Identifier
-
Mp00202:
Integer partitions
—first row removal⟶
Integer partitions
St000620: Integer partitions ⟶ ℤ
Values
[1,1,1] => [1,1] => 1
[2,2] => [2] => 0
[2,1,1] => [1,1] => 1
[1,1,1,1] => [1,1,1] => 1
[3,2] => [2] => 0
[3,1,1] => [1,1] => 1
[2,2,1] => [2,1] => 1
[2,1,1,1] => [1,1,1] => 1
[1,1,1,1,1] => [1,1,1,1] => 1
[4,2] => [2] => 0
[4,1,1] => [1,1] => 1
[3,3] => [3] => 1
[3,2,1] => [2,1] => 1
[3,1,1,1] => [1,1,1] => 1
[2,2,2] => [2,2] => 1
[2,2,1,1] => [2,1,1] => 2
[2,1,1,1,1] => [1,1,1,1] => 1
[1,1,1,1,1,1] => [1,1,1,1,1] => 1
[5,2] => [2] => 0
[5,1,1] => [1,1] => 1
[4,3] => [3] => 1
[4,2,1] => [2,1] => 1
[4,1,1,1] => [1,1,1] => 1
[3,3,1] => [3,1] => 2
[3,2,2] => [2,2] => 1
[3,2,1,1] => [2,1,1] => 2
[3,1,1,1,1] => [1,1,1,1] => 1
[2,2,2,1] => [2,2,1] => 3
[2,2,1,1,1] => [2,1,1,1] => 3
[2,1,1,1,1,1] => [1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[6,2] => [2] => 0
[6,1,1] => [1,1] => 1
[5,3] => [3] => 1
[5,2,1] => [2,1] => 1
[5,1,1,1] => [1,1,1] => 1
[4,4] => [4] => 0
[4,3,1] => [3,1] => 2
[4,2,2] => [2,2] => 1
[4,2,1,1] => [2,1,1] => 2
[4,1,1,1,1] => [1,1,1,1] => 1
[3,3,2] => [3,2] => 3
[3,3,1,1] => [3,1,1] => 4
[3,2,2,1] => [2,2,1] => 3
[3,2,1,1,1] => [2,1,1,1] => 3
[3,1,1,1,1,1] => [1,1,1,1,1] => 1
[2,2,2,2] => [2,2,2] => 3
[2,2,2,1,1] => [2,2,1,1] => 6
[2,2,1,1,1,1] => [2,1,1,1,1] => 4
[2,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[7,2] => [2] => 0
[7,1,1] => [1,1] => 1
[6,3] => [3] => 1
[6,2,1] => [2,1] => 1
[6,1,1,1] => [1,1,1] => 1
[5,4] => [4] => 0
[5,3,1] => [3,1] => 2
[5,2,2] => [2,2] => 1
[5,2,1,1] => [2,1,1] => 2
[5,1,1,1,1] => [1,1,1,1] => 1
[4,4,1] => [4,1] => 2
[4,3,2] => [3,2] => 3
[4,3,1,1] => [3,1,1] => 4
[4,2,2,1] => [2,2,1] => 3
[4,2,1,1,1] => [2,1,1,1] => 3
[4,1,1,1,1,1] => [1,1,1,1,1] => 1
[3,3,3] => [3,3] => 3
[3,3,2,1] => [3,2,1] => 10
[3,3,1,1,1] => [3,1,1,1] => 7
[3,2,2,2] => [2,2,2] => 3
[3,2,2,1,1] => [2,2,1,1] => 6
[3,2,1,1,1,1] => [2,1,1,1,1] => 4
[3,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[2,2,2,2,1] => [2,2,2,1] => 9
[2,2,2,1,1,1] => [2,2,1,1,1] => 10
[2,2,1,1,1,1,1] => [2,1,1,1,1,1] => 5
[2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 1
[8,2] => [2] => 0
[8,1,1] => [1,1] => 1
[7,3] => [3] => 1
[7,2,1] => [2,1] => 1
[7,1,1,1] => [1,1,1] => 1
[6,4] => [4] => 0
[6,3,1] => [3,1] => 2
[6,2,2] => [2,2] => 1
[6,2,1,1] => [2,1,1] => 2
[6,1,1,1,1] => [1,1,1,1] => 1
[5,5] => [5] => 1
[5,4,1] => [4,1] => 2
[5,3,2] => [3,2] => 3
[5,3,1,1] => [3,1,1] => 4
[5,2,2,1] => [2,2,1] => 3
[5,2,1,1,1] => [2,1,1,1] => 3
[5,1,1,1,1,1] => [1,1,1,1,1] => 1
[4,4,2] => [4,2] => 5
[4,4,1,1] => [4,1,1] => 6
[4,3,3] => [3,3] => 3
[4,3,2,1] => [3,2,1] => 10
[4,3,1,1,1] => [3,1,1,1] => 7
>>> Load all 1115 entries. <<<[4,2,2,2] => [2,2,2] => 3
[4,2,2,1,1] => [2,2,1,1] => 6
[4,2,1,1,1,1] => [2,1,1,1,1] => 4
[4,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[3,3,3,1] => [3,3,1] => 13
[3,3,2,2] => [3,2,2] => 13
[3,3,2,1,1] => [3,2,1,1] => 23
[3,3,1,1,1,1] => [3,1,1,1,1] => 11
[3,2,2,2,1] => [2,2,2,1] => 9
[3,2,2,1,1,1] => [2,2,1,1,1] => 10
[3,2,1,1,1,1,1] => [2,1,1,1,1,1] => 5
[3,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[2,2,2,2,2] => [2,2,2,2] => 9
[2,2,2,2,1,1] => [2,2,2,1,1] => 19
[2,2,2,1,1,1,1] => [2,2,1,1,1,1] => 15
[2,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 6
[2,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[9,2] => [2] => 0
[9,1,1] => [1,1] => 1
[8,3] => [3] => 1
[8,2,1] => [2,1] => 1
[8,1,1,1] => [1,1,1] => 1
[7,4] => [4] => 0
[7,3,1] => [3,1] => 2
[7,2,2] => [2,2] => 1
[7,2,1,1] => [2,1,1] => 2
[7,1,1,1,1] => [1,1,1,1] => 1
[6,5] => [5] => 1
[6,4,1] => [4,1] => 2
[6,3,2] => [3,2] => 3
[6,3,1,1] => [3,1,1] => 4
[6,2,2,1] => [2,2,1] => 3
[6,2,1,1,1] => [2,1,1,1] => 3
[6,1,1,1,1,1] => [1,1,1,1,1] => 1
[5,5,1] => [5,1] => 3
[5,4,2] => [4,2] => 5
[5,4,1,1] => [4,1,1] => 6
[5,3,3] => [3,3] => 3
[5,3,2,1] => [3,2,1] => 10
[5,3,1,1,1] => [3,1,1,1] => 7
[5,2,2,2] => [2,2,2] => 3
[5,2,2,1,1] => [2,2,1,1] => 6
[5,2,1,1,1,1] => [2,1,1,1,1] => 4
[5,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[4,4,3] => [4,3] => 8
[4,4,2,1] => [4,2,1] => 21
[4,4,1,1,1] => [4,1,1,1] => 13
[4,3,3,1] => [3,3,1] => 13
[4,3,2,2] => [3,2,2] => 13
[4,3,2,1,1] => [3,2,1,1] => 23
[4,3,1,1,1,1] => [3,1,1,1,1] => 11
[4,2,2,2,1] => [2,2,2,1] => 9
[4,2,2,1,1,1] => [2,2,1,1,1] => 10
[4,2,1,1,1,1,1] => [2,1,1,1,1,1] => 5
[4,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[3,3,3,2] => [3,3,2] => 26
[3,3,3,1,1] => [3,3,1,1] => 36
[3,3,2,2,1] => [3,2,2,1] => 45
[3,3,2,1,1,1] => [3,2,1,1,1] => 44
[3,3,1,1,1,1,1] => [3,1,1,1,1,1] => 16
[3,2,2,2,2] => [2,2,2,2] => 9
[3,2,2,2,1,1] => [2,2,2,1,1] => 19
[3,2,2,1,1,1,1] => [2,2,1,1,1,1] => 15
[3,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 6
[3,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,1] => [2,2,2,2,1] => 28
[2,2,2,2,1,1,1] => [2,2,2,1,1,1] => 34
[2,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 21
[2,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 7
[2,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 1
[10,2] => [2] => 0
[10,1,1] => [1,1] => 1
[9,3] => [3] => 1
[9,2,1] => [2,1] => 1
[9,1,1,1] => [1,1,1] => 1
[8,4] => [4] => 0
[8,3,1] => [3,1] => 2
[8,2,2] => [2,2] => 1
[8,2,1,1] => [2,1,1] => 2
[8,1,1,1,1] => [1,1,1,1] => 1
[7,5] => [5] => 1
[7,4,1] => [4,1] => 2
[7,3,2] => [3,2] => 3
[7,3,1,1] => [3,1,1] => 4
[7,2,2,1] => [2,2,1] => 3
[7,2,1,1,1] => [2,1,1,1] => 3
[7,1,1,1,1,1] => [1,1,1,1,1] => 1
[6,6] => [6] => 0
[6,5,1] => [5,1] => 3
[6,4,2] => [4,2] => 5
[6,4,1,1] => [4,1,1] => 6
[6,3,3] => [3,3] => 3
[6,3,2,1] => [3,2,1] => 10
[6,3,1,1,1] => [3,1,1,1] => 7
[6,2,2,2] => [2,2,2] => 3
[6,2,2,1,1] => [2,2,1,1] => 6
[6,2,1,1,1,1] => [2,1,1,1,1] => 4
[6,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[5,5,2] => [5,2] => 8
[5,5,1,1] => [5,1,1] => 9
[5,4,3] => [4,3] => 8
[5,4,2,1] => [4,2,1] => 21
[5,4,1,1,1] => [4,1,1,1] => 13
[5,3,3,1] => [3,3,1] => 13
[5,3,2,2] => [3,2,2] => 13
[5,3,2,1,1] => [3,2,1,1] => 23
[5,3,1,1,1,1] => [3,1,1,1,1] => 11
[5,2,2,2,1] => [2,2,2,1] => 9
[5,2,2,1,1,1] => [2,2,1,1,1] => 10
[5,2,1,1,1,1,1] => [2,1,1,1,1,1] => 5
[5,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[4,4,4] => [4,4] => 8
[4,4,3,1] => [4,3,1] => 42
[4,4,2,2] => [4,2,2] => 34
[4,4,2,1,1] => [4,2,1,1] => 57
[4,4,1,1,1,1] => [4,1,1,1,1] => 24
[4,3,3,2] => [3,3,2] => 26
[4,3,3,1,1] => [3,3,1,1] => 36
[4,3,2,2,1] => [3,2,2,1] => 45
[4,3,2,1,1,1] => [3,2,1,1,1] => 44
[4,3,1,1,1,1,1] => [3,1,1,1,1,1] => 16
[4,2,2,2,2] => [2,2,2,2] => 9
[4,2,2,2,1,1] => [2,2,2,1,1] => 19
[4,2,2,1,1,1,1] => [2,2,1,1,1,1] => 15
[4,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 6
[4,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 1
[3,3,3,3] => [3,3,3] => 26
[3,3,3,2,1] => [3,3,2,1] => 107
[3,3,3,1,1,1] => [3,3,1,1,1] => 80
[3,3,2,2,2] => [3,2,2,2] => 54
[3,3,2,2,1,1] => [3,2,2,1,1] => 108
[3,3,2,1,1,1,1] => [3,2,1,1,1,1] => 75
[3,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 22
[3,2,2,2,2,1] => [2,2,2,2,1] => 28
[3,2,2,2,1,1,1] => [2,2,2,1,1,1] => 34
[3,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 21
[3,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 7
[3,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2] => [2,2,2,2,2] => 28
[2,2,2,2,2,1,1] => [2,2,2,2,1,1] => 62
[2,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => 55
[2,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 28
[2,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 8
[2,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[11,2] => [2] => 0
[11,1,1] => [1,1] => 1
[10,3] => [3] => 1
[10,2,1] => [2,1] => 1
[10,1,1,1] => [1,1,1] => 1
[9,4] => [4] => 0
[9,3,1] => [3,1] => 2
[9,2,2] => [2,2] => 1
[9,2,1,1] => [2,1,1] => 2
[9,1,1,1,1] => [1,1,1,1] => 1
[8,5] => [5] => 1
[8,4,1] => [4,1] => 2
[8,3,2] => [3,2] => 3
[8,3,1,1] => [3,1,1] => 4
[8,2,2,1] => [2,2,1] => 3
[8,2,1,1,1] => [2,1,1,1] => 3
[8,1,1,1,1,1] => [1,1,1,1,1] => 1
[7,6] => [6] => 0
[7,5,1] => [5,1] => 3
[7,4,2] => [4,2] => 5
[7,4,1,1] => [4,1,1] => 6
[7,3,3] => [3,3] => 3
[7,3,2,1] => [3,2,1] => 10
[7,3,1,1,1] => [3,1,1,1] => 7
[7,2,2,2] => [2,2,2] => 3
[7,2,2,1,1] => [2,2,1,1] => 6
[7,2,1,1,1,1] => [2,1,1,1,1] => 4
[7,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[6,6,1] => [6,1] => 3
[6,5,2] => [5,2] => 8
[6,5,1,1] => [5,1,1] => 9
[6,4,3] => [4,3] => 8
[6,4,2,1] => [4,2,1] => 21
[6,4,1,1,1] => [4,1,1,1] => 13
[6,3,3,1] => [3,3,1] => 13
[6,3,2,2] => [3,2,2] => 13
[6,3,2,1,1] => [3,2,1,1] => 23
[6,3,1,1,1,1] => [3,1,1,1,1] => 11
[6,2,2,2,1] => [2,2,2,1] => 9
[6,2,2,1,1,1] => [2,2,1,1,1] => 10
[6,2,1,1,1,1,1] => [2,1,1,1,1,1] => 5
[6,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[5,5,3] => [5,3] => 16
[5,5,2,1] => [5,2,1] => 38
[5,5,1,1,1] => [5,1,1,1] => 22
[5,4,4] => [4,4] => 8
[5,4,3,1] => [4,3,1] => 42
[5,4,2,2] => [4,2,2] => 34
[5,4,2,1,1] => [4,2,1,1] => 57
[5,4,1,1,1,1] => [4,1,1,1,1] => 24
[5,3,3,2] => [3,3,2] => 26
[5,3,3,1,1] => [3,3,1,1] => 36
[5,3,2,2,1] => [3,2,2,1] => 45
[5,3,2,1,1,1] => [3,2,1,1,1] => 44
[5,3,1,1,1,1,1] => [3,1,1,1,1,1] => 16
[5,2,2,2,2] => [2,2,2,2] => 9
[5,2,2,2,1,1] => [2,2,2,1,1] => 19
[5,2,2,1,1,1,1] => [2,2,1,1,1,1] => 15
[5,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 6
[5,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 1
[4,4,4,1] => [4,4,1] => 50
[4,4,3,2] => [4,3,2] => 102
[4,4,3,1,1] => [4,3,1,1] => 135
[4,4,2,2,1] => [4,2,2,1] => 136
[4,4,2,1,1,1] => [4,2,1,1,1] => 125
[4,4,1,1,1,1,1] => [4,1,1,1,1,1] => 40
[4,3,3,3] => [3,3,3] => 26
[4,3,3,2,1] => [3,3,2,1] => 107
[4,3,3,1,1,1] => [3,3,1,1,1] => 80
[4,3,2,2,2] => [3,2,2,2] => 54
[4,3,2,2,1,1] => [3,2,2,1,1] => 108
[4,3,2,1,1,1,1] => [3,2,1,1,1,1] => 75
[4,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 22
[4,2,2,2,2,1] => [2,2,2,2,1] => 28
[4,2,2,2,1,1,1] => [2,2,2,1,1,1] => 34
[4,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 21
[4,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 7
[4,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[3,3,3,3,1] => [3,3,3,1] => 133
[3,3,3,2,2] => [3,3,2,2] => 161
[3,3,3,2,1,1] => [3,3,2,1,1] => 295
[3,3,3,1,1,1,1] => [3,3,1,1,1,1] => 155
[3,3,2,2,2,1] => [3,2,2,2,1] => 190
[3,3,2,2,1,1,1] => [3,2,2,1,1,1] => 217
[3,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => 118
[3,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 29
[3,2,2,2,2,2] => [2,2,2,2,2] => 28
[3,2,2,2,2,1,1] => [2,2,2,2,1,1] => 62
[3,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => 55
[3,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 28
[3,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 8
[3,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2,1] => [2,2,2,2,2,1] => 90
[2,2,2,2,2,1,1,1] => [2,2,2,2,1,1,1] => 117
[2,2,2,2,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => 83
[2,2,2,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => 36
[2,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => 9
[2,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
[12,2] => [2] => 0
[12,1,1] => [1,1] => 1
[11,3] => [3] => 1
[11,2,1] => [2,1] => 1
[11,1,1,1] => [1,1,1] => 1
[10,4] => [4] => 0
[10,3,1] => [3,1] => 2
[10,2,2] => [2,2] => 1
[10,2,1,1] => [2,1,1] => 2
[10,1,1,1,1] => [1,1,1,1] => 1
[9,5] => [5] => 1
[9,4,1] => [4,1] => 2
[9,3,2] => [3,2] => 3
[9,3,1,1] => [3,1,1] => 4
[9,2,2,1] => [2,2,1] => 3
[9,2,1,1,1] => [2,1,1,1] => 3
[9,1,1,1,1,1] => [1,1,1,1,1] => 1
[8,6] => [6] => 0
[8,5,1] => [5,1] => 3
[8,4,2] => [4,2] => 5
[8,4,1,1] => [4,1,1] => 6
[8,3,3] => [3,3] => 3
[8,3,2,1] => [3,2,1] => 10
[8,3,1,1,1] => [3,1,1,1] => 7
[8,2,2,2] => [2,2,2] => 3
[8,2,2,1,1] => [2,2,1,1] => 6
[8,2,1,1,1,1] => [2,1,1,1,1] => 4
[8,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[7,7] => [7] => 1
[7,6,1] => [6,1] => 3
[7,5,2] => [5,2] => 8
[7,5,1,1] => [5,1,1] => 9
[7,4,3] => [4,3] => 8
[7,4,2,1] => [4,2,1] => 21
[7,4,1,1,1] => [4,1,1,1] => 13
[7,3,3,1] => [3,3,1] => 13
[7,3,2,2] => [3,2,2] => 13
[7,3,2,1,1] => [3,2,1,1] => 23
[7,3,1,1,1,1] => [3,1,1,1,1] => 11
[7,2,2,2,1] => [2,2,2,1] => 9
[7,2,2,1,1,1] => [2,2,1,1,1] => 10
[7,2,1,1,1,1,1] => [2,1,1,1,1,1] => 5
[7,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[6,6,2] => [6,2] => 11
[6,6,1,1] => [6,1,1] => 12
[6,5,3] => [5,3] => 16
[6,5,2,1] => [5,2,1] => 38
[6,5,1,1,1] => [5,1,1,1] => 22
[6,4,4] => [4,4] => 8
[6,4,3,1] => [4,3,1] => 42
[6,4,2,2] => [4,2,2] => 34
[6,4,2,1,1] => [4,2,1,1] => 57
[6,4,1,1,1,1] => [4,1,1,1,1] => 24
[6,3,3,2] => [3,3,2] => 26
[6,3,3,1,1] => [3,3,1,1] => 36
[6,3,2,2,1] => [3,2,2,1] => 45
[6,3,2,1,1,1] => [3,2,1,1,1] => 44
[6,3,1,1,1,1,1] => [3,1,1,1,1,1] => 16
[6,2,2,2,2] => [2,2,2,2] => 9
[6,2,2,2,1,1] => [2,2,2,1,1] => 19
[6,2,2,1,1,1,1] => [2,2,1,1,1,1] => 15
[6,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 6
[6,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 1
[5,5,4] => [5,4] => 24
[5,5,3,1] => [5,3,1] => 96
[5,5,2,2] => [5,2,2] => 72
[5,5,2,1,1] => [5,2,1,1] => 117
[5,5,1,1,1,1] => [5,1,1,1,1] => 46
[5,4,4,1] => [4,4,1] => 50
[5,4,3,2] => [4,3,2] => 102
[5,4,3,1,1] => [4,3,1,1] => 135
[5,4,2,2,1] => [4,2,2,1] => 136
[5,4,2,1,1,1] => [4,2,1,1,1] => 125
[5,4,1,1,1,1,1] => [4,1,1,1,1,1] => 40
[5,3,3,3] => [3,3,3] => 26
[5,3,3,2,1] => [3,3,2,1] => 107
[5,3,3,1,1,1] => [3,3,1,1,1] => 80
[5,3,2,2,2] => [3,2,2,2] => 54
[5,3,2,2,1,1] => [3,2,2,1,1] => 108
[5,3,2,1,1,1,1] => [3,2,1,1,1,1] => 75
[5,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 22
[5,2,2,2,2,1] => [2,2,2,2,1] => 28
[5,2,2,2,1,1,1] => [2,2,2,1,1,1] => 34
[5,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 21
[5,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 7
[5,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[4,4,4,2] => [4,4,2] => 152
[4,4,4,1,1] => [4,4,1,1] => 185
[4,4,3,3] => [4,3,3] => 128
[4,4,3,2,1] => [4,3,2,1] => 480
[4,4,3,1,1,1] => [4,3,1,1,1] => 340
[4,4,2,2,2] => [4,2,2,2] => 190
[4,4,2,2,1,1] => [4,2,2,1,1] => 369
[4,4,2,1,1,1,1] => [4,2,1,1,1,1] => 240
[4,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => 62
[4,3,3,3,1] => [3,3,3,1] => 133
[4,3,3,2,2] => [3,3,2,2] => 161
[4,3,3,2,1,1] => [3,3,2,1,1] => 295
[4,3,3,1,1,1,1] => [3,3,1,1,1,1] => 155
[4,3,2,2,2,1] => [3,2,2,2,1] => 190
[4,3,2,2,1,1,1] => [3,2,2,1,1,1] => 217
[4,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => 118
[4,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 29
[4,2,2,2,2,2] => [2,2,2,2,2] => 28
[4,2,2,2,2,1,1] => [2,2,2,2,1,1] => 62
[4,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => 55
[4,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 28
[4,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 8
[4,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 1
[3,3,3,3,2] => [3,3,3,2] => 294
[3,3,3,3,1,1] => [3,3,3,1,1] => 428
[3,3,3,2,2,1] => [3,3,2,2,1] => 646
[3,3,3,2,1,1,1] => [3,3,2,1,1,1] => 667
[3,3,3,1,1,1,1,1] => [3,3,1,1,1,1,1] => 273
[3,3,2,2,2,2] => [3,2,2,2,2] => 218
[3,3,2,2,2,1,1] => [3,2,2,2,1,1] => 469
[3,3,2,2,1,1,1,1] => [3,2,2,1,1,1,1] => 390
[3,3,2,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1] => 175
[3,3,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1] => 37
[3,2,2,2,2,2,1] => [2,2,2,2,2,1] => 90
[3,2,2,2,2,1,1,1] => [2,2,2,2,1,1,1] => 117
[3,2,2,2,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => 83
[3,2,2,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => 36
[3,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => 9
[3,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2,2] => [2,2,2,2,2,2] => 90
[2,2,2,2,2,2,1,1] => [2,2,2,2,2,1,1] => 207
[2,2,2,2,2,1,1,1,1] => [2,2,2,2,1,1,1,1] => 200
[2,2,2,2,1,1,1,1,1,1] => [2,2,2,1,1,1,1,1,1] => 119
[2,2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => 45
[2,2,1,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1] => 10
[2,1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
[13,2] => [2] => 0
[13,1,1] => [1,1] => 1
[12,3] => [3] => 1
[12,2,1] => [2,1] => 1
[12,1,1,1] => [1,1,1] => 1
[11,4] => [4] => 0
[11,3,1] => [3,1] => 2
[11,2,2] => [2,2] => 1
[11,2,1,1] => [2,1,1] => 2
[11,1,1,1,1] => [1,1,1,1] => 1
[10,5] => [5] => 1
[10,4,1] => [4,1] => 2
[10,3,2] => [3,2] => 3
[10,3,1,1] => [3,1,1] => 4
[10,2,2,1] => [2,2,1] => 3
[10,2,1,1,1] => [2,1,1,1] => 3
[10,1,1,1,1,1] => [1,1,1,1,1] => 1
[9,6] => [6] => 0
[9,5,1] => [5,1] => 3
[9,4,2] => [4,2] => 5
[9,4,1,1] => [4,1,1] => 6
[9,3,3] => [3,3] => 3
[9,3,2,1] => [3,2,1] => 10
[9,3,1,1,1] => [3,1,1,1] => 7
[9,2,2,2] => [2,2,2] => 3
[9,2,2,1,1] => [2,2,1,1] => 6
[9,2,1,1,1,1] => [2,1,1,1,1] => 4
[9,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[8,7] => [7] => 1
[8,6,1] => [6,1] => 3
[8,5,2] => [5,2] => 8
[8,5,1,1] => [5,1,1] => 9
[8,4,3] => [4,3] => 8
[8,4,2,1] => [4,2,1] => 21
[8,4,1,1,1] => [4,1,1,1] => 13
[8,3,3,1] => [3,3,1] => 13
[8,3,2,2] => [3,2,2] => 13
[8,3,2,1,1] => [3,2,1,1] => 23
[8,3,1,1,1,1] => [3,1,1,1,1] => 11
[8,2,2,2,1] => [2,2,2,1] => 9
[8,2,2,1,1,1] => [2,2,1,1,1] => 10
[8,2,1,1,1,1,1] => [2,1,1,1,1,1] => 5
[8,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[7,7,1] => [7,1] => 4
[7,6,2] => [6,2] => 11
[7,6,1,1] => [6,1,1] => 12
[7,5,3] => [5,3] => 16
[7,5,2,1] => [5,2,1] => 38
[7,5,1,1,1] => [5,1,1,1] => 22
[7,4,4] => [4,4] => 8
[7,4,3,1] => [4,3,1] => 42
[7,4,2,2] => [4,2,2] => 34
[7,4,2,1,1] => [4,2,1,1] => 57
[7,4,1,1,1,1] => [4,1,1,1,1] => 24
[7,3,3,2] => [3,3,2] => 26
[7,3,3,1,1] => [3,3,1,1] => 36
[7,3,2,2,1] => [3,2,2,1] => 45
[7,3,2,1,1,1] => [3,2,1,1,1] => 44
[7,3,1,1,1,1,1] => [3,1,1,1,1,1] => 16
[7,2,2,2,2] => [2,2,2,2] => 9
[7,2,2,2,1,1] => [2,2,2,1,1] => 19
[7,2,2,1,1,1,1] => [2,2,1,1,1,1] => 15
[7,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 6
[7,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 1
[6,6,3] => [6,3] => 27
[6,6,2,1] => [6,2,1] => 61
[6,6,1,1,1] => [6,1,1,1] => 34
[6,5,4] => [5,4] => 24
[6,5,3,1] => [5,3,1] => 96
[6,5,2,2] => [5,2,2] => 72
[6,5,2,1,1] => [5,2,1,1] => 117
[6,5,1,1,1,1] => [5,1,1,1,1] => 46
[6,4,4,1] => [4,4,1] => 50
[6,4,3,2] => [4,3,2] => 102
[6,4,3,1,1] => [4,3,1,1] => 135
[6,4,2,2,1] => [4,2,2,1] => 136
[6,4,2,1,1,1] => [4,2,1,1,1] => 125
[6,4,1,1,1,1,1] => [4,1,1,1,1,1] => 40
[6,3,3,3] => [3,3,3] => 26
[6,3,3,2,1] => [3,3,2,1] => 107
[6,3,3,1,1,1] => [3,3,1,1,1] => 80
[6,3,2,2,2] => [3,2,2,2] => 54
[6,3,2,2,1,1] => [3,2,2,1,1] => 108
[6,3,2,1,1,1,1] => [3,2,1,1,1,1] => 75
[6,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 22
[6,2,2,2,2,1] => [2,2,2,2,1] => 28
[6,2,2,2,1,1,1] => [2,2,2,1,1,1] => 34
[6,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 21
[6,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 7
[6,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[5,5,5] => [5,5] => 24
[5,5,4,1] => [5,4,1] => 170
[5,5,3,2] => [5,3,2] => 270
[5,5,3,1,1] => [5,3,1,1] => 348
[5,5,2,2,1] => [5,2,2,1] => 325
[5,5,2,1,1,1] => [5,2,1,1,1] => 288
[5,5,1,1,1,1,1] => [5,1,1,1,1,1] => 86
[5,4,4,2] => [4,4,2] => 152
[5,4,4,1,1] => [4,4,1,1] => 185
[5,4,3,3] => [4,3,3] => 128
[5,4,3,2,1] => [4,3,2,1] => 480
[5,4,3,1,1,1] => [4,3,1,1,1] => 340
[5,4,2,2,2] => [4,2,2,2] => 190
[5,4,2,2,1,1] => [4,2,2,1,1] => 369
[5,4,2,1,1,1,1] => [4,2,1,1,1,1] => 240
[5,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => 62
[5,3,3,3,1] => [3,3,3,1] => 133
[5,3,3,2,2] => [3,3,2,2] => 161
[5,3,3,2,1,1] => [3,3,2,1,1] => 295
[5,3,3,1,1,1,1] => [3,3,1,1,1,1] => 155
[5,3,2,2,2,1] => [3,2,2,2,1] => 190
[5,3,2,2,1,1,1] => [3,2,2,1,1,1] => 217
[5,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => 118
[5,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 29
[5,2,2,2,2,2] => [2,2,2,2,2] => 28
[5,2,2,2,2,1,1] => [2,2,2,2,1,1] => 62
[5,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => 55
[5,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 28
[5,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 8
[5,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 1
[4,4,4,3] => [4,4,3] => 280
[4,4,4,2,1] => [4,4,2,1] => 817
[4,4,4,1,1,1] => [4,4,1,1,1] => 525
[4,4,3,3,1] => [4,3,3,1] => 741
[4,4,3,2,2] => [4,3,2,2] => 831
[4,4,3,2,1,1] => [4,3,2,1,1] => 1484
[4,4,3,1,1,1,1] => [4,3,1,1,1,1] => 735
[4,4,2,2,2,1] => [4,2,2,2,1] => 749
[4,4,2,2,1,1,1] => [4,2,2,1,1,1] => 826
[4,4,2,1,1,1,1,1] => [4,2,1,1,1,1,1] => 420
[4,4,1,1,1,1,1,1,1] => [4,1,1,1,1,1,1,1] => 91
[4,3,3,3,2] => [3,3,3,2] => 294
[4,3,3,3,1,1] => [3,3,3,1,1] => 428
[4,3,3,2,2,1] => [3,3,2,2,1] => 646
[4,3,3,2,1,1,1] => [3,3,2,1,1,1] => 667
[4,3,3,1,1,1,1,1] => [3,3,1,1,1,1,1] => 273
[4,3,2,2,2,2] => [3,2,2,2,2] => 218
[4,3,2,2,2,1,1] => [3,2,2,2,1,1] => 469
[4,3,2,2,1,1,1,1] => [3,2,2,1,1,1,1] => 390
[4,3,2,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1] => 175
[4,3,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1] => 37
[4,2,2,2,2,2,1] => [2,2,2,2,2,1] => 90
[4,2,2,2,2,1,1,1] => [2,2,2,2,1,1,1] => 117
[4,2,2,2,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => 83
[4,2,2,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => 36
[4,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => 9
[4,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[3,3,3,3,3] => [3,3,3,3] => 294
[3,3,3,3,2,1] => [3,3,3,2,1] => 1368
[3,3,3,3,1,1,1] => [3,3,3,1,1,1] => 1095
[3,3,3,2,2,2] => [3,3,2,2,2] => 864
[3,3,3,2,2,1,1] => [3,3,2,2,1,1] => 1782
[3,3,3,2,1,1,1,1] => [3,3,2,1,1,1,1] => 1330
[3,3,3,1,1,1,1,1,1] => [3,3,1,1,1,1,1,1] => 448
[3,3,2,2,2,2,1] => [3,2,2,2,2,1] => 777
[3,3,2,2,2,1,1,1] => [3,2,2,2,1,1,1] => 976
[3,3,2,2,1,1,1,1,1] => [3,2,2,1,1,1,1,1] => 648
[3,3,2,1,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1,1] => 248
[3,3,1,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1,1] => 46
[3,2,2,2,2,2,2] => [2,2,2,2,2,2] => 90
[3,2,2,2,2,2,1,1] => [2,2,2,2,2,1,1] => 207
[3,2,2,2,2,1,1,1,1] => [2,2,2,2,1,1,1,1] => 200
[3,2,2,2,1,1,1,1,1,1] => [2,2,2,1,1,1,1,1,1] => 119
[3,2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => 45
[3,2,1,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1] => 10
[3,1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
[14,2] => [2] => 0
[14,1,1] => [1,1] => 1
[13,3] => [3] => 1
[13,2,1] => [2,1] => 1
[13,1,1,1] => [1,1,1] => 1
[12,4] => [4] => 0
[12,3,1] => [3,1] => 2
[12,2,2] => [2,2] => 1
[12,2,1,1] => [2,1,1] => 2
[12,1,1,1,1] => [1,1,1,1] => 1
[11,5] => [5] => 1
[11,4,1] => [4,1] => 2
[11,3,2] => [3,2] => 3
[11,3,1,1] => [3,1,1] => 4
[11,2,2,1] => [2,2,1] => 3
[11,2,1,1,1] => [2,1,1,1] => 3
[11,1,1,1,1,1] => [1,1,1,1,1] => 1
[10,6] => [6] => 0
[10,5,1] => [5,1] => 3
[10,4,2] => [4,2] => 5
[10,4,1,1] => [4,1,1] => 6
[10,3,3] => [3,3] => 3
[10,3,2,1] => [3,2,1] => 10
[10,3,1,1,1] => [3,1,1,1] => 7
[10,2,2,2] => [2,2,2] => 3
[10,2,2,1,1] => [2,2,1,1] => 6
[10,2,1,1,1,1] => [2,1,1,1,1] => 4
[10,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[9,7] => [7] => 1
[9,6,1] => [6,1] => 3
[9,5,2] => [5,2] => 8
[9,5,1,1] => [5,1,1] => 9
[9,4,3] => [4,3] => 8
[9,4,2,1] => [4,2,1] => 21
[9,4,1,1,1] => [4,1,1,1] => 13
[9,3,3,1] => [3,3,1] => 13
[9,3,2,2] => [3,2,2] => 13
[9,3,2,1,1] => [3,2,1,1] => 23
[9,3,1,1,1,1] => [3,1,1,1,1] => 11
[9,2,2,2,1] => [2,2,2,1] => 9
[9,2,2,1,1,1] => [2,2,1,1,1] => 10
[9,2,1,1,1,1,1] => [2,1,1,1,1,1] => 5
[9,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[8,8] => [8] => 0
[8,7,1] => [7,1] => 4
[8,6,2] => [6,2] => 11
[8,6,1,1] => [6,1,1] => 12
[8,5,3] => [5,3] => 16
[8,5,2,1] => [5,2,1] => 38
[8,5,1,1,1] => [5,1,1,1] => 22
[8,4,4] => [4,4] => 8
[8,4,3,1] => [4,3,1] => 42
[8,4,2,2] => [4,2,2] => 34
[8,4,2,1,1] => [4,2,1,1] => 57
[8,4,1,1,1,1] => [4,1,1,1,1] => 24
[8,3,3,2] => [3,3,2] => 26
[8,3,3,1,1] => [3,3,1,1] => 36
[8,3,2,2,1] => [3,2,2,1] => 45
[8,3,2,1,1,1] => [3,2,1,1,1] => 44
[8,3,1,1,1,1,1] => [3,1,1,1,1,1] => 16
[8,2,2,2,2] => [2,2,2,2] => 9
[8,2,2,2,1,1] => [2,2,2,1,1] => 19
[8,2,2,1,1,1,1] => [2,2,1,1,1,1] => 15
[8,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 6
[8,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 1
[7,7,2] => [7,2] => 15
[7,7,1,1] => [7,1,1] => 16
[7,6,3] => [6,3] => 27
[7,6,2,1] => [6,2,1] => 61
[7,6,1,1,1] => [6,1,1,1] => 34
[7,5,4] => [5,4] => 24
[7,5,3,1] => [5,3,1] => 96
[7,5,2,2] => [5,2,2] => 72
[7,5,2,1,1] => [5,2,1,1] => 117
[7,5,1,1,1,1] => [5,1,1,1,1] => 46
[7,4,4,1] => [4,4,1] => 50
[7,4,3,2] => [4,3,2] => 102
[7,4,3,1,1] => [4,3,1,1] => 135
[7,4,2,2,1] => [4,2,2,1] => 136
[7,4,2,1,1,1] => [4,2,1,1,1] => 125
[7,4,1,1,1,1,1] => [4,1,1,1,1,1] => 40
[7,3,3,3] => [3,3,3] => 26
[7,3,3,2,1] => [3,3,2,1] => 107
[7,3,3,1,1,1] => [3,3,1,1,1] => 80
[7,3,2,2,2] => [3,2,2,2] => 54
[7,3,2,2,1,1] => [3,2,2,1,1] => 108
[7,3,2,1,1,1,1] => [3,2,1,1,1,1] => 75
[7,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 22
[7,2,2,2,2,1] => [2,2,2,2,1] => 28
[7,2,2,2,1,1,1] => [2,2,2,1,1,1] => 34
[7,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 21
[7,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 7
[7,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[6,6,4] => [6,4] => 51
[6,6,3,1] => [6,3,1] => 184
[6,6,2,2] => [6,2,2] => 133
[6,6,2,1,1] => [6,2,1,1] => 212
[6,6,1,1,1,1] => [6,1,1,1,1] => 80
[6,5,5] => [5,5] => 24
[6,5,4,1] => [5,4,1] => 170
[6,5,3,2] => [5,3,2] => 270
[6,5,3,1,1] => [5,3,1,1] => 348
[6,5,2,2,1] => [5,2,2,1] => 325
[6,5,2,1,1,1] => [5,2,1,1,1] => 288
[6,5,1,1,1,1,1] => [5,1,1,1,1,1] => 86
[6,4,4,2] => [4,4,2] => 152
[6,4,4,1,1] => [4,4,1,1] => 185
[6,4,3,3] => [4,3,3] => 128
[6,4,3,2,1] => [4,3,2,1] => 480
[6,4,3,1,1,1] => [4,3,1,1,1] => 340
[6,4,2,2,2] => [4,2,2,2] => 190
[6,4,2,2,1,1] => [4,2,2,1,1] => 369
[6,4,2,1,1,1,1] => [4,2,1,1,1,1] => 240
[6,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => 62
[6,3,3,3,1] => [3,3,3,1] => 133
[6,3,3,2,2] => [3,3,2,2] => 161
[6,3,3,2,1,1] => [3,3,2,1,1] => 295
[6,3,3,1,1,1,1] => [3,3,1,1,1,1] => 155
[6,3,2,2,2,1] => [3,2,2,2,1] => 190
[6,3,2,2,1,1,1] => [3,2,2,1,1,1] => 217
[6,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => 118
[6,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 29
[6,2,2,2,2,2] => [2,2,2,2,2] => 28
[6,2,2,2,2,1,1] => [2,2,2,2,1,1] => 62
[6,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => 55
[6,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 28
[6,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 8
[6,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 1
[5,5,5,1] => [5,5,1] => 194
[5,5,4,2] => [5,4,2] => 592
[5,5,4,1,1] => [5,4,1,1] => 703
[5,5,3,3] => [5,3,3] => 398
[5,5,3,2,1] => [5,3,2,1] => 1423
[5,5,3,1,1,1] => [5,3,1,1,1] => 976
[5,5,2,2,2] => [5,2,2,2] => 515
[5,5,2,2,1,1] => [5,2,2,1,1] => 982
[5,5,2,1,1,1,1] => [5,2,1,1,1,1] => 614
[5,5,1,1,1,1,1,1] => [5,1,1,1,1,1,1] => 148
[5,4,4,3] => [4,4,3] => 280
[5,4,4,2,1] => [4,4,2,1] => 817
[5,4,4,1,1,1] => [4,4,1,1,1] => 525
[5,4,3,3,1] => [4,3,3,1] => 741
[5,4,3,2,2] => [4,3,2,2] => 831
[5,4,3,2,1,1] => [4,3,2,1,1] => 1484
[5,4,3,1,1,1,1] => [4,3,1,1,1,1] => 735
[5,4,2,2,2,1] => [4,2,2,2,1] => 749
[5,4,2,2,1,1,1] => [4,2,2,1,1,1] => 826
[5,4,2,1,1,1,1,1] => [4,2,1,1,1,1,1] => 420
[5,4,1,1,1,1,1,1,1] => [4,1,1,1,1,1,1,1] => 91
[5,3,3,3,2] => [3,3,3,2] => 294
[5,3,3,3,1,1] => [3,3,3,1,1] => 428
[5,3,3,2,2,1] => [3,3,2,2,1] => 646
[5,3,3,2,1,1,1] => [3,3,2,1,1,1] => 667
[5,3,3,1,1,1,1,1] => [3,3,1,1,1,1,1] => 273
[5,3,2,2,2,2] => [3,2,2,2,2] => 218
[5,3,2,2,2,1,1] => [3,2,2,2,1,1] => 469
[5,3,2,2,1,1,1,1] => [3,2,2,1,1,1,1] => 390
[5,3,2,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1] => 175
[5,3,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1] => 37
[5,2,2,2,2,2,1] => [2,2,2,2,2,1] => 90
[5,2,2,2,2,1,1,1] => [2,2,2,2,1,1,1] => 117
[5,2,2,2,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => 83
[5,2,2,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => 36
[5,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => 9
[5,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[4,4,4,4] => [4,4,4] => 280
[4,4,4,3,1] => [4,4,3,1] => 1838
[4,4,4,2,2] => [4,4,2,2] => 1648
[4,4,4,2,1,1] => [4,4,2,1,1] => 2826
[4,4,4,1,1,1,1] => [4,4,1,1,1,1] => 1260
[4,4,3,3,2] => [4,3,3,2] => 1866
[4,4,3,3,1,1] => [4,3,3,1,1] => 2653
[4,4,3,2,2,1] => [4,3,2,2,1] => 3710
[4,4,3,2,1,1,1] => [4,3,2,1,1,1] => 3712
[4,4,3,1,1,1,1,1] => [4,3,1,1,1,1,1] => 1428
[4,4,2,2,2,2] => [4,2,2,2,2] => 967
[4,4,2,2,2,1,1] => [4,2,2,2,1,1] => 2044
[4,4,2,2,1,1,1,1] => [4,2,2,1,1,1,1] => 1636
[4,4,2,1,1,1,1,1,1] => [4,2,1,1,1,1,1,1] => 686
[4,4,1,1,1,1,1,1,1,1] => [4,1,1,1,1,1,1,1,1] => 128
[4,3,3,3,3] => [3,3,3,3] => 294
[4,3,3,3,2,1] => [3,3,3,2,1] => 1368
[4,3,3,3,1,1,1] => [3,3,3,1,1,1] => 1095
[4,3,3,2,2,2] => [3,3,2,2,2] => 864
[4,3,3,2,2,1,1] => [3,3,2,2,1,1] => 1782
[4,3,3,2,1,1,1,1] => [3,3,2,1,1,1,1] => 1330
[4,3,3,1,1,1,1,1,1] => [3,3,1,1,1,1,1,1] => 448
[4,3,2,2,2,2,1] => [3,2,2,2,2,1] => 777
[4,3,2,2,2,1,1,1] => [3,2,2,2,1,1,1] => 976
[4,3,2,2,1,1,1,1,1] => [3,2,2,1,1,1,1,1] => 648
[4,3,2,1,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1,1] => 248
[4,3,1,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1,1] => 46
[4,2,2,2,2,2,2] => [2,2,2,2,2,2] => 90
[4,2,2,2,2,2,1,1] => [2,2,2,2,2,1,1] => 207
[4,2,2,2,2,1,1,1,1] => [2,2,2,2,1,1,1,1] => 200
[4,2,2,2,1,1,1,1,1,1] => [2,2,2,1,1,1,1,1,1] => 119
[4,2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => 45
[4,2,1,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1] => 10
[4,1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
[15,2] => [2] => 0
[15,1,1] => [1,1] => 1
[14,3] => [3] => 1
[14,2,1] => [2,1] => 1
[14,1,1,1] => [1,1,1] => 1
[13,4] => [4] => 0
[13,3,1] => [3,1] => 2
[13,2,2] => [2,2] => 1
[13,2,1,1] => [2,1,1] => 2
[13,1,1,1,1] => [1,1,1,1] => 1
[12,5] => [5] => 1
[12,4,1] => [4,1] => 2
[12,3,2] => [3,2] => 3
[12,3,1,1] => [3,1,1] => 4
[12,2,2,1] => [2,2,1] => 3
[12,2,1,1,1] => [2,1,1,1] => 3
[12,1,1,1,1,1] => [1,1,1,1,1] => 1
[11,6] => [6] => 0
[11,5,1] => [5,1] => 3
[11,4,2] => [4,2] => 5
[11,4,1,1] => [4,1,1] => 6
[11,3,3] => [3,3] => 3
[11,3,2,1] => [3,2,1] => 10
[11,3,1,1,1] => [3,1,1,1] => 7
[11,2,2,2] => [2,2,2] => 3
[11,2,2,1,1] => [2,2,1,1] => 6
[11,2,1,1,1,1] => [2,1,1,1,1] => 4
[11,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[10,7] => [7] => 1
[10,6,1] => [6,1] => 3
[10,5,2] => [5,2] => 8
[10,5,1,1] => [5,1,1] => 9
[10,4,3] => [4,3] => 8
[10,4,2,1] => [4,2,1] => 21
[10,4,1,1,1] => [4,1,1,1] => 13
[10,3,3,1] => [3,3,1] => 13
[10,3,2,2] => [3,2,2] => 13
[10,3,2,1,1] => [3,2,1,1] => 23
[10,3,1,1,1,1] => [3,1,1,1,1] => 11
[10,2,2,2,1] => [2,2,2,1] => 9
[10,2,2,1,1,1] => [2,2,1,1,1] => 10
[10,2,1,1,1,1,1] => [2,1,1,1,1,1] => 5
[10,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[9,8] => [8] => 0
[9,7,1] => [7,1] => 4
[9,6,2] => [6,2] => 11
[9,6,1,1] => [6,1,1] => 12
[9,5,3] => [5,3] => 16
[9,5,2,1] => [5,2,1] => 38
[9,5,1,1,1] => [5,1,1,1] => 22
[9,4,4] => [4,4] => 8
[9,4,3,1] => [4,3,1] => 42
[9,4,2,2] => [4,2,2] => 34
[9,4,2,1,1] => [4,2,1,1] => 57
[9,4,1,1,1,1] => [4,1,1,1,1] => 24
[9,3,3,2] => [3,3,2] => 26
[9,3,3,1,1] => [3,3,1,1] => 36
[9,3,2,2,1] => [3,2,2,1] => 45
[9,3,2,1,1,1] => [3,2,1,1,1] => 44
[9,3,1,1,1,1,1] => [3,1,1,1,1,1] => 16
[9,2,2,2,2] => [2,2,2,2] => 9
[9,2,2,2,1,1] => [2,2,2,1,1] => 19
[9,2,2,1,1,1,1] => [2,2,1,1,1,1] => 15
[9,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 6
[9,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 1
[8,8,1] => [8,1] => 4
[8,7,2] => [7,2] => 15
[8,7,1,1] => [7,1,1] => 16
[8,6,3] => [6,3] => 27
[8,6,2,1] => [6,2,1] => 61
[8,6,1,1,1] => [6,1,1,1] => 34
[8,5,4] => [5,4] => 24
[8,5,3,1] => [5,3,1] => 96
[8,5,2,2] => [5,2,2] => 72
[8,5,2,1,1] => [5,2,1,1] => 117
[8,5,1,1,1,1] => [5,1,1,1,1] => 46
[8,4,4,1] => [4,4,1] => 50
[8,4,3,2] => [4,3,2] => 102
[8,4,3,1,1] => [4,3,1,1] => 135
[8,4,2,2,1] => [4,2,2,1] => 136
[8,4,2,1,1,1] => [4,2,1,1,1] => 125
[8,4,1,1,1,1,1] => [4,1,1,1,1,1] => 40
[8,3,3,3] => [3,3,3] => 26
[8,3,3,2,1] => [3,3,2,1] => 107
[8,3,3,1,1,1] => [3,3,1,1,1] => 80
[8,3,2,2,2] => [3,2,2,2] => 54
[8,3,2,2,1,1] => [3,2,2,1,1] => 108
[8,3,2,1,1,1,1] => [3,2,1,1,1,1] => 75
[8,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 22
[8,2,2,2,2,1] => [2,2,2,2,1] => 28
[8,2,2,2,1,1,1] => [2,2,2,1,1,1] => 34
[8,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 21
[8,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 7
[8,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1
[7,7,3] => [7,3] => 42
[7,7,2,1] => [7,2,1] => 92
[7,7,1,1,1] => [7,1,1,1] => 50
[7,6,4] => [6,4] => 51
[7,6,3,1] => [6,3,1] => 184
[7,6,2,2] => [6,2,2] => 133
[7,6,2,1,1] => [6,2,1,1] => 212
[7,6,1,1,1,1] => [6,1,1,1,1] => 80
[7,5,5] => [5,5] => 24
[7,5,4,1] => [5,4,1] => 170
[7,5,3,2] => [5,3,2] => 270
[7,5,3,1,1] => [5,3,1,1] => 348
[7,5,2,2,1] => [5,2,2,1] => 325
[7,5,2,1,1,1] => [5,2,1,1,1] => 288
[7,5,1,1,1,1,1] => [5,1,1,1,1,1] => 86
[7,4,4,2] => [4,4,2] => 152
[7,4,4,1,1] => [4,4,1,1] => 185
[7,4,3,3] => [4,3,3] => 128
[7,4,3,2,1] => [4,3,2,1] => 480
[7,4,3,1,1,1] => [4,3,1,1,1] => 340
[7,4,2,2,2] => [4,2,2,2] => 190
[7,4,2,2,1,1] => [4,2,2,1,1] => 369
[7,4,2,1,1,1,1] => [4,2,1,1,1,1] => 240
[7,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => 62
[7,3,3,3,1] => [3,3,3,1] => 133
[7,3,3,2,2] => [3,3,2,2] => 161
[7,3,3,2,1,1] => [3,3,2,1,1] => 295
[7,3,3,1,1,1,1] => [3,3,1,1,1,1] => 155
[7,3,2,2,2,1] => [3,2,2,2,1] => 190
[7,3,2,2,1,1,1] => [3,2,2,1,1,1] => 217
[7,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => 118
[7,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 29
[7,2,2,2,2,2] => [2,2,2,2,2] => 28
[7,2,2,2,2,1,1] => [2,2,2,2,1,1] => 62
[7,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => 55
[7,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 28
[7,2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 8
[7,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 1
[6,6,5] => [6,5] => 75
[6,6,4,1] => [6,4,1] => 405
[6,6,3,2] => [6,3,2] => 587
[6,6,3,1,1] => [6,3,1,1] => 744
[6,6,2,2,1] => [6,2,2,1] => 670
[6,6,2,1,1,1] => [6,2,1,1,1] => 580
[6,6,1,1,1,1,1] => [6,1,1,1,1,1] => 166
[6,5,5,1] => [5,5,1] => 194
[6,5,4,2] => [5,4,2] => 592
[6,5,4,1,1] => [5,4,1,1] => 703
[6,5,3,3] => [5,3,3] => 398
[6,5,3,2,1] => [5,3,2,1] => 1423
[6,5,3,1,1,1] => [5,3,1,1,1] => 976
[6,5,2,2,2] => [5,2,2,2] => 515
[6,5,2,2,1,1] => [5,2,2,1,1] => 982
[6,5,2,1,1,1,1] => [5,2,1,1,1,1] => 614
[6,5,1,1,1,1,1,1] => [5,1,1,1,1,1,1] => 148
[6,4,4,3] => [4,4,3] => 280
[6,4,4,2,1] => [4,4,2,1] => 817
[6,4,4,1,1,1] => [4,4,1,1,1] => 525
[6,4,3,3,1] => [4,3,3,1] => 741
[6,4,3,2,2] => [4,3,2,2] => 831
[6,4,3,2,1,1] => [4,3,2,1,1] => 1484
[6,4,3,1,1,1,1] => [4,3,1,1,1,1] => 735
[6,4,2,2,2,1] => [4,2,2,2,1] => 749
[6,4,2,2,1,1,1] => [4,2,2,1,1,1] => 826
[6,4,2,1,1,1,1,1] => [4,2,1,1,1,1,1] => 420
[6,4,1,1,1,1,1,1,1] => [4,1,1,1,1,1,1,1] => 91
[6,3,3,3,2] => [3,3,3,2] => 294
[6,3,3,3,1,1] => [3,3,3,1,1] => 428
[6,3,3,2,2,1] => [3,3,2,2,1] => 646
[6,3,3,2,1,1,1] => [3,3,2,1,1,1] => 667
[6,3,3,1,1,1,1,1] => [3,3,1,1,1,1,1] => 273
[6,3,2,2,2,2] => [3,2,2,2,2] => 218
[6,3,2,2,2,1,1] => [3,2,2,2,1,1] => 469
[6,3,2,2,1,1,1,1] => [3,2,2,1,1,1,1] => 390
[6,3,2,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1] => 175
[6,3,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1] => 37
[6,2,2,2,2,2,1] => [2,2,2,2,2,1] => 90
[6,2,2,2,2,1,1,1] => [2,2,2,2,1,1,1] => 117
[6,2,2,2,1,1,1,1,1] => [2,2,2,1,1,1,1,1] => 83
[6,2,2,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1] => 36
[6,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => 9
[6,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[5,5,5,2] => [5,5,2] => 786
[5,5,5,1,1] => [5,5,1,1] => 897
[5,5,4,3] => [5,4,3] => 1270
[5,5,4,2,1] => [5,4,2,1] => 3535
[5,5,4,1,1,1] => [5,4,1,1,1] => 2204
[5,5,3,3,1] => [5,3,3,1] => 2562
[5,5,3,2,2] => [5,3,2,2] => 2769
[5,5,3,2,1,1] => [5,3,2,1,1] => 4865
[5,5,3,1,1,1,1] => [5,3,1,1,1,1] => 2325
[5,5,2,2,2,1] => [5,2,2,2,1] => 2246
[5,5,2,2,1,1,1] => [5,2,2,1,1,1] => 2422
[5,5,2,1,1,1,1,1] => [5,2,1,1,1,1,1] => 1182
[5,5,1,1,1,1,1,1,1] => [5,1,1,1,1,1,1,1] => 239
[5,4,4,4] => [4,4,4] => 280
[5,4,4,3,1] => [4,4,3,1] => 1838
[5,4,4,2,2] => [4,4,2,2] => 1648
[5,4,4,2,1,1] => [4,4,2,1,1] => 2826
[5,4,4,1,1,1,1] => [4,4,1,1,1,1] => 1260
[5,4,3,3,2] => [4,3,3,2] => 1866
[5,4,3,3,1,1] => [4,3,3,1,1] => 2653
[5,4,3,2,2,1] => [4,3,2,2,1] => 3710
[5,4,3,2,1,1,1] => [4,3,2,1,1,1] => 3712
[5,4,3,1,1,1,1,1] => [4,3,1,1,1,1,1] => 1428
[5,4,2,2,2,2] => [4,2,2,2,2] => 967
[5,4,2,2,2,1,1] => [4,2,2,2,1,1] => 2044
[5,4,2,2,1,1,1,1] => [4,2,2,1,1,1,1] => 1636
[5,4,2,1,1,1,1,1,1] => [4,2,1,1,1,1,1,1] => 686
[5,4,1,1,1,1,1,1,1,1] => [4,1,1,1,1,1,1,1,1] => 128
[5,3,3,3,3] => [3,3,3,3] => 294
[5,3,3,3,2,1] => [3,3,3,2,1] => 1368
[5,3,3,3,1,1,1] => [3,3,3,1,1,1] => 1095
[5,3,3,2,2,2] => [3,3,2,2,2] => 864
[5,3,3,2,2,1,1] => [3,3,2,2,1,1] => 1782
[5,3,3,2,1,1,1,1] => [3,3,2,1,1,1,1] => 1330
[5,3,3,1,1,1,1,1,1] => [3,3,1,1,1,1,1,1] => 448
[5,3,2,2,2,2,1] => [3,2,2,2,2,1] => 777
[5,3,2,2,2,1,1,1] => [3,2,2,2,1,1,1] => 976
[5,3,2,2,1,1,1,1,1] => [3,2,2,1,1,1,1,1] => 648
[5,3,2,1,1,1,1,1,1,1] => [3,2,1,1,1,1,1,1,1] => 248
[5,3,1,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1,1,1] => 46
[5,2,2,2,2,2,2] => [2,2,2,2,2,2] => 90
[5,2,2,2,2,2,1,1] => [2,2,2,2,2,1,1] => 207
[5,2,2,2,2,1,1,1,1] => [2,2,2,2,1,1,1,1] => 200
[5,2,2,2,1,1,1,1,1,1] => [2,2,2,1,1,1,1,1,1] => 119
[5,2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => 45
[5,2,1,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1] => 10
[5,1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
[6,3,3,3,2,1] => [3,3,3,2,1] => 1368
[6,4,3,2,2,1] => [4,3,2,2,1] => 3710
[6,5,2,2,2,1] => [5,2,2,2,1] => 2246
[6,4,3,3,1,1] => [4,3,3,1,1] => 2653
[6,4,4,2,1,1] => [4,4,2,1,1] => 2826
[6,5,3,2,1,1] => [5,3,2,1,1] => 4865
[6,5,4,1,1,1] => [5,4,1,1,1] => 2204
[6,4,3,3,2] => [4,3,3,2] => 1866
[6,4,4,2,2] => [4,4,2,2] => 1648
[6,5,3,2,2] => [5,3,2,2] => 2769
[6,4,4,3,1] => [4,4,3,1] => 1838
[6,5,3,3,1] => [5,3,3,1] => 2562
[6,5,4,2,1] => [5,4,2,1] => 3535
[6,5,4,3] => [5,4,3] => 1270
[7,2,2,2,2,2,1] => [2,2,2,2,2,1] => 90
[7,4,3,2,1,1,1] => [4,3,2,1,1,1] => 3712
[6,4,3,2,1,1,1] => [4,3,2,1,1,1] => 3712
[6,6,2,1,1,1,1] => [6,2,1,1,1,1] => 1360
[7,6,1,1,1,1,1] => [6,1,1,1,1,1] => 166
[7,3,3,3,2,1] => [3,3,3,2,1] => 1368
[7,4,3,2,1,1] => [4,3,2,1,1] => 1484
[7,3,3,3,3] => [3,3,3,3] => 294
[7,5,4,2,1] => [5,4,2,1] => 3535
[7,6,3,2,1] => [6,3,2,1] => 3424
[6,6,3,2,1] => [6,3,2,1] => 3424
[7,5,4,3] => [5,4,3] => 1270
[7,6,5] => [6,5] => 75
[6,2,2,2,2,2,2] => [2,2,2,2,2,2] => 90
[6,6,6] => [6,6] => 75
[7,6,6] => [6,6] => 75
[10,4,4] => [4,4] => 8
[10,10] => [10] => 0
[11,7,3] => [7,3] => 42
[8,2,2,2,2,2,2] => [2,2,2,2,2,2] => 90
[9,6,3] => [6,3] => 27
[8,6,4,2] => [6,4,2] => 1584
[8,6,4] => [6,4] => 51
[10,6,4] => [6,4] => 51
[15,5,5] => [5,5] => 24
[9,6,4] => [6,4] => 51
[10,7,3] => [7,3] => 42
[8,5,4,2] => [5,4,2] => 592
[9,7,4] => [7,4] => 93
[8,5,5,1] => [5,5,1] => 194
[9,5,5,1] => [5,5,1] => 194
[9,9] => [9] => 1
[7,7,1,1,1,1] => [7,1,1,1,1] => 130
[8,7,5] => [7,5] => 168
search for individual values
searching the database for the individual values of this statistic
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd.
The case of an even minimum is St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even..
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd.
The case of an even minimum is St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even..
Map
first row removal
Description
Removes the first entry of an integer partition
searching the database
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