Identifier
-
Mp00202:
Integer partitions
—first row removal⟶
Integer partitions
St000146: Integer partitions ⟶ ℤ
Values
[1] => [] => 0
[2] => [] => 0
[1,1] => [1] => -1
[3] => [] => 0
[2,1] => [1] => -1
[1,1,1] => [1,1] => -2
[4] => [] => 0
[3,1] => [1] => -1
[2,2] => [2] => 1
[2,1,1] => [1,1] => -2
[1,1,1,1] => [1,1,1] => -3
[5] => [] => 0
[4,1] => [1] => -1
[3,2] => [2] => 1
[3,1,1] => [1,1] => -2
[2,2,1] => [2,1] => 0
[2,1,1,1] => [1,1,1] => -3
[1,1,1,1,1] => [1,1,1,1] => -4
[6] => [] => 0
[5,1] => [1] => -1
[4,2] => [2] => 1
[4,1,1] => [1,1] => -2
[3,3] => [3] => 1
[3,2,1] => [2,1] => 0
[3,1,1,1] => [1,1,1] => -3
[2,2,2] => [2,2] => 2
[2,2,1,1] => [2,1,1] => -2
[2,1,1,1,1] => [1,1,1,1] => -4
[1,1,1,1,1,1] => [1,1,1,1,1] => -5
[7] => [] => 0
[6,1] => [1] => -1
[5,2] => [2] => 1
[5,1,1] => [1,1] => -2
[4,3] => [3] => 1
[4,2,1] => [2,1] => 0
[4,1,1,1] => [1,1,1] => -3
[3,3,1] => [3,1] => 0
[3,2,2] => [2,2] => 2
[3,2,1,1] => [2,1,1] => -2
[3,1,1,1,1] => [1,1,1,1] => -4
[2,2,2,1] => [2,2,1] => 1
[2,2,1,1,1] => [2,1,1,1] => -3
[2,1,1,1,1,1] => [1,1,1,1,1] => -5
[1,1,1,1,1,1,1] => [1,1,1,1,1,1] => -6
[8] => [] => 0
[7,1] => [1] => -1
[6,2] => [2] => 1
[6,1,1] => [1,1] => -2
[5,3] => [3] => 1
[5,2,1] => [2,1] => 0
[5,1,1,1] => [1,1,1] => -3
[4,4] => [4] => 1
[4,3,1] => [3,1] => 0
[4,2,2] => [2,2] => 2
[4,2,1,1] => [2,1,1] => -2
[4,1,1,1,1] => [1,1,1,1] => -4
[3,3,2] => [3,2] => 2
[3,3,1,1] => [3,1,1] => -1
[3,2,2,1] => [2,2,1] => 1
[3,2,1,1,1] => [2,1,1,1] => -3
[3,1,1,1,1,1] => [1,1,1,1,1] => -5
[2,2,2,2] => [2,2,2] => 3
[2,2,2,1,1] => [2,2,1,1] => -2
[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] => -6
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => -7
[9] => [] => 0
[8,1] => [1] => -1
[7,2] => [2] => 1
[7,1,1] => [1,1] => -2
[6,3] => [3] => 1
[6,2,1] => [2,1] => 0
[6,1,1,1] => [1,1,1] => -3
[5,4] => [4] => 1
[5,3,1] => [3,1] => 0
[5,2,2] => [2,2] => 2
[5,2,1,1] => [2,1,1] => -2
[5,1,1,1,1] => [1,1,1,1] => -4
[4,4,1] => [4,1] => 0
[4,3,2] => [3,2] => 2
[4,3,1,1] => [3,1,1] => -1
[4,2,2,1] => [2,2,1] => 1
[4,2,1,1,1] => [2,1,1,1] => -3
[4,1,1,1,1,1] => [1,1,1,1,1] => -5
[3,3,3] => [3,3] => 2
[3,3,2,1] => [3,2,1] => 1
[3,3,1,1,1] => [3,1,1,1] => -3
[3,2,2,2] => [2,2,2] => 3
[3,2,2,1,1] => [2,2,1,1] => -2
[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] => -6
[2,2,2,2,1] => [2,2,2,1] => 2
[2,2,2,1,1,1] => [2,2,1,1,1] => -3
[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] => -7
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => -8
[10] => [] => 0
[9,1] => [1] => -1
[8,2] => [2] => 1
[8,1,1] => [1,1] => -2
[7,3] => [3] => 1
>>> Load all 1170 entries. <<<[7,2,1] => [2,1] => 0
[7,1,1,1] => [1,1,1] => -3
[6,4] => [4] => 1
[6,3,1] => [3,1] => 0
[6,2,2] => [2,2] => 2
[6,2,1,1] => [2,1,1] => -2
[6,1,1,1,1] => [1,1,1,1] => -4
[5,5] => [5] => 1
[5,4,1] => [4,1] => 0
[5,3,2] => [3,2] => 2
[5,3,1,1] => [3,1,1] => -1
[5,2,2,1] => [2,2,1] => 1
[5,2,1,1,1] => [2,1,1,1] => -3
[5,1,1,1,1,1] => [1,1,1,1,1] => -5
[4,4,2] => [4,2] => 2
[4,4,1,1] => [4,1,1] => -1
[4,3,3] => [3,3] => 2
[4,3,2,1] => [3,2,1] => 1
[4,3,1,1,1] => [3,1,1,1] => -3
[4,2,2,2] => [2,2,2] => 3
[4,2,2,1,1] => [2,2,1,1] => -2
[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] => -6
[3,3,3,1] => [3,3,1] => 1
[3,3,2,2] => [3,2,2] => 3
[3,3,2,1,1] => [3,2,1,1] => -1
[3,3,1,1,1,1] => [3,1,1,1,1] => -4
[3,2,2,2,1] => [2,2,2,1] => 2
[3,2,2,1,1,1] => [2,2,1,1,1] => -3
[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] => -7
[2,2,2,2,2] => [2,2,2,2] => 4
[2,2,2,2,1,1] => [2,2,2,1,1] => -2
[2,2,2,1,1,1,1] => [2,2,1,1,1,1] => -4
[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] => -8
[1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => -9
[11] => [] => 0
[10,1] => [1] => -1
[9,2] => [2] => 1
[9,1,1] => [1,1] => -2
[8,3] => [3] => 1
[8,2,1] => [2,1] => 0
[8,1,1,1] => [1,1,1] => -3
[7,4] => [4] => 1
[7,3,1] => [3,1] => 0
[7,2,2] => [2,2] => 2
[7,2,1,1] => [2,1,1] => -2
[7,1,1,1,1] => [1,1,1,1] => -4
[6,5] => [5] => 1
[6,4,1] => [4,1] => 0
[6,3,2] => [3,2] => 2
[6,3,1,1] => [3,1,1] => -1
[6,2,2,1] => [2,2,1] => 1
[6,2,1,1,1] => [2,1,1,1] => -3
[6,1,1,1,1,1] => [1,1,1,1,1] => -5
[5,5,1] => [5,1] => 0
[5,4,2] => [4,2] => 2
[5,4,1,1] => [4,1,1] => -1
[5,3,3] => [3,3] => 2
[5,3,2,1] => [3,2,1] => 1
[5,3,1,1,1] => [3,1,1,1] => -3
[5,2,2,2] => [2,2,2] => 3
[5,2,2,1,1] => [2,2,1,1] => -2
[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] => -6
[4,4,3] => [4,3] => 2
[4,4,2,1] => [4,2,1] => 1
[4,4,1,1,1] => [4,1,1,1] => -2
[4,3,3,1] => [3,3,1] => 1
[4,3,2,2] => [3,2,2] => 3
[4,3,2,1,1] => [3,2,1,1] => -1
[4,3,1,1,1,1] => [3,1,1,1,1] => -4
[4,2,2,2,1] => [2,2,2,1] => 2
[4,2,2,1,1,1] => [2,2,1,1,1] => -3
[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] => -7
[3,3,3,2] => [3,3,2] => 3
[3,3,3,1,1] => [3,3,1,1] => 0
[3,3,2,2,1] => [3,2,2,1] => 2
[3,3,2,1,1,1] => [3,2,1,1,1] => -3
[3,3,1,1,1,1,1] => [3,1,1,1,1,1] => -5
[3,2,2,2,2] => [2,2,2,2] => 4
[3,2,2,2,1,1] => [2,2,2,1,1] => -2
[3,2,2,1,1,1,1] => [2,2,1,1,1,1] => -4
[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] => -8
[2,2,2,2,2,1] => [2,2,2,2,1] => 3
[2,2,2,2,1,1,1] => [2,2,2,1,1,1] => -3
[2,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => -5
[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] => -9
[1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => -10
[12] => [] => 0
[11,1] => [1] => -1
[10,2] => [2] => 1
[10,1,1] => [1,1] => -2
[9,3] => [3] => 1
[9,2,1] => [2,1] => 0
[9,1,1,1] => [1,1,1] => -3
[8,4] => [4] => 1
[8,3,1] => [3,1] => 0
[8,2,2] => [2,2] => 2
[8,2,1,1] => [2,1,1] => -2
[8,1,1,1,1] => [1,1,1,1] => -4
[7,5] => [5] => 1
[7,4,1] => [4,1] => 0
[7,3,2] => [3,2] => 2
[7,3,1,1] => [3,1,1] => -1
[7,2,2,1] => [2,2,1] => 1
[7,2,1,1,1] => [2,1,1,1] => -3
[7,1,1,1,1,1] => [1,1,1,1,1] => -5
[6,6] => [6] => 1
[6,5,1] => [5,1] => 0
[6,4,2] => [4,2] => 2
[6,4,1,1] => [4,1,1] => -1
[6,3,3] => [3,3] => 2
[6,3,2,1] => [3,2,1] => 1
[6,3,1,1,1] => [3,1,1,1] => -3
[6,2,2,2] => [2,2,2] => 3
[6,2,2,1,1] => [2,2,1,1] => -2
[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] => -6
[5,5,2] => [5,2] => 2
[5,5,1,1] => [5,1,1] => -1
[5,4,3] => [4,3] => 2
[5,4,2,1] => [4,2,1] => 1
[5,4,1,1,1] => [4,1,1,1] => -2
[5,3,3,1] => [3,3,1] => 1
[5,3,2,2] => [3,2,2] => 3
[5,3,2,1,1] => [3,2,1,1] => -1
[5,3,1,1,1,1] => [3,1,1,1,1] => -4
[5,2,2,2,1] => [2,2,2,1] => 2
[5,2,2,1,1,1] => [2,2,1,1,1] => -3
[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] => -7
[4,4,4] => [4,4] => 2
[4,4,3,1] => [4,3,1] => 1
[4,4,2,2] => [4,2,2] => 3
[4,4,2,1,1] => [4,2,1,1] => -1
[4,4,1,1,1,1] => [4,1,1,1,1] => -4
[4,3,3,2] => [3,3,2] => 3
[4,3,3,1,1] => [3,3,1,1] => 0
[4,3,2,2,1] => [3,2,2,1] => 2
[4,3,2,1,1,1] => [3,2,1,1,1] => -3
[4,3,1,1,1,1,1] => [3,1,1,1,1,1] => -5
[4,2,2,2,2] => [2,2,2,2] => 4
[4,2,2,2,1,1] => [2,2,2,1,1] => -2
[4,2,2,1,1,1,1] => [2,2,1,1,1,1] => -4
[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] => -8
[3,3,3,3] => [3,3,3] => 3
[3,3,3,2,1] => [3,3,2,1] => 2
[3,3,3,1,1,1] => [3,3,1,1,1] => -3
[3,3,2,2,2] => [3,2,2,2] => 4
[3,3,2,2,1,1] => [3,2,2,1,1] => -1
[3,3,2,1,1,1,1] => [3,2,1,1,1,1] => -4
[3,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => -6
[3,2,2,2,2,1] => [2,2,2,2,1] => 3
[3,2,2,2,1,1,1] => [2,2,2,1,1,1] => -3
[3,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => -5
[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] => -9
[2,2,2,2,2,2] => [2,2,2,2,2] => 5
[2,2,2,2,2,1,1] => [2,2,2,2,1,1] => -2
[2,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => -4
[2,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => -6
[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] => -10
[13] => [] => 0
[12,1] => [1] => -1
[11,2] => [2] => 1
[11,1,1] => [1,1] => -2
[10,3] => [3] => 1
[10,2,1] => [2,1] => 0
[10,1,1,1] => [1,1,1] => -3
[9,4] => [4] => 1
[9,3,1] => [3,1] => 0
[9,2,2] => [2,2] => 2
[9,2,1,1] => [2,1,1] => -2
[9,1,1,1,1] => [1,1,1,1] => -4
[8,5] => [5] => 1
[8,4,1] => [4,1] => 0
[8,3,2] => [3,2] => 2
[8,3,1,1] => [3,1,1] => -1
[8,2,2,1] => [2,2,1] => 1
[8,2,1,1,1] => [2,1,1,1] => -3
[8,1,1,1,1,1] => [1,1,1,1,1] => -5
[7,6] => [6] => 1
[7,5,1] => [5,1] => 0
[7,4,2] => [4,2] => 2
[7,4,1,1] => [4,1,1] => -1
[7,3,3] => [3,3] => 2
[7,3,2,1] => [3,2,1] => 1
[7,3,1,1,1] => [3,1,1,1] => -3
[7,2,2,2] => [2,2,2] => 3
[7,2,2,1,1] => [2,2,1,1] => -2
[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] => -6
[6,6,1] => [6,1] => 0
[6,5,2] => [5,2] => 2
[6,5,1,1] => [5,1,1] => -1
[6,4,3] => [4,3] => 2
[6,4,2,1] => [4,2,1] => 1
[6,4,1,1,1] => [4,1,1,1] => -2
[6,3,3,1] => [3,3,1] => 1
[6,3,2,2] => [3,2,2] => 3
[6,3,2,1,1] => [3,2,1,1] => -1
[6,3,1,1,1,1] => [3,1,1,1,1] => -4
[6,2,2,2,1] => [2,2,2,1] => 2
[6,2,2,1,1,1] => [2,2,1,1,1] => -3
[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] => -7
[5,5,3] => [5,3] => 2
[5,5,2,1] => [5,2,1] => 1
[5,5,1,1,1] => [5,1,1,1] => -2
[5,4,4] => [4,4] => 2
[5,4,3,1] => [4,3,1] => 1
[5,4,2,2] => [4,2,2] => 3
[5,4,2,1,1] => [4,2,1,1] => -1
[5,4,1,1,1,1] => [4,1,1,1,1] => -4
[5,3,3,2] => [3,3,2] => 3
[5,3,3,1,1] => [3,3,1,1] => 0
[5,3,2,2,1] => [3,2,2,1] => 2
[5,3,2,1,1,1] => [3,2,1,1,1] => -3
[5,3,1,1,1,1,1] => [3,1,1,1,1,1] => -5
[5,2,2,2,2] => [2,2,2,2] => 4
[5,2,2,2,1,1] => [2,2,2,1,1] => -2
[5,2,2,1,1,1,1] => [2,2,1,1,1,1] => -4
[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] => -8
[4,4,4,1] => [4,4,1] => 1
[4,4,3,2] => [4,3,2] => 3
[4,4,3,1,1] => [4,3,1,1] => 0
[4,4,2,2,1] => [4,2,2,1] => 2
[4,4,2,1,1,1] => [4,2,1,1,1] => -2
[4,4,1,1,1,1,1] => [4,1,1,1,1,1] => -5
[4,3,3,3] => [3,3,3] => 3
[4,3,3,2,1] => [3,3,2,1] => 2
[4,3,3,1,1,1] => [3,3,1,1,1] => -3
[4,3,2,2,2] => [3,2,2,2] => 4
[4,3,2,2,1,1] => [3,2,2,1,1] => -1
[4,3,2,1,1,1,1] => [3,2,1,1,1,1] => -4
[4,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => -6
[4,2,2,2,2,1] => [2,2,2,2,1] => 3
[4,2,2,2,1,1,1] => [2,2,2,1,1,1] => -3
[4,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => -5
[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] => -9
[3,3,3,3,1] => [3,3,3,1] => 2
[3,3,3,2,2] => [3,3,2,2] => 4
[3,3,3,2,1,1] => [3,3,2,1,1] => 0
[3,3,3,1,1,1,1] => [3,3,1,1,1,1] => -4
[3,3,2,2,2,1] => [3,2,2,2,1] => 3
[3,3,2,2,1,1,1] => [3,2,2,1,1,1] => -3
[3,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => -5
[3,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => -7
[3,2,2,2,2,2] => [2,2,2,2,2] => 5
[3,2,2,2,2,1,1] => [2,2,2,2,1,1] => -2
[3,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => -4
[3,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => -6
[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] => -10
[2,2,2,2,2,2,1] => [2,2,2,2,2,1] => 4
[14] => [] => 0
[13,1] => [1] => -1
[12,2] => [2] => 1
[12,1,1] => [1,1] => -2
[11,3] => [3] => 1
[11,2,1] => [2,1] => 0
[11,1,1,1] => [1,1,1] => -3
[10,4] => [4] => 1
[10,3,1] => [3,1] => 0
[10,2,2] => [2,2] => 2
[10,2,1,1] => [2,1,1] => -2
[10,1,1,1,1] => [1,1,1,1] => -4
[9,5] => [5] => 1
[9,4,1] => [4,1] => 0
[9,3,2] => [3,2] => 2
[9,3,1,1] => [3,1,1] => -1
[9,2,2,1] => [2,2,1] => 1
[9,2,1,1,1] => [2,1,1,1] => -3
[9,1,1,1,1,1] => [1,1,1,1,1] => -5
[8,6] => [6] => 1
[8,5,1] => [5,1] => 0
[8,4,2] => [4,2] => 2
[8,4,1,1] => [4,1,1] => -1
[8,3,3] => [3,3] => 2
[8,3,2,1] => [3,2,1] => 1
[8,3,1,1,1] => [3,1,1,1] => -3
[8,2,2,2] => [2,2,2] => 3
[8,2,2,1,1] => [2,2,1,1] => -2
[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] => -6
[7,7] => [7] => 1
[7,6,1] => [6,1] => 0
[7,5,2] => [5,2] => 2
[7,5,1,1] => [5,1,1] => -1
[7,4,3] => [4,3] => 2
[7,4,2,1] => [4,2,1] => 1
[7,4,1,1,1] => [4,1,1,1] => -2
[7,3,3,1] => [3,3,1] => 1
[7,3,2,2] => [3,2,2] => 3
[7,3,2,1,1] => [3,2,1,1] => -1
[7,3,1,1,1,1] => [3,1,1,1,1] => -4
[7,2,2,2,1] => [2,2,2,1] => 2
[7,2,2,1,1,1] => [2,2,1,1,1] => -3
[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] => -7
[6,6,2] => [6,2] => 2
[6,6,1,1] => [6,1,1] => -1
[6,5,3] => [5,3] => 2
[6,5,2,1] => [5,2,1] => 1
[6,5,1,1,1] => [5,1,1,1] => -2
[6,4,4] => [4,4] => 2
[6,4,3,1] => [4,3,1] => 1
[6,4,2,2] => [4,2,2] => 3
[6,4,2,1,1] => [4,2,1,1] => -1
[6,4,1,1,1,1] => [4,1,1,1,1] => -4
[6,3,3,2] => [3,3,2] => 3
[6,3,3,1,1] => [3,3,1,1] => 0
[6,3,2,2,1] => [3,2,2,1] => 2
[6,3,2,1,1,1] => [3,2,1,1,1] => -3
[6,3,1,1,1,1,1] => [3,1,1,1,1,1] => -5
[6,2,2,2,2] => [2,2,2,2] => 4
[6,2,2,2,1,1] => [2,2,2,1,1] => -2
[6,2,2,1,1,1,1] => [2,2,1,1,1,1] => -4
[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] => -8
[5,5,4] => [5,4] => 2
[5,5,3,1] => [5,3,1] => 1
[5,5,2,2] => [5,2,2] => 3
[5,5,2,1,1] => [5,2,1,1] => -1
[5,5,1,1,1,1] => [5,1,1,1,1] => -3
[5,4,4,1] => [4,4,1] => 1
[5,4,3,2] => [4,3,2] => 3
[5,4,3,1,1] => [4,3,1,1] => 0
[5,4,2,2,1] => [4,2,2,1] => 2
[5,4,2,1,1,1] => [4,2,1,1,1] => -2
[5,4,1,1,1,1,1] => [4,1,1,1,1,1] => -5
[5,3,3,3] => [3,3,3] => 3
[5,3,3,2,1] => [3,3,2,1] => 2
[5,3,3,1,1,1] => [3,3,1,1,1] => -3
[5,3,2,2,2] => [3,2,2,2] => 4
[5,3,2,2,1,1] => [3,2,2,1,1] => -1
[5,3,2,1,1,1,1] => [3,2,1,1,1,1] => -4
[5,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => -6
[5,2,2,2,2,1] => [2,2,2,2,1] => 3
[5,2,2,2,1,1,1] => [2,2,2,1,1,1] => -3
[5,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => -5
[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] => -9
[4,4,4,2] => [4,4,2] => 3
[4,4,4,1,1] => [4,4,1,1] => 0
[4,4,3,3] => [4,3,3] => 3
[4,4,3,2,1] => [4,3,2,1] => 2
[4,4,3,1,1,1] => [4,3,1,1,1] => -2
[4,4,2,2,2] => [4,2,2,2] => 4
[4,4,2,2,1,1] => [4,2,2,1,1] => -1
[4,4,2,1,1,1,1] => [4,2,1,1,1,1] => -4
[4,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => -6
[4,3,3,3,1] => [3,3,3,1] => 2
[4,3,3,2,2] => [3,3,2,2] => 4
[4,3,3,2,1,1] => [3,3,2,1,1] => 0
[4,3,3,1,1,1,1] => [3,3,1,1,1,1] => -4
[4,3,2,2,2,1] => [3,2,2,2,1] => 3
[4,3,2,2,1,1,1] => [3,2,2,1,1,1] => -3
[4,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => -5
[4,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => -7
[4,2,2,2,2,2] => [2,2,2,2,2] => 5
[4,2,2,2,2,1,1] => [2,2,2,2,1,1] => -2
[4,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => -4
[4,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => -6
[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] => -10
[3,3,3,3,2] => [3,3,3,2] => 4
[3,3,3,3,1,1] => [3,3,3,1,1] => 1
[3,3,3,2,2,1] => [3,3,2,2,1] => 3
[3,3,2,2,2,2] => [3,2,2,2,2] => 5
[3,2,2,2,2,2,1] => [2,2,2,2,2,1] => 4
[2,2,2,2,2,2,2] => [2,2,2,2,2,2] => 6
[15] => [] => 0
[14,1] => [1] => -1
[13,2] => [2] => 1
[13,1,1] => [1,1] => -2
[12,3] => [3] => 1
[12,2,1] => [2,1] => 0
[12,1,1,1] => [1,1,1] => -3
[11,4] => [4] => 1
[11,3,1] => [3,1] => 0
[11,2,2] => [2,2] => 2
[11,2,1,1] => [2,1,1] => -2
[11,1,1,1,1] => [1,1,1,1] => -4
[10,5] => [5] => 1
[10,4,1] => [4,1] => 0
[10,3,2] => [3,2] => 2
[10,3,1,1] => [3,1,1] => -1
[10,2,2,1] => [2,2,1] => 1
[10,2,1,1,1] => [2,1,1,1] => -3
[10,1,1,1,1,1] => [1,1,1,1,1] => -5
[9,6] => [6] => 1
[9,5,1] => [5,1] => 0
[9,4,2] => [4,2] => 2
[9,4,1,1] => [4,1,1] => -1
[9,3,3] => [3,3] => 2
[9,3,2,1] => [3,2,1] => 1
[9,3,1,1,1] => [3,1,1,1] => -3
[9,2,2,2] => [2,2,2] => 3
[9,2,2,1,1] => [2,2,1,1] => -2
[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] => -6
[8,7] => [7] => 1
[8,6,1] => [6,1] => 0
[8,5,2] => [5,2] => 2
[8,5,1,1] => [5,1,1] => -1
[8,4,3] => [4,3] => 2
[8,4,2,1] => [4,2,1] => 1
[8,4,1,1,1] => [4,1,1,1] => -2
[8,3,3,1] => [3,3,1] => 1
[8,3,2,2] => [3,2,2] => 3
[8,3,2,1,1] => [3,2,1,1] => -1
[8,3,1,1,1,1] => [3,1,1,1,1] => -4
[8,2,2,2,1] => [2,2,2,1] => 2
[8,2,2,1,1,1] => [2,2,1,1,1] => -3
[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] => -7
[7,7,1] => [7,1] => 0
[7,6,2] => [6,2] => 2
[7,6,1,1] => [6,1,1] => -1
[7,5,3] => [5,3] => 2
[7,5,2,1] => [5,2,1] => 1
[7,5,1,1,1] => [5,1,1,1] => -2
[7,4,4] => [4,4] => 2
[7,4,3,1] => [4,3,1] => 1
[7,4,2,2] => [4,2,2] => 3
[7,4,2,1,1] => [4,2,1,1] => -1
[7,4,1,1,1,1] => [4,1,1,1,1] => -4
[7,3,3,2] => [3,3,2] => 3
[7,3,3,1,1] => [3,3,1,1] => 0
[7,3,2,2,1] => [3,2,2,1] => 2
[7,3,2,1,1,1] => [3,2,1,1,1] => -3
[7,3,1,1,1,1,1] => [3,1,1,1,1,1] => -5
[7,2,2,2,2] => [2,2,2,2] => 4
[7,2,2,2,1,1] => [2,2,2,1,1] => -2
[7,2,2,1,1,1,1] => [2,2,1,1,1,1] => -4
[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] => -8
[6,6,3] => [6,3] => 2
[6,6,2,1] => [6,2,1] => 1
[6,6,1,1,1] => [6,1,1,1] => -2
[6,5,4] => [5,4] => 2
[6,5,3,1] => [5,3,1] => 1
[6,5,2,2] => [5,2,2] => 3
[6,5,2,1,1] => [5,2,1,1] => -1
[6,5,1,1,1,1] => [5,1,1,1,1] => -3
[6,4,4,1] => [4,4,1] => 1
[6,4,3,2] => [4,3,2] => 3
[6,4,3,1,1] => [4,3,1,1] => 0
[6,4,2,2,1] => [4,2,2,1] => 2
[6,4,2,1,1,1] => [4,2,1,1,1] => -2
[6,4,1,1,1,1,1] => [4,1,1,1,1,1] => -5
[6,3,3,3] => [3,3,3] => 3
[6,3,3,2,1] => [3,3,2,1] => 2
[6,3,3,1,1,1] => [3,3,1,1,1] => -3
[6,3,2,2,2] => [3,2,2,2] => 4
[6,3,2,2,1,1] => [3,2,2,1,1] => -1
[6,3,2,1,1,1,1] => [3,2,1,1,1,1] => -4
[6,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => -6
[6,2,2,2,2,1] => [2,2,2,2,1] => 3
[6,2,2,2,1,1,1] => [2,2,2,1,1,1] => -3
[6,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => -5
[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] => -9
[5,5,5] => [5,5] => 2
[5,5,4,1] => [5,4,1] => 1
[5,5,3,2] => [5,3,2] => 3
[5,5,3,1,1] => [5,3,1,1] => 0
[5,5,2,2,1] => [5,2,2,1] => 2
[5,5,2,1,1,1] => [5,2,1,1,1] => -2
[5,5,1,1,1,1,1] => [5,1,1,1,1,1] => -5
[5,4,4,2] => [4,4,2] => 3
[5,4,4,1,1] => [4,4,1,1] => 0
[5,4,3,3] => [4,3,3] => 3
[5,4,3,2,1] => [4,3,2,1] => 2
[5,4,3,1,1,1] => [4,3,1,1,1] => -2
[5,4,2,2,2] => [4,2,2,2] => 4
[5,4,2,2,1,1] => [4,2,2,1,1] => -1
[5,4,2,1,1,1,1] => [4,2,1,1,1,1] => -4
[5,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => -6
[5,3,3,3,1] => [3,3,3,1] => 2
[5,3,3,2,2] => [3,3,2,2] => 4
[5,3,3,2,1,1] => [3,3,2,1,1] => 0
[5,3,3,1,1,1,1] => [3,3,1,1,1,1] => -4
[5,3,2,2,2,1] => [3,2,2,2,1] => 3
[5,3,2,2,1,1,1] => [3,2,2,1,1,1] => -3
[5,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => -5
[5,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => -7
[5,2,2,2,2,2] => [2,2,2,2,2] => 5
[5,2,2,2,2,1,1] => [2,2,2,2,1,1] => -2
[5,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => -4
[5,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => -6
[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] => -10
[4,4,4,3] => [4,4,3] => 3
[4,4,4,2,1] => [4,4,2,1] => 2
[4,4,4,1,1,1] => [4,4,1,1,1] => -1
[4,4,3,3,1] => [4,3,3,1] => 2
[4,4,3,2,2] => [4,3,2,2] => 4
[4,4,3,2,1,1] => [4,3,2,1,1] => 0
[4,4,2,2,2,1] => [4,2,2,2,1] => 3
[4,3,3,3,2] => [3,3,3,2] => 4
[4,3,3,3,1,1] => [3,3,3,1,1] => 1
[4,3,3,2,2,1] => [3,3,2,2,1] => 3
[4,3,2,2,2,2] => [3,2,2,2,2] => 5
[4,2,2,2,2,2,1] => [2,2,2,2,2,1] => 4
[3,3,3,3,3] => [3,3,3,3] => 4
[3,3,3,3,2,1] => [3,3,3,2,1] => 3
[3,3,3,2,2,2] => [3,3,2,2,2] => 5
[3,3,3,2,2,1,1] => [3,3,2,2,1,1] => 0
[3,3,2,2,2,2,1] => [3,2,2,2,2,1] => 4
[3,2,2,2,2,2,2] => [2,2,2,2,2,2] => 6
[16] => [] => 0
[15,1] => [1] => -1
[14,2] => [2] => 1
[14,1,1] => [1,1] => -2
[13,3] => [3] => 1
[13,2,1] => [2,1] => 0
[13,1,1,1] => [1,1,1] => -3
[12,4] => [4] => 1
[12,3,1] => [3,1] => 0
[12,2,2] => [2,2] => 2
[12,2,1,1] => [2,1,1] => -2
[12,1,1,1,1] => [1,1,1,1] => -4
[11,5] => [5] => 1
[11,4,1] => [4,1] => 0
[11,3,2] => [3,2] => 2
[11,3,1,1] => [3,1,1] => -1
[11,2,2,1] => [2,2,1] => 1
[11,2,1,1,1] => [2,1,1,1] => -3
[11,1,1,1,1,1] => [1,1,1,1,1] => -5
[10,6] => [6] => 1
[10,5,1] => [5,1] => 0
[10,4,2] => [4,2] => 2
[10,4,1,1] => [4,1,1] => -1
[10,3,3] => [3,3] => 2
[10,3,2,1] => [3,2,1] => 1
[10,3,1,1,1] => [3,1,1,1] => -3
[10,2,2,2] => [2,2,2] => 3
[10,2,2,1,1] => [2,2,1,1] => -2
[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] => -6
[9,7] => [7] => 1
[9,6,1] => [6,1] => 0
[9,5,2] => [5,2] => 2
[9,5,1,1] => [5,1,1] => -1
[9,4,3] => [4,3] => 2
[9,4,2,1] => [4,2,1] => 1
[9,4,1,1,1] => [4,1,1,1] => -2
[9,3,3,1] => [3,3,1] => 1
[9,3,2,2] => [3,2,2] => 3
[9,3,2,1,1] => [3,2,1,1] => -1
[9,3,1,1,1,1] => [3,1,1,1,1] => -4
[9,2,2,2,1] => [2,2,2,1] => 2
[9,2,2,1,1,1] => [2,2,1,1,1] => -3
[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] => -7
[8,8] => [8] => 1
[8,7,1] => [7,1] => 0
[8,6,2] => [6,2] => 2
[8,6,1,1] => [6,1,1] => -1
[8,5,3] => [5,3] => 2
[8,5,2,1] => [5,2,1] => 1
[8,5,1,1,1] => [5,1,1,1] => -2
[8,4,4] => [4,4] => 2
[8,4,3,1] => [4,3,1] => 1
[8,4,2,2] => [4,2,2] => 3
[8,4,2,1,1] => [4,2,1,1] => -1
[8,4,1,1,1,1] => [4,1,1,1,1] => -4
[8,3,3,2] => [3,3,2] => 3
[8,3,3,1,1] => [3,3,1,1] => 0
[8,3,2,2,1] => [3,2,2,1] => 2
[8,3,2,1,1,1] => [3,2,1,1,1] => -3
[8,3,1,1,1,1,1] => [3,1,1,1,1,1] => -5
[8,2,2,2,2] => [2,2,2,2] => 4
[8,2,2,2,1,1] => [2,2,2,1,1] => -2
[8,2,2,1,1,1,1] => [2,2,1,1,1,1] => -4
[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] => -8
[7,7,2] => [7,2] => 2
[7,7,1,1] => [7,1,1] => -1
[7,6,3] => [6,3] => 2
[7,6,2,1] => [6,2,1] => 1
[7,6,1,1,1] => [6,1,1,1] => -2
[7,5,4] => [5,4] => 2
[7,5,3,1] => [5,3,1] => 1
[7,5,2,2] => [5,2,2] => 3
[7,5,2,1,1] => [5,2,1,1] => -1
[7,5,1,1,1,1] => [5,1,1,1,1] => -3
[7,4,4,1] => [4,4,1] => 1
[7,4,3,2] => [4,3,2] => 3
[7,4,3,1,1] => [4,3,1,1] => 0
[7,4,2,2,1] => [4,2,2,1] => 2
[7,4,2,1,1,1] => [4,2,1,1,1] => -2
[7,4,1,1,1,1,1] => [4,1,1,1,1,1] => -5
[7,3,3,3] => [3,3,3] => 3
[7,3,3,2,1] => [3,3,2,1] => 2
[7,3,3,1,1,1] => [3,3,1,1,1] => -3
[7,3,2,2,2] => [3,2,2,2] => 4
[7,3,2,2,1,1] => [3,2,2,1,1] => -1
[7,3,2,1,1,1,1] => [3,2,1,1,1,1] => -4
[7,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => -6
[7,2,2,2,2,1] => [2,2,2,2,1] => 3
[7,2,2,2,1,1,1] => [2,2,2,1,1,1] => -3
[7,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => -5
[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] => -9
[6,6,4] => [6,4] => 2
[6,6,3,1] => [6,3,1] => 1
[6,6,2,2] => [6,2,2] => 3
[6,6,2,1,1] => [6,2,1,1] => -1
[6,6,1,1,1,1] => [6,1,1,1,1] => -3
[6,5,5] => [5,5] => 2
[6,5,4,1] => [5,4,1] => 1
[6,5,3,2] => [5,3,2] => 3
[6,5,3,1,1] => [5,3,1,1] => 0
[6,5,2,2,1] => [5,2,2,1] => 2
[6,5,2,1,1,1] => [5,2,1,1,1] => -2
[6,5,1,1,1,1,1] => [5,1,1,1,1,1] => -5
[6,4,4,2] => [4,4,2] => 3
[6,4,4,1,1] => [4,4,1,1] => 0
[6,4,3,3] => [4,3,3] => 3
[6,4,3,2,1] => [4,3,2,1] => 2
[6,4,3,1,1,1] => [4,3,1,1,1] => -2
[6,4,2,2,2] => [4,2,2,2] => 4
[6,4,2,2,1,1] => [4,2,2,1,1] => -1
[6,4,2,1,1,1,1] => [4,2,1,1,1,1] => -4
[6,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => -6
[6,3,3,3,1] => [3,3,3,1] => 2
[6,3,3,2,2] => [3,3,2,2] => 4
[6,3,3,2,1,1] => [3,3,2,1,1] => 0
[6,3,3,1,1,1,1] => [3,3,1,1,1,1] => -4
[6,3,2,2,2,1] => [3,2,2,2,1] => 3
[6,3,2,2,1,1,1] => [3,2,2,1,1,1] => -3
[6,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => -5
[6,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => -7
[6,2,2,2,2,2] => [2,2,2,2,2] => 5
[6,2,2,2,2,1,1] => [2,2,2,2,1,1] => -2
[6,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => -4
[6,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => -6
[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] => -10
[5,5,5,1] => [5,5,1] => 1
[5,5,4,2] => [5,4,2] => 3
[5,5,4,1,1] => [5,4,1,1] => 0
[5,5,3,3] => [5,3,3] => 3
[5,5,3,2,1] => [5,3,2,1] => 2
[5,5,3,1,1,1] => [5,3,1,1,1] => -2
[5,5,2,2,2] => [5,2,2,2] => 4
[5,5,2,2,1,1] => [5,2,2,1,1] => -1
[5,5,2,1,1,1,1] => [5,2,1,1,1,1] => -3
[5,4,4,3] => [4,4,3] => 3
[5,4,4,2,1] => [4,4,2,1] => 2
[5,4,4,1,1,1] => [4,4,1,1,1] => -1
[5,4,3,3,1] => [4,3,3,1] => 2
[5,4,3,2,2] => [4,3,2,2] => 4
[5,4,3,2,1,1] => [4,3,2,1,1] => 0
[5,4,2,2,2,1] => [4,2,2,2,1] => 3
[5,3,3,3,2] => [3,3,3,2] => 4
[5,3,3,3,1,1] => [3,3,3,1,1] => 1
[5,3,3,2,2,1] => [3,3,2,2,1] => 3
[5,3,2,2,2,2] => [3,2,2,2,2] => 5
[5,2,2,2,2,2,1] => [2,2,2,2,2,1] => 4
[4,4,4,4] => [4,4,4] => 3
[4,4,4,3,1] => [4,4,3,1] => 2
[4,4,4,2,2] => [4,4,2,2] => 4
[4,4,4,2,1,1] => [4,4,2,1,1] => 0
[4,4,3,3,2] => [4,3,3,2] => 4
[4,4,3,3,1,1] => [4,3,3,1,1] => 1
[4,4,3,2,2,1] => [4,3,2,2,1] => 3
[4,3,3,3,3] => [3,3,3,3] => 4
[4,3,3,3,2,1] => [3,3,3,2,1] => 3
[4,3,3,2,2,2] => [3,3,2,2,2] => 5
[4,3,3,2,2,1,1] => [3,3,2,2,1,1] => 0
[4,3,2,2,2,2,1] => [3,2,2,2,2,1] => 4
[4,2,2,2,2,2,2] => [2,2,2,2,2,2] => 6
[3,3,3,3,3,1] => [3,3,3,3,1] => 3
[3,3,3,3,2,2] => [3,3,3,2,2] => 5
[3,3,3,2,2,2,1] => [3,3,2,2,2,1] => 4
[17] => [] => 0
[16,1] => [1] => -1
[15,2] => [2] => 1
[15,1,1] => [1,1] => -2
[14,3] => [3] => 1
[14,2,1] => [2,1] => 0
[14,1,1,1] => [1,1,1] => -3
[13,4] => [4] => 1
[13,3,1] => [3,1] => 0
[13,2,2] => [2,2] => 2
[13,2,1,1] => [2,1,1] => -2
[13,1,1,1,1] => [1,1,1,1] => -4
[12,5] => [5] => 1
[12,4,1] => [4,1] => 0
[12,3,2] => [3,2] => 2
[12,3,1,1] => [3,1,1] => -1
[12,2,2,1] => [2,2,1] => 1
[12,2,1,1,1] => [2,1,1,1] => -3
[12,1,1,1,1,1] => [1,1,1,1,1] => -5
[11,6] => [6] => 1
[11,5,1] => [5,1] => 0
[11,4,2] => [4,2] => 2
[11,4,1,1] => [4,1,1] => -1
[11,3,3] => [3,3] => 2
[11,3,2,1] => [3,2,1] => 1
[11,3,1,1,1] => [3,1,1,1] => -3
[11,2,2,2] => [2,2,2] => 3
[11,2,2,1,1] => [2,2,1,1] => -2
[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] => -6
[10,7] => [7] => 1
[10,6,1] => [6,1] => 0
[10,5,2] => [5,2] => 2
[10,5,1,1] => [5,1,1] => -1
[10,4,3] => [4,3] => 2
[10,4,2,1] => [4,2,1] => 1
[10,4,1,1,1] => [4,1,1,1] => -2
[10,3,3,1] => [3,3,1] => 1
[10,3,2,2] => [3,2,2] => 3
[10,3,2,1,1] => [3,2,1,1] => -1
[10,3,1,1,1,1] => [3,1,1,1,1] => -4
[10,2,2,2,1] => [2,2,2,1] => 2
[10,2,2,1,1,1] => [2,2,1,1,1] => -3
[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] => -7
[9,8] => [8] => 1
[9,7,1] => [7,1] => 0
[9,6,2] => [6,2] => 2
[9,6,1,1] => [6,1,1] => -1
[9,5,3] => [5,3] => 2
[9,5,2,1] => [5,2,1] => 1
[9,5,1,1,1] => [5,1,1,1] => -2
[9,4,4] => [4,4] => 2
[9,4,3,1] => [4,3,1] => 1
[9,4,2,2] => [4,2,2] => 3
[9,4,2,1,1] => [4,2,1,1] => -1
[9,4,1,1,1,1] => [4,1,1,1,1] => -4
[9,3,3,2] => [3,3,2] => 3
[9,3,3,1,1] => [3,3,1,1] => 0
[9,3,2,2,1] => [3,2,2,1] => 2
[9,3,2,1,1,1] => [3,2,1,1,1] => -3
[9,3,1,1,1,1,1] => [3,1,1,1,1,1] => -5
[9,2,2,2,2] => [2,2,2,2] => 4
[9,2,2,2,1,1] => [2,2,2,1,1] => -2
[9,2,2,1,1,1,1] => [2,2,1,1,1,1] => -4
[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] => -8
[8,8,1] => [8,1] => 0
[8,7,2] => [7,2] => 2
[8,7,1,1] => [7,1,1] => -1
[8,6,3] => [6,3] => 2
[8,6,2,1] => [6,2,1] => 1
[8,6,1,1,1] => [6,1,1,1] => -2
[8,5,4] => [5,4] => 2
[8,5,3,1] => [5,3,1] => 1
[8,5,2,2] => [5,2,2] => 3
[8,5,2,1,1] => [5,2,1,1] => -1
[8,5,1,1,1,1] => [5,1,1,1,1] => -3
[8,4,4,1] => [4,4,1] => 1
[8,4,3,2] => [4,3,2] => 3
[8,4,3,1,1] => [4,3,1,1] => 0
[8,4,2,2,1] => [4,2,2,1] => 2
[8,4,2,1,1,1] => [4,2,1,1,1] => -2
[8,4,1,1,1,1,1] => [4,1,1,1,1,1] => -5
[8,3,3,3] => [3,3,3] => 3
[8,3,3,2,1] => [3,3,2,1] => 2
[8,3,3,1,1,1] => [3,3,1,1,1] => -3
[8,3,2,2,2] => [3,2,2,2] => 4
[8,3,2,2,1,1] => [3,2,2,1,1] => -1
[8,3,2,1,1,1,1] => [3,2,1,1,1,1] => -4
[8,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => -6
[8,2,2,2,2,1] => [2,2,2,2,1] => 3
[8,2,2,2,1,1,1] => [2,2,2,1,1,1] => -3
[8,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => -5
[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] => -9
[7,7,3] => [7,3] => 2
[7,7,2,1] => [7,2,1] => 1
[7,7,1,1,1] => [7,1,1,1] => -2
[7,6,4] => [6,4] => 2
[7,6,3,1] => [6,3,1] => 1
[7,6,2,2] => [6,2,2] => 3
[7,6,2,1,1] => [6,2,1,1] => -1
[7,6,1,1,1,1] => [6,1,1,1,1] => -3
[7,5,5] => [5,5] => 2
[7,5,4,1] => [5,4,1] => 1
[7,5,3,2] => [5,3,2] => 3
[7,5,3,1,1] => [5,3,1,1] => 0
[7,5,2,2,1] => [5,2,2,1] => 2
[7,5,2,1,1,1] => [5,2,1,1,1] => -2
[7,5,1,1,1,1,1] => [5,1,1,1,1,1] => -5
[7,4,4,2] => [4,4,2] => 3
[7,4,4,1,1] => [4,4,1,1] => 0
[7,4,3,3] => [4,3,3] => 3
[7,4,3,2,1] => [4,3,2,1] => 2
[7,4,3,1,1,1] => [4,3,1,1,1] => -2
[7,4,2,2,2] => [4,2,2,2] => 4
[7,4,2,2,1,1] => [4,2,2,1,1] => -1
[7,4,2,1,1,1,1] => [4,2,1,1,1,1] => -4
[7,4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => -6
[7,3,3,3,1] => [3,3,3,1] => 2
[7,3,3,2,2] => [3,3,2,2] => 4
[7,3,3,2,1,1] => [3,3,2,1,1] => 0
[7,3,3,1,1,1,1] => [3,3,1,1,1,1] => -4
[7,3,2,2,2,1] => [3,2,2,2,1] => 3
[7,3,2,2,1,1,1] => [3,2,2,1,1,1] => -3
[7,3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => -5
[7,3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => -7
[7,2,2,2,2,2] => [2,2,2,2,2] => 5
[7,2,2,2,2,1,1] => [2,2,2,2,1,1] => -2
[7,2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => -4
[7,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => -6
[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] => -10
[6,6,5] => [6,5] => 2
[6,6,4,1] => [6,4,1] => 1
[6,6,1,1,1,1,1] => [6,1,1,1,1,1] => -4
[6,5,5,1] => [5,5,1] => 1
[6,5,4,2] => [5,4,2] => 3
[6,5,4,1,1] => [5,4,1,1] => 0
[6,5,3,3] => [5,3,3] => 3
[6,5,3,2,1] => [5,3,2,1] => 2
[6,5,3,1,1,1] => [5,3,1,1,1] => -2
[6,5,2,2,2] => [5,2,2,2] => 4
[6,5,2,2,1,1] => [5,2,2,1,1] => -1
[6,5,2,1,1,1,1] => [5,2,1,1,1,1] => -3
[6,4,4,3] => [4,4,3] => 3
[6,4,4,2,1] => [4,4,2,1] => 2
[6,4,4,1,1,1] => [4,4,1,1,1] => -1
[6,4,3,3,1] => [4,3,3,1] => 2
[6,4,3,2,2] => [4,3,2,2] => 4
[6,4,3,2,1,1] => [4,3,2,1,1] => 0
[6,4,2,2,2,1] => [4,2,2,2,1] => 3
[6,3,3,3,2] => [3,3,3,2] => 4
[6,3,3,3,1,1] => [3,3,3,1,1] => 1
[6,3,3,2,2,1] => [3,3,2,2,1] => 3
[6,3,2,2,2,2] => [3,2,2,2,2] => 5
[6,2,2,2,2,2,1] => [2,2,2,2,2,1] => 4
[5,5,5,2] => [5,5,2] => 3
[5,5,4,3] => [5,4,3] => 3
[5,5,4,2,1] => [5,4,2,1] => 2
[5,5,4,1,1,1] => [5,4,1,1,1] => -1
[5,5,3,3,1] => [5,3,3,1] => 2
[5,5,3,2,2] => [5,3,2,2] => 4
[5,5,3,2,1,1] => [5,3,2,1,1] => 0
[5,5,2,2,2,1] => [5,2,2,2,1] => 3
[5,5,2,2,1,1,1] => [5,2,2,1,1,1] => -2
[5,4,4,4] => [4,4,4] => 3
[5,4,4,3,1] => [4,4,3,1] => 2
[5,4,4,2,2] => [4,4,2,2] => 4
[5,4,4,2,1,1] => [4,4,2,1,1] => 0
[5,4,3,3,2] => [4,3,3,2] => 4
[5,4,3,3,1,1] => [4,3,3,1,1] => 1
[5,4,3,2,2,1] => [4,3,2,2,1] => 3
[5,3,3,3,3] => [3,3,3,3] => 4
[5,3,3,3,2,1] => [3,3,3,2,1] => 3
[5,3,3,2,2,2] => [3,3,2,2,2] => 5
[5,3,3,2,2,1,1] => [3,3,2,2,1,1] => 0
[5,3,2,2,2,2,1] => [3,2,2,2,2,1] => 4
[5,2,2,2,2,2,2] => [2,2,2,2,2,2] => 6
[4,4,4,4,1] => [4,4,4,1] => 2
[4,4,4,3,2] => [4,4,3,2] => 4
[4,4,4,3,1,1] => [4,4,3,1,1] => 1
[4,4,4,2,2,1] => [4,4,2,2,1] => 3
[4,4,3,3,3] => [4,3,3,3] => 4
[4,4,3,3,2,1] => [4,3,3,2,1] => 3
[4,3,3,3,3,1] => [3,3,3,3,1] => 3
[4,3,3,3,2,2] => [3,3,3,2,2] => 5
[4,3,3,2,2,2,1] => [3,3,2,2,2,1] => 4
[3,3,3,3,3,2] => [3,3,3,3,2] => 5
[3,3,3,3,3,1,1] => [3,3,3,3,1,1] => 2
[6,5,4,3,2,1] => [5,4,3,2,1] => 3
[5,5,4,3,2,1] => [5,4,3,2,1] => 3
[6,4,4,3,2,1] => [4,4,3,2,1] => 3
[5,4,4,3,2,1] => [4,4,3,2,1] => 3
[4,4,4,3,2,1] => [4,4,3,2,1] => 3
[6,5,3,3,2,1] => [5,3,3,2,1] => 3
[5,5,3,3,2,1] => [5,3,3,2,1] => 3
[6,4,3,3,2,1] => [4,3,3,2,1] => 3
[5,4,3,3,2,1] => [4,3,3,2,1] => 3
[6,3,3,3,2,1] => [3,3,3,2,1] => 3
[6,5,4,2,2,1] => [5,4,2,2,1] => 3
[5,5,4,2,2,1] => [5,4,2,2,1] => 3
[6,4,4,2,2,1] => [4,4,2,2,1] => 3
[5,4,4,2,2,1] => [4,4,2,2,1] => 3
[6,5,3,2,2,1] => [5,3,2,2,1] => 3
[5,5,3,2,2,1] => [5,3,2,2,1] => 3
[6,4,3,2,2,1] => [4,3,2,2,1] => 3
[6,5,2,2,2,1] => [5,2,2,2,1] => 3
[6,5,4,3,1,1] => [5,4,3,1,1] => 1
[5,5,4,3,1,1] => [5,4,3,1,1] => 1
[6,4,4,3,1,1] => [4,4,3,1,1] => 1
[5,4,4,3,1,1] => [4,4,3,1,1] => 1
[6,5,3,3,1,1] => [5,3,3,1,1] => 1
[5,5,3,3,1,1] => [5,3,3,1,1] => 1
[6,4,3,3,1,1] => [4,3,3,1,1] => 1
[6,5,4,2,1,1] => [5,4,2,1,1] => 0
[5,5,4,2,1,1] => [5,4,2,1,1] => 0
[6,4,4,2,1,1] => [4,4,2,1,1] => 0
[6,5,3,2,1,1] => [5,3,2,1,1] => 0
[6,5,4,1,1,1] => [5,4,1,1,1] => -1
[6,5,4,3,2] => [5,4,3,2] => 4
[5,5,4,3,2] => [5,4,3,2] => 4
[6,4,4,3,2] => [4,4,3,2] => 4
[5,4,4,3,2] => [4,4,3,2] => 4
[6,5,3,3,2] => [5,3,3,2] => 4
[5,5,3,3,2] => [5,3,3,2] => 4
[6,4,3,3,2] => [4,3,3,2] => 4
[6,5,4,2,2] => [5,4,2,2] => 4
[5,5,4,2,2] => [5,4,2,2] => 4
[6,4,4,2,2] => [4,4,2,2] => 4
[6,5,3,2,2] => [5,3,2,2] => 4
[6,5,4,3,1] => [5,4,3,1] => 2
[5,5,4,3,1] => [5,4,3,1] => 2
[6,4,4,3,1] => [4,4,3,1] => 2
[6,5,3,3,1] => [5,3,3,1] => 2
[6,5,4,2,1] => [5,4,2,1] => 2
[6,5,4,3] => [5,4,3] => 3
[7,6,5,4,3,2,1] => [6,5,4,3,2,1] => 4
[6,6,5,4,3,2,1] => [6,5,4,3,2,1] => 4
[7,5,5,4,3,2,1] => [5,5,4,3,2,1] => 4
[6,5,5,4,3,2,1] => [5,5,4,3,2,1] => 4
[5,5,5,4,3,2,1] => [5,5,4,3,2,1] => 4
[7,6,4,4,3,2,1] => [6,4,4,3,2,1] => 4
[6,6,4,4,3,2,1] => [6,4,4,3,2,1] => 4
[7,5,4,4,3,2,1] => [5,4,4,3,2,1] => 4
[6,5,4,4,3,2,1] => [5,4,4,3,2,1] => 4
[5,5,4,4,3,2,1] => [5,4,4,3,2,1] => 4
[7,4,4,4,3,2,1] => [4,4,4,3,2,1] => 4
[6,4,4,4,3,2,1] => [4,4,4,3,2,1] => 4
[5,4,4,4,3,2,1] => [4,4,4,3,2,1] => 4
[4,4,4,4,3,2,1] => [4,4,4,3,2,1] => 4
[6,6,5,3,3,2,1] => [6,5,3,3,2,1] => 4
[6,5,4,3,3,2,1] => [5,4,3,3,2,1] => 4
[6,5,3,3,3,2,1] => [5,3,3,3,2,1] => 4
[6,4,3,3,3,2,1] => [4,3,3,3,2,1] => 4
[5,4,3,3,3,2,1] => [4,3,3,3,2,1] => 4
[6,3,3,3,3,2,1] => [3,3,3,3,2,1] => 4
[3,3,3,3,3,2,1] => [3,3,3,3,2,1] => 4
[6,6,5,4,2,2,1] => [6,5,4,2,2,1] => 4
[6,5,5,4,2,2,1] => [5,5,4,2,2,1] => 4
[6,5,4,3,2,2,1] => [5,4,3,2,2,1] => 4
[5,5,4,3,2,2,1] => [5,4,3,2,2,1] => 4
[7,6,5,2,2,2,1] => [6,5,2,2,2,1] => 4
[6,5,4,2,2,2,1] => [5,4,2,2,2,1] => 4
[6,5,3,2,2,2,1] => [5,3,2,2,2,1] => 4
[6,4,3,2,2,2,1] => [4,3,2,2,2,1] => 4
[5,4,3,2,2,2,1] => [4,3,2,2,2,1] => 4
[6,5,2,2,2,2,1] => [5,2,2,2,2,1] => 4
[7,2,2,2,2,2,1] => [2,2,2,2,2,1] => 4
[7,6,5,4,3,1,1] => [6,5,4,3,1,1] => 2
[6,6,5,4,3,1,1] => [6,5,4,3,1,1] => 2
[6,5,5,4,3,1,1] => [5,5,4,3,1,1] => 2
[6,5,4,3,2,1,1] => [5,4,3,2,1,1] => 1
[5,5,4,3,2,1,1] => [5,4,3,2,1,1] => 1
[7,6,5,4,1,1,1] => [6,5,4,1,1,1] => 0
[6,5,5,4,1,1,1] => [5,5,4,1,1,1] => 0
[6,5,4,3,1,1,1] => [5,4,3,1,1,1] => -1
[7,5,4,2,1,1,1] => [5,4,2,1,1,1] => -1
[6,5,4,2,1,1,1] => [5,4,2,1,1,1] => -1
[6,5,4,1,1,1,1] => [5,4,1,1,1,1] => -3
[7,6,1,1,1,1,1] => [6,1,1,1,1,1] => -4
[7,6,5,4,3,2] => [6,5,4,3,2] => 5
[6,6,5,4,3,2] => [6,5,4,3,2] => 5
[7,5,5,4,3,2] => [5,5,4,3,2] => 5
[6,5,5,4,3,2] => [5,5,4,3,2] => 5
[5,5,5,4,3,2] => [5,5,4,3,2] => 5
[7,6,4,4,3,2] => [6,4,4,3,2] => 5
[4,3,3,3,3,2] => [3,3,3,3,2] => 5
[7,6,5,2,2,2] => [6,5,2,2,2] => 5
[6,6,2,2,2,2] => [6,2,2,2,2] => 5
[7,6,5,4,3,1] => [6,5,4,3,1] => 3
[6,6,5,4,3,1] => [6,5,4,3,1] => 3
[7,5,4,3,2,1] => [5,4,3,2,1] => 3
[7,4,4,3,2,1] => [4,4,3,2,1] => 3
[7,4,3,3,2,1] => [4,3,3,2,1] => 3
[7,3,3,3,2,1] => [3,3,3,2,1] => 3
[7,4,3,2,1,1] => [4,3,2,1,1] => 0
[7,6,5,4,3] => [6,5,4,3] => 4
[6,6,5,4,3] => [6,5,4,3] => 4
[6,5,5,4,3] => [5,5,4,3] => 4
[5,4,4,4,3] => [4,4,4,3] => 4
[5,4,3,3,3] => [4,3,3,3] => 4
[7,3,3,3,3] => [3,3,3,3] => 4
[4,4,4,4,2] => [4,4,4,2] => 4
[7,5,4,3,2] => [5,4,3,2] => 4
[7,5,4,3,1] => [5,4,3,1] => 2
[7,5,4,2,1] => [5,4,2,1] => 2
[7,6,5,4] => [6,5,4] => 3
[6,5,5,4] => [5,5,4] => 3
[7,5,4,3] => [5,4,3] => 3
[7,6,5] => [6,5] => 2
[3,3,3,3,3,3] => [3,3,3,3,3] => 5
[4,3,3,3,3,3] => [3,3,3,3,3] => 5
[4,4,4,4,4] => [4,4,4,4] => 4
[5,5,5,5] => [5,5,5] => 3
[6,2,2,2,2,2,2] => [2,2,2,2,2,2] => 6
[6,6,6] => [6,6] => 2
[7,6,6] => [6,6] => 2
[10,4,4] => [4,4] => 2
[10,10] => [10] => 1
[11,7,3] => [7,3] => 2
[8,2,2,2,2,2,2] => [2,2,2,2,2,2] => 6
[9,6,3] => [6,3] => 2
[8,6,4,2] => [6,4,2] => 3
[8,6,4] => [6,4] => 2
[10,6,4] => [6,4] => 2
[15,5,5] => [5,5] => 2
[11,7,5,1] => [7,5,1] => 1
[9,7,5,3,1] => [7,5,3,1] => 2
[11,9,7,5,3,1] => [9,7,5,3,1] => 3
[8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 5
[7,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 5
[7,6,6,5,4,3,2,1] => [6,6,5,4,3,2,1] => 5
[7,6,5,5,4,3,2,1] => [6,5,5,4,3,2,1] => 5
[7,6,5,4,4,3,2,1] => [6,5,4,4,3,2,1] => 5
[7,6,5,4,3,3,2,1] => [6,5,4,3,3,2,1] => 5
[7,6,5,4,3,2,2,1] => [6,5,4,3,2,2,1] => 5
[7,6,5,4,3,2,1,1] => [6,5,4,3,2,1,1] => 2
[9,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 6
[8,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 6
[6,6,6,5,4,3,2,1] => [6,6,5,4,3,2,1] => 5
[8,7,7,6,5,4,3,2,1] => [7,7,6,5,4,3,2,1] => 6
[7,5,5,5,4,3,2,1] => [5,5,5,4,3,2,1] => 5
[8,7,6,6,5,4,3,2,1] => [7,6,6,5,4,3,2,1] => 6
[7,6,4,4,4,3,2,1] => [6,4,4,4,3,2,1] => 5
[8,7,6,5,5,4,3,2,1] => [7,6,5,5,4,3,2,1] => 6
[7,6,5,3,3,3,2,1] => [6,5,3,3,3,2,1] => 5
[8,7,6,5,4,4,3,2,1] => [7,6,5,4,4,3,2,1] => 6
[7,6,5,4,2,2,2,1] => [6,5,4,2,2,2,1] => 5
[8,7,6,5,4,3,3,2,1] => [7,6,5,4,3,3,2,1] => 6
[7,6,5,4,3,1,1,1] => [6,5,4,3,1,1,1] => 0
[8,7,6,5,4,3,2,2,1] => [7,6,5,4,3,2,2,1] => 6
[8,7,6,5,4,3,2,1,1] => [7,6,5,4,3,2,1,1] => 3
[8,7,6,5,4,3,2] => [7,6,5,4,3,2] => 6
[8,6,5,4,3,2,1] => [6,5,4,3,2,1] => 4
[8,5,5,4,3,2,1] => [5,5,4,3,2,1] => 4
[9,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 5
[9,6,4] => [6,4] => 2
[10,7,3] => [7,3] => 2
[8,5,4,2] => [5,4,2] => 3
[10,7,5,3] => [7,5,3] => 3
[9,7,4] => [7,4] => 2
[10,8,6,4,2] => [8,6,4,2] => 4
[8,5,5,1] => [5,5,1] => 1
[9,5,5,1] => [5,5,1] => 1
[9,7,5,3] => [7,5,3] => 3
[9,9] => [9] => 1
[8,5,4,3,2,1] => [5,4,3,2,1] => 3
[8,6,5,4,3,2] => [6,5,4,3,2] => 5
[7,7,6,5,4,3,2] => [7,6,5,4,3,2] => 6
[6,5,5,5,4,3,2,1] => [5,5,5,4,3,2,1] => 5
[6,5,4,4,4,3,2,1] => [5,4,4,4,3,2,1] => 5
[6,5,5,5] => [5,5,5] => 3
[5,4,4,4,4] => [4,4,4,4] => 4
[9,8,8,7,6,5,4,3,2,1] => [8,8,7,6,5,4,3,2,1] => 7
[9,9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 7
[9,7,6,5,5,4,3,2,1] => [7,6,5,5,4,3,2,1] => 6
[8,7,5] => [7,5] => 2
[8,5,4,3,2,1,1] => [5,4,3,2,1,1] => 1
[6,6,5,4,3,2,1,1] => [6,5,4,3,2,1,1] => 2
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Generating function
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Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
1,1 1,1,1 1,1,1,1,1 1,1,1,1,2,1 1,1,1,2,1,2,2,1 1,1,1,2,2,1,3,3,1 1,1,1,2,2,3,2,3,4,2,1 1,1,1,2,2,4,3,2,4,5,4,1 1,1,1,2,2,4,4,4,4,4,7,5,2,1 1,1,1,2,2,4,4,6,5,4,6,8,7,4,1
$F_{1} = 1$
$F_{2} = q^{-1} + 1$
$F_{3} = q^{-2} + q^{-1} + 1$
$F_{4} = q^{-3} + q^{-2} + q^{-1} + 1 + q$
$F_{5} = q^{-4} + q^{-3} + q^{-2} + q^{-1} + 2 + q$
$F_{6} = q^{-5} + q^{-4} + q^{-3} + 2\ q^{-2} + q^{-1} + 2 + 2\ q + q^{2}$
$F_{7} = q^{-6} + q^{-5} + q^{-4} + 2\ q^{-3} + 2\ q^{-2} + q^{-1} + 3 + 3\ q + q^{2}$
$F_{8} = q^{-7} + q^{-6} + q^{-5} + 2\ q^{-4} + 2\ q^{-3} + 3\ q^{-2} + 2\ q^{-1} + 3 + 4\ q + 2\ q^{2} + q^{3}$
$F_{9} = q^{-8} + q^{-7} + q^{-6} + 2\ q^{-5} + 2\ q^{-4} + 4\ q^{-3} + 3\ q^{-2} + 2\ q^{-1} + 4 + 5\ q + 4\ q^{2} + q^{3}$
$F_{10} = q^{-9} + q^{-8} + q^{-7} + 2\ q^{-6} + 2\ q^{-5} + 4\ q^{-4} + 4\ q^{-3} + 4\ q^{-2} + 4\ q^{-1} + 4 + 7\ q + 5\ q^{2} + 2\ q^{3} + q^{4}$
$F_{11} = q^{-10} + q^{-9} + q^{-8} + 2\ q^{-7} + 2\ q^{-6} + 4\ q^{-5} + 4\ q^{-4} + 6\ q^{-3} + 5\ q^{-2} + 4\ q^{-1} + 6 + 8\ q + 7\ q^{2} + 4\ q^{3} + q^{4}$
Description
The Andrews-Garvan crank of a partition.
If $\pi$ is a partition, let $l(\pi)$ be its length (number of parts), $\omega(\pi)$ be the number of parts equal to 1, and $\mu(\pi)$ be the number of parts larger than $\omega(\pi)$. The crank is then defined by
$$ c(\pi) = \begin{cases} l(\pi) &\text{if \(\omega(\pi)=0\)}\\ \mu(\pi) - \omega(\pi) &\text{otherwise}. \end{cases} $$
This statistic was defined in [1] to explain Ramanujan's partition congruence $$p(11n+6) \equiv 0 \pmod{11}$$ in the same way as the Dyson rank (St000145The Dyson rank of a partition.) explains the congruences $$p(5n+4) \equiv 0 \pmod{5}$$ and $$p(7n+5) \equiv 0 \pmod{7}.$$
If $\pi$ is a partition, let $l(\pi)$ be its length (number of parts), $\omega(\pi)$ be the number of parts equal to 1, and $\mu(\pi)$ be the number of parts larger than $\omega(\pi)$. The crank is then defined by
$$ c(\pi) = \begin{cases} l(\pi) &\text{if \(\omega(\pi)=0\)}\\ \mu(\pi) - \omega(\pi) &\text{otherwise}. \end{cases} $$
This statistic was defined in [1] to explain Ramanujan's partition congruence $$p(11n+6) \equiv 0 \pmod{11}$$ in the same way as the Dyson rank (St000145The Dyson rank of a partition.) explains the congruences $$p(5n+4) \equiv 0 \pmod{5}$$ and $$p(7n+5) \equiv 0 \pmod{7}.$$
Map
first row removal
Description
Removes the first entry of an integer partition
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