2,1,1,1,1,1] => 47
[8,2,2,1] => [4,3,1,1,1,1,1,1] => 48
[8,2,1,1,1] => [5,2,1,1,1,1,1,1] => 49
[8,1,1,1,1,1] => [6,1,1,1,1,1,1,1] => 50
[7,6] => [2,2,2,2,2,2,1] => 42
[7,5,1] => [3,2,2,2,2,1,1] => 47
[7,4,2] => [3,3,2,2,1,1,1] => 50
[7,4,1,1] => [4,2,2,2,1,1,1] => 51
[7,3,3] => [3,3,3,1,1,1,1] => 51
[7,3,2,1] => [4,3,2,1,1,1,1] => 53
[7,3,1,1,1] => [5,2,2,1,1,1,1] => 54
[7,2,2,2] => [4,4,1,1,1,1,1] => 54
[7,2,2,1,1] => [5,3,1,1,1,1,1] => 55
[7,2,1,1,1,1] => [6,2,1,1,1,1,1] => 56
[7,1,1,1,1,1,1] => [7,1,1,1,1,1,1] => 57
[6,6,1] => [3,2,2,2,2,2] => 48
[6,5,2] => [3,3,2,2,2,1] => 52
[6,5,1,1] => [4,2,2,2,2,1] => 53
[6,4,3] => [3,3,3,2,1,1] => 54
[6,4,2,1] => [4,3,2,2,1,1] => 56
[6,4,1,1,1] => [5,2,2,2,1,1] => 57
[6,3,3,1] => [4,3,3,1,1,1] => 57
[6,3,2,2] => [4,4,2,1,1,1] => 58
[6,3,2,1,1] => [5,3,2,1,1,1] => 59
[6,3,1,1,1,1] => [6,2,2,1,1,1] => 60
[6,2,2,2,1] => [5,4,1,1,1,1] => 60
[6,2,2,1,1,1] => [6,3,1,1,1,1] => 61
[6,2,1,1,1,1,1] => [7,2,1,1,1,1] => 62
[6,1,1,1,1,1,1,1] => [8,1,1,1,1,1] => 63
[5,5,3] => [3,3,3,2,2] => 55
[5,5,2,1] => [4,3,2,2,2] => 57
[5,5,1,1,1] => [5,2,2,2,2] => 58
[5,4,4] => [3,3,3,3,1] => 56
[5,4,3,1] => [4,3,3,2,1] => 59
[5,4,2,2] => [4,4,2,2,1] => 60
[5,4,2,1,1] => [5,3,2,2,1] => 61
[5,4,1,1,1,1] => [6,2,2,2,1] => 62
[5,3,3,2] => [4,4,3,1,1] => 61
[5,3,3,1,1] => [5,3,3,1,1] => 62
[5,3,2,2,1] => [5,4,2,1,1] => 63
[5,3,2,1,1,1] => [6,3,2,1,1] => 64
[5,3,1,1,1,1,1] => [7,2,2,1,1] => 65
[5,2,2,2,2] => [5,5,1,1,1] => 64
[5,2,2,2,1,1] => [6,4,1,1,1] => 65
[5,2,2,1,1,1,1] => [7,3,1,1,1] => 66
[5,2,1,1,1,1,1,1] => [8,2,1,1,1] => 67
[5,1,1,1,1,1,1,1,1] => [9,1,1,1,1] => 68
[4,4,4,1] => [4,3,3,3] => 60
[4,4,3,2] => [4,4,3,2] => 62
[4,4,3,1,1] => [5,3,3,2] => 63
[4,4,2,2,1] => [5,4,2,2] => 64
[4,4,2,1,1,1] => [6,3,2,2] => 65
[4,4,1,1,1,1,1] => [7,2,2,2] => 66
[4,3,3,3] => [4,4,4,1] => 63
[4,3,3,2,1] => [5,4,3,1] => 65
[4,3,3,1,1,1] => [6,3,3,1] => 66
[4,3,2,2,2] => [5,5,2,1] => 66
[4,3,2,2,1,1] => [6,4,2,1] => 67
[4,3,2,1,1,1,1] => [7,3,2,1] => 68
[4,3,1,1,1,1,1,1] => [8,2,2,1] => 69
[4,2,2,2,2,1] => [6,5,1,1] => 68
[4,2,2,2,1,1,1] => [7,4,1,1] => 69
[4,2,2,1,1,1,1,1] => [8,3,1,1] => 70
[4,2,1,1,1,1,1,1,1] => [9,2,1,1] => 71
[4,1,1,1,1,1,1,1,1,1] => [10,1,1,1] => 72
[3,3,3,3,1] => [5,4,4] => 66
[3,3,3,2,2] => [5,5,3] => 67
[3,3,3,2,1,1] => [6,4,3] => 68
[3,3,3,1,1,1,1] => [7,3,3] => 69
[3,3,2,2,2,1] => [6,5,2] => 69
[3,3,2,2,1,1,1] => [7,4,2] => 70
[3,3,2,1,1,1,1,1] => [8,3,2] => 71
[3,3,1,1,1,1,1,1,1] => [9,2,2] => 72
[3,2,2,2,2,2] => [6,6,1] => 70
[3,2,2,2,2,1,1] => [7,5,1] => 71
[3,2,2,2,1,1,1,1] => [8,4,1] => 72
[3,2,2,1,1,1,1,1,1] => [9,3,1] => 73
[3,2,1,1,1,1,1,1,1,1] => [10,2,1] => 74
[3,1,1,1,1,1,1,1,1,1,1] => [11,1,1] => 75
[2,2,2,2,2,2,1] => [7,6] => 72
[2,2,2,2,2,1,1,1] => [8,5] => 73
[2,2,2,2,1,1,1,1,1] => [9,4] => 74
[2,2,2,1,1,1,1,1,1,1] => [10,3] => 75
[2,2,1,1,1,1,1,1,1,1,1] => [11,2] => 76
[2,1,1,1,1,1,1,1,1,1,1,1] => [12,1] => 77
[1,1,1,1,1,1,1,1,1,1,1,1,1] => [13] => 78
[14] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[13,1] => [2,1,1,1,1,1,1,1,1,1,1,1,1] => 13
[12,2] => [2,2,1,1,1,1,1,1,1,1,1,1] => 24
[12,1,1] => [3,1,1,1,1,1,1,1,1,1,1,1] => 25
[11,3] => [2,2,2,1,1,1,1,1,1,1,1] => 33
[11,2,1] => [3,2,1,1,1,1,1,1,1,1,1] => 35
[11,1,1,1] => [4,1,1,1,1,1,1,1,1,1,1] => 36
[10,4] => [2,2,2,2,1,1,1,1,1,1] => 40
[10,3,1] => [3,2,2,1,1,1,1,1,1,1] => 43
[10,2,2] => [3,3,1,1,1,1,1,1,1,1] => 44
[10,2,1,1] => [4,2,1,1,1,1,1,1,1,1] => 45
[10,1,1,1,1] => [5,1,1,1,1,1,1,1,1,1] => 46
[9,5] => [2,2,2,2,2,1,1,1,1] => 45
[9,4,1] => [3,2,2,2,1,1,1,1,1] => 49
[9,3,2] => [3,3,2,1,1,1,1,1,1] => 51
[9,3,1,1] => [4,2,2,1,1,1,1,1,1] => 52
[9,2,2,1] => [4,3,1,1,1,1,1,1,1] => 53
[9,2,1,1,1] => [5,2,1,1,1,1,1,1,1] => 54
[9,1,1,1,1,1] => [6,1,1,1,1,1,1,1,1] => 55
[8,6] => [2,2,2,2,2,2,1,1] => 48
[8,5,1] => [3,2,2,2,2,1,1,1] => 53
[8,4,2] => [3,3,2,2,1,1,1,1] => 56
[8,4,1,1] => [4,2,2,2,1,1,1,1] => 57
[8,3,3] => [3,3,3,1,1,1,1,1] => 57
[8,3,2,1] => [4,3,2,1,1,1,1,1] => 59
[8,3,1,1,1] => [5,2,2,1,1,1,1,1] => 60
[8,2,2,2] => [4,4,1,1,1,1,1,1] => 60
[8,2,2,1,1] => [5,3,1,1,1,1,1,1] => 61
[8,2,1,1,1,1] => [6,2,1,1,1,1,1,1] => 62
[8,1,1,1,1,1,1] => [7,1,1,1,1,1,1,1] => 63
[7,7] => [2,2,2,2,2,2,2] => 49
[7,6,1] => [3,2,2,2,2,2,1] => 55
[7,5,2] => [3,3,2,2,2,1,1] => 59
[7,5,1,1] => [4,2,2,2,2,1,1] => 60
[7,4,3] => [3,3,3,2,1,1,1] => 61
[7,4,2,1] => [4,3,2,2,1,1,1] => 63
[7,4,1,1,1] => [5,2,2,2,1,1,1] => 64
[7,3,3,1] => [4,3,3,1,1,1,1] => 64
[7,3,2,2] => [4,4,2,1,1,1,1] => 65
[7,3,2,1,1] => [5,3,2,1,1,1,1] => 66
[7,3,1,1,1,1] => [6,2,2,1,1,1,1] => 67
[7,2,2,2,1] => [5,4,1,1,1,1,1] => 67
[7,2,2,1,1,1] => [6,3,1,1,1,1,1] => 68
[7,2,1,1,1,1,1] => [7,2,1,1,1,1,1] => 69
[7,1,1,1,1,1,1,1] => [8,1,1,1,1,1,1] => 70
[6,6,2] => [3,3,2,2,2,2] => 60
[6,6,1,1] => [4,2,2,2,2,2] => 61
[6,5,3] => [3,3,3,2,2,1] => 63
[6,5,2,1] => [4,3,2,2,2,1] => 65
[6,5,1,1,1] => [5,2,2,2,2,1] => 66
[6,4,4] => [3,3,3,3,1,1] => 64
[6,4,3,1] => [4,3,3,2,1,1] => 67
[6,4,2,2] => [4,4,2,2,1,1] => 68
[6,4,2,1,1] => [5,3,2,2,1,1] => 69
[6,4,1,1,1,1] => [6,2,2,2,1,1] => 70
[6,3,3,2] => [4,4,3,1,1,1] => 69
[6,3,3,1,1] => [5,3,3,1,1,1] => 70
[6,3,2,2,1] => [5,4,2,1,1,1] => 71
[6,3,2,1,1,1] => [6,3,2,1,1,1] => 72
[6,3,1,1,1,1,1] => [7,2,2,1,1,1] => 73
[6,2,2,2,2] => [5,5,1,1,1,1] => 72
[6,2,2,2,1,1] => [6,4,1,1,1,1] => 73
[6,2,2,1,1,1,1] => [7,3,1,1,1,1] => 74
[6,2,1,1,1,1,1,1] => [8,2,1,1,1,1] => 75
[6,1,1,1,1,1,1,1,1] => [9,1,1,1,1,1] => 76
[5,5,4] => [3,3,3,3,2] => 65
[5,5,3,1] => [4,3,3,2,2] => 68
[5,5,2,2] => [4,4,2,2,2] => 69
[5,5,2,1,1] => [5,3,2,2,2] => 70
[5,5,1,1,1,1] => [6,2,2,2,2] => 71
[5,4,4,1] => [4,3,3,3,1] => 69
[5,4,3,2] => [4,4,3,2,1] => 71
[5,4,3,1,1] => [5,3,3,2,1] => 72
[5,4,2,2,1] => [5,4,2,2,1] => 73
[5,4,2,1,1,1] => [6,3,2,2,1] => 74
[5,4,1,1,1,1,1] => [7,2,2,2,1] => 75
[5,3,3,3] => [4,4,4,1,1] => 72
[5,3,3,2,1] => [5,4,3,1,1] => 74
[5,3,3,1,1,1] => [6,3,3,1,1] => 75
[5,3,2,2,2] => [5,5,2,1,1] => 75
[5,3,2,2,1,1] => [6,4,2,1,1] => 76
[5,3,2,1,1,1,1] => [7,3,2,1,1] => 77
[5,3,1,1,1,1,1,1] => [8,2,2,1,1] => 78
[5,2,2,2,2,1] => [6,5,1,1,1] => 77
[5,2,2,2,1,1,1] => [7,4,1,1,1] => 78
[5,2,2,1,1,1,1,1] => [8,3,1,1,1] => 79
[5,2,1,1,1,1,1,1,1] => [9,2,1,1,1] => 80
[5,1,1,1,1,1,1,1,1,1] => [10,1,1,1,1] => 81
[4,4,4,2] => [4,4,3,3] => 72
[4,4,4,1,1] => [5,3,3,3] => 73
[4,4,3,3] => [4,4,4,2] => 73
[4,4,3,2,1] => [5,4,3,2] => 75
[4,4,3,1,1,1] => [6,3,3,2] => 76
[4,4,2,2,2] => [5,5,2,2] => 76
[4,4,2,2,1,1] => [6,4,2,2] => 77
[4,4,2,1,1,1,1] => [7,3,2,2] => 78
[4,4,1,1,1,1,1,1] => [8,2,2,2] => 79
[4,3,3,3,1] => [5,4,4,1] => 76
[4,3,3,2,2] => [5,5,3,1] => 77
[4,3,3,2,1,1] => [6,4,3,1] => 78
[4,3,3,1,1,1,1] => [7,3,3,1] => 79
[4,3,2,2,2,1] => [6,5,2,1] => 79
[4,3,2,2,1,1,1] => [7,4,2,1] => 80
[4,3,2,1,1,1,1,1] => [8,3,2,1] => 81
[4,3,1,1,1,1,1,1,1] => [9,2,2,1] => 82
[4,2,2,2,2,2] => [6,6,1,1] => 80
[4,2,2,2,2,1,1] => [7,5,1,1] => 81
[4,2,2,2,1,1,1,1] => [8,4,1,1] => 82
[4,2,2,1,1,1,1,1,1] => [9,3,1,1] => 83
[4,2,1,1,1,1,1,1,1,1] => [10,2,1,1] => 84
[4,1,1,1,1,1,1,1,1,1,1] => [11,1,1,1] => 85
[3,3,3,3,2] => [5,5,4] => 78
[3,3,3,3,1,1] => [6,4,4] => 79
[3,3,3,2,2,1] => [6,5,3] => 80
[3,3,3,2,1,1,1] => [7,4,3] => 81
[3,3,3,1,1,1,1,1] => [8,3,3] => 82
[3,3,2,2,2,2] => [6,6,2] => 81
[3,3,2,2,2,1,1] => [7,5,2] => 82
[3,3,2,2,1,1,1,1] => [8,4,2] => 83
[3,3,2,1,1,1,1,1,1] => [9,3,2] => 84
[3,3,1,1,1,1,1,1,1,1] => [10,2,2] => 85
[3,2,2,2,2,2,1] => [7,6,1] => 83
[3,2,2,2,2,1,1,1] => [8,5,1] => 84
[3,2,2,2,1,1,1,1,1] => [9,4,1] => 85
[3,2,2,1,1,1,1,1,1,1] => [10,3,1] => 86
[3,2,1,1,1,1,1,1,1,1,1] => [11,2,1] => 87
[3,1,1,1,1,1,1,1,1,1,1,1] => [12,1,1] => 88
[2,2,2,2,2,2,2] => [7,7] => 84
[2,2,2,2,2,2,1,1] => [8,6] => 85
[2,2,2,2,2,1,1,1,1] => [9,5] => 86
[2,2,2,2,1,1,1,1,1,1] => [10,4] => 87
[2,2,2,1,1,1,1,1,1,1,1] => [11,3] => 88
[2,2,1,1,1,1,1,1,1,1,1,1] => [12,2] => 89
[2,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1] => 90
[1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14] => 91
[15] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[14,1] => [2,1,1,1,1,1,1,1,1,1,1,1,1,1] => 14
[13,2] => [2,2,1,1,1,1,1,1,1,1,1,1,1] => 26
[13,1,1] => [3,1,1,1,1,1,1,1,1,1,1,1,1] => 27
[12,3] => [2,2,2,1,1,1,1,1,1,1,1,1] => 36
[12,2,1] => [3,2,1,1,1,1,1,1,1,1,1,1] => 38
[12,1,1,1] => [4,1,1,1,1,1,1,1,1,1,1,1] => 39
[11,4] => [2,2,2,2,1,1,1,1,1,1,1] => 44
[11,3,1] => [3,2,2,1,1,1,1,1,1,1,1] => 47
[11,2,2] => [3,3,1,1,1,1,1,1,1,1,1] => 48
[11,2,1,1] => [4,2,1,1,1,1,1,1,1,1,1] => 49
[11,1,1,1,1] => [5,1,1,1,1,1,1,1,1,1,1] => 50
[10,5] => [2,2,2,2,2,1,1,1,1,1] => 50
[10,4,1] => [3,2,2,2,1,1,1,1,1,1] => 54
[10,3,2] => [3,3,2,1,1,1,1,1,1,1] => 56
[10,3,1,1] => [4,2,2,1,1,1,1,1,1,1] => 57
[10,2,2,1] => [4,3,1,1,1,1,1,1,1,1] => 58
[10,2,1,1,1] => [5,2,1,1,1,1,1,1,1,1] => 59
[10,1,1,1,1,1] => [6,1,1,1,1,1,1,1,1,1] => 60
[9,6] => [2,2,2,2,2,2,1,1,1] => 54
[9,5,1] => [3,2,2,2,2,1,1,1,1] => 59
[9,4,2] => [3,3,2,2,1,1,1,1,1] => 62
[9,4,1,1] => [4,2,2,2,1,1,1,1,1] => 63
[9,3,3] => [3,3,3,1,1,1,1,1,1] => 63
[9,3,2,1] => [4,3,2,1,1,1,1,1,1] => 65
[9,3,1,1,1] => [5,2,2,1,1,1,1,1,1] => 66
[9,2,2,2] => [4,4,1,1,1,1,1,1,1] => 66
[9,2,2,1,1] => [5,3,1,1,1,1,1,1,1] => 67
[9,2,1,1,1,1] => [6,2,1,1,1,1,1,1,1] => 68
[9,1,1,1,1,1,1] => [7,1,1,1,1,1,1,1,1] => 69
[8,7] => [2,2,2,2,2,2,2,1] => 56
[8,6,1] => [3,2,2,2,2,2,1,1] => 62
[8,5,2] => [3,3,2,2,2,1,1,1] => 66
[8,5,1,1] => [4,2,2,2,2,1,1,1] => 67
[8,4,3] => [3,3,3,2,1,1,1,1] => 68
[8,4,2,1] => [4,3,2,2,1,1,1,1] => 70
[8,4,1,1,1] => [5,2,2,2,1,1,1,1] => 71
[8,3,3,1] => [4,3,3,1,1,1,1,1] => 71
[8,3,2,2] => [4,4,2,1,1,1,1,1] => 72
[8,3,2,1,1] => [5,3,2,1,1,1,1,1] => 73
[8,3,1,1,1,1] => [6,2,2,1,1,1,1,1] => 74
[8,2,2,2,1] => [5,4,1,1,1,1,1,1] => 74
[8,2,2,1,1,1] => [6,3,1,1,1,1,1,1] => 75
[8,2,1,1,1,1,1] => [7,2,1,1,1,1,1,1] => 76
[8,1,1,1,1,1,1,1] => [8,1,1,1,1,1,1,1] => 77
[7,7,1] => [3,2,2,2,2,2,2] => 63
[7,6,2] => [3,3,2,2,2,2,1] => 68
[7,6,1,1] => [4,2,2,2,2,2,1] => 69
[7,5,3] => [3,3,3,2,2,1,1] => 71
[7,5,2,1] => [4,3,2,2,2,1,1] => 73
[7,5,1,1,1] => [5,2,2,2,2,1,1] => 74
[7,4,4] => [3,3,3,3,1,1,1] => 72
[7,4,3,1] => [4,3,3,2,1,1,1] => 75
[7,4,2,2] => [4,4,2,2,1,1,1] => 76
[7,4,2,1,1] => [5,3,2,2,1,1,1] => 77
[7,4,1,1,1,1] => [6,2,2,2,1,1,1] => 78
[7,3,3,2] => [4,4,3,1,1,1,1] => 77
[7,3,3,1,1] => [5,3,3,1,1,1,1] => 78
[7,3,2,2,1] => [5,4,2,1,1,1,1] => 79
[7,3,2,1,1,1] => [6,3,2,1,1,1,1] => 80
[7,3,1,1,1,1,1] => [7,2,2,1,1,1,1] => 81
[7,2,2,2,2] => [5,5,1,1,1,1,1] => 80
[7,2,2,2,1,1] => [6,4,1,1,1,1,1] => 81
[7,2,2,1,1,1,1] => [7,3,1,1,1,1,1] => 82
[7,2,1,1,1,1,1,1] => [8,2,1,1,1,1,1] => 83
[7,1,1,1,1,1,1,1,1] => [9,1,1,1,1,1,1] => 84
[6,6,3] => [3,3,3,2,2,2] => 72
[6,6,2,1] => [4,3,2,2,2,2] => 74
[6,6,1,1,1] => [5,2,2,2,2,2] => 75
[6,5,4] => [3,3,3,3,2,1] => 74
[6,5,3,1] => [4,3,3,2,2,1] => 77
[6,5,2,2] => [4,4,2,2,2,1] => 78
[6,5,2,1,1] => [5,3,2,2,2,1] => 79
[6,5,1,1,1,1] => [6,2,2,2,2,1] => 80
[6,4,4,1] => [4,3,3,3,1,1] => 78
[6,4,3,2] => [4,4,3,2,1,1] => 80
[6,4,3,1,1] => [5,3,3,2,1,1] => 81
[6,4,2,2,1] => [5,4,2,2,1,1] => 82
[6,4,2,1,1,1] => [6,3,2,2,1,1] => 83
[6,4,1,1,1,1,1] => [7,2,2,2,1,1] => 84
[6,3,3,3] => [4,4,4,1,1,1] => 81
[6,3,3,2,1] => [5,4,3,1,1,1] => 83
[6,3,3,1,1,1] => [6,3,3,1,1,1] => 84
[6,3,2,2,2] => [5,5,2,1,1,1] => 84
[6,3,2,2,1,1] => [6,4,2,1,1,1] => 85
[6,3,2,1,1,1,1] => [7,3,2,1,1,1] => 86
[6,3,1,1,1,1,1,1] => [8,2,2,1,1,1] => 87
[6,2,2,2,2,1] => [6,5,1,1,1,1] => 86
[6,2,2,2,1,1,1] => [7,4,1,1,1,1] => 87
[6,2,2,1,1,1,1,1] => [8,3,1,1,1,1] => 88
[6,2,1,1,1,1,1,1,1] => [9,2,1,1,1,1] => 89
[6,1,1,1,1,1,1,1,1,1] => [10,1,1,1,1,1] => 90
[5,5,5] => [3,3,3,3,3] => 75
[5,5,4,1] => [4,3,3,3,2] => 79
[5,5,3,2] => [4,4,3,2,2] => 81
[5,5,3,1,1] => [5,3,3,2,2] => 82
[5,5,2,2,1] => [5,4,2,2,2] => 83
[5,5,2,1,1,1] => [6,3,2,2,2] => 84
[5,5,1,1,1,1,1] => [7,2,2,2,2] => 85
[5,4,4,2] => [4,4,3,3,1] => 82
[5,4,4,1,1] => [5,3,3,3,1] => 83
[5,4,3,3] => [4,4,4,2,1] => 83
[5,4,3,2,1] => [5,4,3,2,1] => 85
[5,4,3,1,1,1] => [6,3,3,2,1] => 86
[5,4,2,2,2] => [5,5,2,2,1] => 86
[5,4,2,2,1,1] => [6,4,2,2,1] => 87
[5,4,2,1,1,1,1] => [7,3,2,2,1] => 88
[5,4,1,1,1,1,1,1] => [8,2,2,2,1] => 89
[5,3,3,3,1] => [5,4,4,1,1] => 86
[5,3,3,2,2] => [5,5,3,1,1] => 87
[5,3,3,2,1,1] => [6,4,3,1,1] => 88
[5,3,3,1,1,1,1] => [7,3,3,1,1] => 89
[5,3,2,2,2,1] => [6,5,2,1,1] => 89
[5,3,2,2,1,1,1] => [7,4,2,1,1] => 90
[5,3,2,1,1,1,1,1] => [8,3,2,1,1] => 91
[5,3,1,1,1,1,1,1,1] => [9,2,2,1,1] => 92
[5,2,2,2,2,2] => [6,6,1,1,1] => 90
[5,2,2,2,2,1,1] => [7,5,1,1,1] => 91
[5,2,2,2,1,1,1,1] => [8,4,1,1,1] => 92
[5,2,2,1,1,1,1,1,1] => [9,3,1,1,1] => 93
[5,2,1,1,1,1,1,1,1,1] => [10,2,1,1,1] => 94
[5,1,1,1,1,1,1,1,1,1,1] => [11,1,1,1,1] => 95
[4,4,4,3] => [4,4,4,3] => 84
[4,4,4,2,1] => [5,4,3,3] => 86
[4,4,4,1,1,1] => [6,3,3,3] => 87
[4,4,3,3,1] => [5,4,4,2] => 87
[4,4,3,2,2] => [5,5,3,2] => 88
[4,4,3,2,1,1] => [6,4,3,2] => 89
[4,4,3,1,1,1,1] => [7,3,3,2] => 90
[4,4,2,2,2,1] => [6,5,2,2] => 90
[4,4,2,2,1,1,1] => [7,4,2,2] => 91
[4,4,2,1,1,1,1,1] => [8,3,2,2] => 92
[4,4,1,1,1,1,1,1,1] => [9,2,2,2] => 93
[4,3,3,3,2] => [5,5,4,1] => 89
[4,3,3,3,1,1] => [6,4,4,1] => 90
[4,3,3,2,2,1] => [6,5,3,1] => 91
[4,3,3,2,1,1,1] => [7,4,3,1] => 92
[4,3,3,1,1,1,1,1] => [8,3,3,1] => 93
[4,3,2,2,2,2] => [6,6,2,1] => 92
[4,3,2,2,2,1,1] => [7,5,2,1] => 93
[4,3,2,2,1,1,1,1] => [8,4,2,1] => 94
[4,3,2,1,1,1,1,1,1] => [9,3,2,1] => 95
[4,3,1,1,1,1,1,1,1,1] => [10,2,2,1] => 96
[4,2,2,2,2,2,1] => [7,6,1,1] => 94
[4,2,2,2,2,1,1,1] => [8,5,1,1] => 95
[4,2,2,2,1,1,1,1,1] => [9,4,1,1] => 96
[4,2,2,1,1,1,1,1,1,1] => [10,3,1,1] => 97
[4,2,1,1,1,1,1,1,1,1,1] => [11,2,1,1] => 98
[4,1,1,1,1,1,1,1,1,1,1,1] => [12,1,1,1] => 99
[3,3,3,3,3] => [5,5,5] => 90
[3,3,3,3,2,1] => [6,5,4] => 92
[3,3,3,3,1,1,1] => [7,4,4] => 93
[3,3,3,2,2,2] => [6,6,3] => 93
[3,3,3,2,2,1,1] => [7,5,3] => 94
[3,3,3,2,1,1,1,1] => [8,4,3] => 95
[3,3,3,1,1,1,1,1,1] => [9,3,3] => 96
[3,3,2,2,2,2,1] => [7,6,2] => 95
[3,3,2,2,2,1,1,1] => [8,5,2] => 96
[3,3,2,2,1,1,1,1,1] => [9,4,2] => 97
[3,3,2,1,1,1,1,1,1,1] => [10,3,2] => 98
[3,3,1,1,1,1,1,1,1,1,1] => [11,2,2] => 99
[3,2,2,2,2,2,2] => [7,7,1] => 96
[3,2,2,2,2,2,1,1] => [8,6,1] => 97
[3,2,2,2,2,1,1,1,1] => [9,5,1] => 98
[3,2,2,2,1,1,1,1,1,1] => [10,4,1] => 99
[3,2,2,1,1,1,1,1,1,1,1] => [11,3,1] => 100
[3,2,1,1,1,1,1,1,1,1,1,1] => [12,2,1] => 101
[3,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1] => 102
[2,2,2,2,2,2,2,1] => [8,7] => 98
[2,2,2,2,2,2,1,1,1] => [9,6] => 99
[2,2,2,2,2,1,1,1,1,1] => [10,5] => 100
[2,2,2,2,1,1,1,1,1,1,1] => [11,4] => 101
[2,2,2,1,1,1,1,1,1,1,1,1] => [12,3] => 102
[2,2,1,1,1,1,1,1,1,1,1,1,1] => [13,2] => 103
[2,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1] => 104
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15] => 105
[16] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[15,1] => [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 15
[14,2] => [2,2,1,1,1,1,1,1,1,1,1,1,1,1] => 28
[14,1,1] => [3,1,1,1,1,1,1,1,1,1,1,1,1,1] => 29
[13,3] => [2,2,2,1,1,1,1,1,1,1,1,1,1] => 39
[13,2,1] => [3,2,1,1,1,1,1,1,1,1,1,1,1] => 41
[13,1,1,1] => [4,1,1,1,1,1,1,1,1,1,1,1,1] => 42
[12,4] => [2,2,2,2,1,1,1,1,1,1,1,1] => 48
[12,3,1] => [3,2,2,1,1,1,1,1,1,1,1,1] => 51
[12,2,2] => [3,3,1,1,1,1,1,1,1,1,1,1] => 52
[12,2,1,1] => [4,2,1,1,1,1,1,1,1,1,1,1] => 53
[12,1,1,1,1] => [5,1,1,1,1,1,1,1,1,1,1,1] => 54
[11,5] => [2,2,2,2,2,1,1,1,1,1,1] => 55
[11,4,1] => [3,2,2,2,1,1,1,1,1,1,1] => 59
[11,3,2] => [3,3,2,1,1,1,1,1,1,1,1] => 61
[11,3,1,1] => [4,2,2,1,1,1,1,1,1,1,1] => 62
[11,2,2,1] => [4,3,1,1,1,1,1,1,1,1,1] => 63
[11,2,1,1,1] => [5,2,1,1,1,1,1,1,1,1,1] => 64
[11,1,1,1,1,1] => [6,1,1,1,1,1,1,1,1,1,1] => 65
[10,6] => [2,2,2,2,2,2,1,1,1,1] => 60
[10,5,1] => [3,2,2,2,2,1,1,1,1,1] => 65
[10,4,2] => [3,3,2,2,1,1,1,1,1,1] => 68
[10,4,1,1] => [4,2,2,2,1,1,1,1,1,1] => 69
[10,3,3] => [3,3,3,1,1,1,1,1,1,1] => 69
[10,3,2,1] => [4,3,2,1,1,1,1,1,1,1] => 71
[10,3,1,1,1] => [5,2,2,1,1,1,1,1,1,1] => 72
[10,2,2,2] => [4,4,1,1,1,1,1,1,1,1] => 72
[10,2,2,1,1] => [5,3,1,1,1,1,1,1,1,1] => 73
[10,2,1,1,1,1] => [6,2,1,1,1,1,1,1,1,1] => 74
[10,1,1,1,1,1,1] => [7,1,1,1,1,1,1,1,1,1] => 75
[9,7] => [2,2,2,2,2,2,2,1,1] => 63
[9,6,1] => [3,2,2,2,2,2,1,1,1] => 69
[9,5,2] => [3,3,2,2,2,1,1,1,1] => 73
[9,5,1,1] => [4,2,2,2,2,1,1,1,1] => 74
[9,4,3] => [3,3,3,2,1,1,1,1,1] => 75
[9,4,2,1] => [4,3,2,2,1,1,1,1,1] => 77
[9,4,1,1,1] => [5,2,2,2,1,1,1,1,1] => 78
[9,3,3,1] => [4,3,3,1,1,1,1,1,1] => 78
[9,3,2,2] => [4,4,2,1,1,1,1,1,1] => 79
[9,3,2,1,1] => [5,3,2,1,1,1,1,1,1] => 80
[9,3,1,1,1,1] => [6,2,2,1,1,1,1,1,1] => 81
[9,2,2,2,1] => [5,4,1,1,1,1,1,1,1] => 81
[9,2,2,1,1,1] => [6,3,1,1,1,1,1,1,1] => 82
[9,2,1,1,1,1,1] => [7,2,1,1,1,1,1,1,1] => 83
[9,1,1,1,1,1,1,1] => [8,1,1,1,1,1,1,1,1] => 84
[8,8] => [2,2,2,2,2,2,2,2] => 64
[8,7,1] => [3,2,2,2,2,2,2,1] => 71
[8,6,2] => [3,3,2,2,2,2,1,1] => 76
[8,6,1,1] => [4,2,2,2,2,2,1,1] => 77
[8,5,3] => [3,3,3,2,2,1,1,1] => 79
[8,5,2,1] => [4,3,2,2,2,1,1,1] => 81
[8,5,1,1,1] => [5,2,2,2,2,1,1,1] => 82
[8,4,4] => [3,3,3,3,1,1,1,1] => 80
[8,4,3,1] => [4,3,3,2,1,1,1,1] => 83
[8,4,2,2] => [4,4,2,2,1,1,1,1] => 84
[8,4,2,1,1] => [5,3,2,2,1,1,1,1] => 85
[8,4,1,1,1,1] => [6,2,2,2,1,1,1,1] => 86
[8,3,3,2] => [4,4,3,1,1,1,1,1] => 85
[8,3,3,1,1] => [5,3,3,1,1,1,1,1] => 86
[8,3,2,2,1] => [5,4,2,1,1,1,1,1] => 87
[8,3,2,1,1,1] => [6,3,2,1,1,1,1,1] => 88
[8,3,1,1,1,1,1] => [7,2,2,1,1,1,1,1] => 89
[8,2,2,2,2] => [5,5,1,1,1,1,1,1] => 88
[8,2,2,2,1,1] => [6,4,1,1,1,1,1,1] => 89
[8,2,2,1,1,1,1] => [7,3,1,1,1,1,1,1] => 90
[8,2,1,1,1,1,1,1] => [8,2,1,1,1,1,1,1] => 91
[8,1,1,1,1,1,1,1,1] => [9,1,1,1,1,1,1,1] => 92
[7,7,2] => [3,3,2,2,2,2,2] => 77
[7,7,1,1] => [4,2,2,2,2,2,2] => 78
[7,6,3] => [3,3,3,2,2,2,1] => 81
[7,6,2,1] => [4,3,2,2,2,2,1] => 83
[7,6,1,1,1] => [5,2,2,2,2,2,1] => 84
[7,5,4] => [3,3,3,3,2,1,1] => 83
[7,5,3,1] => [4,3,3,2,2,1,1] => 86
[7,5,2,2] => [4,4,2,2,2,1,1] => 87
[7,5,2,1,1] => [5,3,2,2,2,1,1] => 88
[7,5,1,1,1,1] => [6,2,2,2,2,1,1] => 89
[7,4,4,1] => [4,3,3,3,1,1,1] => 87
[7,4,3,2] => [4,4,3,2,1,1,1] => 89
[7,4,3,1,1] => [5,3,3,2,1,1,1] => 90
[7,4,2,2,1] => [5,4,2,2,1,1,1] => 91
[7,4,2,1,1,1] => [6,3,2,2,1,1,1] => 92
[7,4,1,1,1,1,1] => [7,2,2,2,1,1,1] => 93
[7,3,3,3] => [4,4,4,1,1,1,1] => 90
[7,3,3,2,1] => [5,4,3,1,1,1,1] => 92
[7,3,3,1,1,1] => [6,3,3,1,1,1,1] => 93
[7,3,2,2,2] => [5,5,2,1,1,1,1] => 93
[7,3,2,2,1,1] => [6,4,2,1,1,1,1] => 94
[7,3,2,1,1,1,1] => [7,3,2,1,1,1,1] => 95
[7,3,1,1,1,1,1,1] => [8,2,2,1,1,1,1] => 96
[7,2,2,2,2,1] => [6,5,1,1,1,1,1] => 95
[7,2,2,2,1,1,1] => [7,4,1,1,1,1,1] => 96
[7,2,2,1,1,1,1,1] => [8,3,1,1,1,1,1] => 97
[7,2,1,1,1,1,1,1,1] => [9,2,1,1,1,1,1] => 98
[7,1,1,1,1,1,1,1,1,1] => [10,1,1,1,1,1,1] => 99
[6,6,4] => [3,3,3,3,2,2] => 84
[6,6,3,1] => [4,3,3,2,2,2] => 87
[6,6,2,2] => [4,4,2,2,2,2] => 88
[6,6,2,1,1] => [5,3,2,2,2,2] => 89
[6,6,1,1,1,1] => [6,2,2,2,2,2] => 90
[6,5,5] => [3,3,3,3,3,1] => 85
[6,5,4,1] => [4,3,3,3,2,1] => 89
[6,5,3,2] => [4,4,3,2,2,1] => 91
[6,5,3,1,1] => [5,3,3,2,2,1] => 92
[6,5,2,2,1] => [5,4,2,2,2,1] => 93
[6,5,2,1,1,1] => [6,3,2,2,2,1] => 94
[6,5,1,1,1,1,1] => [7,2,2,2,2,1] => 95
[6,4,4,2] => [4,4,3,3,1,1] => 92
[6,4,4,1,1] => [5,3,3,3,1,1] => 93
[6,4,3,3] => [4,4,4,2,1,1] => 93
[6,4,3,2,1] => [5,4,3,2,1,1] => 95
[6,4,3,1,1,1] => [6,3,3,2,1,1] => 96
[6,4,2,2,2] => [5,5,2,2,1,1] => 96
[6,4,2,2,1,1] => [6,4,2,2,1,1] => 97
[6,4,2,1,1,1,1] => [7,3,2,2,1,1] => 98
[6,4,1,1,1,1,1,1] => [8,2,2,2,1,1] => 99
[6,3,3,3,1] => [5,4,4,1,1,1] => 96
[6,3,3,2,2] => [5,5,3,1,1,1] => 97
[6,3,3,2,1,1] => [6,4,3,1,1,1] => 98
[6,3,3,1,1,1,1] => [7,3,3,1,1,1] => 99
[6,3,2,2,2,1] => [6,5,2,1,1,1] => 99
[6,3,2,2,1,1,1] => [7,4,2,1,1,1] => 100
[6,3,2,1,1,1,1,1] => [8,3,2,1,1,1] => 101
[6,3,1,1,1,1,1,1,1] => [9,2,2,1,1,1] => 102
[6,2,2,2,2,2] => [6,6,1,1,1,1] => 100
[6,2,2,2,2,1,1] => [7,5,1,1,1,1] => 101
[6,2,2,2,1,1,1,1] => [8,4,1,1,1,1] => 102
[6,2,2,1,1,1,1,1,1] => [9,3,1,1,1,1] => 103
[6,2,1,1,1,1,1,1,1,1] => [10,2,1,1,1,1] => 104
[6,1,1,1,1,1,1,1,1,1,1] => [11,1,1,1,1,1] => 105
[5,5,5,1] => [4,3,3,3,3] => 90
[5,5,4,2] => [4,4,3,3,2] => 93
[5,5,4,1,1] => [5,3,3,3,2] => 94
[5,5,3,3] => [4,4,4,2,2] => 94
[5,5,3,2,1] => [5,4,3,2,2] => 96
[5,5,3,1,1,1] => [6,3,3,2,2] => 97
[5,5,2,2,2] => [5,5,2,2,2] => 97
[5,5,2,2,1,1] => [6,4,2,2,2] => 98
[5,5,2,1,1,1,1] => [7,3,2,2,2] => 99
[5,5,1,1,1,1,1,1] => [8,2,2,2,2] => 100
[5,4,4,3] => [4,4,4,3,1] => 95
[5,4,4,2,1] => [5,4,3,3,1] => 97
[5,4,4,1,1,1] => [6,3,3,3,1] => 98
[5,4,3,3,1] => [5,4,4,2,1] => 98
[5,4,3,2,2] => [5,5,3,2,1] => 99
[5,4,3,2,1,1] => [6,4,3,2,1] => 100
[5,4,3,1,1,1,1] => [7,3,3,2,1] => 101
[5,4,2,2,2,1] => [6,5,2,2,1] => 101
[5,4,2,2,1,1,1] => [7,4,2,2,1] => 102
[5,4,2,1,1,1,1,1] => [8,3,2,2,1] => 103
[5,4,1,1,1,1,1,1,1] => [9,2,2,2,1] => 104
[5,3,3,3,2] => [5,5,4,1,1] => 100
[5,3,3,3,1,1] => [6,4,4,1,1] => 101
[5,3,3,2,2,1] => [6,5,3,1,1] => 102
[5,3,3,2,1,1,1] => [7,4,3,1,1] => 103
[5,3,3,1,1,1,1,1] => [8,3,3,1,1] => 104
[5,3,2,2,2,2] => [6,6,2,1,1] => 103
[5,3,2,2,2,1,1] => [7,5,2,1,1] => 104
[5,3,2,2,1,1,1,1] => [8,4,2,1,1] => 105
[5,3,2,1,1,1,1,1,1] => [9,3,2,1,1] => 106
[5,3,1,1,1,1,1,1,1,1] => [10,2,2,1,1] => 107
[5,2,2,2,2,2,1] => [7,6,1,1,1] => 105
[5,2,2,2,2,1,1,1] => [8,5,1,1,1] => 106
[5,2,2,2,1,1,1,1,1] => [9,4,1,1,1] => 107
[5,2,2,1,1,1,1,1,1,1] => [10,3,1,1,1] => 108
[5,2,1,1,1,1,1,1,1,1,1] => [11,2,1,1,1] => 109
[5,1,1,1,1,1,1,1,1,1,1,1] => [12,1,1,1,1] => 110
[4,4,4,4] => [4,4,4,4] => 96
[4,4,4,3,1] => [5,4,4,3] => 99
[4,4,4,2,2] => [5,5,3,3] => 100
[4,4,4,2,1,1] => [6,4,3,3] => 101
[4,4,4,1,1,1,1] => [7,3,3,3] => 102
[4,4,3,3,2] => [5,5,4,2] => 101
[4,4,3,3,1,1] => [6,4,4,2] => 102
[4,4,3,2,2,1] => [6,5,3,2] => 103
[4,4,3,2,1,1,1] => [7,4,3,2] => 104
[4,4,3,1,1,1,1,1] => [8,3,3,2] => 105
[4,4,2,2,2,2] => [6,6,2,2] => 104
[4,4,2,2,2,1,1] => [7,5,2,2] => 105
[4,4,2,2,1,1,1,1] => [8,4,2,2] => 106
[4,4,2,1,1,1,1,1,1] => [9,3,2,2] => 107
[4,4,1,1,1,1,1,1,1,1] => [10,2,2,2] => 108
[4,3,3,3,3] => [5,5,5,1] => 102
[4,3,3,3,2,1] => [6,5,4,1] => 104
[4,3,3,3,1,1,1] => [7,4,4,1] => 105
[4,3,3,2,2,2] => [6,6,3,1] => 105
[4,3,3,2,2,1,1] => [7,5,3,1] => 106
[4,3,3,2,1,1,1,1] => [8,4,3,1] => 107
[4,3,3,1,1,1,1,1,1] => [9,3,3,1] => 108
[4,3,2,2,2,2,1] => [7,6,2,1] => 107
[4,3,2,2,2,1,1,1] => [8,5,2,1] => 108
[4,3,2,2,1,1,1,1,1] => [9,4,2,1] => 109
[4,3,2,1,1,1,1,1,1,1] => [10,3,2,1] => 110
[4,3,1,1,1,1,1,1,1,1,1] => [11,2,2,1] => 111
[4,2,2,2,2,2,2] => [7,7,1,1] => 108
[4,2,2,2,2,2,1,1] => [8,6,1,1] => 109
[4,2,2,2,2,1,1,1,1] => [9,5,1,1] => 110
[4,2,2,2,1,1,1,1,1,1] => [10,4,1,1] => 111
[4,2,2,1,1,1,1,1,1,1,1] => [11,3,1,1] => 112
[4,2,1,1,1,1,1,1,1,1,1,1] => [12,2,1,1] => 113
[4,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1,1] => 114
[3,3,3,3,3,1] => [6,5,5] => 105
[3,3,3,3,2,2] => [6,6,4] => 106
[3,3,3,3,2,1,1] => [7,5,4] => 107
[3,3,3,3,1,1,1,1] => [8,4,4] => 108
[3,3,3,2,2,2,1] => [7,6,3] => 108
[3,3,3,2,2,1,1,1] => [8,5,3] => 109
[3,3,3,2,1,1,1,1,1] => [9,4,3] => 110
[3,3,3,1,1,1,1,1,1,1] => [10,3,3] => 111
[3,3,2,2,2,2,2] => [7,7,2] => 109
[3,3,2,2,2,2,1,1] => [8,6,2] => 110
[3,3,2,2,2,1,1,1,1] => [9,5,2] => 111
[3,3,2,2,1,1,1,1,1,1] => [10,4,2] => 112
[3,3,2,1,1,1,1,1,1,1,1] => [11,3,2] => 113
[3,3,1,1,1,1,1,1,1,1,1,1] => [12,2,2] => 114
[3,2,2,2,2,2,2,1] => [8,7,1] => 111
[3,2,2,2,2,2,1,1,1] => [9,6,1] => 112
[3,2,2,2,2,1,1,1,1,1] => [10,5,1] => 113
[3,2,2,2,1,1,1,1,1,1,1] => [11,4,1] => 114
[3,2,2,1,1,1,1,1,1,1,1,1] => [12,3,1] => 115
[3,2,1,1,1,1,1,1,1,1,1,1,1] => [13,2,1] => 116
[3,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1,1] => 117
[2,2,2,2,2,2,2,2] => [8,8] => 112
[2,2,2,2,2,2,2,1,1] => [9,7] => 113
[2,2,2,2,2,2,1,1,1,1] => [10,6] => 114
[2,2,2,2,2,1,1,1,1,1,1] => [11,5] => 115
[2,2,2,2,1,1,1,1,1,1,1,1] => [12,4] => 116
[2,2,2,1,1,1,1,1,1,1,1,1,1] => [13,3] => 117
[2,2,1,1,1,1,1,1,1,1,1,1,1,1] => [14,2] => 118
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,1] => 119
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [16] => 120
[14,1,1,1] => [4,1,1,1,1,1,1,1,1,1,1,1,1,1] => 45
[13,3,1] => [3,2,2,1,1,1,1,1,1,1,1,1,1] => 55
[13,2,2] => [3,3,1,1,1,1,1,1,1,1,1,1,1] => 56
[13,2,1,1] => [4,2,1,1,1,1,1,1,1,1,1,1,1] => 57
[13,1,1,1,1] => [5,1,1,1,1,1,1,1,1,1,1,1,1] => 58
[12,4,1] => [3,2,2,2,1,1,1,1,1,1,1,1] => 64
[12,3,2] => [3,3,2,1,1,1,1,1,1,1,1,1] => 66
[12,3,1,1] => [4,2,2,1,1,1,1,1,1,1,1,1] => 67
[12,2,2,1] => [4,3,1,1,1,1,1,1,1,1,1,1] => 68
[12,2,1,1,1] => [5,2,1,1,1,1,1,1,1,1,1,1] => 69
[12,1,1,1,1,1] => [6,1,1,1,1,1,1,1,1,1,1,1] => 70
[11,5,1] => [3,2,2,2,2,1,1,1,1,1,1] => 71
[11,4,2] => [3,3,2,2,1,1,1,1,1,1,1] => 74
[11,4,1,1] => [4,2,2,2,1,1,1,1,1,1,1] => 75
[11,3,3] => [3,3,3,1,1,1,1,1,1,1,1] => 75
[11,3,2,1] => [4,3,2,1,1,1,1,1,1,1,1] => 77
[11,3,1,1,1] => [5,2,2,1,1,1,1,1,1,1,1] => 78
[11,2,2,2] => [4,4,1,1,1,1,1,1,1,1,1] => 78
[11,2,2,1,1] => [5,3,1,1,1,1,1,1,1,1,1] => 79
[11,2,1,1,1,1] => [6,2,1,1,1,1,1,1,1,1,1] => 80
[11,1,1,1,1,1,1] => [7,1,1,1,1,1,1,1,1,1,1] => 81
[10,6,1] => [3,2,2,2,2,2,1,1,1,1] => 76
[10,5,2] => [3,3,2,2,2,1,1,1,1,1] => 80
[10,5,1,1] => [4,2,2,2,2,1,1,1,1,1] => 81
[10,4,3] => [3,3,3,2,1,1,1,1,1,1] => 82
[10,4,2,1] => [4,3,2,2,1,1,1,1,1,1] => 84
[10,4,1,1,1] => [5,2,2,2,1,1,1,1,1,1] => 85
[10,3,3,1] => [4,3,3,1,1,1,1,1,1,1] => 85
[10,3,2,2] => [4,4,2,1,1,1,1,1,1,1] => 86
[10,3,2,1,1] => [5,3,2,1,1,1,1,1,1,1] => 87
[10,3,1,1,1,1] => [6,2,2,1,1,1,1,1,1,1] => 88
[10,2,2,2,1] => [5,4,1,1,1,1,1,1,1,1] => 88
[10,2,2,1,1,1] => [6,3,1,1,1,1,1,1,1,1] => 89
[10,2,1,1,1,1,1] => [7,2,1,1,1,1,1,1,1,1] => 90
[10,1,1,1,1,1,1,1] => [8,1,1,1,1,1,1,1,1,1] => 91
[9,7,1] => [3,2,2,2,2,2,2,1,1] => 79
[9,6,2] => [3,3,2,2,2,2,1,1,1] => 84
[9,6,1,1] => [4,2,2,2,2,2,1,1,1] => 85
[9,5,3] => [3,3,3,2,2,1,1,1,1] => 87
[9,5,2,1] => [4,3,2,2,2,1,1,1,1] => 89
[9,5,1,1,1] => [5,2,2,2,2,1,1,1,1] => 90
[9,4,4] => [3,3,3,3,1,1,1,1,1] => 88
[9,4,3,1] => [4,3,3,2,1,1,1,1,1] => 91
[9,4,2,2] => [4,4,2,2,1,1,1,1,1] => 92
[9,4,2,1,1] => [5,3,2,2,1,1,1,1,1] => 93
[9,4,1,1,1,1] => [6,2,2,2,1,1,1,1,1] => 94
[9,3,3,2] => [4,4,3,1,1,1,1,1,1] => 93
[9,3,3,1,1] => [5,3,3,1,1,1,1,1,1] => 94
[9,3,2,2,1] => [5,4,2,1,1,1,1,1,1] => 95
[9,3,2,1,1,1] => [6,3,2,1,1,1,1,1,1] => 96
[9,3,1,1,1,1,1] => [7,2,2,1,1,1,1,1,1] => 97
[9,2,2,2,2] => [5,5,1,1,1,1,1,1,1] => 96
[9,2,2,2,1,1] => [6,4,1,1,1,1,1,1,1] => 97
[9,2,2,1,1,1,1] => [7,3,1,1,1,1,1,1,1] => 98
[9,2,1,1,1,1,1,1] => [8,2,1,1,1,1,1,1,1] => 99
[9,1,1,1,1,1,1,1,1] => [9,1,1,1,1,1,1,1,1] => 100
[8,8,1] => [3,2,2,2,2,2,2,2] => 80
[8,7,2] => [3,3,2,2,2,2,2,1] => 86
[8,7,1,1] => [4,2,2,2,2,2,2,1] => 87
[8,6,3] => [3,3,3,2,2,2,1,1] => 90
[8,6,2,1] => [4,3,2,2,2,2,1,1] => 92
[8,6,1,1,1] => [5,2,2,2,2,2,1,1] => 93
[8,5,4] => [3,3,3,3,2,1,1,1] => 92
[8,5,3,1] => [4,3,3,2,2,1,1,1] => 95
[8,5,2,2] => [4,4,2,2,2,1,1,1] => 96
[8,5,2,1,1] => [5,3,2,2,2,1,1,1] => 97
[8,5,1,1,1,1] => [6,2,2,2,2,1,1,1] => 98
[8,4,4,1] => [4,3,3,3,1,1,1,1] => 96
[8,4,3,2] => [4,4,3,2,1,1,1,1] => 98
[8,4,3,1,1] => [5,3,3,2,1,1,1,1] => 99
[8,4,2,2,1] => [5,4,2,2,1,1,1,1] => 100
[8,4,2,1,1,1] => [6,3,2,2,1,1,1,1] => 101
[8,4,1,1,1,1,1] => [7,2,2,2,1,1,1,1] => 102
[8,3,3,3] => [4,4,4,1,1,1,1,1] => 99
[8,3,3,2,1] => [5,4,3,1,1,1,1,1] => 101
[8,3,3,1,1,1] => [6,3,3,1,1,1,1,1] => 102
[8,3,2,2,2] => [5,5,2,1,1,1,1,1] => 102
[8,3,2,2,1,1] => [6,4,2,1,1,1,1,1] => 103
[8,3,2,1,1,1,1] => [7,3,2,1,1,1,1,1] => 104
[8,3,1,1,1,1,1,1] => [8,2,2,1,1,1,1,1] => 105
[8,2,2,2,2,1] => [6,5,1,1,1,1,1,1] => 104
[8,2,2,2,1,1,1] => [7,4,1,1,1,1,1,1] => 105
[8,2,2,1,1,1,1,1] => [8,3,1,1,1,1,1,1] => 106
[8,2,1,1,1,1,1,1,1] => [9,2,1,1,1,1,1,1] => 107
[8,1,1,1,1,1,1,1,1,1] => [10,1,1,1,1,1,1,1] => 108
[7,7,3] => [3,3,3,2,2,2,2] => 91
[7,7,2,1] => [4,3,2,2,2,2,2] => 93
[7,7,1,1,1] => [5,2,2,2,2,2,2] => 94
[7,6,4] => [3,3,3,3,2,2,1] => 94
[7,6,3,1] => [4,3,3,2,2,2,1] => 97
[7,6,2,2] => [4,4,2,2,2,2,1] => 98
[7,6,2,1,1] => [5,3,2,2,2,2,1] => 99
[7,6,1,1,1,1] => [6,2,2,2,2,2,1] => 100
[7,5,5] => [3,3,3,3,3,1,1] => 95
[7,5,4,1] => [4,3,3,3,2,1,1] => 99
[7,5,3,2] => [4,4,3,2,2,1,1] => 101
[7,5,3,1,1] => [5,3,3,2,2,1,1] => 102
[7,5,2,2,1] => [5,4,2,2,2,1,1] => 103
[7,5,2,1,1,1] => [6,3,2,2,2,1,1] => 104
[7,5,1,1,1,1,1] => [7,2,2,2,2,1,1] => 105
[7,4,4,2] => [4,4,3,3,1,1,1] => 102
[7,4,4,1,1] => [5,3,3,3,1,1,1] => 103
[7,4,3,3] => [4,4,4,2,1,1,1] => 103
[7,4,3,2,1] => [5,4,3,2,1,1,1] => 105
[7,4,3,1,1,1] => [6,3,3,2,1,1,1] => 106
[7,4,2,2,2] => [5,5,2,2,1,1,1] => 106
[7,4,2,2,1,1] => [6,4,2,2,1,1,1] => 107
[7,4,2,1,1,1,1] => [7,3,2,2,1,1,1] => 108
[7,4,1,1,1,1,1,1] => [8,2,2,2,1,1,1] => 109
[7,3,3,3,1] => [5,4,4,1,1,1,1] => 106
[7,3,3,2,2] => [5,5,3,1,1,1,1] => 107
[7,3,3,2,1,1] => [6,4,3,1,1,1,1] => 108
[7,3,3,1,1,1,1] => [7,3,3,1,1,1,1] => 109
[7,3,2,2,2,1] => [6,5,2,1,1,1,1] => 109
[7,3,2,2,1,1,1] => [7,4,2,1,1,1,1] => 110
[7,3,2,1,1,1,1,1] => [8,3,2,1,1,1,1] => 111
[7,3,1,1,1,1,1,1,1] => [9,2,2,1,1,1,1] => 112
[7,2,2,2,2,2] => [6,6,1,1,1,1,1] => 110
[7,2,2,2,2,1,1] => [7,5,1,1,1,1,1] => 111
[7,2,2,2,1,1,1,1] => [8,4,1,1,1,1,1] => 112
[7,2,2,1,1,1,1,1,1] => [9,3,1,1,1,1,1] => 113
[7,2,1,1,1,1,1,1,1,1] => [10,2,1,1,1,1,1] => 114
[7,1,1,1,1,1,1,1,1,1,1] => [11,1,1,1,1,1,1] => 115
[6,6,5] => [3,3,3,3,3,2] => 96
[6,6,4,1] => [4,3,3,3,2,2] => 100
[6,6,3,2] => [4,4,3,2,2,2] => 102
[6,6,3,1,1] => [5,3,3,2,2,2] => 103
[6,6,2,2,1] => [5,4,2,2,2,2] => 104
[6,6,2,1,1,1] => [6,3,2,2,2,2] => 105
[6,6,1,1,1,1,1] => [7,2,2,2,2,2] => 106
[6,5,5,1] => [4,3,3,3,3,1] => 101
[6,5,4,2] => [4,4,3,3,2,1] => 104
[6,5,4,1,1] => [5,3,3,3,2,1] => 105
[6,5,3,3] => [4,4,4,2,2,1] => 105
[6,5,3,2,1] => [5,4,3,2,2,1] => 107
[6,5,3,1,1,1] => [6,3,3,2,2,1] => 108
[6,5,2,2,2] => [5,5,2,2,2,1] => 108
[6,5,2,2,1,1] => [6,4,2,2,2,1] => 109
[6,5,2,1,1,1,1] => [7,3,2,2,2,1] => 110
[6,5,1,1,1,1,1,1] => [8,2,2,2,2,1] => 111
[6,4,4,3] => [4,4,4,3,1,1] => 106
[6,4,4,2,1] => [5,4,3,3,1,1] => 108
[6,4,4,1,1,1] => [6,3,3,3,1,1] => 109
[6,4,3,3,1] => [5,4,4,2,1,1] => 109
[6,4,3,2,2] => [5,5,3,2,1,1] => 110
[6,4,3,2,1,1] => [6,4,3,2,1,1] => 111
[6,4,3,1,1,1,1] => [7,3,3,2,1,1] => 112
[6,4,2,2,2,1] => [6,5,2,2,1,1] => 112
[6,4,2,2,1,1,1] => [7,4,2,2,1,1] => 113
[6,4,2,1,1,1,1,1] => [8,3,2,2,1,1] => 114
[6,4,1,1,1,1,1,1,1] => [9,2,2,2,1,1] => 115
[6,3,3,3,2] => [5,5,4,1,1,1] => 111
[6,3,3,3,1,1] => [6,4,4,1,1,1] => 112
[6,3,3,2,2,1] => [6,5,3,1,1,1] => 113
[6,3,3,2,1,1,1] => [7,4,3,1,1,1] => 114
[6,3,3,1,1,1,1,1] => [8,3,3,1,1,1] => 115
[6,3,2,2,2,2] => [6,6,2,1,1,1] => 114
[6,3,2,2,2,1,1] => [7,5,2,1,1,1] => 115
[6,3,2,2,1,1,1,1] => [8,4,2,1,1,1] => 116
[6,3,2,1,1,1,1,1,1] => [9,3,2,1,1,1] => 117
[6,3,1,1,1,1,1,1,1,1] => [10,2,2,1,1,1] => 118
[6,2,2,2,2,2,1] => [7,6,1,1,1,1] => 116
[6,2,2,2,2,1,1,1] => [8,5,1,1,1,1] => 117
[6,2,2,2,1,1,1,1,1] => [9,4,1,1,1,1] => 118
[6,2,2,1,1,1,1,1,1,1] => [10,3,1,1,1,1] => 119
[6,2,1,1,1,1,1,1,1,1,1] => [11,2,1,1,1,1] => 120
[6,1,1,1,1,1,1,1,1,1,1,1] => [12,1,1,1,1,1] => 121
[5,5,5,2] => [4,4,3,3,3] => 105
[5,5,5,1,1] => [5,3,3,3,3] => 106
[5,5,4,3] => [4,4,4,3,2] => 107
[5,5,4,2,1] => [5,4,3,3,2] => 109
[5,5,4,1,1,1] => [6,3,3,3,2] => 110
[5,5,3,3,1] => [5,4,4,2,2] => 110
[5,5,3,2,2] => [5,5,3,2,2] => 111
[5,5,3,2,1,1] => [6,4,3,2,2] => 112
[5,5,3,1,1,1,1] => [7,3,3,2,2] => 113
[5,5,2,2,2,1] => [6,5,2,2,2] => 113
[5,5,2,2,1,1,1] => [7,4,2,2,2] => 114
[5,5,2,1,1,1,1,1] => [8,3,2,2,2] => 115
[5,5,1,1,1,1,1,1,1] => [9,2,2,2,2] => 116
[5,4,4,4] => [4,4,4,4,1] => 108
[5,4,4,3,1] => [5,4,4,3,1] => 111
[5,4,4,2,2] => [5,5,3,3,1] => 112
[5,4,4,2,1,1] => [6,4,3,3,1] => 113
[5,4,4,1,1,1,1] => [7,3,3,3,1] => 114
[5,4,3,3,2] => [5,5,4,2,1] => 113
[5,4,3,3,1,1] => [6,4,4,2,1] => 114
[5,4,3,2,2,1] => [6,5,3,2,1] => 115
[5,4,3,2,1,1,1] => [7,4,3,2,1] => 116
[5,4,3,1,1,1,1,1] => [8,3,3,2,1] => 117
[5,4,2,2,2,2] => [6,6,2,2,1] => 116
[5,4,2,2,2,1,1] => [7,5,2,2,1] => 117
[5,4,2,2,1,1,1,1] => [8,4,2,2,1] => 118
[5,4,2,1,1,1,1,1,1] => [9,3,2,2,1] => 119
[5,4,1,1,1,1,1,1,1,1] => [10,2,2,2,1] => 120
[5,3,3,3,3] => [5,5,5,1,1] => 114
[5,3,3,3,2,1] => [6,5,4,1,1] => 116
[5,3,3,3,1,1,1] => [7,4,4,1,1] => 117
[5,3,3,2,2,2] => [6,6,3,1,1] => 117
[5,3,3,2,2,1,1] => [7,5,3,1,1] => 118
[5,3,3,2,1,1,1,1] => [8,4,3,1,1] => 119
[5,3,3,1,1,1,1,1,1] => [9,3,3,1,1] => 120
[5,3,2,2,2,2,1] => [7,6,2,1,1] => 119
[5,3,2,2,2,1,1,1] => [8,5,2,1,1] => 120
[5,3,2,2,1,1,1,1,1] => [9,4,2,1,1] => 121
[5,3,2,1,1,1,1,1,1,1] => [10,3,2,1,1] => 122
[5,3,1,1,1,1,1,1,1,1,1] => [11,2,2,1,1] => 123
[5,2,2,2,2,2,2] => [7,7,1,1,1] => 120
[5,2,2,2,2,2,1,1] => [8,6,1,1,1] => 121
[5,2,2,2,2,1,1,1,1] => [9,5,1,1,1] => 122
[5,2,2,2,1,1,1,1,1,1] => [10,4,1,1,1] => 123
[5,2,2,1,1,1,1,1,1,1,1] => [11,3,1,1,1] => 124
[5,2,1,1,1,1,1,1,1,1,1,1] => [12,2,1,1,1] => 125
[5,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1,1,1] => 126
[4,4,4,4,1] => [5,4,4,4] => 112
[4,4,4,3,2] => [5,5,4,3] => 114
[4,4,4,3,1,1] => [6,4,4,3] => 115
[4,4,4,2,2,1] => [6,5,3,3] => 116
[4,4,4,2,1,1,1] => [7,4,3,3] => 117
[4,4,4,1,1,1,1,1] => [8,3,3,3] => 118
[4,4,3,3,3] => [5,5,5,2] => 115
[4,4,3,3,2,1] => [6,5,4,2] => 117
[4,4,3,3,1,1,1] => [7,4,4,2] => 118
[4,4,3,2,2,2] => [6,6,3,2] => 118
[4,4,3,2,2,1,1] => [7,5,3,2] => 119
[4,4,3,2,1,1,1,1] => [8,4,3,2] => 120
[4,4,3,1,1,1,1,1,1] => [9,3,3,2] => 121
[4,4,2,2,2,2,1] => [7,6,2,2] => 120
[4,4,2,2,2,1,1,1] => [8,5,2,2] => 121
[4,4,2,2,1,1,1,1,1] => [9,4,2,2] => 122
[4,4,2,1,1,1,1,1,1,1] => [10,3,2,2] => 123
[4,4,1,1,1,1,1,1,1,1,1] => [11,2,2,2] => 124
[4,3,3,3,3,1] => [6,5,5,1] => 118
[4,3,3,3,2,2] => [6,6,4,1] => 119
[4,3,3,3,2,1,1] => [7,5,4,1] => 120
[4,3,3,3,1,1,1,1] => [8,4,4,1] => 121
[4,3,3,2,2,2,1] => [7,6,3,1] => 121
[4,3,3,2,2,1,1,1] => [8,5,3,1] => 122
[4,3,3,2,1,1,1,1,1] => [9,4,3,1] => 123
[4,3,3,1,1,1,1,1,1,1] => [10,3,3,1] => 124
[4,3,2,2,2,2,2] => [7,7,2,1] => 122
[4,3,2,2,2,2,1,1] => [8,6,2,1] => 123
[4,3,2,2,2,1,1,1,1] => [9,5,2,1] => 124
[4,3,2,2,1,1,1,1,1,1] => [10,4,2,1] => 125
[4,3,2,1,1,1,1,1,1,1,1] => [11,3,2,1] => 126
[4,3,1,1,1,1,1,1,1,1,1,1] => [12,2,2,1] => 127
[4,2,2,2,2,2,2,1] => [8,7,1,1] => 124
[4,2,2,2,2,2,1,1,1] => [9,6,1,1] => 125
[4,2,2,2,2,1,1,1,1,1] => [10,5,1,1] => 126
[4,2,2,2,1,1,1,1,1,1,1] => [11,4,1,1] => 127
[4,2,2,1,1,1,1,1,1,1,1,1] => [12,3,1,1] => 128
[4,2,1,1,1,1,1,1,1,1,1,1,1] => [13,2,1,1] => 129
[4,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1,1,1] => 130
[3,3,3,3,3,2] => [6,6,5] => 120
[3,3,3,3,3,1,1] => [7,5,5] => 121
[3,3,3,3,2,2,1] => [7,6,4] => 122
[3,3,3,3,2,1,1,1] => [8,5,4] => 123
[3,3,3,3,1,1,1,1,1] => [9,4,4] => 124
[3,3,3,2,2,2,2] => [7,7,3] => 123
[3,3,3,2,2,2,1,1] => [8,6,3] => 124
[3,3,3,2,2,1,1,1,1] => [9,5,3] => 125
[3,3,3,2,1,1,1,1,1,1] => [10,4,3] => 126
[3,3,3,1,1,1,1,1,1,1,1] => [11,3,3] => 127
[3,3,2,2,2,2,2,1] => [8,7,2] => 125
[3,3,2,2,2,2,1,1,1] => [9,6,2] => 126
[3,3,2,2,2,1,1,1,1,1] => [10,5,2] => 127
[3,3,2,2,1,1,1,1,1,1,1] => [11,4,2] => 128
[3,3,2,1,1,1,1,1,1,1,1,1] => [12,3,2] => 129
[3,3,1,1,1,1,1,1,1,1,1,1,1] => [13,2,2] => 130
[3,2,2,2,2,2,2,2] => [8,8,1] => 126
[3,2,2,2,2,2,2,1,1] => [9,7,1] => 127
[3,2,2,2,2,2,1,1,1,1] => [10,6,1] => 128
[3,2,2,2,2,1,1,1,1,1,1] => [11,5,1] => 129
[3,2,2,2,1,1,1,1,1,1,1,1] => [12,4,1] => 130
[3,2,2,1,1,1,1,1,1,1,1,1,1] => [13,3,1] => 131
[3,2,1,1,1,1,1,1,1,1,1,1,1,1] => [14,2,1] => 132
[3,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,1,1] => 133
[2,2,2,2,2,2,2,2,1] => [9,8] => 128
[2,2,2,2,2,2,2,1,1,1] => [10,7] => 129
[2,2,2,2,2,2,1,1,1,1,1] => [11,6] => 130
[2,2,2,2,2,1,1,1,1,1,1,1] => [12,5] => 131
[2,2,2,2,1,1,1,1,1,1,1,1,1] => [13,4] => 132
[2,2,2,1,1,1,1,1,1,1,1,1,1,1] => [14,3] => 133
[2,2,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,2] => 134
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [16,1] => 135
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [17] => 136
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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 1,0,1,1 1,0,0,1,1,1,1 1,0,0,0,1,0,1,1,1,1,1 1,0,0,0,0,1,0,0,1,2,0,1,2,1,1,1 1,0,0,0,0,0,1,0,0,0,1,1,1,0,1,2,1,1,2,1,1,1 1,0,0,0,0,0,0,1,0,0,0,0,1,1,0,1,1,1,1,1,1,2,2,1,2,2,1,1,1
$F_{1} = 1$
$F_{2} = 1 + q$
$F_{3} = 1 + q^{2} + q^{3}$
$F_{4} = 1 + q^{3} + q^{4} + q^{5} + q^{6}$
$F_{5} = 1 + q^{4} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10}$
$F_{6} = 1 + q^{5} + q^{8} + 2\ q^{9} + q^{11} + 2\ q^{12} + q^{13} + q^{14} + q^{15}$
$F_{7} = 1 + q^{6} + q^{10} + q^{11} + q^{12} + q^{14} + 2\ q^{15} + q^{16} + q^{17} + 2\ q^{18} + q^{19} + q^{20} + q^{21}$
$F_{8} = 1 + q^{7} + q^{12} + q^{13} + q^{15} + q^{16} + q^{17} + q^{18} + q^{19} + q^{20} + 2\ q^{21} + 2\ q^{22} + q^{23} + 2\ q^{24} + 2\ q^{25} + q^{26} + q^{27} + q^{28}$
$F_{9} = 1 + q^{8} + q^{14} + q^{15} + q^{18} + 2\ q^{20} + q^{21} + q^{23} + 2\ q^{24} + q^{25} + 2\ q^{26} + 2\ q^{27} + q^{28} + 2\ q^{29} + 3\ q^{30} + q^{31} + 2\ q^{32} + 2\ q^{33} + q^{34} + q^{35} + q^{36}$
$F_{10} = 1 + q^{9} + q^{16} + q^{17} + q^{21} + q^{23} + 2\ q^{24} + q^{25} + q^{27} + q^{28} + 2\ q^{29} + q^{30} + q^{31} + 2\ q^{32} + 3\ q^{33} + q^{34} + 2\ q^{35} + 3\ q^{36} + 2\ q^{37} + 2\ q^{38} + 3\ q^{39} + 2\ q^{40} + 2\ q^{41} + 2\ q^{42} + q^{43} + q^{44} + q^{45}$
$F_{11} = 1 + q^{10} + q^{18} + q^{19} + q^{24} + q^{26} + q^{27} + q^{28} + q^{30} + q^{31} + q^{32} + q^{33} + 2\ q^{34} + q^{35} + q^{36} + q^{37} + 2\ q^{38} + 3\ q^{39} + 2\ q^{40} + q^{41} + 3\ q^{42} + 3\ q^{43} + 2\ q^{44} + 3\ q^{45} + 3\ q^{46} + 2\ q^{47} + 3\ q^{48} + 3\ q^{49} + 2\ q^{50} + 2\ q^{51} + 2\ q^{52} + q^{53} + q^{54} + q^{55}$
$F_{12} = 1 + q^{11} + q^{20} + q^{21} + q^{27} + q^{29} + q^{30} + q^{32} + 2\ q^{35} + 2\ q^{36} + q^{37} + q^{38} + q^{39} + 2\ q^{41} + q^{42} + q^{43} + 2\ q^{44} + 4\ q^{45} + q^{46} + 2\ q^{47} + 3\ q^{48} + 2\ q^{49} + 3\ q^{50} + 3\ q^{51} + 2\ q^{52} + 4\ q^{53} + 4\ q^{54} + 2\ q^{55} + 4\ q^{56} + 4\ q^{57} + 2\ q^{58} + 3\ q^{59} + 4\ q^{60} + 2\ q^{61} + 2\ q^{62} + 2\ q^{63} + q^{64} + q^{65} + q^{66}$
$F_{13} = 1 + q^{12} + q^{22} + q^{23} + q^{30} + q^{32} + q^{33} + q^{36} + q^{39} + 2\ q^{40} + q^{41} + 2\ q^{42} + q^{44} + q^{46} + 2\ q^{47} + 2\ q^{48} + q^{49} + 2\ q^{50} + 2\ q^{51} + q^{52} + 2\ q^{53} + 3\ q^{54} + 2\ q^{55} + 3\ q^{56} + 4\ q^{57} + 2\ q^{58} + 2\ q^{59} + 4\ q^{60} + 3\ q^{61} + 4\ q^{62} + 4\ q^{63} + 3\ q^{64} + 4\ q^{65} + 5\ q^{66} + 3\ q^{67} + 4\ q^{68} + 4\ q^{69} + 3\ q^{70} + 3\ q^{71} + 4\ q^{72} + 2\ q^{73} + 2\ q^{74} + 2\ q^{75} + q^{76} + q^{77} + q^{78}$
$F_{14} = 1 + q^{13} + q^{24} + q^{25} + q^{33} + q^{35} + q^{36} + q^{40} + q^{43} + q^{44} + 2\ q^{45} + q^{46} + q^{48} + 2\ q^{49} + q^{51} + q^{52} + 2\ q^{53} + q^{54} + 2\ q^{55} + q^{56} + 2\ q^{57} + 2\ q^{59} + 4\ q^{60} + 3\ q^{61} + q^{62} + 3\ q^{63} + 3\ q^{64} + 3\ q^{65} + 2\ q^{66} + 3\ q^{67} + 3\ q^{68} + 5\ q^{69} + 4\ q^{70} + 3\ q^{71} + 5\ q^{72} + 5\ q^{73} + 3\ q^{74} + 5\ q^{75} + 5\ q^{76} + 4\ q^{77} + 5\ q^{78} + 5\ q^{79} + 4\ q^{80} + 5\ q^{81} + 4\ q^{82} + 3\ q^{83} + 4\ q^{84} + 4\ q^{85} + 2\ q^{86} + 2\ q^{87} + 2\ q^{88} + q^{89} + q^{90} + q^{91}$
$F_{15} = 1 + q^{14} + q^{26} + q^{27} + q^{36} + q^{38} + q^{39} + q^{44} + q^{47} + q^{48} + q^{49} + 2\ q^{50} + 2\ q^{54} + 2\ q^{56} + q^{57} + q^{58} + 2\ q^{59} + q^{60} + 2\ q^{62} + 3\ q^{63} + q^{65} + 3\ q^{66} + 2\ q^{67} + 3\ q^{68} + 2\ q^{69} + q^{70} + 3\ q^{71} + 3\ q^{72} + 2\ q^{73} + 5\ q^{74} + 4\ q^{75} + 2\ q^{76} + 4\ q^{77} + 4\ q^{78} + 3\ q^{79} + 4\ q^{80} + 5\ q^{81} + 4\ q^{82} + 6\ q^{83} + 6\ q^{84} + 3\ q^{85} + 6\ q^{86} + 6\ q^{87} + 4\ q^{88} + 6\ q^{89} + 7\ q^{90} + 4\ q^{91} + 6\ q^{92} + 6\ q^{93} + 4\ q^{94} + 5\ q^{95} + 5\ q^{96} + 3\ q^{97} + 4\ q^{98} + 4\ q^{99} + 2\ q^{100} + 2\ q^{101} + 2\ q^{102} + q^{103} + q^{104} + q^{105}$
$F_{16} = 1 + q^{15} + q^{28} + q^{29} + q^{39} + q^{41} + q^{42} + q^{48} + q^{51} + q^{52} + q^{53} + q^{54} + q^{55} + q^{59} + q^{60} + q^{61} + q^{62} + 2\ q^{63} + 2\ q^{64} + 2\ q^{65} + q^{68} + 3\ q^{69} + 2\ q^{71} + 2\ q^{72} + 2\ q^{73} + 2\ q^{74} + 2\ q^{75} + q^{76} + 3\ q^{77} + 3\ q^{78} + 2\ q^{79} + 2\ q^{80} + 4\ q^{81} + 2\ q^{82} + 4\ q^{83} + 4\ q^{84} + 3\ q^{85} + 3\ q^{86} + 4\ q^{87} + 4\ q^{88} + 6\ q^{89} + 5\ q^{90} + 3\ q^{91} + 5\ q^{92} + 7\ q^{93} + 4\ q^{94} + 5\ q^{95} + 7\ q^{96} + 6\ q^{97} + 6\ q^{98} + 7\ q^{99} + 6\ q^{100} + 7\ q^{101} + 7\ q^{102} + 5\ q^{103} + 7\ q^{104} + 8\ q^{105} + 5\ q^{106} + 6\ q^{107} + 7\ q^{108} + 5\ q^{109} + 5\ q^{110} + 5\ q^{111} + 4\ q^{112} + 4\ q^{113} + 4\ q^{114} + 2\ q^{115} + 2\ q^{116} + 2\ q^{117} + q^{118} + q^{119} + q^{120}$
Description
The Gini index of an integer partition.
As discussed in [1], this statistic is equal to St000567The sum of the products of all pairs of parts. applied to the conjugate partition.
As discussed in [1], this statistic is equal to St000567The sum of the products of all pairs of parts. applied to the conjugate partition.
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
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