Identifier
Values
[] => 0
[1] => 0
[2] => 1
[1,1] => 0
[3] => 1
[2,1] => 1
[1,1,1] => 0
[4] => 0
[3,1] => 1
[2,2] => 2
[2,1,1] => 1
[1,1,1,1] => 0
[5] => 1
[4,1] => 0
[3,2] => 2
[3,1,1] => 1
[2,2,1] => 2
[2,1,1,1] => 1
[1,1,1,1,1] => 0
[6] => 1
[5,1] => 1
[4,2] => 1
[4,1,1] => 0
[3,3] => 2
[3,2,1] => 2
[3,1,1,1] => 1
[2,2,2] => 3
[2,2,1,1] => 2
[2,1,1,1,1] => 1
[1,1,1,1,1,1] => 0
[7] => 0
[6,1] => 1
[5,2] => 2
[5,1,1] => 1
[4,3] => 1
[4,2,1] => 1
[4,1,1,1] => 0
[3,3,1] => 2
[3,2,2] => 3
[3,2,1,1] => 2
[3,1,1,1,1] => 1
[2,2,2,1] => 3
[2,2,1,1,1] => 2
[2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => 0
[8] => 1
[7,1] => 0
[6,2] => 2
[6,1,1] => 1
[5,3] => 2
[5,2,1] => 2
[5,1,1,1] => 1
[4,4] => 0
[4,3,1] => 1
[4,2,2] => 2
[4,2,1,1] => 1
[4,1,1,1,1] => 0
[3,3,2] => 3
[3,3,1,1] => 2
[3,2,2,1] => 3
[3,2,1,1,1] => 2
[3,1,1,1,1,1] => 1
[2,2,2,2] => 4
[2,2,2,1,1] => 3
[2,2,1,1,1,1] => 2
[2,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1] => 0
[9] => 1
[8,1] => 1
[7,2] => 1
[7,1,1] => 0
[6,3] => 2
[6,2,1] => 2
[6,1,1,1] => 1
[5,4] => 1
[5,3,1] => 2
[5,2,2] => 3
[5,2,1,1] => 2
[5,1,1,1,1] => 1
[4,4,1] => 0
[4,3,2] => 2
[4,3,1,1] => 1
[4,2,2,1] => 2
[4,2,1,1,1] => 1
[4,1,1,1,1,1] => 0
[3,3,3] => 3
[3,3,2,1] => 3
[3,3,1,1,1] => 2
[3,2,2,2] => 4
[3,2,2,1,1] => 3
[3,2,1,1,1,1] => 2
[3,1,1,1,1,1,1] => 1
[2,2,2,2,1] => 4
[2,2,2,1,1,1] => 3
[2,2,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1] => 0
[10] => 0
[9,1] => 1
[8,2] => 2
[8,1,1] => 1
>>> Load all 1485 entries. <<<[7,3] => 1
[7,2,1] => 1
[7,1,1,1] => 0
[6,4] => 1
[6,3,1] => 2
[6,2,2] => 3
[6,2,1,1] => 2
[6,1,1,1,1] => 1
[5,5] => 2
[5,4,1] => 1
[5,3,2] => 3
[5,3,1,1] => 2
[5,2,2,1] => 3
[5,2,1,1,1] => 2
[5,1,1,1,1,1] => 1
[4,4,2] => 1
[4,4,1,1] => 0
[4,3,3] => 2
[4,3,2,1] => 2
[4,3,1,1,1] => 1
[4,2,2,2] => 3
[4,2,2,1,1] => 2
[4,2,1,1,1,1] => 1
[4,1,1,1,1,1,1] => 0
[3,3,3,1] => 3
[3,3,2,2] => 4
[3,3,2,1,1] => 3
[3,3,1,1,1,1] => 2
[3,2,2,2,1] => 4
[3,2,2,1,1,1] => 3
[3,2,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1] => 1
[2,2,2,2,2] => 5
[2,2,2,2,1,1] => 4
[2,2,2,1,1,1,1] => 3
[2,2,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1] => 0
[11] => 1
[10,1] => 0
[9,2] => 2
[9,1,1] => 1
[8,3] => 2
[8,2,1] => 2
[8,1,1,1] => 1
[7,4] => 0
[7,3,1] => 1
[7,2,2] => 2
[7,2,1,1] => 1
[7,1,1,1,1] => 0
[6,5] => 2
[6,4,1] => 1
[6,3,2] => 3
[6,3,1,1] => 2
[6,2,2,1] => 3
[6,2,1,1,1] => 2
[6,1,1,1,1,1] => 1
[5,5,1] => 2
[5,4,2] => 2
[5,4,1,1] => 1
[5,3,3] => 3
[5,3,2,1] => 3
[5,3,1,1,1] => 2
[5,2,2,2] => 4
[5,2,2,1,1] => 3
[5,2,1,1,1,1] => 2
[5,1,1,1,1,1,1] => 1
[4,4,3] => 1
[4,4,2,1] => 1
[4,4,1,1,1] => 0
[4,3,3,1] => 2
[4,3,2,2] => 3
[4,3,2,1,1] => 2
[4,3,1,1,1,1] => 1
[4,2,2,2,1] => 3
[4,2,2,1,1,1] => 2
[4,2,1,1,1,1,1] => 1
[4,1,1,1,1,1,1,1] => 0
[3,3,3,2] => 4
[3,3,3,1,1] => 3
[3,3,2,2,1] => 4
[3,3,2,1,1,1] => 3
[3,3,1,1,1,1,1] => 2
[3,2,2,2,2] => 5
[3,2,2,2,1,1] => 4
[3,2,2,1,1,1,1] => 3
[3,2,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,1] => 5
[2,2,2,2,1,1,1] => 4
[2,2,2,1,1,1,1,1] => 3
[2,2,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1] => 0
[12] => 1
[11,1] => 1
[10,2] => 1
[10,1,1] => 0
[9,3] => 2
[9,2,1] => 2
[9,1,1,1] => 1
[8,4] => 1
[8,3,1] => 2
[8,2,2] => 3
[8,2,1,1] => 2
[8,1,1,1,1] => 1
[7,5] => 1
[7,4,1] => 0
[7,3,2] => 2
[7,3,1,1] => 1
[7,2,2,1] => 2
[7,2,1,1,1] => 1
[7,1,1,1,1,1] => 0
[6,6] => 2
[6,5,1] => 2
[6,4,2] => 2
[6,4,1,1] => 1
[6,3,3] => 3
[6,3,2,1] => 3
[6,3,1,1,1] => 2
[6,2,2,2] => 4
[6,2,2,1,1] => 3
[6,2,1,1,1,1] => 2
[6,1,1,1,1,1,1] => 1
[5,5,2] => 3
[5,5,1,1] => 2
[5,4,3] => 2
[5,4,2,1] => 2
[5,4,1,1,1] => 1
[5,3,3,1] => 3
[5,3,2,2] => 4
[5,3,2,1,1] => 3
[5,3,1,1,1,1] => 2
[5,2,2,2,1] => 4
[5,2,2,1,1,1] => 3
[5,2,1,1,1,1,1] => 2
[5,1,1,1,1,1,1,1] => 1
[4,4,4] => 0
[4,4,3,1] => 1
[4,4,2,2] => 2
[4,4,2,1,1] => 1
[4,4,1,1,1,1] => 0
[4,3,3,2] => 3
[4,3,3,1,1] => 2
[4,3,2,2,1] => 3
[4,3,2,1,1,1] => 2
[4,3,1,1,1,1,1] => 1
[4,2,2,2,2] => 4
[4,2,2,2,1,1] => 3
[4,2,2,1,1,1,1] => 2
[4,2,1,1,1,1,1,1] => 1
[4,1,1,1,1,1,1,1,1] => 0
[3,3,3,3] => 4
[3,3,3,2,1] => 4
[3,3,3,1,1,1] => 3
[3,3,2,2,2] => 5
[3,3,2,2,1,1] => 4
[3,3,2,1,1,1,1] => 3
[3,3,1,1,1,1,1,1] => 2
[3,2,2,2,2,1] => 5
[3,2,2,2,1,1,1] => 4
[3,2,2,1,1,1,1,1] => 3
[3,2,1,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2] => 6
[2,2,2,2,2,1,1] => 5
[2,2,2,2,1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1] => 3
[2,2,1,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1] => 0
[13] => 0
[12,1] => 1
[11,2] => 2
[11,1,1] => 1
[10,3] => 1
[10,2,1] => 1
[10,1,1,1] => 0
[9,4] => 1
[9,3,1] => 2
[9,2,2] => 3
[9,2,1,1] => 2
[9,1,1,1,1] => 1
[8,5] => 2
[8,4,1] => 1
[8,3,2] => 3
[8,3,1,1] => 2
[8,2,2,1] => 3
[8,2,1,1,1] => 2
[8,1,1,1,1,1] => 1
[7,6] => 1
[7,5,1] => 1
[7,4,2] => 1
[7,4,1,1] => 0
[7,3,3] => 2
[7,3,2,1] => 2
[7,3,1,1,1] => 1
[7,2,2,2] => 3
[7,2,2,1,1] => 2
[7,2,1,1,1,1] => 1
[7,1,1,1,1,1,1] => 0
[6,6,1] => 2
[6,5,2] => 3
[6,5,1,1] => 2
[6,4,3] => 2
[6,4,2,1] => 2
[6,4,1,1,1] => 1
[6,3,3,1] => 3
[6,3,2,2] => 4
[6,3,2,1,1] => 3
[6,3,1,1,1,1] => 2
[6,2,2,2,1] => 4
[6,2,2,1,1,1] => 3
[6,2,1,1,1,1,1] => 2
[6,1,1,1,1,1,1,1] => 1
[5,5,3] => 3
[5,5,2,1] => 3
[5,5,1,1,1] => 2
[5,4,4] => 1
[5,4,3,1] => 2
[5,4,2,2] => 3
[5,4,2,1,1] => 2
[5,4,1,1,1,1] => 1
[5,3,3,2] => 4
[5,3,3,1,1] => 3
[5,3,2,2,1] => 4
[5,3,2,1,1,1] => 3
[5,3,1,1,1,1,1] => 2
[5,2,2,2,2] => 5
[5,2,2,2,1,1] => 4
[5,2,2,1,1,1,1] => 3
[5,2,1,1,1,1,1,1] => 2
[5,1,1,1,1,1,1,1,1] => 1
[4,4,4,1] => 0
[4,4,3,2] => 2
[4,4,3,1,1] => 1
[4,4,2,2,1] => 2
[4,4,2,1,1,1] => 1
[4,4,1,1,1,1,1] => 0
[4,3,3,3] => 3
[4,3,3,2,1] => 3
[4,3,3,1,1,1] => 2
[4,3,2,2,2] => 4
[4,3,2,2,1,1] => 3
[4,3,2,1,1,1,1] => 2
[4,3,1,1,1,1,1,1] => 1
[4,2,2,2,2,1] => 4
[4,2,2,2,1,1,1] => 3
[4,2,2,1,1,1,1,1] => 2
[4,2,1,1,1,1,1,1,1] => 1
[4,1,1,1,1,1,1,1,1,1] => 0
[3,3,3,3,1] => 4
[3,3,3,2,2] => 5
[3,3,3,2,1,1] => 4
[3,3,3,1,1,1,1] => 3
[3,3,2,2,2,1] => 5
[3,3,2,2,1,1,1] => 4
[3,3,2,1,1,1,1,1] => 3
[3,3,1,1,1,1,1,1,1] => 2
[3,2,2,2,2,2] => 6
[3,2,2,2,2,1,1] => 5
[3,2,2,2,1,1,1,1] => 4
[3,2,2,1,1,1,1,1,1] => 3
[3,2,1,1,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2,1] => 6
[2,2,2,2,2,1,1,1] => 5
[2,2,2,2,1,1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1,1] => 3
[2,2,1,1,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[14] => 1
[13,1] => 0
[12,2] => 2
[12,1,1] => 1
[11,3] => 2
[11,2,1] => 2
[11,1,1,1] => 1
[10,4] => 0
[10,3,1] => 1
[10,2,2] => 2
[10,2,1,1] => 1
[10,1,1,1,1] => 0
[9,5] => 2
[9,4,1] => 1
[9,3,2] => 3
[9,3,1,1] => 2
[9,2,2,1] => 3
[9,2,1,1,1] => 2
[9,1,1,1,1,1] => 1
[8,6] => 2
[8,5,1] => 2
[8,4,2] => 2
[8,4,1,1] => 1
[8,3,3] => 3
[8,3,2,1] => 3
[8,3,1,1,1] => 2
[8,2,2,2] => 4
[8,2,2,1,1] => 3
[8,2,1,1,1,1] => 2
[8,1,1,1,1,1,1] => 1
[7,7] => 0
[7,6,1] => 1
[7,5,2] => 2
[7,5,1,1] => 1
[7,4,3] => 1
[7,4,2,1] => 1
[7,4,1,1,1] => 0
[7,3,3,1] => 2
[7,3,2,2] => 3
[7,3,2,1,1] => 2
[7,3,1,1,1,1] => 1
[7,2,2,2,1] => 3
[7,2,2,1,1,1] => 2
[7,2,1,1,1,1,1] => 1
[7,1,1,1,1,1,1,1] => 0
[6,6,2] => 3
[6,6,1,1] => 2
[6,5,3] => 3
[6,5,2,1] => 3
[6,5,1,1,1] => 2
[6,4,4] => 1
[6,4,3,1] => 2
[6,4,2,2] => 3
[6,4,2,1,1] => 2
[6,4,1,1,1,1] => 1
[6,3,3,2] => 4
[6,3,3,1,1] => 3
[6,3,2,2,1] => 4
[6,3,2,1,1,1] => 3
[6,3,1,1,1,1,1] => 2
[6,2,2,2,2] => 5
[6,2,2,2,1,1] => 4
[6,2,2,1,1,1,1] => 3
[6,2,1,1,1,1,1,1] => 2
[6,1,1,1,1,1,1,1,1] => 1
[5,5,4] => 2
[5,5,3,1] => 3
[5,5,2,2] => 4
[5,5,2,1,1] => 3
[5,5,1,1,1,1] => 2
[5,4,4,1] => 1
[5,4,3,2] => 3
[5,4,3,1,1] => 2
[5,4,2,2,1] => 3
[5,4,2,1,1,1] => 2
[5,4,1,1,1,1,1] => 1
[5,3,3,3] => 4
[5,3,3,2,1] => 4
[5,3,3,1,1,1] => 3
[5,3,2,2,2] => 5
[5,3,2,2,1,1] => 4
[5,3,2,1,1,1,1] => 3
[5,3,1,1,1,1,1,1] => 2
[5,2,2,2,2,1] => 5
[5,2,2,2,1,1,1] => 4
[5,2,2,1,1,1,1,1] => 3
[5,2,1,1,1,1,1,1,1] => 2
[5,1,1,1,1,1,1,1,1,1] => 1
[4,4,4,2] => 1
[4,4,4,1,1] => 0
[4,4,3,3] => 2
[4,4,3,2,1] => 2
[4,4,3,1,1,1] => 1
[4,4,2,2,2] => 3
[4,4,2,2,1,1] => 2
[4,4,2,1,1,1,1] => 1
[4,4,1,1,1,1,1,1] => 0
[4,3,3,3,1] => 3
[4,3,3,2,2] => 4
[4,3,3,2,1,1] => 3
[4,3,3,1,1,1,1] => 2
[4,3,2,2,2,1] => 4
[4,3,2,2,1,1,1] => 3
[4,3,2,1,1,1,1,1] => 2
[4,3,1,1,1,1,1,1,1] => 1
[4,2,2,2,2,2] => 5
[4,2,2,2,2,1,1] => 4
[4,2,2,2,1,1,1,1] => 3
[4,2,2,1,1,1,1,1,1] => 2
[4,2,1,1,1,1,1,1,1,1] => 1
[4,1,1,1,1,1,1,1,1,1,1] => 0
[3,3,3,3,2] => 5
[3,3,3,3,1,1] => 4
[3,3,3,2,2,1] => 5
[3,3,3,2,1,1,1] => 4
[3,3,3,1,1,1,1,1] => 3
[3,3,2,2,2,2] => 6
[3,3,2,2,2,1,1] => 5
[3,3,2,2,1,1,1,1] => 4
[3,3,2,1,1,1,1,1,1] => 3
[3,3,1,1,1,1,1,1,1,1] => 2
[3,2,2,2,2,2,1] => 6
[3,2,2,2,2,1,1,1] => 5
[3,2,2,2,1,1,1,1,1] => 4
[3,2,2,1,1,1,1,1,1,1] => 3
[3,2,1,1,1,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2,2] => 7
[2,2,2,2,2,2,1,1] => 6
[2,2,2,2,2,1,1,1,1] => 5
[2,2,2,2,1,1,1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1,1,1] => 3
[2,2,1,1,1,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[15] => 1
[14,1] => 1
[13,2] => 1
[13,1,1] => 0
[12,3] => 2
[12,2,1] => 2
[12,1,1,1] => 1
[11,4] => 1
[11,3,1] => 2
[11,2,2] => 3
[11,2,1,1] => 2
[11,1,1,1,1] => 1
[10,5] => 1
[10,4,1] => 0
[10,3,2] => 2
[10,3,1,1] => 1
[10,2,2,1] => 2
[10,2,1,1,1] => 1
[10,1,1,1,1,1] => 0
[9,6] => 2
[9,5,1] => 2
[9,4,2] => 2
[9,4,1,1] => 1
[9,3,3] => 3
[9,3,2,1] => 3
[9,3,1,1,1] => 2
[9,2,2,2] => 4
[9,2,2,1,1] => 3
[9,2,1,1,1,1] => 2
[9,1,1,1,1,1,1] => 1
[8,7] => 1
[8,6,1] => 2
[8,5,2] => 3
[8,5,1,1] => 2
[8,4,3] => 2
[8,4,2,1] => 2
[8,4,1,1,1] => 1
[8,3,3,1] => 3
[8,3,2,2] => 4
[8,3,2,1,1] => 3
[8,3,1,1,1,1] => 2
[8,2,2,2,1] => 4
[8,2,2,1,1,1] => 3
[8,2,1,1,1,1,1] => 2
[8,1,1,1,1,1,1,1] => 1
[7,7,1] => 0
[7,6,2] => 2
[7,6,1,1] => 1
[7,5,3] => 2
[7,5,2,1] => 2
[7,5,1,1,1] => 1
[7,4,4] => 0
[7,4,3,1] => 1
[7,4,2,2] => 2
[7,4,2,1,1] => 1
[7,4,1,1,1,1] => 0
[7,3,3,2] => 3
[7,3,3,1,1] => 2
[7,3,2,2,1] => 3
[7,3,2,1,1,1] => 2
[7,3,1,1,1,1,1] => 1
[7,2,2,2,2] => 4
[7,2,2,2,1,1] => 3
[7,2,2,1,1,1,1] => 2
[7,2,1,1,1,1,1,1] => 1
[7,1,1,1,1,1,1,1,1] => 0
[6,6,3] => 3
[6,6,2,1] => 3
[6,6,1,1,1] => 2
[6,5,4] => 2
[6,5,3,1] => 3
[6,5,2,2] => 4
[6,5,2,1,1] => 3
[6,5,1,1,1,1] => 2
[6,4,4,1] => 1
[6,4,3,2] => 3
[6,4,3,1,1] => 2
[6,4,2,2,1] => 3
[6,4,2,1,1,1] => 2
[6,4,1,1,1,1,1] => 1
[6,3,3,3] => 4
[6,3,3,2,1] => 4
[6,3,3,1,1,1] => 3
[6,3,2,2,2] => 5
[6,3,2,2,1,1] => 4
[6,3,2,1,1,1,1] => 3
[6,3,1,1,1,1,1,1] => 2
[6,2,2,2,2,1] => 5
[6,2,2,2,1,1,1] => 4
[6,2,2,1,1,1,1,1] => 3
[6,2,1,1,1,1,1,1,1] => 2
[6,1,1,1,1,1,1,1,1,1] => 1
[5,5,5] => 3
[5,5,4,1] => 2
[5,5,3,2] => 4
[5,5,3,1,1] => 3
[5,5,2,2,1] => 4
[5,5,2,1,1,1] => 3
[5,5,1,1,1,1,1] => 2
[5,4,4,2] => 2
[5,4,4,1,1] => 1
[5,4,3,3] => 3
[5,4,3,2,1] => 3
[5,4,3,1,1,1] => 2
[5,4,2,2,2] => 4
[5,4,2,2,1,1] => 3
[5,4,2,1,1,1,1] => 2
[5,4,1,1,1,1,1,1] => 1
[5,3,3,3,1] => 4
[5,3,3,2,2] => 5
[5,3,3,2,1,1] => 4
[5,3,3,1,1,1,1] => 3
[5,3,2,2,2,1] => 5
[5,3,2,2,1,1,1] => 4
[5,3,2,1,1,1,1,1] => 3
[5,3,1,1,1,1,1,1,1] => 2
[5,2,2,2,2,2] => 6
[5,2,2,2,2,1,1] => 5
[5,2,2,2,1,1,1,1] => 4
[5,2,2,1,1,1,1,1,1] => 3
[5,2,1,1,1,1,1,1,1,1] => 2
[5,1,1,1,1,1,1,1,1,1,1] => 1
[4,4,4,3] => 1
[4,4,4,2,1] => 1
[4,4,4,1,1,1] => 0
[4,4,3,3,1] => 2
[4,4,3,2,2] => 3
[4,4,3,2,1,1] => 2
[4,4,3,1,1,1,1] => 1
[4,4,2,2,2,1] => 3
[4,4,2,2,1,1,1] => 2
[4,4,2,1,1,1,1,1] => 1
[4,4,1,1,1,1,1,1,1] => 0
[4,3,3,3,2] => 4
[4,3,3,3,1,1] => 3
[4,3,3,2,2,1] => 4
[4,3,3,2,1,1,1] => 3
[4,3,3,1,1,1,1,1] => 2
[4,3,2,2,2,2] => 5
[4,3,2,2,2,1,1] => 4
[4,3,2,2,1,1,1,1] => 3
[4,3,2,1,1,1,1,1,1] => 2
[4,3,1,1,1,1,1,1,1,1] => 1
[4,2,2,2,2,2,1] => 5
[4,2,2,2,2,1,1,1] => 4
[4,2,2,2,1,1,1,1,1] => 3
[4,2,2,1,1,1,1,1,1,1] => 2
[4,2,1,1,1,1,1,1,1,1,1] => 1
[4,1,1,1,1,1,1,1,1,1,1,1] => 0
[3,3,3,3,3] => 5
[3,3,3,3,2,1] => 5
[3,3,3,3,1,1,1] => 4
[3,3,3,2,2,2] => 6
[3,3,3,2,2,1,1] => 5
[3,3,3,2,1,1,1,1] => 4
[3,3,3,1,1,1,1,1,1] => 3
[3,3,2,2,2,2,1] => 6
[3,3,2,2,2,1,1,1] => 5
[3,3,2,2,1,1,1,1,1] => 4
[3,3,2,1,1,1,1,1,1,1] => 3
[3,3,1,1,1,1,1,1,1,1,1] => 2
[3,2,2,2,2,2,2] => 7
[3,2,2,2,2,2,1,1] => 6
[3,2,2,2,2,1,1,1,1] => 5
[3,2,2,2,1,1,1,1,1,1] => 4
[3,2,2,1,1,1,1,1,1,1,1] => 3
[3,2,1,1,1,1,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2,2,1] => 7
[2,2,2,2,2,2,1,1,1] => 6
[2,2,2,2,2,1,1,1,1,1] => 5
[2,2,2,2,1,1,1,1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1,1,1,1] => 3
[2,2,1,1,1,1,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[16] => 0
[15,1] => 1
[14,2] => 2
[14,1,1] => 1
[13,3] => 1
[13,2,1] => 1
[13,1,1,1] => 0
[12,4] => 1
[12,3,1] => 2
[12,2,2] => 3
[12,2,1,1] => 2
[12,1,1,1,1] => 1
[11,5] => 2
[11,4,1] => 1
[11,3,2] => 3
[11,3,1,1] => 2
[11,2,2,1] => 3
[11,2,1,1,1] => 2
[11,1,1,1,1,1] => 1
[10,6] => 1
[10,5,1] => 1
[10,4,2] => 1
[10,4,1,1] => 0
[10,3,3] => 2
[10,3,2,1] => 2
[10,3,1,1,1] => 1
[10,2,2,2] => 3
[10,2,2,1,1] => 2
[10,2,1,1,1,1] => 1
[10,1,1,1,1,1,1] => 0
[9,7] => 1
[9,6,1] => 2
[9,5,2] => 3
[9,5,1,1] => 2
[9,4,3] => 2
[9,4,2,1] => 2
[9,4,1,1,1] => 1
[9,3,3,1] => 3
[9,3,2,2] => 4
[9,3,2,1,1] => 3
[9,3,1,1,1,1] => 2
[9,2,2,2,1] => 4
[9,2,2,1,1,1] => 3
[9,2,1,1,1,1,1] => 2
[9,1,1,1,1,1,1,1] => 1
[8,8] => 2
[8,7,1] => 1
[8,6,2] => 3
[8,6,1,1] => 2
[8,5,3] => 3
[8,5,2,1] => 3
[8,5,1,1,1] => 2
[8,4,4] => 1
[8,4,3,1] => 2
[8,4,2,2] => 3
[8,4,2,1,1] => 2
[8,4,1,1,1,1] => 1
[8,3,3,2] => 4
[8,3,3,1,1] => 3
[8,3,2,2,1] => 4
[8,3,2,1,1,1] => 3
[8,3,1,1,1,1,1] => 2
[8,2,2,2,2] => 5
[8,2,2,2,1,1] => 4
[8,2,2,1,1,1,1] => 3
[8,2,1,1,1,1,1,1] => 2
[8,1,1,1,1,1,1,1,1] => 1
[7,7,2] => 1
[7,7,1,1] => 0
[7,6,3] => 2
[7,6,2,1] => 2
[7,6,1,1,1] => 1
[7,5,4] => 1
[7,5,3,1] => 2
[7,5,2,2] => 3
[7,5,2,1,1] => 2
[7,5,1,1,1,1] => 1
[7,4,4,1] => 0
[7,4,3,2] => 2
[7,4,3,1,1] => 1
[7,4,2,2,1] => 2
[7,4,2,1,1,1] => 1
[7,4,1,1,1,1,1] => 0
[7,3,3,3] => 3
[7,3,3,2,1] => 3
[7,3,3,1,1,1] => 2
[7,3,2,2,2] => 4
[7,3,2,2,1,1] => 3
[7,3,2,1,1,1,1] => 2
[7,3,1,1,1,1,1,1] => 1
[7,2,2,2,2,1] => 4
[7,2,2,2,1,1,1] => 3
[7,2,2,1,1,1,1,1] => 2
[7,2,1,1,1,1,1,1,1] => 1
[7,1,1,1,1,1,1,1,1,1] => 0
[6,6,4] => 2
[6,6,3,1] => 3
[6,6,2,2] => 4
[6,6,2,1,1] => 3
[6,6,1,1,1,1] => 2
[6,5,5] => 3
[6,5,4,1] => 2
[6,5,3,2] => 4
[6,5,3,1,1] => 3
[6,5,2,2,1] => 4
[6,5,2,1,1,1] => 3
[6,5,1,1,1,1,1] => 2
[6,4,4,2] => 2
[6,4,4,1,1] => 1
[6,4,3,3] => 3
[6,4,3,2,1] => 3
[6,4,3,1,1,1] => 2
[6,4,2,2,2] => 4
[6,4,2,2,1,1] => 3
[6,4,2,1,1,1,1] => 2
[6,4,1,1,1,1,1,1] => 1
[6,3,3,3,1] => 4
[6,3,3,2,2] => 5
[6,3,3,2,1,1] => 4
[6,3,3,1,1,1,1] => 3
[6,3,2,2,2,1] => 5
[6,3,2,2,1,1,1] => 4
[6,3,2,1,1,1,1,1] => 3
[6,3,1,1,1,1,1,1,1] => 2
[6,2,2,2,2,2] => 6
[6,2,2,2,2,1,1] => 5
[6,2,2,2,1,1,1,1] => 4
[6,2,2,1,1,1,1,1,1] => 3
[6,2,1,1,1,1,1,1,1,1] => 2
[6,1,1,1,1,1,1,1,1,1,1] => 1
[5,5,5,1] => 3
[5,5,4,2] => 3
[5,5,4,1,1] => 2
[5,5,3,3] => 4
[5,5,3,2,1] => 4
[5,5,3,1,1,1] => 3
[5,5,2,2,2] => 5
[5,5,2,2,1,1] => 4
[5,5,2,1,1,1,1] => 3
[5,5,1,1,1,1,1,1] => 2
[5,4,4,3] => 2
[5,4,4,2,1] => 2
[5,4,4,1,1,1] => 1
[5,4,3,3,1] => 3
[5,4,3,2,2] => 4
[5,4,3,2,1,1] => 3
[5,4,3,1,1,1,1] => 2
[5,4,2,2,2,1] => 4
[5,4,2,2,1,1,1] => 3
[5,4,2,1,1,1,1,1] => 2
[5,4,1,1,1,1,1,1,1] => 1
[5,3,3,3,2] => 5
[5,3,3,3,1,1] => 4
[5,3,3,2,2,1] => 5
[5,3,3,2,1,1,1] => 4
[5,3,3,1,1,1,1,1] => 3
[5,3,2,2,2,2] => 6
[5,3,2,2,2,1,1] => 5
[5,3,2,2,1,1,1,1] => 4
[5,3,2,1,1,1,1,1,1] => 3
[5,3,1,1,1,1,1,1,1,1] => 2
[5,2,2,2,2,2,1] => 6
[5,2,2,2,2,1,1,1] => 5
[5,2,2,2,1,1,1,1,1] => 4
[5,2,2,1,1,1,1,1,1,1] => 3
[5,2,1,1,1,1,1,1,1,1,1] => 2
[5,1,1,1,1,1,1,1,1,1,1,1] => 1
[4,4,4,4] => 0
[4,4,4,3,1] => 1
[4,4,4,2,2] => 2
[4,4,4,2,1,1] => 1
[4,4,4,1,1,1,1] => 0
[4,4,3,3,2] => 3
[4,4,3,3,1,1] => 2
[4,4,3,2,2,1] => 3
[4,4,3,2,1,1,1] => 2
[4,4,3,1,1,1,1,1] => 1
[4,4,2,2,2,2] => 4
[4,4,2,2,2,1,1] => 3
[4,4,2,2,1,1,1,1] => 2
[4,4,2,1,1,1,1,1,1] => 1
[4,4,1,1,1,1,1,1,1,1] => 0
[4,3,3,3,3] => 4
[4,3,3,3,2,1] => 4
[4,3,3,3,1,1,1] => 3
[4,3,3,2,2,2] => 5
[4,3,3,2,2,1,1] => 4
[4,3,3,2,1,1,1,1] => 3
[4,3,3,1,1,1,1,1,1] => 2
[4,3,2,2,2,2,1] => 5
[4,3,2,2,2,1,1,1] => 4
[4,3,2,2,1,1,1,1,1] => 3
[4,3,2,1,1,1,1,1,1,1] => 2
[4,3,1,1,1,1,1,1,1,1,1] => 1
[4,2,2,2,2,2,2] => 6
[4,2,2,2,2,2,1,1] => 5
[4,2,2,2,2,1,1,1,1] => 4
[4,2,2,2,1,1,1,1,1,1] => 3
[4,2,2,1,1,1,1,1,1,1,1] => 2
[4,2,1,1,1,1,1,1,1,1,1,1] => 1
[4,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[3,3,3,3,3,1] => 5
[3,3,3,3,2,2] => 6
[3,3,3,3,2,1,1] => 5
[3,3,3,3,1,1,1,1] => 4
[3,3,3,2,2,2,1] => 6
[3,3,3,2,2,1,1,1] => 5
[3,3,3,2,1,1,1,1,1] => 4
[3,3,3,1,1,1,1,1,1,1] => 3
[3,3,2,2,2,2,2] => 7
[3,3,2,2,2,2,1,1] => 6
[3,3,2,2,2,1,1,1,1] => 5
[3,3,2,2,1,1,1,1,1,1] => 4
[3,3,2,1,1,1,1,1,1,1,1] => 3
[3,3,1,1,1,1,1,1,1,1,1,1] => 2
[3,2,2,2,2,2,2,1] => 7
[3,2,2,2,2,2,1,1,1] => 6
[3,2,2,2,2,1,1,1,1,1] => 5
[3,2,2,2,1,1,1,1,1,1,1] => 4
[3,2,2,1,1,1,1,1,1,1,1,1] => 3
[3,2,1,1,1,1,1,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2,2,2] => 8
[2,2,2,2,2,2,2,1,1] => 7
[2,2,2,2,2,2,1,1,1,1] => 6
[2,2,2,2,2,1,1,1,1,1,1] => 5
[2,2,2,2,1,1,1,1,1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1,1,1,1,1] => 3
[2,2,1,1,1,1,1,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[17] => 1
[16,1] => 0
[15,2] => 2
[15,1,1] => 1
[14,3] => 2
[14,2,1] => 2
[14,1,1,1] => 1
[13,4] => 0
[13,3,1] => 1
[13,2,2] => 2
[13,2,1,1] => 1
[13,1,1,1,1] => 0
[12,5] => 2
[12,4,1] => 1
[12,3,2] => 3
[12,3,1,1] => 2
[12,2,2,1] => 3
[12,2,1,1,1] => 2
[12,1,1,1,1,1] => 1
[11,6] => 2
[11,5,1] => 2
[11,4,2] => 2
[11,4,1,1] => 1
[11,3,3] => 3
[11,3,2,1] => 3
[11,3,1,1,1] => 2
[11,2,2,2] => 4
[11,2,2,1,1] => 3
[11,2,1,1,1,1] => 2
[11,1,1,1,1,1,1] => 1
[10,7] => 0
[10,6,1] => 1
[10,5,2] => 2
[10,5,1,1] => 1
[10,4,3] => 1
[10,4,2,1] => 1
[10,4,1,1,1] => 0
[10,3,3,1] => 2
[10,3,2,2] => 3
[10,3,2,1,1] => 2
[10,3,1,1,1,1] => 1
[10,2,2,2,1] => 3
[10,2,2,1,1,1] => 2
[10,2,1,1,1,1,1] => 1
[10,1,1,1,1,1,1,1] => 0
[9,8] => 2
[9,7,1] => 1
[9,6,2] => 3
[9,6,1,1] => 2
[9,5,3] => 3
[9,5,2,1] => 3
[9,5,1,1,1] => 2
[9,4,4] => 1
[9,4,3,1] => 2
[9,4,2,2] => 3
[9,4,2,1,1] => 2
[9,4,1,1,1,1] => 1
[9,3,3,2] => 4
[9,3,3,1,1] => 3
[9,3,2,2,1] => 4
[9,3,2,1,1,1] => 3
[9,3,1,1,1,1,1] => 2
[9,2,2,2,2] => 5
[9,2,2,2,1,1] => 4
[9,2,2,1,1,1,1] => 3
[9,2,1,1,1,1,1,1] => 2
[9,1,1,1,1,1,1,1,1] => 1
[8,8,1] => 2
[8,7,2] => 2
[8,7,1,1] => 1
[8,6,3] => 3
[8,6,2,1] => 3
[8,6,1,1,1] => 2
[8,5,4] => 2
[8,5,3,1] => 3
[8,5,2,2] => 4
[8,5,2,1,1] => 3
[8,5,1,1,1,1] => 2
[8,4,4,1] => 1
[8,4,3,2] => 3
[8,4,3,1,1] => 2
[8,4,2,2,1] => 3
[8,4,2,1,1,1] => 2
[8,4,1,1,1,1,1] => 1
[8,3,3,3] => 4
[8,3,3,2,1] => 4
[8,3,3,1,1,1] => 3
[8,3,2,2,2] => 5
[8,3,2,2,1,1] => 4
[8,3,2,1,1,1,1] => 3
[8,3,1,1,1,1,1,1] => 2
[8,2,2,2,2,1] => 5
[8,2,2,2,1,1,1] => 4
[8,2,2,1,1,1,1,1] => 3
[8,2,1,1,1,1,1,1,1] => 2
[8,1,1,1,1,1,1,1,1,1] => 1
[7,7,3] => 1
[7,7,2,1] => 1
[7,7,1,1,1] => 0
[7,6,4] => 1
[7,6,3,1] => 2
[7,6,2,2] => 3
[7,6,2,1,1] => 2
[7,6,1,1,1,1] => 1
[7,5,5] => 2
[7,5,4,1] => 1
[7,5,3,2] => 3
[7,5,3,1,1] => 2
[7,5,2,2,1] => 3
[7,5,2,1,1,1] => 2
[7,5,1,1,1,1,1] => 1
[7,4,4,2] => 1
[7,4,4,1,1] => 0
[7,4,3,3] => 2
[7,4,3,2,1] => 2
[7,4,3,1,1,1] => 1
[7,4,2,2,2] => 3
[7,4,2,2,1,1] => 2
[7,4,2,1,1,1,1] => 1
[7,4,1,1,1,1,1,1] => 0
[7,3,3,3,1] => 3
[7,3,3,2,2] => 4
[7,3,3,2,1,1] => 3
[7,3,3,1,1,1,1] => 2
[7,3,2,2,2,1] => 4
[7,3,2,2,1,1,1] => 3
[7,3,2,1,1,1,1,1] => 2
[7,3,1,1,1,1,1,1,1] => 1
[7,2,2,2,2,2] => 5
[7,2,2,2,2,1,1] => 4
[7,2,2,2,1,1,1,1] => 3
[7,2,2,1,1,1,1,1,1] => 2
[7,2,1,1,1,1,1,1,1,1] => 1
[7,1,1,1,1,1,1,1,1,1,1] => 0
[6,6,5] => 3
[6,6,4,1] => 2
[6,6,3,2] => 4
[6,6,3,1,1] => 3
[6,6,2,2,1] => 4
[6,6,2,1,1,1] => 3
[6,6,1,1,1,1,1] => 2
[6,5,5,1] => 3
[6,5,4,2] => 3
[6,5,4,1,1] => 2
[6,5,3,3] => 4
[6,5,3,2,1] => 4
[6,5,3,1,1,1] => 3
[6,5,2,2,2] => 5
[6,5,2,2,1,1] => 4
[6,5,2,1,1,1,1] => 3
[6,5,1,1,1,1,1,1] => 2
[6,4,4,3] => 2
[6,4,4,2,1] => 2
[6,4,4,1,1,1] => 1
[6,4,3,3,1] => 3
[6,4,3,2,2] => 4
[6,4,3,2,1,1] => 3
[6,4,3,1,1,1,1] => 2
[6,4,2,2,2,1] => 4
[6,4,2,2,1,1,1] => 3
[6,4,2,1,1,1,1,1] => 2
[6,4,1,1,1,1,1,1,1] => 1
[6,3,3,3,2] => 5
[6,3,3,3,1,1] => 4
[6,3,3,2,2,1] => 5
[6,3,3,2,1,1,1] => 4
[6,3,3,1,1,1,1,1] => 3
[6,3,2,2,2,2] => 6
[6,3,2,2,2,1,1] => 5
[6,3,2,2,1,1,1,1] => 4
[6,3,2,1,1,1,1,1,1] => 3
[6,3,1,1,1,1,1,1,1,1] => 2
[6,2,2,2,2,2,1] => 6
[6,2,2,2,2,1,1,1] => 5
[6,2,2,2,1,1,1,1,1] => 4
[6,2,2,1,1,1,1,1,1,1] => 3
[6,2,1,1,1,1,1,1,1,1,1] => 2
[6,1,1,1,1,1,1,1,1,1,1,1] => 1
[5,5,5,2] => 4
[5,5,5,1,1] => 3
[5,5,4,3] => 3
[5,5,4,2,1] => 3
[5,5,4,1,1,1] => 2
[5,5,3,3,1] => 4
[5,5,3,2,2] => 5
[5,5,3,2,1,1] => 4
[5,5,3,1,1,1,1] => 3
[5,5,2,2,2,1] => 5
[5,5,2,2,1,1,1] => 4
[5,5,2,1,1,1,1,1] => 3
[5,5,1,1,1,1,1,1,1] => 2
[5,4,4,4] => 1
[5,4,4,3,1] => 2
[5,4,4,2,2] => 3
[5,4,4,2,1,1] => 2
[5,4,4,1,1,1,1] => 1
[5,4,3,3,2] => 4
[5,4,3,3,1,1] => 3
[5,4,3,2,2,1] => 4
[5,4,3,2,1,1,1] => 3
[5,4,3,1,1,1,1,1] => 2
[5,4,2,2,2,2] => 5
[5,4,2,2,2,1,1] => 4
[5,4,2,2,1,1,1,1] => 3
[5,4,2,1,1,1,1,1,1] => 2
[5,4,1,1,1,1,1,1,1,1] => 1
[5,3,3,3,3] => 5
[5,3,3,3,2,1] => 5
[5,3,3,3,1,1,1] => 4
[5,3,3,2,2,2] => 6
[5,3,3,2,2,1,1] => 5
[5,3,3,2,1,1,1,1] => 4
[5,3,3,1,1,1,1,1,1] => 3
[5,3,2,2,2,2,1] => 6
[5,3,2,2,2,1,1,1] => 5
[5,3,2,2,1,1,1,1,1] => 4
[5,3,2,1,1,1,1,1,1,1] => 3
[5,3,1,1,1,1,1,1,1,1,1] => 2
[5,2,2,2,2,2,2] => 7
[5,2,2,2,2,2,1,1] => 6
[5,2,2,2,2,1,1,1,1] => 5
[5,2,2,2,1,1,1,1,1,1] => 4
[5,2,2,1,1,1,1,1,1,1,1] => 3
[5,2,1,1,1,1,1,1,1,1,1,1] => 2
[5,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[4,4,4,4,1] => 0
[4,4,4,3,2] => 2
[4,4,4,3,1,1] => 1
[4,4,4,2,2,1] => 2
[4,4,4,2,1,1,1] => 1
[4,4,4,1,1,1,1,1] => 0
[4,4,3,3,3] => 3
[4,4,3,3,2,1] => 3
[4,4,3,3,1,1,1] => 2
[4,4,3,2,2,2] => 4
[4,4,3,2,2,1,1] => 3
[4,4,3,2,1,1,1,1] => 2
[4,4,3,1,1,1,1,1,1] => 1
[4,4,2,2,2,2,1] => 4
[4,4,2,2,2,1,1,1] => 3
[4,4,2,2,1,1,1,1,1] => 2
[4,4,2,1,1,1,1,1,1,1] => 1
[4,4,1,1,1,1,1,1,1,1,1] => 0
[4,3,3,3,3,1] => 4
[4,3,3,3,2,2] => 5
[4,3,3,3,2,1,1] => 4
[4,3,3,3,1,1,1,1] => 3
[4,3,3,2,2,2,1] => 5
[4,3,3,2,2,1,1,1] => 4
[4,3,3,2,1,1,1,1,1] => 3
[4,3,3,1,1,1,1,1,1,1] => 2
[4,3,2,2,2,2,2] => 6
[4,3,2,2,2,2,1,1] => 5
[4,3,2,2,2,1,1,1,1] => 4
[4,3,2,2,1,1,1,1,1,1] => 3
[4,3,2,1,1,1,1,1,1,1,1] => 2
[4,3,1,1,1,1,1,1,1,1,1,1] => 1
[4,2,2,2,2,2,2,1] => 6
[4,2,2,2,2,2,1,1,1] => 5
[4,2,2,2,2,1,1,1,1,1] => 4
[4,2,2,2,1,1,1,1,1,1,1] => 3
[4,2,2,1,1,1,1,1,1,1,1,1] => 2
[4,2,1,1,1,1,1,1,1,1,1,1,1] => 1
[4,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[3,3,3,3,3,2] => 6
[3,3,3,3,3,1,1] => 5
[3,3,3,3,2,2,1] => 6
[3,3,3,3,2,1,1,1] => 5
[3,3,3,3,1,1,1,1,1] => 4
[3,3,3,2,2,2,2] => 7
[3,3,3,2,2,2,1,1] => 6
[3,3,3,2,2,1,1,1,1] => 5
[3,3,3,2,1,1,1,1,1,1] => 4
[3,3,3,1,1,1,1,1,1,1,1] => 3
[3,3,2,2,2,2,2,1] => 7
[3,3,2,2,2,2,1,1,1] => 6
[3,3,2,2,2,1,1,1,1,1] => 5
[3,3,2,2,1,1,1,1,1,1,1] => 4
[3,3,2,1,1,1,1,1,1,1,1,1] => 3
[3,3,1,1,1,1,1,1,1,1,1,1,1] => 2
[3,2,2,2,2,2,2,2] => 8
[3,2,2,2,2,2,2,1,1] => 7
[3,2,2,2,2,2,1,1,1,1] => 6
[3,2,2,2,2,1,1,1,1,1,1] => 5
[3,2,2,2,1,1,1,1,1,1,1,1] => 4
[3,2,2,1,1,1,1,1,1,1,1,1,1] => 3
[3,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[4,4,4,3,2,1] => 2
[5,4,3,3,2,1] => 4
[6,3,3,3,2,1] => 5
[6,5,2,2,2,1] => 5
[5,5,3,3,1,1] => 4
[6,5,4,1,1,1] => 2
[5,5,3,3,2] => 5
[5,5,4,2,2] => 4
[6,4,4,2,2] => 3
[6,5,4,3] => 3
[4,4,4,3,3] => 2
[4,4,4,4,2] => 1
[5,5,4,4] => 2
[6,4,4,4] => 1
[2,2,2,2,2,2,2,2,2] => 9
[3,3,3,3,3,3] => 6
[18] => 1
[6,2,2,2,2,2,2] => 7
[6,6,6] => 3
[4,2,2,2,2,2,2,2] => 7
[10,4,4] => 0
[9,6,3] => 3
[8,6,4] => 2
[10,8] => 1
[3,3,3,3,1,1,1,1,1,1] => 4
[9,9] => 2
[7,7,1,1,1,1] => 0
[5,4,4,3,2,1] => 3
[5,5,3,3,2,1] => 5
[5,5,4,2,2,1] => 4
[6,4,4,2,2,1] => 3
[5,5,4,3,1,1] => 3
[6,4,4,3,1,1] => 2
[6,5,3,3,1,1] => 4
[5,5,4,3,2] => 4
[6,4,4,3,2] => 3
[6,5,3,3,2] => 5
[6,5,4,2,2] => 4
[6,5,4,3,1] => 3
[6,5,4,1,1,1,1] => 2
[4,4,4,4,3] => 1
[4,3,3,3,3,3] => 5
[19] => 0
[7,2,2,2,2,2,2] => 6
[7,6,6] => 2
[5,2,2,2,2,2,2,2] => 8
[13,6] => 1
[11,4,4] => 1
[9,6,4] => 2
[8,5,4,2] => 3
[8,5,5,1] => 3
[5,5,4,3,2,1] => 4
[6,4,4,3,2,1] => 3
[6,5,3,3,2,1] => 5
[6,5,4,2,2,1] => 4
[6,5,4,3,1,1] => 3
[6,5,4,3,2] => 4
[6,5,2,2,2,2,1] => 6
[6,5,4,2,1,1,1] => 3
[6,6,2,2,2,2] => 6
[7,5,4,3,1] => 2
[2,2,2,2,2,2,2,2,2,2] => 10
[3,3,3,3,3,3,2] => 7
[4,4,3,3,3,3] => 4
[4,4,4,4,4] => 0
[5,5,5,5] => 4
[20] => 1
[10,10] => 0
[8,2,2,2,2,2,2] => 7
[8,6,4,2] => 3
[8,6,6] => 3
[10,6,4] => 1
[9,8,3] => 3
[6,6,6,2] => 4
[10,7,3] => 1
[9,7,4] => 1
[9,5,5,1] => 3
[6,5,4,3,2,1] => 4
[6,3,3,3,3,2,1] => 6
[6,5,3,2,2,2,1] => 6
[6,5,4,3,1,1,1] => 3
[7,6,2,2,2,2] => 5
[6,6,5,4] => 3
[3,3,3,3,3,3,3] => 7
[4,4,4,3,3,3] => 3
[21] => 1
[11,7,3] => 2
[5,4,2,2,2,2,2,2] => 7
[11,10] => 1
[13,4,4] => 0
[7,6,6,2] => 3
[4,4,4,4,3,2,1] => 2
[6,4,3,3,3,2,1] => 5
[6,5,4,2,2,2,1] => 5
[6,5,4,3,2,1,1] => 4
[4,4,4,4,3,3] => 2
[8,6,2,2,2,2] => 6
[22] => 0
[9,6,4,3] => 3
[8,7,7] => 1
[5,4,4,4,3,2,1] => 3
[6,5,3,3,3,2,1] => 6
[6,5,4,3,2,2,1] => 5
[15,2,2,2,2] => 5
[9,6,5,3] => 4
[8,6,5,3,1] => 4
[6,4,4,4,3,2,1] => 3
[6,5,4,3,3,2,1] => 5
[3,3,3,3,3,3,3,3] => 8
[4,4,4,4,4,4] => 0
[6,6,6,6] => 4
[24] => 1
[11,7,5,1] => 2
[9,7,5,3] => 3
[8,8,8] => 3
[17,7] => 1
[12,9,3] => 3
[5,5,5,4,3,2,1] => 5
[6,5,4,4,3,2,1] => 4
[7,6,6,6] => 3
[5,4,4,4,2,2,2,2] => 5
[13,12] => 1
[15,5,5] => 3
[9,7,5,3,1] => 3
[10,7,5,3] => 2
[3,3,3,3,3,1,1,1,1,1,1,1,1,1,1] => 5
[5,5,5,5,5] => 5
[12,10,3] => 2
[6,5,5,4,3,2,1] => 5
[6,6,4,3,3,2,2] => 6
[8,6,6,6] => 4
[9,7,5,4,1] => 2
[16,8,2] => 2
[6,6,5,4,3,2,1] => 5
[7,6,5,4,3,2] => 4
[3,3,3,3,3,3,3,3,3] => 9
[15,6,6] => 3
[7,6,5,4,3,2,1] => 4
[6,6,6,4,3,3] => 5
[4,3,3,3,3,3,3,3,3] => 8
[7,6,5,4,3,1,1,1] => 3
[10,7,6,4,1] => 1
[9,7,6,4,2] => 3
[10,8,5,4,1] => 2
[15,14] => 2
[7,6,5,4,3,2,1,1] => 4
[7,6,5,4,2,2,2,1] => 5
[10,8,6,4,1] => 2
[6,6,6,6,6] => 5
[15,15] => 2
[5,5,5,5,5,5] => 6
[6,6,6,6,3,3] => 6
[10,10,10] => 0
[9,7,5,5,3,1] => 4
[7,6,5,4,3,2,2,1] => 5
[7,6,5,3,3,3,2,1] => 6
[11,8,6,4,1] => 3
[10,8,6,4,2] => 3
[2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] => 15
[7,6,6,6,6] => 4
[7,6,5,4,3,3,2,1] => 5
[7,6,4,4,4,3,2,1] => 3
[11,8,6,5,1] => 4
[10,10,9,2] => 2
[14,9,8] => 3
[4,4,4,4,4,4,4,4] => 0
[2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] => 16
[7,6,5,4,4,3,2,1] => 4
[7,5,5,5,4,3,2,1] => 5
[16,16] => 0
[10,10,10,2] => 1
[16,8,4,2,2] => 3
[3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] => 16
[5,4,4,4,4,4,4,4] => 1
[7,6,5,5,4,3,2,1] => 5
[6,6,6,5,4,3,2,1] => 6
[22,7,4] => 0
[12,12,4,3,2] => 4
[4,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] => 15
[7,6,6,5,4,3,2,1] => 5
[12,9,7,5,1] => 3
[7,2,2,2,2,2,2,2,2,2,2,2,2,2,2] => 14
[7,7,6,5,4,3,2,1] => 4
[13,9,7,5,1] => 2
[18,18] => 2
[24,12] => 2
[11,9,7,5,3,1] => 4
[11,8,7,5,4,1] => 3
[8,7,6,5,4,3,2,1] => 5
[8,7,7,7,7] => 1
[7,6,2,2,2,2,2,2,2,2,2,2,2,2] => 13
[7,6,6,6,6,6] => 5
[8,7,6,5,4,3,2,1,1] => 5
[8,7,6,5,4,3,2,2,1] => 6
[8,7,6,5,4,3,3,2,1] => 6
[10,10,10,10] => 0
[12,12,12,4] => 3
[8,7,6,5,4,4,3,2,1] => 5
[11,9,7,5,5,3] => 5
[4,4,4,4,4,4,4,4,4,4] => 0
[14,14,8,4] => 3
[7,6,6,6,2,2,2,2,2,2,2,2] => 11
[8,7,6,5,5,4,3,2,1] => 6
[9,9,9,9,3,3] => 6
[8,7,6,6,5,4,3,2,1] => 6
[10,10,10,5,5,2] => 3
[16,10,8,8] => 2
[26,9,7] => 2
[8,7,7,6,5,4,3,2,1] => 5
[8,8,7,6,5,4,3,2,1] => 6
[18,18,9] => 3
[9,8,7,6,5,4,3,2,1] => 6
[11,9,7,7,5,3,3] => 5
[48] => 1
[7,6,6,6,6,6,6,6] => 7
[22,9,8,5,5] => 4
[6,6,6,6,6,6,6,3,3,2] => 10
[27,14,9] => 3
[17,14,13,8] => 3
[5,5,5,5,5,5,5,5,5,5,5] => 11
[27,20,8] => 3
[26,13,7,7,2] => 2
[31,12,10,4] => 1
[6,6,6,6,6,6,6,6,6,6] => 10
[5,5,5,5,5,5,5,5,5,5,5,5] => 12
[12,12,12,12,12] => 5
[12,12,12,12,11,4] => 5
[12,12,12,12,12,4] => 5
[3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] => 32
[11,11,11,11,11,10] => 5
[11,11,11,11,11,11] => 6
[16,14,9,9,9,9] => 5
[8,8,8,8,8,8,8,8,4,2] => 9
[24,24,24] => 3
[16,16,16,16,8,4,2] => 2
[12,12,12,12,12,12,4,4] => 6
[22,20,20,12,8] => 4
[20,20,8,4,4,4,4,4,4,4,4,4] => 3
[33,32,9,9,5] => 5
[18,18,18,18,18] => 5
[18,18,18,18,9,9] => 6
[6,6,6,6,6,6,6,6,6,6,6,6,6,6,3,3] => 16
[56,17,11,10,5] => 4
[33,26,23,12,9] => 5
[22,22,22,10,10,9,4,4] => 1
[22,22,22,10,10,10,4,4] => 0
[12,12,12,12,12,12,12,12,4,3,2] => 10
[24,24,24,24,14] => 5
[76,22,9,6] => 2
[5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5] => 23
[5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5] => 24
[6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6] => 20
[12,12,12,12,12,12,12,12,12,12] => 10
[24,24,24,24,24] => 5
[12,12,12,12,12,12,12,12,12,6,6,6,4,2] => 13
[54,48,18,10,8] => 4
[24,24,24,24,24,24] => 6
[37,33,20,20,13,10,10,10,7] => 3
[42,34,30,22,22,14,2] => 4
[54,52,34,18,11,7] => 3
[65,27,27,25,13,9,6,6] => 6
[18,18,18,18,18,18,18,18,18,18] => 10
[6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6] => 30
[16,16,16,16,16,16,16,8,8,8,8,8,8,8,8,4,2] => 9
[58,38,38,38,30] => 4
[12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12] => 20
[10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,2] => 1
[74,58,34,34,22,22,13,11,2] => 3
[61,48,48,48,45,22,12,2] => 6
[15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3] => 22
[98,62,36,26,14,14,14,14,14,14,12,2] => 12
[14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,13,2] => 25
[14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,2] => 26
[74,74,26,26,26,26,12,12,12,12,12,12,6,6,4,4,3,3,3,3] => 18
[6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6] => 60
[64,57,40,39,28,23,19,19,16,9,8,8,8,8,8,8,6,6] => 12
[70,70,46,28,22,22,10,9,9,9,9,8,8,8,8,8,7,7,7,3,3,3] => 12
[14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,7,7,7,7,7,2] => 29
[170,92,36,28,27,24,20,14,13,10,8,8] => 9
[12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,6,6,6,6,4,4,4,3,3] => 41
[145,83,62,44,44,42,19,16,16,10,8] => 6
[98,98,37,37,37,37,24,24,24,24,24,24,24,24,10,10,8,8] => 12
[6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6] => 120
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Generating function
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Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
1,1 1,2 2,2,1 2,3,2 2,5,3,1 3,6,4,2 4,7,7,3,1 4,10,9,5,2 5,13,12,8,3,1 6,15,17,11,5,2 7,19,23,15,9,3,1 8,24,28,22,12,5,2 10,28,37,30,17,9,3,1 11,34,47,39,25,13,5,2 13,42,58,52,35,18,9,3,1
$F_{1} = 1$
$F_{2} = 1 + q$
$F_{3} = 1 + 2\ q$
$F_{4} = 2 + 2\ q + q^{2}$
$F_{5} = 2 + 3\ q + 2\ q^{2}$
$F_{6} = 2 + 5\ q + 3\ q^{2} + q^{3}$
$F_{7} = 3 + 6\ q + 4\ q^{2} + 2\ q^{3}$
$F_{8} = 4 + 7\ q + 7\ q^{2} + 3\ q^{3} + q^{4}$
$F_{9} = 4 + 10\ q + 9\ q^{2} + 5\ q^{3} + 2\ q^{4}$
$F_{10} = 5 + 13\ q + 12\ q^{2} + 8\ q^{3} + 3\ q^{4} + q^{5}$
$F_{11} = 6 + 15\ q + 17\ q^{2} + 11\ q^{3} + 5\ q^{4} + 2\ q^{5}$
$F_{12} = 7 + 19\ q + 23\ q^{2} + 15\ q^{3} + 9\ q^{4} + 3\ q^{5} + q^{6}$
$F_{13} = 8 + 24\ q + 28\ q^{2} + 22\ q^{3} + 12\ q^{4} + 5\ q^{5} + 2\ q^{6}$
$F_{14} = 10 + 28\ q + 37\ q^{2} + 30\ q^{3} + 17\ q^{4} + 9\ q^{5} + 3\ q^{6} + q^{7}$
$F_{15} = 11 + 34\ q + 47\ q^{2} + 39\ q^{3} + 25\ q^{4} + 13\ q^{5} + 5\ q^{6} + 2\ q^{7}$
$F_{16} = 13 + 42\ q + 58\ q^{2} + 52\ q^{3} + 35\ q^{4} + 18\ q^{5} + 9\ q^{6} + 3\ q^{7} + q^{8}$
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Code
def statistic(L):
L = list(L)
return sum(1 for i in L if (i % 3) != 1)
Created
Sep 10, 2018 at 09:54 by Christian Stump
Updated
Feb 22, 2024 at 12:13 by Martin Rubey
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