Identifier
-
Mp00044:
Integer partitions
—conjugate⟶
Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000288: Binary words ⟶ ℤ
Values
[1] => [1] => 1 => 1 => 1
[2] => [1,1] => 11 => 11 => 2
[1,1] => [2] => 0 => 0 => 0
[3] => [1,1,1] => 111 => 111 => 3
[2,1] => [2,1] => 01 => 01 => 1
[1,1,1] => [3] => 1 => 1 => 1
[4] => [1,1,1,1] => 1111 => 1111 => 4
[3,1] => [2,1,1] => 011 => 011 => 2
[2,2] => [2,2] => 00 => 00 => 0
[2,1,1] => [3,1] => 11 => 11 => 2
[1,1,1,1] => [4] => 0 => 0 => 0
[5] => [1,1,1,1,1] => 11111 => 11111 => 5
[4,1] => [2,1,1,1] => 0111 => 0111 => 3
[3,2] => [2,2,1] => 001 => 001 => 1
[3,1,1] => [3,1,1] => 111 => 111 => 3
[2,2,1] => [3,2] => 10 => 01 => 1
[2,1,1,1] => [4,1] => 01 => 01 => 1
[1,1,1,1,1] => [5] => 1 => 1 => 1
[6] => [1,1,1,1,1,1] => 111111 => 111111 => 6
[5,1] => [2,1,1,1,1] => 01111 => 01111 => 4
[4,2] => [2,2,1,1] => 0011 => 0011 => 2
[4,1,1] => [3,1,1,1] => 1111 => 1111 => 4
[3,3] => [2,2,2] => 000 => 000 => 0
[3,2,1] => [3,2,1] => 101 => 011 => 2
[3,1,1,1] => [4,1,1] => 011 => 011 => 2
[2,2,2] => [3,3] => 11 => 11 => 2
[2,2,1,1] => [4,2] => 00 => 00 => 0
[2,1,1,1,1] => [5,1] => 11 => 11 => 2
[1,1,1,1,1,1] => [6] => 0 => 0 => 0
[7] => [1,1,1,1,1,1,1] => 1111111 => 1111111 => 7
[6,1] => [2,1,1,1,1,1] => 011111 => 011111 => 5
[5,2] => [2,2,1,1,1] => 00111 => 00111 => 3
[5,1,1] => [3,1,1,1,1] => 11111 => 11111 => 5
[4,3] => [2,2,2,1] => 0001 => 0001 => 1
[4,2,1] => [3,2,1,1] => 1011 => 0111 => 3
[4,1,1,1] => [4,1,1,1] => 0111 => 0111 => 3
[3,3,1] => [3,2,2] => 100 => 001 => 1
[3,2,2] => [3,3,1] => 111 => 111 => 3
[3,2,1,1] => [4,2,1] => 001 => 001 => 1
[3,1,1,1,1] => [5,1,1] => 111 => 111 => 3
[2,2,2,1] => [4,3] => 01 => 01 => 1
[2,2,1,1,1] => [5,2] => 10 => 01 => 1
[2,1,1,1,1,1] => [6,1] => 01 => 01 => 1
[1,1,1,1,1,1,1] => [7] => 1 => 1 => 1
[8] => [1,1,1,1,1,1,1,1] => 11111111 => 11111111 => 8
[7,1] => [2,1,1,1,1,1,1] => 0111111 => 0111111 => 6
[6,2] => [2,2,1,1,1,1] => 001111 => 001111 => 4
[6,1,1] => [3,1,1,1,1,1] => 111111 => 111111 => 6
[5,3] => [2,2,2,1,1] => 00011 => 00011 => 2
[5,2,1] => [3,2,1,1,1] => 10111 => 01111 => 4
[5,1,1,1] => [4,1,1,1,1] => 01111 => 01111 => 4
[4,4] => [2,2,2,2] => 0000 => 0000 => 0
[4,3,1] => [3,2,2,1] => 1001 => 0011 => 2
[4,2,2] => [3,3,1,1] => 1111 => 1111 => 4
[4,2,1,1] => [4,2,1,1] => 0011 => 0011 => 2
[4,1,1,1,1] => [5,1,1,1] => 1111 => 1111 => 4
[3,3,2] => [3,3,2] => 110 => 011 => 2
[3,3,1,1] => [4,2,2] => 000 => 000 => 0
[3,2,2,1] => [4,3,1] => 011 => 011 => 2
[3,2,1,1,1] => [5,2,1] => 101 => 011 => 2
[3,1,1,1,1,1] => [6,1,1] => 011 => 011 => 2
[2,2,2,2] => [4,4] => 00 => 00 => 0
[2,2,2,1,1] => [5,3] => 11 => 11 => 2
[2,2,1,1,1,1] => [6,2] => 00 => 00 => 0
[2,1,1,1,1,1,1] => [7,1] => 11 => 11 => 2
[1,1,1,1,1,1,1,1] => [8] => 0 => 0 => 0
[9] => [1,1,1,1,1,1,1,1,1] => 111111111 => 111111111 => 9
[8,1] => [2,1,1,1,1,1,1,1] => 01111111 => 01111111 => 7
[7,2] => [2,2,1,1,1,1,1] => 0011111 => 0011111 => 5
[7,1,1] => [3,1,1,1,1,1,1] => 1111111 => 1111111 => 7
[6,3] => [2,2,2,1,1,1] => 000111 => 000111 => 3
[6,2,1] => [3,2,1,1,1,1] => 101111 => 011111 => 5
[6,1,1,1] => [4,1,1,1,1,1] => 011111 => 011111 => 5
[5,4] => [2,2,2,2,1] => 00001 => 00001 => 1
[5,3,1] => [3,2,2,1,1] => 10011 => 00111 => 3
[5,2,2] => [3,3,1,1,1] => 11111 => 11111 => 5
[5,2,1,1] => [4,2,1,1,1] => 00111 => 00111 => 3
[5,1,1,1,1] => [5,1,1,1,1] => 11111 => 11111 => 5
[4,4,1] => [3,2,2,2] => 1000 => 0001 => 1
[4,3,2] => [3,3,2,1] => 1101 => 0111 => 3
[4,3,1,1] => [4,2,2,1] => 0001 => 0001 => 1
[4,2,2,1] => [4,3,1,1] => 0111 => 0111 => 3
[4,2,1,1,1] => [5,2,1,1] => 1011 => 0111 => 3
[4,1,1,1,1,1] => [6,1,1,1] => 0111 => 0111 => 3
[3,3,3] => [3,3,3] => 111 => 111 => 3
[3,3,2,1] => [4,3,2] => 010 => 001 => 1
[3,3,1,1,1] => [5,2,2] => 100 => 001 => 1
[3,2,2,2] => [4,4,1] => 001 => 001 => 1
[3,2,2,1,1] => [5,3,1] => 111 => 111 => 3
[3,2,1,1,1,1] => [6,2,1] => 001 => 001 => 1
[3,1,1,1,1,1,1] => [7,1,1] => 111 => 111 => 3
[2,2,2,2,1] => [5,4] => 10 => 01 => 1
[2,2,2,1,1,1] => [6,3] => 01 => 01 => 1
[2,2,1,1,1,1,1] => [7,2] => 10 => 01 => 1
[2,1,1,1,1,1,1,1] => [8,1] => 01 => 01 => 1
[1,1,1,1,1,1,1,1,1] => [9] => 1 => 1 => 1
[9,1] => [2,1,1,1,1,1,1,1,1] => 011111111 => 011111111 => 8
[8,2] => [2,2,1,1,1,1,1,1] => 00111111 => 00111111 => 6
[8,1,1] => [3,1,1,1,1,1,1,1] => 11111111 => 11111111 => 8
[7,3] => [2,2,2,1,1,1,1] => 0001111 => 0001111 => 4
[7,2,1] => [3,2,1,1,1,1,1] => 1011111 => 0111111 => 6
>>> Load all 1107 entries. <<<[7,1,1,1] => [4,1,1,1,1,1,1] => 0111111 => 0111111 => 6
[6,4] => [2,2,2,2,1,1] => 000011 => 000011 => 2
[6,3,1] => [3,2,2,1,1,1] => 100111 => 001111 => 4
[6,2,2] => [3,3,1,1,1,1] => 111111 => 111111 => 6
[6,2,1,1] => [4,2,1,1,1,1] => 001111 => 001111 => 4
[6,1,1,1,1] => [5,1,1,1,1,1] => 111111 => 111111 => 6
[5,5] => [2,2,2,2,2] => 00000 => 00000 => 0
[5,4,1] => [3,2,2,2,1] => 10001 => 00011 => 2
[5,3,2] => [3,3,2,1,1] => 11011 => 01111 => 4
[5,3,1,1] => [4,2,2,1,1] => 00011 => 00011 => 2
[5,2,2,1] => [4,3,1,1,1] => 01111 => 01111 => 4
[5,2,1,1,1] => [5,2,1,1,1] => 10111 => 01111 => 4
[5,1,1,1,1,1] => [6,1,1,1,1] => 01111 => 01111 => 4
[4,4,2] => [3,3,2,2] => 1100 => 0011 => 2
[4,4,1,1] => [4,2,2,2] => 0000 => 0000 => 0
[4,3,3] => [3,3,3,1] => 1111 => 1111 => 4
[4,3,2,1] => [4,3,2,1] => 0101 => 0101 => 2
[4,3,1,1,1] => [5,2,2,1] => 1001 => 0011 => 2
[4,2,2,2] => [4,4,1,1] => 0011 => 0011 => 2
[4,2,2,1,1] => [5,3,1,1] => 1111 => 1111 => 4
[4,2,1,1,1,1] => [6,2,1,1] => 0011 => 0011 => 2
[4,1,1,1,1,1,1] => [7,1,1,1] => 1111 => 1111 => 4
[3,3,3,1] => [4,3,3] => 011 => 011 => 2
[3,3,2,2] => [4,4,2] => 000 => 000 => 0
[3,3,2,1,1] => [5,3,2] => 110 => 011 => 2
[3,3,1,1,1,1] => [6,2,2] => 000 => 000 => 0
[3,2,2,2,1] => [5,4,1] => 101 => 011 => 2
[3,2,2,1,1,1] => [6,3,1] => 011 => 011 => 2
[3,2,1,1,1,1,1] => [7,2,1] => 101 => 011 => 2
[3,1,1,1,1,1,1,1] => [8,1,1] => 011 => 011 => 2
[2,2,2,2,2] => [5,5] => 11 => 11 => 2
[2,2,2,2,1,1] => [6,4] => 00 => 00 => 0
[2,2,2,1,1,1,1] => [7,3] => 11 => 11 => 2
[2,2,1,1,1,1,1,1] => [8,2] => 00 => 00 => 0
[2,1,1,1,1,1,1,1,1] => [9,1] => 11 => 11 => 2
[1,1,1,1,1,1,1,1,1,1] => [10] => 0 => 0 => 0
[10,1] => [2,1,1,1,1,1,1,1,1,1] => 0111111111 => 0111111111 => 9
[9,2] => [2,2,1,1,1,1,1,1,1] => 001111111 => 001111111 => 7
[9,1,1] => [3,1,1,1,1,1,1,1,1] => 111111111 => 111111111 => 9
[8,3] => [2,2,2,1,1,1,1,1] => 00011111 => 00011111 => 5
[8,2,1] => [3,2,1,1,1,1,1,1] => 10111111 => 01111111 => 7
[8,1,1,1] => [4,1,1,1,1,1,1,1] => 01111111 => 01111111 => 7
[7,4] => [2,2,2,2,1,1,1] => 0000111 => 0000111 => 3
[7,3,1] => [3,2,2,1,1,1,1] => 1001111 => 0011111 => 5
[7,2,2] => [3,3,1,1,1,1,1] => 1111111 => 1111111 => 7
[7,2,1,1] => [4,2,1,1,1,1,1] => 0011111 => 0011111 => 5
[7,1,1,1,1] => [5,1,1,1,1,1,1] => 1111111 => 1111111 => 7
[6,5] => [2,2,2,2,2,1] => 000001 => 000001 => 1
[6,4,1] => [3,2,2,2,1,1] => 100011 => 000111 => 3
[6,3,2] => [3,3,2,1,1,1] => 110111 => 011111 => 5
[6,3,1,1] => [4,2,2,1,1,1] => 000111 => 000111 => 3
[6,2,2,1] => [4,3,1,1,1,1] => 011111 => 011111 => 5
[6,2,1,1,1] => [5,2,1,1,1,1] => 101111 => 011111 => 5
[6,1,1,1,1,1] => [6,1,1,1,1,1] => 011111 => 011111 => 5
[5,5,1] => [3,2,2,2,2] => 10000 => 00001 => 1
[5,4,2] => [3,3,2,2,1] => 11001 => 00111 => 3
[5,4,1,1] => [4,2,2,2,1] => 00001 => 00001 => 1
[5,3,3] => [3,3,3,1,1] => 11111 => 11111 => 5
[5,3,2,1] => [4,3,2,1,1] => 01011 => 01011 => 3
[5,3,1,1,1] => [5,2,2,1,1] => 10011 => 00111 => 3
[5,2,2,2] => [4,4,1,1,1] => 00111 => 00111 => 3
[5,2,2,1,1] => [5,3,1,1,1] => 11111 => 11111 => 5
[5,2,1,1,1,1] => [6,2,1,1,1] => 00111 => 00111 => 3
[5,1,1,1,1,1,1] => [7,1,1,1,1] => 11111 => 11111 => 5
[4,4,3] => [3,3,3,2] => 1110 => 0111 => 3
[4,4,2,1] => [4,3,2,2] => 0100 => 0001 => 1
[4,4,1,1,1] => [5,2,2,2] => 1000 => 0001 => 1
[4,3,3,1] => [4,3,3,1] => 0111 => 0111 => 3
[4,3,2,2] => [4,4,2,1] => 0001 => 0001 => 1
[4,3,2,1,1] => [5,3,2,1] => 1101 => 0111 => 3
[4,3,1,1,1,1] => [6,2,2,1] => 0001 => 0001 => 1
[4,2,2,2,1] => [5,4,1,1] => 1011 => 0111 => 3
[4,2,2,1,1,1] => [6,3,1,1] => 0111 => 0111 => 3
[4,2,1,1,1,1,1] => [7,2,1,1] => 1011 => 0111 => 3
[4,1,1,1,1,1,1,1] => [8,1,1,1] => 0111 => 0111 => 3
[3,3,3,2] => [4,4,3] => 001 => 001 => 1
[3,3,3,1,1] => [5,3,3] => 111 => 111 => 3
[3,3,2,2,1] => [5,4,2] => 100 => 001 => 1
[3,3,2,1,1,1] => [6,3,2] => 010 => 001 => 1
[3,3,1,1,1,1,1] => [7,2,2] => 100 => 001 => 1
[3,2,2,2,2] => [5,5,1] => 111 => 111 => 3
[3,2,2,2,1,1] => [6,4,1] => 001 => 001 => 1
[3,2,2,1,1,1,1] => [7,3,1] => 111 => 111 => 3
[3,2,1,1,1,1,1,1] => [8,2,1] => 001 => 001 => 1
[3,1,1,1,1,1,1,1,1] => [9,1,1] => 111 => 111 => 3
[2,2,2,2,2,1] => [6,5] => 01 => 01 => 1
[2,2,2,2,1,1,1] => [7,4] => 10 => 01 => 1
[2,2,2,1,1,1,1,1] => [8,3] => 01 => 01 => 1
[2,2,1,1,1,1,1,1,1] => [9,2] => 10 => 01 => 1
[2,1,1,1,1,1,1,1,1,1] => [10,1] => 01 => 01 => 1
[1,1,1,1,1,1,1,1,1,1,1] => [11] => 1 => 1 => 1
[11,1] => [2,1,1,1,1,1,1,1,1,1,1] => 01111111111 => 01111111111 => 10
[9,3] => [2,2,2,1,1,1,1,1,1] => 000111111 => 000111111 => 6
[9,2,1] => [3,2,1,1,1,1,1,1,1] => 101111111 => 011111111 => 8
[9,1,1,1] => [4,1,1,1,1,1,1,1,1] => 011111111 => 011111111 => 8
[8,4] => [2,2,2,2,1,1,1,1] => 00001111 => 00001111 => 4
[8,3,1] => [3,2,2,1,1,1,1,1] => 10011111 => 00111111 => 6
[8,2,2] => [3,3,1,1,1,1,1,1] => 11111111 => 11111111 => 8
[8,2,1,1] => [4,2,1,1,1,1,1,1] => 00111111 => 00111111 => 6
[8,1,1,1,1] => [5,1,1,1,1,1,1,1] => 11111111 => 11111111 => 8
[7,5] => [2,2,2,2,2,1,1] => 0000011 => 0000011 => 2
[7,4,1] => [3,2,2,2,1,1,1] => 1000111 => 0001111 => 4
[7,3,2] => [3,3,2,1,1,1,1] => 1101111 => 0111111 => 6
[7,3,1,1] => [4,2,2,1,1,1,1] => 0001111 => 0001111 => 4
[7,2,2,1] => [4,3,1,1,1,1,1] => 0111111 => 0111111 => 6
[7,2,1,1,1] => [5,2,1,1,1,1,1] => 1011111 => 0111111 => 6
[7,1,1,1,1,1] => [6,1,1,1,1,1,1] => 0111111 => 0111111 => 6
[6,6] => [2,2,2,2,2,2] => 000000 => 000000 => 0
[6,5,1] => [3,2,2,2,2,1] => 100001 => 000011 => 2
[6,4,2] => [3,3,2,2,1,1] => 110011 => 001111 => 4
[6,4,1,1] => [4,2,2,2,1,1] => 000011 => 000011 => 2
[6,3,3] => [3,3,3,1,1,1] => 111111 => 111111 => 6
[6,3,2,1] => [4,3,2,1,1,1] => 010111 => 010111 => 4
[6,3,1,1,1] => [5,2,2,1,1,1] => 100111 => 001111 => 4
[6,2,2,2] => [4,4,1,1,1,1] => 001111 => 001111 => 4
[6,2,2,1,1] => [5,3,1,1,1,1] => 111111 => 111111 => 6
[6,2,1,1,1,1] => [6,2,1,1,1,1] => 001111 => 001111 => 4
[6,1,1,1,1,1,1] => [7,1,1,1,1,1] => 111111 => 111111 => 6
[5,5,2] => [3,3,2,2,2] => 11000 => 00011 => 2
[5,5,1,1] => [4,2,2,2,2] => 00000 => 00000 => 0
[5,4,3] => [3,3,3,2,1] => 11101 => 01111 => 4
[5,4,2,1] => [4,3,2,2,1] => 01001 => 00101 => 2
[5,4,1,1,1] => [5,2,2,2,1] => 10001 => 00011 => 2
[5,3,3,1] => [4,3,3,1,1] => 01111 => 01111 => 4
[5,3,2,2] => [4,4,2,1,1] => 00011 => 00011 => 2
[5,3,2,1,1] => [5,3,2,1,1] => 11011 => 01111 => 4
[5,3,1,1,1,1] => [6,2,2,1,1] => 00011 => 00011 => 2
[5,2,2,2,1] => [5,4,1,1,1] => 10111 => 01111 => 4
[5,2,2,1,1,1] => [6,3,1,1,1] => 01111 => 01111 => 4
[5,2,1,1,1,1,1] => [7,2,1,1,1] => 10111 => 01111 => 4
[5,1,1,1,1,1,1,1] => [8,1,1,1,1] => 01111 => 01111 => 4
[4,4,4] => [3,3,3,3] => 1111 => 1111 => 4
[4,4,3,1] => [4,3,3,2] => 0110 => 0011 => 2
[4,4,2,2] => [4,4,2,2] => 0000 => 0000 => 0
[4,4,2,1,1] => [5,3,2,2] => 1100 => 0011 => 2
[4,4,1,1,1,1] => [6,2,2,2] => 0000 => 0000 => 0
[4,3,3,2] => [4,4,3,1] => 0011 => 0011 => 2
[4,3,3,1,1] => [5,3,3,1] => 1111 => 1111 => 4
[4,3,2,2,1] => [5,4,2,1] => 1001 => 0011 => 2
[4,3,2,1,1,1] => [6,3,2,1] => 0101 => 0101 => 2
[4,3,1,1,1,1,1] => [7,2,2,1] => 1001 => 0011 => 2
[4,2,2,2,2] => [5,5,1,1] => 1111 => 1111 => 4
[4,2,2,2,1,1] => [6,4,1,1] => 0011 => 0011 => 2
[4,2,2,1,1,1,1] => [7,3,1,1] => 1111 => 1111 => 4
[4,2,1,1,1,1,1,1] => [8,2,1,1] => 0011 => 0011 => 2
[4,1,1,1,1,1,1,1,1] => [9,1,1,1] => 1111 => 1111 => 4
[3,3,3,3] => [4,4,4] => 000 => 000 => 0
[3,3,3,2,1] => [5,4,3] => 101 => 011 => 2
[3,3,3,1,1,1] => [6,3,3] => 011 => 011 => 2
[3,3,2,2,2] => [5,5,2] => 110 => 011 => 2
[3,3,2,2,1,1] => [6,4,2] => 000 => 000 => 0
[3,3,2,1,1,1,1] => [7,3,2] => 110 => 011 => 2
[3,3,1,1,1,1,1,1] => [8,2,2] => 000 => 000 => 0
[3,2,2,2,2,1] => [6,5,1] => 011 => 011 => 2
[3,2,2,2,1,1,1] => [7,4,1] => 101 => 011 => 2
[3,2,2,1,1,1,1,1] => [8,3,1] => 011 => 011 => 2
[3,2,1,1,1,1,1,1,1] => [9,2,1] => 101 => 011 => 2
[3,1,1,1,1,1,1,1,1,1] => [10,1,1] => 011 => 011 => 2
[2,2,2,2,2,2] => [6,6] => 00 => 00 => 0
[2,2,2,2,2,1,1] => [7,5] => 11 => 11 => 2
[2,2,2,2,1,1,1,1] => [8,4] => 00 => 00 => 0
[2,2,2,1,1,1,1,1,1] => [9,3] => 11 => 11 => 2
[2,2,1,1,1,1,1,1,1,1] => [10,2] => 00 => 00 => 0
[2,1,1,1,1,1,1,1,1,1,1] => [11,1] => 11 => 11 => 2
[1,1,1,1,1,1,1,1,1,1,1,1] => [12] => 0 => 0 => 0
[10,1,1,1] => [4,1,1,1,1,1,1,1,1,1] => 0111111111 => 0111111111 => 9
[9,4] => [2,2,2,2,1,1,1,1,1] => 000011111 => 000011111 => 5
[9,3,1] => [3,2,2,1,1,1,1,1,1] => 100111111 => 001111111 => 7
[9,2,2] => [3,3,1,1,1,1,1,1,1] => 111111111 => 111111111 => 9
[9,2,1,1] => [4,2,1,1,1,1,1,1,1] => 001111111 => 001111111 => 7
[9,1,1,1,1] => [5,1,1,1,1,1,1,1,1] => 111111111 => 111111111 => 9
[8,5] => [2,2,2,2,2,1,1,1] => 00000111 => 00000111 => 3
[8,4,1] => [3,2,2,2,1,1,1,1] => 10001111 => 00011111 => 5
[8,3,2] => [3,3,2,1,1,1,1,1] => 11011111 => 01111111 => 7
[8,3,1,1] => [4,2,2,1,1,1,1,1] => 00011111 => 00011111 => 5
[8,2,2,1] => [4,3,1,1,1,1,1,1] => 01111111 => 01111111 => 7
[8,2,1,1,1] => [5,2,1,1,1,1,1,1] => 10111111 => 01111111 => 7
[8,1,1,1,1,1] => [6,1,1,1,1,1,1,1] => 01111111 => 01111111 => 7
[7,6] => [2,2,2,2,2,2,1] => 0000001 => 0000001 => 1
[7,5,1] => [3,2,2,2,2,1,1] => 1000011 => 0000111 => 3
[7,4,2] => [3,3,2,2,1,1,1] => 1100111 => 0011111 => 5
[7,4,1,1] => [4,2,2,2,1,1,1] => 0000111 => 0000111 => 3
[7,3,3] => [3,3,3,1,1,1,1] => 1111111 => 1111111 => 7
[7,3,2,1] => [4,3,2,1,1,1,1] => 0101111 => 0101111 => 5
[7,3,1,1,1] => [5,2,2,1,1,1,1] => 1001111 => 0011111 => 5
[7,2,2,2] => [4,4,1,1,1,1,1] => 0011111 => 0011111 => 5
[7,2,2,1,1] => [5,3,1,1,1,1,1] => 1111111 => 1111111 => 7
[7,2,1,1,1,1] => [6,2,1,1,1,1,1] => 0011111 => 0011111 => 5
[7,1,1,1,1,1,1] => [7,1,1,1,1,1,1] => 1111111 => 1111111 => 7
[6,6,1] => [3,2,2,2,2,2] => 100000 => 000001 => 1
[6,5,2] => [3,3,2,2,2,1] => 110001 => 000111 => 3
[6,5,1,1] => [4,2,2,2,2,1] => 000001 => 000001 => 1
[6,4,3] => [3,3,3,2,1,1] => 111011 => 011111 => 5
[6,4,2,1] => [4,3,2,2,1,1] => 010011 => 001101 => 3
[6,4,1,1,1] => [5,2,2,2,1,1] => 100011 => 000111 => 3
[6,3,3,1] => [4,3,3,1,1,1] => 011111 => 011111 => 5
[6,3,2,2] => [4,4,2,1,1,1] => 000111 => 000111 => 3
[6,3,2,1,1] => [5,3,2,1,1,1] => 110111 => 011111 => 5
[6,3,1,1,1,1] => [6,2,2,1,1,1] => 000111 => 000111 => 3
[6,2,2,2,1] => [5,4,1,1,1,1] => 101111 => 011111 => 5
[6,2,2,1,1,1] => [6,3,1,1,1,1] => 011111 => 011111 => 5
[6,2,1,1,1,1,1] => [7,2,1,1,1,1] => 101111 => 011111 => 5
[6,1,1,1,1,1,1,1] => [8,1,1,1,1,1] => 011111 => 011111 => 5
[5,5,3] => [3,3,3,2,2] => 11100 => 00111 => 3
[5,5,2,1] => [4,3,2,2,2] => 01000 => 00001 => 1
[5,5,1,1,1] => [5,2,2,2,2] => 10000 => 00001 => 1
[5,4,4] => [3,3,3,3,1] => 11111 => 11111 => 5
[5,4,3,1] => [4,3,3,2,1] => 01101 => 01011 => 3
[5,4,2,2] => [4,4,2,2,1] => 00001 => 00001 => 1
[5,4,2,1,1] => [5,3,2,2,1] => 11001 => 00111 => 3
[5,4,1,1,1,1] => [6,2,2,2,1] => 00001 => 00001 => 1
[5,3,3,2] => [4,4,3,1,1] => 00111 => 00111 => 3
[5,3,3,1,1] => [5,3,3,1,1] => 11111 => 11111 => 5
[5,3,2,2,1] => [5,4,2,1,1] => 10011 => 00111 => 3
[5,3,2,1,1,1] => [6,3,2,1,1] => 01011 => 01011 => 3
[5,3,1,1,1,1,1] => [7,2,2,1,1] => 10011 => 00111 => 3
[5,2,2,2,2] => [5,5,1,1,1] => 11111 => 11111 => 5
[5,2,2,2,1,1] => [6,4,1,1,1] => 00111 => 00111 => 3
[5,2,2,1,1,1,1] => [7,3,1,1,1] => 11111 => 11111 => 5
[5,2,1,1,1,1,1,1] => [8,2,1,1,1] => 00111 => 00111 => 3
[5,1,1,1,1,1,1,1,1] => [9,1,1,1,1] => 11111 => 11111 => 5
[4,4,4,1] => [4,3,3,3] => 0111 => 0111 => 3
[4,4,3,2] => [4,4,3,2] => 0010 => 0001 => 1
[4,4,3,1,1] => [5,3,3,2] => 1110 => 0111 => 3
[4,4,2,2,1] => [5,4,2,2] => 1000 => 0001 => 1
[4,4,2,1,1,1] => [6,3,2,2] => 0100 => 0001 => 1
[4,4,1,1,1,1,1] => [7,2,2,2] => 1000 => 0001 => 1
[4,3,3,3] => [4,4,4,1] => 0001 => 0001 => 1
[4,3,3,2,1] => [5,4,3,1] => 1011 => 0111 => 3
[4,3,3,1,1,1] => [6,3,3,1] => 0111 => 0111 => 3
[4,3,2,2,2] => [5,5,2,1] => 1101 => 0111 => 3
[4,3,2,2,1,1] => [6,4,2,1] => 0001 => 0001 => 1
[4,3,2,1,1,1,1] => [7,3,2,1] => 1101 => 0111 => 3
[4,3,1,1,1,1,1,1] => [8,2,2,1] => 0001 => 0001 => 1
[4,2,2,2,2,1] => [6,5,1,1] => 0111 => 0111 => 3
[4,2,2,2,1,1,1] => [7,4,1,1] => 1011 => 0111 => 3
[4,2,2,1,1,1,1,1] => [8,3,1,1] => 0111 => 0111 => 3
[4,2,1,1,1,1,1,1,1] => [9,2,1,1] => 1011 => 0111 => 3
[4,1,1,1,1,1,1,1,1,1] => [10,1,1,1] => 0111 => 0111 => 3
[3,3,3,3,1] => [5,4,4] => 100 => 001 => 1
[3,3,3,2,2] => [5,5,3] => 111 => 111 => 3
[3,3,3,2,1,1] => [6,4,3] => 001 => 001 => 1
[3,3,3,1,1,1,1] => [7,3,3] => 111 => 111 => 3
[3,3,2,2,2,1] => [6,5,2] => 010 => 001 => 1
[3,3,2,2,1,1,1] => [7,4,2] => 100 => 001 => 1
[3,3,2,1,1,1,1,1] => [8,3,2] => 010 => 001 => 1
[3,3,1,1,1,1,1,1,1] => [9,2,2] => 100 => 001 => 1
[3,2,2,2,2,2] => [6,6,1] => 001 => 001 => 1
[3,2,2,2,2,1,1] => [7,5,1] => 111 => 111 => 3
[3,2,2,2,1,1,1,1] => [8,4,1] => 001 => 001 => 1
[3,2,2,1,1,1,1,1,1] => [9,3,1] => 111 => 111 => 3
[3,2,1,1,1,1,1,1,1,1] => [10,2,1] => 001 => 001 => 1
[3,1,1,1,1,1,1,1,1,1,1] => [11,1,1] => 111 => 111 => 3
[2,2,2,2,2,2,1] => [7,6] => 10 => 01 => 1
[2,2,2,2,2,1,1,1] => [8,5] => 01 => 01 => 1
[2,2,2,2,1,1,1,1,1] => [9,4] => 10 => 01 => 1
[2,2,2,1,1,1,1,1,1,1] => [10,3] => 01 => 01 => 1
[2,2,1,1,1,1,1,1,1,1,1] => [11,2] => 10 => 01 => 1
[2,1,1,1,1,1,1,1,1,1,1,1] => [12,1] => 01 => 01 => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1] => [13] => 1 => 1 => 1
[11,1,1,1] => [4,1,1,1,1,1,1,1,1,1,1] => 01111111111 => 01111111111 => 10
[9,5] => [2,2,2,2,2,1,1,1,1] => 000001111 => 000001111 => 4
[9,4,1] => [3,2,2,2,1,1,1,1,1] => 100011111 => 000111111 => 6
[9,3,2] => [3,3,2,1,1,1,1,1,1] => 110111111 => 011111111 => 8
[9,3,1,1] => [4,2,2,1,1,1,1,1,1] => 000111111 => 000111111 => 6
[9,2,2,1] => [4,3,1,1,1,1,1,1,1] => 011111111 => 011111111 => 8
[9,2,1,1,1] => [5,2,1,1,1,1,1,1,1] => 101111111 => 011111111 => 8
[9,1,1,1,1,1] => [6,1,1,1,1,1,1,1,1] => 011111111 => 011111111 => 8
[8,6] => [2,2,2,2,2,2,1,1] => 00000011 => 00000011 => 2
[8,5,1] => [3,2,2,2,2,1,1,1] => 10000111 => 00001111 => 4
[8,4,2] => [3,3,2,2,1,1,1,1] => 11001111 => 00111111 => 6
[8,4,1,1] => [4,2,2,2,1,1,1,1] => 00001111 => 00001111 => 4
[8,3,3] => [3,3,3,1,1,1,1,1] => 11111111 => 11111111 => 8
[8,3,2,1] => [4,3,2,1,1,1,1,1] => 01011111 => 01011111 => 6
[8,3,1,1,1] => [5,2,2,1,1,1,1,1] => 10011111 => 00111111 => 6
[8,2,2,2] => [4,4,1,1,1,1,1,1] => 00111111 => 00111111 => 6
[8,2,2,1,1] => [5,3,1,1,1,1,1,1] => 11111111 => 11111111 => 8
[8,2,1,1,1,1] => [6,2,1,1,1,1,1,1] => 00111111 => 00111111 => 6
[8,1,1,1,1,1,1] => [7,1,1,1,1,1,1,1] => 11111111 => 11111111 => 8
[7,7] => [2,2,2,2,2,2,2] => 0000000 => 0000000 => 0
[7,6,1] => [3,2,2,2,2,2,1] => 1000001 => 0000011 => 2
[7,5,2] => [3,3,2,2,2,1,1] => 1100011 => 0001111 => 4
[7,5,1,1] => [4,2,2,2,2,1,1] => 0000011 => 0000011 => 2
[7,4,3] => [3,3,3,2,1,1,1] => 1110111 => 0111111 => 6
[7,4,2,1] => [4,3,2,2,1,1,1] => 0100111 => 0011101 => 4
[7,4,1,1,1] => [5,2,2,2,1,1,1] => 1000111 => 0001111 => 4
[7,3,3,1] => [4,3,3,1,1,1,1] => 0111111 => 0111111 => 6
[7,3,2,2] => [4,4,2,1,1,1,1] => 0001111 => 0001111 => 4
[7,3,2,1,1] => [5,3,2,1,1,1,1] => 1101111 => 0111111 => 6
[7,3,1,1,1,1] => [6,2,2,1,1,1,1] => 0001111 => 0001111 => 4
[7,2,2,2,1] => [5,4,1,1,1,1,1] => 1011111 => 0111111 => 6
[7,2,2,1,1,1] => [6,3,1,1,1,1,1] => 0111111 => 0111111 => 6
[7,2,1,1,1,1,1] => [7,2,1,1,1,1,1] => 1011111 => 0111111 => 6
[7,1,1,1,1,1,1,1] => [8,1,1,1,1,1,1] => 0111111 => 0111111 => 6
[6,6,2] => [3,3,2,2,2,2] => 110000 => 000011 => 2
[6,6,1,1] => [4,2,2,2,2,2] => 000000 => 000000 => 0
[6,5,3] => [3,3,3,2,2,1] => 111001 => 001111 => 4
[6,5,2,1] => [4,3,2,2,2,1] => 010001 => 000101 => 2
[6,5,1,1,1] => [5,2,2,2,2,1] => 100001 => 000011 => 2
[6,4,4] => [3,3,3,3,1,1] => 111111 => 111111 => 6
[6,4,3,1] => [4,3,3,2,1,1] => 011011 => 011011 => 4
[6,4,2,2] => [4,4,2,2,1,1] => 000011 => 000011 => 2
[6,4,2,1,1] => [5,3,2,2,1,1] => 110011 => 001111 => 4
[6,4,1,1,1,1] => [6,2,2,2,1,1] => 000011 => 000011 => 2
[6,3,3,2] => [4,4,3,1,1,1] => 001111 => 001111 => 4
[6,3,3,1,1] => [5,3,3,1,1,1] => 111111 => 111111 => 6
[6,3,2,2,1] => [5,4,2,1,1,1] => 100111 => 001111 => 4
[6,3,2,1,1,1] => [6,3,2,1,1,1] => 010111 => 010111 => 4
[6,3,1,1,1,1,1] => [7,2,2,1,1,1] => 100111 => 001111 => 4
[6,2,2,2,2] => [5,5,1,1,1,1] => 111111 => 111111 => 6
[6,2,2,2,1,1] => [6,4,1,1,1,1] => 001111 => 001111 => 4
[6,2,2,1,1,1,1] => [7,3,1,1,1,1] => 111111 => 111111 => 6
[6,2,1,1,1,1,1,1] => [8,2,1,1,1,1] => 001111 => 001111 => 4
[6,1,1,1,1,1,1,1,1] => [9,1,1,1,1,1] => 111111 => 111111 => 6
[5,5,4] => [3,3,3,3,2] => 11110 => 01111 => 4
[5,5,3,1] => [4,3,3,2,2] => 01100 => 00011 => 2
[5,5,2,2] => [4,4,2,2,2] => 00000 => 00000 => 0
[5,5,2,1,1] => [5,3,2,2,2] => 11000 => 00011 => 2
[5,5,1,1,1,1] => [6,2,2,2,2] => 00000 => 00000 => 0
[5,4,4,1] => [4,3,3,3,1] => 01111 => 01111 => 4
[5,4,3,2] => [4,4,3,2,1] => 00101 => 00101 => 2
[5,4,3,1,1] => [5,3,3,2,1] => 11101 => 01111 => 4
[5,4,2,2,1] => [5,4,2,2,1] => 10001 => 00011 => 2
[5,4,2,1,1,1] => [6,3,2,2,1] => 01001 => 00101 => 2
[5,4,1,1,1,1,1] => [7,2,2,2,1] => 10001 => 00011 => 2
[5,3,3,3] => [4,4,4,1,1] => 00011 => 00011 => 2
[5,3,3,2,1] => [5,4,3,1,1] => 10111 => 01111 => 4
[5,3,3,1,1,1] => [6,3,3,1,1] => 01111 => 01111 => 4
[5,3,2,2,2] => [5,5,2,1,1] => 11011 => 01111 => 4
[5,3,2,2,1,1] => [6,4,2,1,1] => 00011 => 00011 => 2
[5,3,2,1,1,1,1] => [7,3,2,1,1] => 11011 => 01111 => 4
[5,3,1,1,1,1,1,1] => [8,2,2,1,1] => 00011 => 00011 => 2
[5,2,2,2,2,1] => [6,5,1,1,1] => 01111 => 01111 => 4
[5,2,2,2,1,1,1] => [7,4,1,1,1] => 10111 => 01111 => 4
[5,2,2,1,1,1,1,1] => [8,3,1,1,1] => 01111 => 01111 => 4
[5,2,1,1,1,1,1,1,1] => [9,2,1,1,1] => 10111 => 01111 => 4
[5,1,1,1,1,1,1,1,1,1] => [10,1,1,1,1] => 01111 => 01111 => 4
[4,4,4,2] => [4,4,3,3] => 0011 => 0011 => 2
[4,4,4,1,1] => [5,3,3,3] => 1111 => 1111 => 4
[4,4,3,3] => [4,4,4,2] => 0000 => 0000 => 0
[4,4,3,2,1] => [5,4,3,2] => 1010 => 0011 => 2
[4,4,3,1,1,1] => [6,3,3,2] => 0110 => 0011 => 2
[4,4,2,2,2] => [5,5,2,2] => 1100 => 0011 => 2
[4,4,2,2,1,1] => [6,4,2,2] => 0000 => 0000 => 0
[4,4,2,1,1,1,1] => [7,3,2,2] => 1100 => 0011 => 2
[4,4,1,1,1,1,1,1] => [8,2,2,2] => 0000 => 0000 => 0
[4,3,3,3,1] => [5,4,4,1] => 1001 => 0011 => 2
[4,3,3,2,2] => [5,5,3,1] => 1111 => 1111 => 4
[4,3,3,2,1,1] => [6,4,3,1] => 0011 => 0011 => 2
[4,3,3,1,1,1,1] => [7,3,3,1] => 1111 => 1111 => 4
[4,3,2,2,2,1] => [6,5,2,1] => 0101 => 0101 => 2
[4,3,2,2,1,1,1] => [7,4,2,1] => 1001 => 0011 => 2
[4,3,2,1,1,1,1,1] => [8,3,2,1] => 0101 => 0101 => 2
[4,3,1,1,1,1,1,1,1] => [9,2,2,1] => 1001 => 0011 => 2
[4,2,2,2,2,2] => [6,6,1,1] => 0011 => 0011 => 2
[4,2,2,2,2,1,1] => [7,5,1,1] => 1111 => 1111 => 4
[4,2,2,2,1,1,1,1] => [8,4,1,1] => 0011 => 0011 => 2
[4,2,2,1,1,1,1,1,1] => [9,3,1,1] => 1111 => 1111 => 4
[4,2,1,1,1,1,1,1,1,1] => [10,2,1,1] => 0011 => 0011 => 2
[4,1,1,1,1,1,1,1,1,1,1] => [11,1,1,1] => 1111 => 1111 => 4
[3,3,3,3,2] => [5,5,4] => 110 => 011 => 2
[3,3,3,3,1,1] => [6,4,4] => 000 => 000 => 0
[3,3,3,2,2,1] => [6,5,3] => 011 => 011 => 2
[3,3,3,2,1,1,1] => [7,4,3] => 101 => 011 => 2
[3,3,3,1,1,1,1,1] => [8,3,3] => 011 => 011 => 2
[3,3,2,2,2,2] => [6,6,2] => 000 => 000 => 0
[3,3,2,2,2,1,1] => [7,5,2] => 110 => 011 => 2
[3,3,2,2,1,1,1,1] => [8,4,2] => 000 => 000 => 0
[3,3,2,1,1,1,1,1,1] => [9,3,2] => 110 => 011 => 2
[3,3,1,1,1,1,1,1,1,1] => [10,2,2] => 000 => 000 => 0
[3,2,2,2,2,2,1] => [7,6,1] => 101 => 011 => 2
[3,2,2,2,2,1,1,1] => [8,5,1] => 011 => 011 => 2
[3,2,2,2,1,1,1,1,1] => [9,4,1] => 101 => 011 => 2
[3,2,2,1,1,1,1,1,1,1] => [10,3,1] => 011 => 011 => 2
[3,2,1,1,1,1,1,1,1,1,1] => [11,2,1] => 101 => 011 => 2
[3,1,1,1,1,1,1,1,1,1,1,1] => [12,1,1] => 011 => 011 => 2
[2,2,2,2,2,2,2] => [7,7] => 11 => 11 => 2
[2,2,2,2,2,2,1,1] => [8,6] => 00 => 00 => 0
[2,2,2,2,2,1,1,1,1] => [9,5] => 11 => 11 => 2
[2,2,2,2,1,1,1,1,1,1] => [10,4] => 00 => 00 => 0
[2,2,2,1,1,1,1,1,1,1,1] => [11,3] => 11 => 11 => 2
[2,2,1,1,1,1,1,1,1,1,1,1] => [12,2] => 00 => 00 => 0
[2,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1] => 11 => 11 => 2
[1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14] => 0 => 0 => 0
[10,5] => [2,2,2,2,2,1,1,1,1,1] => 0000011111 => 0000011111 => 5
[10,2,2,1] => [4,3,1,1,1,1,1,1,1,1] => 0111111111 => 0111111111 => 9
[10,1,1,1,1,1] => [6,1,1,1,1,1,1,1,1,1] => 0111111111 => 0111111111 => 9
[9,6] => [2,2,2,2,2,2,1,1,1] => 000000111 => 000000111 => 3
[9,5,1] => [3,2,2,2,2,1,1,1,1] => 100001111 => 000011111 => 5
[9,4,2] => [3,3,2,2,1,1,1,1,1] => 110011111 => 001111111 => 7
[9,4,1,1] => [4,2,2,2,1,1,1,1,1] => 000011111 => 000011111 => 5
[9,3,3] => [3,3,3,1,1,1,1,1,1] => 111111111 => 111111111 => 9
[9,3,2,1] => [4,3,2,1,1,1,1,1,1] => 010111111 => 010111111 => 7
[9,3,1,1,1] => [5,2,2,1,1,1,1,1,1] => 100111111 => 001111111 => 7
[9,2,2,2] => [4,4,1,1,1,1,1,1,1] => 001111111 => 001111111 => 7
[9,2,2,1,1] => [5,3,1,1,1,1,1,1,1] => 111111111 => 111111111 => 9
[9,2,1,1,1,1] => [6,2,1,1,1,1,1,1,1] => 001111111 => 001111111 => 7
[9,1,1,1,1,1,1] => [7,1,1,1,1,1,1,1,1] => 111111111 => 111111111 => 9
[8,7] => [2,2,2,2,2,2,2,1] => 00000001 => 00000001 => 1
[8,6,1] => [3,2,2,2,2,2,1,1] => 10000011 => 00000111 => 3
[8,5,2] => [3,3,2,2,2,1,1,1] => 11000111 => 00011111 => 5
[8,5,1,1] => [4,2,2,2,2,1,1,1] => 00000111 => 00000111 => 3
[8,4,3] => [3,3,3,2,1,1,1,1] => 11101111 => 01111111 => 7
[8,4,2,1] => [4,3,2,2,1,1,1,1] => 01001111 => 00111101 => 5
[8,4,1,1,1] => [5,2,2,2,1,1,1,1] => 10001111 => 00011111 => 5
[8,3,3,1] => [4,3,3,1,1,1,1,1] => 01111111 => 01111111 => 7
[8,3,2,2] => [4,4,2,1,1,1,1,1] => 00011111 => 00011111 => 5
[8,3,2,1,1] => [5,3,2,1,1,1,1,1] => 11011111 => 01111111 => 7
[8,3,1,1,1,1] => [6,2,2,1,1,1,1,1] => 00011111 => 00011111 => 5
[8,2,2,2,1] => [5,4,1,1,1,1,1,1] => 10111111 => 01111111 => 7
[8,2,2,1,1,1] => [6,3,1,1,1,1,1,1] => 01111111 => 01111111 => 7
[8,2,1,1,1,1,1] => [7,2,1,1,1,1,1,1] => 10111111 => 01111111 => 7
[8,1,1,1,1,1,1,1] => [8,1,1,1,1,1,1,1] => 01111111 => 01111111 => 7
[7,7,1] => [3,2,2,2,2,2,2] => 1000000 => 0000001 => 1
[7,6,2] => [3,3,2,2,2,2,1] => 1100001 => 0000111 => 3
[7,6,1,1] => [4,2,2,2,2,2,1] => 0000001 => 0000001 => 1
[7,5,3] => [3,3,3,2,2,1,1] => 1110011 => 0011111 => 5
[7,5,2,1] => [4,3,2,2,2,1,1] => 0100011 => 0001101 => 3
[7,5,1,1,1] => [5,2,2,2,2,1,1] => 1000011 => 0000111 => 3
[7,4,4] => [3,3,3,3,1,1,1] => 1111111 => 1111111 => 7
[7,4,3,1] => [4,3,3,2,1,1,1] => 0110111 => 0110111 => 5
[7,4,2,2] => [4,4,2,2,1,1,1] => 0000111 => 0000111 => 3
[7,4,2,1,1] => [5,3,2,2,1,1,1] => 1100111 => 0011111 => 5
[7,4,1,1,1,1] => [6,2,2,2,1,1,1] => 0000111 => 0000111 => 3
[7,3,3,2] => [4,4,3,1,1,1,1] => 0011111 => 0011111 => 5
[7,3,3,1,1] => [5,3,3,1,1,1,1] => 1111111 => 1111111 => 7
[7,3,2,2,1] => [5,4,2,1,1,1,1] => 1001111 => 0011111 => 5
[7,3,2,1,1,1] => [6,3,2,1,1,1,1] => 0101111 => 0101111 => 5
[7,3,1,1,1,1,1] => [7,2,2,1,1,1,1] => 1001111 => 0011111 => 5
[7,2,2,2,2] => [5,5,1,1,1,1,1] => 1111111 => 1111111 => 7
[7,2,2,2,1,1] => [6,4,1,1,1,1,1] => 0011111 => 0011111 => 5
[7,2,2,1,1,1,1] => [7,3,1,1,1,1,1] => 1111111 => 1111111 => 7
[7,2,1,1,1,1,1,1] => [8,2,1,1,1,1,1] => 0011111 => 0011111 => 5
[7,1,1,1,1,1,1,1,1] => [9,1,1,1,1,1,1] => 1111111 => 1111111 => 7
[6,6,3] => [3,3,3,2,2,2] => 111000 => 000111 => 3
[6,6,2,1] => [4,3,2,2,2,2] => 010000 => 000001 => 1
[6,6,1,1,1] => [5,2,2,2,2,2] => 100000 => 000001 => 1
[6,5,4] => [3,3,3,3,2,1] => 111101 => 011111 => 5
[6,5,3,1] => [4,3,3,2,2,1] => 011001 => 001011 => 3
[6,5,2,2] => [4,4,2,2,2,1] => 000001 => 000001 => 1
[6,5,2,1,1] => [5,3,2,2,2,1] => 110001 => 000111 => 3
[6,5,1,1,1,1] => [6,2,2,2,2,1] => 000001 => 000001 => 1
[6,4,4,1] => [4,3,3,3,1,1] => 011111 => 011111 => 5
[6,4,3,2] => [4,4,3,2,1,1] => 001011 => 001011 => 3
[6,4,3,1,1] => [5,3,3,2,1,1] => 111011 => 011111 => 5
[6,4,2,2,1] => [5,4,2,2,1,1] => 100011 => 000111 => 3
[6,4,2,1,1,1] => [6,3,2,2,1,1] => 010011 => 001101 => 3
[6,4,1,1,1,1,1] => [7,2,2,2,1,1] => 100011 => 000111 => 3
[6,3,3,3] => [4,4,4,1,1,1] => 000111 => 000111 => 3
[6,3,3,2,1] => [5,4,3,1,1,1] => 101111 => 011111 => 5
[6,3,3,1,1,1] => [6,3,3,1,1,1] => 011111 => 011111 => 5
[6,3,2,2,2] => [5,5,2,1,1,1] => 110111 => 011111 => 5
[6,3,2,2,1,1] => [6,4,2,1,1,1] => 000111 => 000111 => 3
[6,3,2,1,1,1,1] => [7,3,2,1,1,1] => 110111 => 011111 => 5
[6,3,1,1,1,1,1,1] => [8,2,2,1,1,1] => 000111 => 000111 => 3
[6,2,2,2,2,1] => [6,5,1,1,1,1] => 011111 => 011111 => 5
[6,2,2,2,1,1,1] => [7,4,1,1,1,1] => 101111 => 011111 => 5
[6,2,2,1,1,1,1,1] => [8,3,1,1,1,1] => 011111 => 011111 => 5
[6,2,1,1,1,1,1,1,1] => [9,2,1,1,1,1] => 101111 => 011111 => 5
[6,1,1,1,1,1,1,1,1,1] => [10,1,1,1,1,1] => 011111 => 011111 => 5
[5,5,5] => [3,3,3,3,3] => 11111 => 11111 => 5
[5,5,4,1] => [4,3,3,3,2] => 01110 => 00111 => 3
[5,5,3,2] => [4,4,3,2,2] => 00100 => 00001 => 1
[5,5,3,1,1] => [5,3,3,2,2] => 11100 => 00111 => 3
[5,5,2,2,1] => [5,4,2,2,2] => 10000 => 00001 => 1
[5,5,2,1,1,1] => [6,3,2,2,2] => 01000 => 00001 => 1
[5,5,1,1,1,1,1] => [7,2,2,2,2] => 10000 => 00001 => 1
[5,4,4,2] => [4,4,3,3,1] => 00111 => 00111 => 3
[5,4,4,1,1] => [5,3,3,3,1] => 11111 => 11111 => 5
[5,4,3,3] => [4,4,4,2,1] => 00001 => 00001 => 1
[5,4,3,2,1] => [5,4,3,2,1] => 10101 => 01011 => 3
[5,4,3,1,1,1] => [6,3,3,2,1] => 01101 => 01011 => 3
[5,4,2,2,2] => [5,5,2,2,1] => 11001 => 00111 => 3
[5,4,2,2,1,1] => [6,4,2,2,1] => 00001 => 00001 => 1
[5,4,2,1,1,1,1] => [7,3,2,2,1] => 11001 => 00111 => 3
[5,4,1,1,1,1,1,1] => [8,2,2,2,1] => 00001 => 00001 => 1
[5,3,3,3,1] => [5,4,4,1,1] => 10011 => 00111 => 3
[5,3,3,2,2] => [5,5,3,1,1] => 11111 => 11111 => 5
[5,3,3,2,1,1] => [6,4,3,1,1] => 00111 => 00111 => 3
[5,3,3,1,1,1,1] => [7,3,3,1,1] => 11111 => 11111 => 5
[5,3,2,2,2,1] => [6,5,2,1,1] => 01011 => 01011 => 3
[5,3,2,2,1,1,1] => [7,4,2,1,1] => 10011 => 00111 => 3
[5,3,2,1,1,1,1,1] => [8,3,2,1,1] => 01011 => 01011 => 3
[5,3,1,1,1,1,1,1,1] => [9,2,2,1,1] => 10011 => 00111 => 3
[5,2,2,2,2,2] => [6,6,1,1,1] => 00111 => 00111 => 3
[5,2,2,2,2,1,1] => [7,5,1,1,1] => 11111 => 11111 => 5
[5,2,2,2,1,1,1,1] => [8,4,1,1,1] => 00111 => 00111 => 3
[5,2,2,1,1,1,1,1,1] => [9,3,1,1,1] => 11111 => 11111 => 5
[5,2,1,1,1,1,1,1,1,1] => [10,2,1,1,1] => 00111 => 00111 => 3
[5,1,1,1,1,1,1,1,1,1,1] => [11,1,1,1,1] => 11111 => 11111 => 5
[4,4,4,3] => [4,4,4,3] => 0001 => 0001 => 1
[4,4,4,2,1] => [5,4,3,3] => 1011 => 0111 => 3
[4,4,4,1,1,1] => [6,3,3,3] => 0111 => 0111 => 3
[4,4,3,3,1] => [5,4,4,2] => 1000 => 0001 => 1
[4,4,3,2,2] => [5,5,3,2] => 1110 => 0111 => 3
[4,4,3,2,1,1] => [6,4,3,2] => 0010 => 0001 => 1
[4,4,3,1,1,1,1] => [7,3,3,2] => 1110 => 0111 => 3
[4,4,2,2,2,1] => [6,5,2,2] => 0100 => 0001 => 1
[4,4,2,2,1,1,1] => [7,4,2,2] => 1000 => 0001 => 1
[4,4,2,1,1,1,1,1] => [8,3,2,2] => 0100 => 0001 => 1
[4,4,1,1,1,1,1,1,1] => [9,2,2,2] => 1000 => 0001 => 1
[4,3,3,3,2] => [5,5,4,1] => 1101 => 0111 => 3
[4,3,3,3,1,1] => [6,4,4,1] => 0001 => 0001 => 1
[4,3,3,2,2,1] => [6,5,3,1] => 0111 => 0111 => 3
[4,3,3,2,1,1,1] => [7,4,3,1] => 1011 => 0111 => 3
[4,3,3,1,1,1,1,1] => [8,3,3,1] => 0111 => 0111 => 3
[4,3,2,2,2,2] => [6,6,2,1] => 0001 => 0001 => 1
[4,3,2,2,2,1,1] => [7,5,2,1] => 1101 => 0111 => 3
[4,3,2,2,1,1,1,1] => [8,4,2,1] => 0001 => 0001 => 1
[4,3,2,1,1,1,1,1,1] => [9,3,2,1] => 1101 => 0111 => 3
[4,3,1,1,1,1,1,1,1,1] => [10,2,2,1] => 0001 => 0001 => 1
[4,2,2,2,2,2,1] => [7,6,1,1] => 1011 => 0111 => 3
[4,2,2,2,2,1,1,1] => [8,5,1,1] => 0111 => 0111 => 3
[4,2,2,2,1,1,1,1,1] => [9,4,1,1] => 1011 => 0111 => 3
[4,2,2,1,1,1,1,1,1,1] => [10,3,1,1] => 0111 => 0111 => 3
[4,2,1,1,1,1,1,1,1,1,1] => [11,2,1,1] => 1011 => 0111 => 3
[4,1,1,1,1,1,1,1,1,1,1,1] => [12,1,1,1] => 0111 => 0111 => 3
[3,3,3,3,3] => [5,5,5] => 111 => 111 => 3
[3,3,3,3,2,1] => [6,5,4] => 010 => 001 => 1
[3,3,3,3,1,1,1] => [7,4,4] => 100 => 001 => 1
[3,3,3,2,2,2] => [6,6,3] => 001 => 001 => 1
[3,3,3,2,2,1,1] => [7,5,3] => 111 => 111 => 3
[3,3,3,2,1,1,1,1] => [8,4,3] => 001 => 001 => 1
[3,3,3,1,1,1,1,1,1] => [9,3,3] => 111 => 111 => 3
[3,3,2,2,2,2,1] => [7,6,2] => 100 => 001 => 1
[3,3,2,2,2,1,1,1] => [8,5,2] => 010 => 001 => 1
[3,3,2,2,1,1,1,1,1] => [9,4,2] => 100 => 001 => 1
[3,3,2,1,1,1,1,1,1,1] => [10,3,2] => 010 => 001 => 1
[3,3,1,1,1,1,1,1,1,1,1] => [11,2,2] => 100 => 001 => 1
[3,2,2,2,2,2,2] => [7,7,1] => 111 => 111 => 3
[3,2,2,2,2,2,1,1] => [8,6,1] => 001 => 001 => 1
[3,2,2,2,2,1,1,1,1] => [9,5,1] => 111 => 111 => 3
[3,2,2,2,1,1,1,1,1,1] => [10,4,1] => 001 => 001 => 1
[3,2,2,1,1,1,1,1,1,1,1] => [11,3,1] => 111 => 111 => 3
[3,2,1,1,1,1,1,1,1,1,1,1] => [12,2,1] => 001 => 001 => 1
[3,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1] => 111 => 111 => 3
[2,2,2,2,2,2,2,1] => [8,7] => 01 => 01 => 1
[2,2,2,2,2,2,1,1,1] => [9,6] => 10 => 01 => 1
[2,2,2,2,2,1,1,1,1,1] => [10,5] => 01 => 01 => 1
[2,2,2,2,1,1,1,1,1,1,1] => [11,4] => 10 => 01 => 1
[2,2,2,1,1,1,1,1,1,1,1,1] => [12,3] => 01 => 01 => 1
[2,2,1,1,1,1,1,1,1,1,1,1,1] => [13,2] => 10 => 01 => 1
[2,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1] => 01 => 01 => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15] => 1 => 1 => 1
[11,2,2,1] => [4,3,1,1,1,1,1,1,1,1,1] => 01111111111 => 01111111111 => 10
[11,1,1,1,1,1] => [6,1,1,1,1,1,1,1,1,1,1] => 01111111111 => 01111111111 => 10
[10,3,2,1] => [4,3,2,1,1,1,1,1,1,1] => 0101111111 => 0101111111 => 8
[9,7] => [2,2,2,2,2,2,2,1,1] => 000000011 => 000000011 => 2
[9,6,1] => [3,2,2,2,2,2,1,1,1] => 100000111 => 000001111 => 4
[9,5,2] => [3,3,2,2,2,1,1,1,1] => 110001111 => 000111111 => 6
[9,5,1,1] => [4,2,2,2,2,1,1,1,1] => 000001111 => 000001111 => 4
[9,4,3] => [3,3,3,2,1,1,1,1,1] => 111011111 => 011111111 => 8
[9,4,2,1] => [4,3,2,2,1,1,1,1,1] => 010011111 => 001111101 => 6
[9,4,1,1,1] => [5,2,2,2,1,1,1,1,1] => 100011111 => 000111111 => 6
[9,3,3,1] => [4,3,3,1,1,1,1,1,1] => 011111111 => 011111111 => 8
[9,3,2,2] => [4,4,2,1,1,1,1,1,1] => 000111111 => 000111111 => 6
[9,3,2,1,1] => [5,3,2,1,1,1,1,1,1] => 110111111 => 011111111 => 8
[9,3,1,1,1,1] => [6,2,2,1,1,1,1,1,1] => 000111111 => 000111111 => 6
[9,2,2,2,1] => [5,4,1,1,1,1,1,1,1] => 101111111 => 011111111 => 8
[9,2,2,1,1,1] => [6,3,1,1,1,1,1,1,1] => 011111111 => 011111111 => 8
[9,2,1,1,1,1,1] => [7,2,1,1,1,1,1,1,1] => 101111111 => 011111111 => 8
[9,1,1,1,1,1,1,1] => [8,1,1,1,1,1,1,1,1] => 011111111 => 011111111 => 8
[8,8] => [2,2,2,2,2,2,2,2] => 00000000 => 00000000 => 0
[8,7,1] => [3,2,2,2,2,2,2,1] => 10000001 => 00000011 => 2
[8,6,2] => [3,3,2,2,2,2,1,1] => 11000011 => 00001111 => 4
[8,6,1,1] => [4,2,2,2,2,2,1,1] => 00000011 => 00000011 => 2
[8,5,3] => [3,3,3,2,2,1,1,1] => 11100111 => 00111111 => 6
[8,5,2,1] => [4,3,2,2,2,1,1,1] => 01000111 => 00011101 => 4
[8,5,1,1,1] => [5,2,2,2,2,1,1,1] => 10000111 => 00001111 => 4
[8,4,4] => [3,3,3,3,1,1,1,1] => 11111111 => 11111111 => 8
[8,4,3,1] => [4,3,3,2,1,1,1,1] => 01101111 => 01101111 => 6
[8,4,2,2] => [4,4,2,2,1,1,1,1] => 00001111 => 00001111 => 4
[8,4,2,1,1] => [5,3,2,2,1,1,1,1] => 11001111 => 00111111 => 6
[8,4,1,1,1,1] => [6,2,2,2,1,1,1,1] => 00001111 => 00001111 => 4
[8,3,3,2] => [4,4,3,1,1,1,1,1] => 00111111 => 00111111 => 6
[8,3,3,1,1] => [5,3,3,1,1,1,1,1] => 11111111 => 11111111 => 8
[8,3,2,2,1] => [5,4,2,1,1,1,1,1] => 10011111 => 00111111 => 6
[8,3,2,1,1,1] => [6,3,2,1,1,1,1,1] => 01011111 => 01011111 => 6
[8,3,1,1,1,1,1] => [7,2,2,1,1,1,1,1] => 10011111 => 00111111 => 6
[8,2,2,2,2] => [5,5,1,1,1,1,1,1] => 11111111 => 11111111 => 8
[8,2,2,2,1,1] => [6,4,1,1,1,1,1,1] => 00111111 => 00111111 => 6
[8,2,2,1,1,1,1] => [7,3,1,1,1,1,1,1] => 11111111 => 11111111 => 8
[8,2,1,1,1,1,1,1] => [8,2,1,1,1,1,1,1] => 00111111 => 00111111 => 6
[8,1,1,1,1,1,1,1,1] => [9,1,1,1,1,1,1,1] => 11111111 => 11111111 => 8
[7,7,2] => [3,3,2,2,2,2,2] => 1100000 => 0000011 => 2
[7,7,1,1] => [4,2,2,2,2,2,2] => 0000000 => 0000000 => 0
[7,6,3] => [3,3,3,2,2,2,1] => 1110001 => 0001111 => 4
[7,6,2,1] => [4,3,2,2,2,2,1] => 0100001 => 0000101 => 2
[7,6,1,1,1] => [5,2,2,2,2,2,1] => 1000001 => 0000011 => 2
[7,5,4] => [3,3,3,3,2,1,1] => 1111011 => 0111111 => 6
[7,5,3,1] => [4,3,3,2,2,1,1] => 0110011 => 0011011 => 4
[7,5,2,2] => [4,4,2,2,2,1,1] => 0000011 => 0000011 => 2
[7,5,2,1,1] => [5,3,2,2,2,1,1] => 1100011 => 0001111 => 4
[7,5,1,1,1,1] => [6,2,2,2,2,1,1] => 0000011 => 0000011 => 2
[7,4,4,1] => [4,3,3,3,1,1,1] => 0111111 => 0111111 => 6
[7,4,3,2] => [4,4,3,2,1,1,1] => 0010111 => 0010111 => 4
[7,4,3,1,1] => [5,3,3,2,1,1,1] => 1110111 => 0111111 => 6
[7,4,2,2,1] => [5,4,2,2,1,1,1] => 1000111 => 0001111 => 4
[7,4,2,1,1,1] => [6,3,2,2,1,1,1] => 0100111 => 0011101 => 4
[7,4,1,1,1,1,1] => [7,2,2,2,1,1,1] => 1000111 => 0001111 => 4
[7,3,3,3] => [4,4,4,1,1,1,1] => 0001111 => 0001111 => 4
[7,3,3,2,1] => [5,4,3,1,1,1,1] => 1011111 => 0111111 => 6
[7,3,3,1,1,1] => [6,3,3,1,1,1,1] => 0111111 => 0111111 => 6
[7,3,2,2,2] => [5,5,2,1,1,1,1] => 1101111 => 0111111 => 6
[7,3,2,2,1,1] => [6,4,2,1,1,1,1] => 0001111 => 0001111 => 4
[7,3,2,1,1,1,1] => [7,3,2,1,1,1,1] => 1101111 => 0111111 => 6
[7,3,1,1,1,1,1,1] => [8,2,2,1,1,1,1] => 0001111 => 0001111 => 4
[7,2,2,2,2,1] => [6,5,1,1,1,1,1] => 0111111 => 0111111 => 6
[7,2,2,2,1,1,1] => [7,4,1,1,1,1,1] => 1011111 => 0111111 => 6
[7,2,2,1,1,1,1,1] => [8,3,1,1,1,1,1] => 0111111 => 0111111 => 6
[7,2,1,1,1,1,1,1,1] => [9,2,1,1,1,1,1] => 1011111 => 0111111 => 6
[7,1,1,1,1,1,1,1,1,1] => [10,1,1,1,1,1,1] => 0111111 => 0111111 => 6
[6,6,4] => [3,3,3,3,2,2] => 111100 => 001111 => 4
[6,6,3,1] => [4,3,3,2,2,2] => 011000 => 000011 => 2
[6,6,2,2] => [4,4,2,2,2,2] => 000000 => 000000 => 0
[6,6,2,1,1] => [5,3,2,2,2,2] => 110000 => 000011 => 2
[6,6,1,1,1,1] => [6,2,2,2,2,2] => 000000 => 000000 => 0
[6,5,5] => [3,3,3,3,3,1] => 111111 => 111111 => 6
[6,5,4,1] => [4,3,3,3,2,1] => 011101 => 010111 => 4
[6,5,3,2] => [4,4,3,2,2,1] => 001001 => 001001 => 2
[6,5,3,1,1] => [5,3,3,2,2,1] => 111001 => 001111 => 4
[6,5,2,2,1] => [5,4,2,2,2,1] => 100001 => 000011 => 2
[6,5,2,1,1,1] => [6,3,2,2,2,1] => 010001 => 000101 => 2
[6,5,1,1,1,1,1] => [7,2,2,2,2,1] => 100001 => 000011 => 2
[6,4,4,2] => [4,4,3,3,1,1] => 001111 => 001111 => 4
[6,4,4,1,1] => [5,3,3,3,1,1] => 111111 => 111111 => 6
[6,4,3,3] => [4,4,4,2,1,1] => 000011 => 000011 => 2
[6,4,3,2,1] => [5,4,3,2,1,1] => 101011 => 010111 => 4
[6,4,3,1,1,1] => [6,3,3,2,1,1] => 011011 => 011011 => 4
[6,4,2,2,2] => [5,5,2,2,1,1] => 110011 => 001111 => 4
[6,4,2,2,1,1] => [6,4,2,2,1,1] => 000011 => 000011 => 2
[6,4,2,1,1,1,1] => [7,3,2,2,1,1] => 110011 => 001111 => 4
[6,4,1,1,1,1,1,1] => [8,2,2,2,1,1] => 000011 => 000011 => 2
[6,3,3,3,1] => [5,4,4,1,1,1] => 100111 => 001111 => 4
[6,3,3,2,2] => [5,5,3,1,1,1] => 111111 => 111111 => 6
[6,3,3,2,1,1] => [6,4,3,1,1,1] => 001111 => 001111 => 4
[6,3,3,1,1,1,1] => [7,3,3,1,1,1] => 111111 => 111111 => 6
[6,3,2,2,2,1] => [6,5,2,1,1,1] => 010111 => 010111 => 4
[6,3,2,2,1,1,1] => [7,4,2,1,1,1] => 100111 => 001111 => 4
[6,3,2,1,1,1,1,1] => [8,3,2,1,1,1] => 010111 => 010111 => 4
[6,3,1,1,1,1,1,1,1] => [9,2,2,1,1,1] => 100111 => 001111 => 4
[6,2,2,2,2,2] => [6,6,1,1,1,1] => 001111 => 001111 => 4
[6,2,2,2,2,1,1] => [7,5,1,1,1,1] => 111111 => 111111 => 6
[6,2,2,2,1,1,1,1] => [8,4,1,1,1,1] => 001111 => 001111 => 4
[6,2,2,1,1,1,1,1,1] => [9,3,1,1,1,1] => 111111 => 111111 => 6
[6,2,1,1,1,1,1,1,1,1] => [10,2,1,1,1,1] => 001111 => 001111 => 4
[6,1,1,1,1,1,1,1,1,1,1] => [11,1,1,1,1,1] => 111111 => 111111 => 6
[5,5,5,1] => [4,3,3,3,3] => 01111 => 01111 => 4
[5,5,4,2] => [4,4,3,3,2] => 00110 => 00011 => 2
[5,5,4,1,1] => [5,3,3,3,2] => 11110 => 01111 => 4
[5,5,3,3] => [4,4,4,2,2] => 00000 => 00000 => 0
[5,5,3,2,1] => [5,4,3,2,2] => 10100 => 00011 => 2
[5,5,3,1,1,1] => [6,3,3,2,2] => 01100 => 00011 => 2
[5,5,2,2,2] => [5,5,2,2,2] => 11000 => 00011 => 2
[5,5,2,2,1,1] => [6,4,2,2,2] => 00000 => 00000 => 0
[5,5,2,1,1,1,1] => [7,3,2,2,2] => 11000 => 00011 => 2
[5,5,1,1,1,1,1,1] => [8,2,2,2,2] => 00000 => 00000 => 0
[5,4,4,3] => [4,4,4,3,1] => 00011 => 00011 => 2
[5,4,4,2,1] => [5,4,3,3,1] => 10111 => 01111 => 4
[5,4,4,1,1,1] => [6,3,3,3,1] => 01111 => 01111 => 4
[5,4,3,3,1] => [5,4,4,2,1] => 10001 => 00011 => 2
[5,4,3,2,2] => [5,5,3,2,1] => 11101 => 01111 => 4
[5,4,3,2,1,1] => [6,4,3,2,1] => 00101 => 00101 => 2
[5,4,3,1,1,1,1] => [7,3,3,2,1] => 11101 => 01111 => 4
[5,4,2,2,2,1] => [6,5,2,2,1] => 01001 => 00101 => 2
[5,4,2,2,1,1,1] => [7,4,2,2,1] => 10001 => 00011 => 2
[5,4,2,1,1,1,1,1] => [8,3,2,2,1] => 01001 => 00101 => 2
[5,4,1,1,1,1,1,1,1] => [9,2,2,2,1] => 10001 => 00011 => 2
[5,3,3,3,2] => [5,5,4,1,1] => 11011 => 01111 => 4
[5,3,3,3,1,1] => [6,4,4,1,1] => 00011 => 00011 => 2
[5,3,3,2,2,1] => [6,5,3,1,1] => 01111 => 01111 => 4
[5,3,3,2,1,1,1] => [7,4,3,1,1] => 10111 => 01111 => 4
[5,3,3,1,1,1,1,1] => [8,3,3,1,1] => 01111 => 01111 => 4
[5,3,2,2,2,2] => [6,6,2,1,1] => 00011 => 00011 => 2
[5,3,2,2,2,1,1] => [7,5,2,1,1] => 11011 => 01111 => 4
[5,3,2,2,1,1,1,1] => [8,4,2,1,1] => 00011 => 00011 => 2
[5,3,2,1,1,1,1,1,1] => [9,3,2,1,1] => 11011 => 01111 => 4
[5,3,1,1,1,1,1,1,1,1] => [10,2,2,1,1] => 00011 => 00011 => 2
[5,2,2,2,2,2,1] => [7,6,1,1,1] => 10111 => 01111 => 4
[5,2,2,2,2,1,1,1] => [8,5,1,1,1] => 01111 => 01111 => 4
[5,2,2,2,1,1,1,1,1] => [9,4,1,1,1] => 10111 => 01111 => 4
[5,2,2,1,1,1,1,1,1,1] => [10,3,1,1,1] => 01111 => 01111 => 4
[5,2,1,1,1,1,1,1,1,1,1] => [11,2,1,1,1] => 10111 => 01111 => 4
[5,1,1,1,1,1,1,1,1,1,1,1] => [12,1,1,1,1] => 01111 => 01111 => 4
[4,4,4,4] => [4,4,4,4] => 0000 => 0000 => 0
[4,4,4,3,1] => [5,4,4,3] => 1001 => 0011 => 2
[4,4,4,2,2] => [5,5,3,3] => 1111 => 1111 => 4
[4,4,4,2,1,1] => [6,4,3,3] => 0011 => 0011 => 2
[4,4,4,1,1,1,1] => [7,3,3,3] => 1111 => 1111 => 4
[4,4,3,3,2] => [5,5,4,2] => 1100 => 0011 => 2
[4,4,3,3,1,1] => [6,4,4,2] => 0000 => 0000 => 0
[4,4,3,2,2,1] => [6,5,3,2] => 0110 => 0011 => 2
[4,4,3,2,1,1,1] => [7,4,3,2] => 1010 => 0011 => 2
[4,4,3,1,1,1,1,1] => [8,3,3,2] => 0110 => 0011 => 2
[4,4,2,2,2,2] => [6,6,2,2] => 0000 => 0000 => 0
[4,4,2,2,2,1,1] => [7,5,2,2] => 1100 => 0011 => 2
[4,4,2,2,1,1,1,1] => [8,4,2,2] => 0000 => 0000 => 0
[4,4,2,1,1,1,1,1,1] => [9,3,2,2] => 1100 => 0011 => 2
[4,4,1,1,1,1,1,1,1,1] => [10,2,2,2] => 0000 => 0000 => 0
[4,3,3,3,3] => [5,5,5,1] => 1111 => 1111 => 4
[4,3,3,3,2,1] => [6,5,4,1] => 0101 => 0101 => 2
[4,3,3,3,1,1,1] => [7,4,4,1] => 1001 => 0011 => 2
[4,3,3,2,2,2] => [6,6,3,1] => 0011 => 0011 => 2
[4,3,3,2,2,1,1] => [7,5,3,1] => 1111 => 1111 => 4
[4,3,3,2,1,1,1,1] => [8,4,3,1] => 0011 => 0011 => 2
[4,3,3,1,1,1,1,1,1] => [9,3,3,1] => 1111 => 1111 => 4
[4,3,2,2,2,2,1] => [7,6,2,1] => 1001 => 0011 => 2
[4,3,2,2,2,1,1,1] => [8,5,2,1] => 0101 => 0101 => 2
[4,3,2,2,1,1,1,1,1] => [9,4,2,1] => 1001 => 0011 => 2
[4,3,2,1,1,1,1,1,1,1] => [10,3,2,1] => 0101 => 0101 => 2
[4,3,1,1,1,1,1,1,1,1,1] => [11,2,2,1] => 1001 => 0011 => 2
[4,2,2,2,2,2,2] => [7,7,1,1] => 1111 => 1111 => 4
[4,2,2,2,2,2,1,1] => [8,6,1,1] => 0011 => 0011 => 2
[4,2,2,2,2,1,1,1,1] => [9,5,1,1] => 1111 => 1111 => 4
[4,2,2,2,1,1,1,1,1,1] => [10,4,1,1] => 0011 => 0011 => 2
[4,2,2,1,1,1,1,1,1,1,1] => [11,3,1,1] => 1111 => 1111 => 4
[4,2,1,1,1,1,1,1,1,1,1,1] => [12,2,1,1] => 0011 => 0011 => 2
[4,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1,1] => 1111 => 1111 => 4
[3,3,3,3,3,1] => [6,5,5] => 011 => 011 => 2
[3,3,3,3,2,2] => [6,6,4] => 000 => 000 => 0
[3,3,3,3,2,1,1] => [7,5,4] => 110 => 011 => 2
[3,3,3,3,1,1,1,1] => [8,4,4] => 000 => 000 => 0
[3,3,3,2,2,2,1] => [7,6,3] => 101 => 011 => 2
[3,3,3,2,2,1,1,1] => [8,5,3] => 011 => 011 => 2
[3,3,3,2,1,1,1,1,1] => [9,4,3] => 101 => 011 => 2
[3,3,3,1,1,1,1,1,1,1] => [10,3,3] => 011 => 011 => 2
[3,3,2,2,2,2,2] => [7,7,2] => 110 => 011 => 2
[3,3,2,2,2,2,1,1] => [8,6,2] => 000 => 000 => 0
[3,3,2,2,2,1,1,1,1] => [9,5,2] => 110 => 011 => 2
[3,3,2,2,1,1,1,1,1,1] => [10,4,2] => 000 => 000 => 0
[3,3,2,1,1,1,1,1,1,1,1] => [11,3,2] => 110 => 011 => 2
[3,3,1,1,1,1,1,1,1,1,1,1] => [12,2,2] => 000 => 000 => 0
[3,2,2,2,2,2,2,1] => [8,7,1] => 011 => 011 => 2
[3,2,2,2,2,2,1,1,1] => [9,6,1] => 101 => 011 => 2
[3,2,2,2,2,1,1,1,1,1] => [10,5,1] => 011 => 011 => 2
[3,2,2,2,1,1,1,1,1,1,1] => [11,4,1] => 101 => 011 => 2
[3,2,2,1,1,1,1,1,1,1,1,1] => [12,3,1] => 011 => 011 => 2
[3,2,1,1,1,1,1,1,1,1,1,1,1] => [13,2,1] => 101 => 011 => 2
[3,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1,1] => 011 => 011 => 2
[2,2,2,2,2,2,2,2] => [8,8] => 00 => 00 => 0
[2,2,2,2,2,2,2,1,1] => [9,7] => 11 => 11 => 2
[2,2,2,2,2,2,1,1,1,1] => [10,6] => 00 => 00 => 0
[2,2,2,2,2,1,1,1,1,1,1] => [11,5] => 11 => 11 => 2
[2,2,2,2,1,1,1,1,1,1,1,1] => [12,4] => 00 => 00 => 0
[2,2,2,1,1,1,1,1,1,1,1,1,1] => [13,3] => 11 => 11 => 2
[2,2,1,1,1,1,1,1,1,1,1,1,1,1] => [14,2] => 00 => 00 => 0
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,1] => 11 => 11 => 2
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [16] => 0 => 0 => 0
[11,3,2,1] => [4,3,2,1,1,1,1,1,1,1,1] => 01011111111 => 01011111111 => 9
[10,5,1,1] => [4,2,2,2,2,1,1,1,1,1] => 0000011111 => 0000011111 => 5
[10,4,2,1] => [4,3,2,2,1,1,1,1,1,1] => 0100111111 => 0011111101 => 7
[10,3,3,1] => [4,3,3,1,1,1,1,1,1,1] => 0111111111 => 0111111111 => 9
[10,2,2,1,1,1] => [6,3,1,1,1,1,1,1,1,1] => 0111111111 => 0111111111 => 9
[10,1,1,1,1,1,1,1] => [8,1,1,1,1,1,1,1,1,1] => 0111111111 => 0111111111 => 9
[9,8] => [2,2,2,2,2,2,2,2,1] => 000000001 => 000000001 => 1
[9,7,1] => [3,2,2,2,2,2,2,1,1] => 100000011 => 000000111 => 3
[9,6,2] => [3,3,2,2,2,2,1,1,1] => 110000111 => 000011111 => 5
[9,6,1,1] => [4,2,2,2,2,2,1,1,1] => 000000111 => 000000111 => 3
[9,5,3] => [3,3,3,2,2,1,1,1,1] => 111001111 => 001111111 => 7
[9,5,2,1] => [4,3,2,2,2,1,1,1,1] => 010001111 => 000111101 => 5
[9,5,1,1,1] => [5,2,2,2,2,1,1,1,1] => 100001111 => 000011111 => 5
[9,4,4] => [3,3,3,3,1,1,1,1,1] => 111111111 => 111111111 => 9
[9,4,3,1] => [4,3,3,2,1,1,1,1,1] => 011011111 => 011011111 => 7
[9,4,2,2] => [4,4,2,2,1,1,1,1,1] => 000011111 => 000011111 => 5
[9,4,2,1,1] => [5,3,2,2,1,1,1,1,1] => 110011111 => 001111111 => 7
[9,4,1,1,1,1] => [6,2,2,2,1,1,1,1,1] => 000011111 => 000011111 => 5
[9,3,3,2] => [4,4,3,1,1,1,1,1,1] => 001111111 => 001111111 => 7
[9,3,3,1,1] => [5,3,3,1,1,1,1,1,1] => 111111111 => 111111111 => 9
[9,3,2,2,1] => [5,4,2,1,1,1,1,1,1] => 100111111 => 001111111 => 7
[9,3,2,1,1,1] => [6,3,2,1,1,1,1,1,1] => 010111111 => 010111111 => 7
[9,3,1,1,1,1,1] => [7,2,2,1,1,1,1,1,1] => 100111111 => 001111111 => 7
[9,2,2,2,2] => [5,5,1,1,1,1,1,1,1] => 111111111 => 111111111 => 9
[9,2,2,2,1,1] => [6,4,1,1,1,1,1,1,1] => 001111111 => 001111111 => 7
[9,2,2,1,1,1,1] => [7,3,1,1,1,1,1,1,1] => 111111111 => 111111111 => 9
[9,2,1,1,1,1,1,1] => [8,2,1,1,1,1,1,1,1] => 001111111 => 001111111 => 7
[9,1,1,1,1,1,1,1,1] => [9,1,1,1,1,1,1,1,1] => 111111111 => 111111111 => 9
[8,8,1] => [3,2,2,2,2,2,2,2] => 10000000 => 00000001 => 1
[8,7,2] => [3,3,2,2,2,2,2,1] => 11000001 => 00000111 => 3
[8,7,1,1] => [4,2,2,2,2,2,2,1] => 00000001 => 00000001 => 1
[8,6,3] => [3,3,3,2,2,2,1,1] => 11100011 => 00011111 => 5
[8,6,2,1] => [4,3,2,2,2,2,1,1] => 01000011 => 00001101 => 3
[8,6,1,1,1] => [5,2,2,2,2,2,1,1] => 10000011 => 00000111 => 3
[8,5,4] => [3,3,3,3,2,1,1,1] => 11110111 => 01111111 => 7
[8,5,3,1] => [4,3,3,2,2,1,1,1] => 01100111 => 00111011 => 5
[8,5,2,2] => [4,4,2,2,2,1,1,1] => 00000111 => 00000111 => 3
[8,5,2,1,1] => [5,3,2,2,2,1,1,1] => 11000111 => 00011111 => 5
[8,5,1,1,1,1] => [6,2,2,2,2,1,1,1] => 00000111 => 00000111 => 3
[8,4,4,1] => [4,3,3,3,1,1,1,1] => 01111111 => 01111111 => 7
[8,4,3,2] => [4,4,3,2,1,1,1,1] => 00101111 => 00101111 => 5
[8,4,3,1,1] => [5,3,3,2,1,1,1,1] => 11101111 => 01111111 => 7
[8,4,2,2,1] => [5,4,2,2,1,1,1,1] => 10001111 => 00011111 => 5
[8,4,2,1,1,1] => [6,3,2,2,1,1,1,1] => 01001111 => 00111101 => 5
[8,4,1,1,1,1,1] => [7,2,2,2,1,1,1,1] => 10001111 => 00011111 => 5
[8,3,3,3] => [4,4,4,1,1,1,1,1] => 00011111 => 00011111 => 5
[8,3,3,2,1] => [5,4,3,1,1,1,1,1] => 10111111 => 01111111 => 7
[8,3,3,1,1,1] => [6,3,3,1,1,1,1,1] => 01111111 => 01111111 => 7
[8,3,2,2,2] => [5,5,2,1,1,1,1,1] => 11011111 => 01111111 => 7
[8,3,2,2,1,1] => [6,4,2,1,1,1,1,1] => 00011111 => 00011111 => 5
[8,3,2,1,1,1,1] => [7,3,2,1,1,1,1,1] => 11011111 => 01111111 => 7
[8,3,1,1,1,1,1,1] => [8,2,2,1,1,1,1,1] => 00011111 => 00011111 => 5
[8,2,2,2,2,1] => [6,5,1,1,1,1,1,1] => 01111111 => 01111111 => 7
[8,2,2,2,1,1,1] => [7,4,1,1,1,1,1,1] => 10111111 => 01111111 => 7
[8,2,2,1,1,1,1,1] => [8,3,1,1,1,1,1,1] => 01111111 => 01111111 => 7
[8,2,1,1,1,1,1,1,1] => [9,2,1,1,1,1,1,1] => 10111111 => 01111111 => 7
[8,1,1,1,1,1,1,1,1,1] => [10,1,1,1,1,1,1,1] => 01111111 => 01111111 => 7
[7,7,3] => [3,3,3,2,2,2,2] => 1110000 => 0000111 => 3
[7,7,2,1] => [4,3,2,2,2,2,2] => 0100000 => 0000001 => 1
[7,7,1,1,1] => [5,2,2,2,2,2,2] => 1000000 => 0000001 => 1
[7,6,4] => [3,3,3,3,2,2,1] => 1111001 => 0011111 => 5
[7,6,3,1] => [4,3,3,2,2,2,1] => 0110001 => 0001011 => 3
[7,6,2,2] => [4,4,2,2,2,2,1] => 0000001 => 0000001 => 1
[7,6,2,1,1] => [5,3,2,2,2,2,1] => 1100001 => 0000111 => 3
[7,6,1,1,1,1] => [6,2,2,2,2,2,1] => 0000001 => 0000001 => 1
[7,5,5] => [3,3,3,3,3,1,1] => 1111111 => 1111111 => 7
[7,5,4,1] => [4,3,3,3,2,1,1] => 0111011 => 0110111 => 5
[7,5,3,2] => [4,4,3,2,2,1,1] => 0010011 => 0010011 => 3
[7,5,3,1,1] => [5,3,3,2,2,1,1] => 1110011 => 0011111 => 5
[7,5,2,2,1] => [5,4,2,2,2,1,1] => 1000011 => 0000111 => 3
[7,5,2,1,1,1] => [6,3,2,2,2,1,1] => 0100011 => 0001101 => 3
[7,5,1,1,1,1,1] => [7,2,2,2,2,1,1] => 1000011 => 0000111 => 3
[7,4,4,2] => [4,4,3,3,1,1,1] => 0011111 => 0011111 => 5
[7,4,4,1,1] => [5,3,3,3,1,1,1] => 1111111 => 1111111 => 7
[7,4,3,3] => [4,4,4,2,1,1,1] => 0000111 => 0000111 => 3
[7,4,3,2,1] => [5,4,3,2,1,1,1] => 1010111 => 0101111 => 5
[7,4,3,1,1,1] => [6,3,3,2,1,1,1] => 0110111 => 0110111 => 5
[7,4,2,2,2] => [5,5,2,2,1,1,1] => 1100111 => 0011111 => 5
[7,4,2,2,1,1] => [6,4,2,2,1,1,1] => 0000111 => 0000111 => 3
[7,4,2,1,1,1,1] => [7,3,2,2,1,1,1] => 1100111 => 0011111 => 5
[7,4,1,1,1,1,1,1] => [8,2,2,2,1,1,1] => 0000111 => 0000111 => 3
[7,3,3,3,1] => [5,4,4,1,1,1,1] => 1001111 => 0011111 => 5
[7,3,3,2,2] => [5,5,3,1,1,1,1] => 1111111 => 1111111 => 7
[7,3,3,2,1,1] => [6,4,3,1,1,1,1] => 0011111 => 0011111 => 5
[7,3,3,1,1,1,1] => [7,3,3,1,1,1,1] => 1111111 => 1111111 => 7
[7,3,2,2,2,1] => [6,5,2,1,1,1,1] => 0101111 => 0101111 => 5
[7,3,2,2,1,1,1] => [7,4,2,1,1,1,1] => 1001111 => 0011111 => 5
[7,3,2,1,1,1,1,1] => [8,3,2,1,1,1,1] => 0101111 => 0101111 => 5
[7,3,1,1,1,1,1,1,1] => [9,2,2,1,1,1,1] => 1001111 => 0011111 => 5
[7,2,2,2,2,2] => [6,6,1,1,1,1,1] => 0011111 => 0011111 => 5
[7,2,2,2,2,1,1] => [7,5,1,1,1,1,1] => 1111111 => 1111111 => 7
[7,2,2,2,1,1,1,1] => [8,4,1,1,1,1,1] => 0011111 => 0011111 => 5
[7,2,2,1,1,1,1,1,1] => [9,3,1,1,1,1,1] => 1111111 => 1111111 => 7
[7,2,1,1,1,1,1,1,1,1] => [10,2,1,1,1,1,1] => 0011111 => 0011111 => 5
[7,1,1,1,1,1,1,1,1,1,1] => [11,1,1,1,1,1,1] => 1111111 => 1111111 => 7
[6,6,5] => [3,3,3,3,3,2] => 111110 => 011111 => 5
[6,6,4,1] => [4,3,3,3,2,2] => 011100 => 000111 => 3
[6,6,3,2] => [4,4,3,2,2,2] => 001000 => 000001 => 1
[6,6,3,1,1] => [5,3,3,2,2,2] => 111000 => 000111 => 3
[6,6,2,2,1] => [5,4,2,2,2,2] => 100000 => 000001 => 1
[6,6,2,1,1,1] => [6,3,2,2,2,2] => 010000 => 000001 => 1
[6,6,1,1,1,1,1] => [7,2,2,2,2,2] => 100000 => 000001 => 1
[6,5,5,1] => [4,3,3,3,3,1] => 011111 => 011111 => 5
[6,5,4,2] => [4,4,3,3,2,1] => 001101 => 001101 => 3
[6,5,4,1,1] => [5,3,3,3,2,1] => 111101 => 011111 => 5
[6,5,3,3] => [4,4,4,2,2,1] => 000001 => 000001 => 1
[6,5,3,2,1] => [5,4,3,2,2,1] => 101001 => 001011 => 3
[6,5,3,1,1,1] => [6,3,3,2,2,1] => 011001 => 001011 => 3
[6,5,2,2,2] => [5,5,2,2,2,1] => 110001 => 000111 => 3
[6,5,2,2,1,1] => [6,4,2,2,2,1] => 000001 => 000001 => 1
[6,5,2,1,1,1,1] => [7,3,2,2,2,1] => 110001 => 000111 => 3
[6,5,1,1,1,1,1,1] => [8,2,2,2,2,1] => 000001 => 000001 => 1
[6,4,4,3] => [4,4,4,3,1,1] => 000111 => 000111 => 3
[6,4,4,2,1] => [5,4,3,3,1,1] => 101111 => 011111 => 5
[6,4,4,1,1,1] => [6,3,3,3,1,1] => 011111 => 011111 => 5
[6,4,3,3,1] => [5,4,4,2,1,1] => 100011 => 000111 => 3
[6,4,3,2,2] => [5,5,3,2,1,1] => 111011 => 011111 => 5
[6,4,3,2,1,1] => [6,4,3,2,1,1] => 001011 => 001011 => 3
[6,4,3,1,1,1,1] => [7,3,3,2,1,1] => 111011 => 011111 => 5
[6,4,2,2,2,1] => [6,5,2,2,1,1] => 010011 => 001101 => 3
[6,4,2,2,1,1,1] => [7,4,2,2,1,1] => 100011 => 000111 => 3
[6,4,2,1,1,1,1,1] => [8,3,2,2,1,1] => 010011 => 001101 => 3
[6,4,1,1,1,1,1,1,1] => [9,2,2,2,1,1] => 100011 => 000111 => 3
[6,3,3,3,2] => [5,5,4,1,1,1] => 110111 => 011111 => 5
[6,3,3,3,1,1] => [6,4,4,1,1,1] => 000111 => 000111 => 3
[6,3,3,2,2,1] => [6,5,3,1,1,1] => 011111 => 011111 => 5
[6,3,3,2,1,1,1] => [7,4,3,1,1,1] => 101111 => 011111 => 5
[6,3,3,1,1,1,1,1] => [8,3,3,1,1,1] => 011111 => 011111 => 5
[6,3,2,2,2,2] => [6,6,2,1,1,1] => 000111 => 000111 => 3
[6,3,2,2,2,1,1] => [7,5,2,1,1,1] => 110111 => 011111 => 5
[6,3,2,2,1,1,1,1] => [8,4,2,1,1,1] => 000111 => 000111 => 3
[6,3,2,1,1,1,1,1,1] => [9,3,2,1,1,1] => 110111 => 011111 => 5
[6,3,1,1,1,1,1,1,1,1] => [10,2,2,1,1,1] => 000111 => 000111 => 3
[6,2,2,2,2,2,1] => [7,6,1,1,1,1] => 101111 => 011111 => 5
[6,2,2,2,2,1,1,1] => [8,5,1,1,1,1] => 011111 => 011111 => 5
[6,2,2,2,1,1,1,1,1] => [9,4,1,1,1,1] => 101111 => 011111 => 5
[6,2,2,1,1,1,1,1,1,1] => [10,3,1,1,1,1] => 011111 => 011111 => 5
[6,2,1,1,1,1,1,1,1,1,1] => [11,2,1,1,1,1] => 101111 => 011111 => 5
[6,1,1,1,1,1,1,1,1,1,1,1] => [12,1,1,1,1,1] => 011111 => 011111 => 5
[5,5,5,2] => [4,4,3,3,3] => 00111 => 00111 => 3
[5,5,5,1,1] => [5,3,3,3,3] => 11111 => 11111 => 5
[5,5,4,3] => [4,4,4,3,2] => 00010 => 00001 => 1
[5,5,4,2,1] => [5,4,3,3,2] => 10110 => 00111 => 3
[5,5,4,1,1,1] => [6,3,3,3,2] => 01110 => 00111 => 3
[5,5,3,3,1] => [5,4,4,2,2] => 10000 => 00001 => 1
[5,5,3,2,2] => [5,5,3,2,2] => 11100 => 00111 => 3
[5,5,3,2,1,1] => [6,4,3,2,2] => 00100 => 00001 => 1
[5,5,3,1,1,1,1] => [7,3,3,2,2] => 11100 => 00111 => 3
[5,5,2,2,2,1] => [6,5,2,2,2] => 01000 => 00001 => 1
[5,5,2,2,1,1,1] => [7,4,2,2,2] => 10000 => 00001 => 1
[5,5,2,1,1,1,1,1] => [8,3,2,2,2] => 01000 => 00001 => 1
[5,5,1,1,1,1,1,1,1] => [9,2,2,2,2] => 10000 => 00001 => 1
[5,4,4,4] => [4,4,4,4,1] => 00001 => 00001 => 1
[5,4,4,3,1] => [5,4,4,3,1] => 10011 => 00111 => 3
[5,4,4,2,2] => [5,5,3,3,1] => 11111 => 11111 => 5
[5,4,4,2,1,1] => [6,4,3,3,1] => 00111 => 00111 => 3
[5,4,4,1,1,1,1] => [7,3,3,3,1] => 11111 => 11111 => 5
[5,4,3,3,2] => [5,5,4,2,1] => 11001 => 00111 => 3
[5,4,3,3,1,1] => [6,4,4,2,1] => 00001 => 00001 => 1
[5,4,3,2,2,1] => [6,5,3,2,1] => 01101 => 01011 => 3
[5,4,3,2,1,1,1] => [7,4,3,2,1] => 10101 => 01011 => 3
[5,4,3,1,1,1,1,1] => [8,3,3,2,1] => 01101 => 01011 => 3
[5,4,2,2,2,2] => [6,6,2,2,1] => 00001 => 00001 => 1
[5,4,2,2,2,1,1] => [7,5,2,2,1] => 11001 => 00111 => 3
[5,4,2,2,1,1,1,1] => [8,4,2,2,1] => 00001 => 00001 => 1
[5,4,2,1,1,1,1,1,1] => [9,3,2,2,1] => 11001 => 00111 => 3
[5,4,1,1,1,1,1,1,1,1] => [10,2,2,2,1] => 00001 => 00001 => 1
[5,3,3,3,3] => [5,5,5,1,1] => 11111 => 11111 => 5
[5,3,3,3,2,1] => [6,5,4,1,1] => 01011 => 01011 => 3
[5,3,3,3,1,1,1] => [7,4,4,1,1] => 10011 => 00111 => 3
[5,3,3,2,2,2] => [6,6,3,1,1] => 00111 => 00111 => 3
[5,3,3,2,2,1,1] => [7,5,3,1,1] => 11111 => 11111 => 5
[5,3,3,2,1,1,1,1] => [8,4,3,1,1] => 00111 => 00111 => 3
[5,3,3,1,1,1,1,1,1] => [9,3,3,1,1] => 11111 => 11111 => 5
[5,3,2,2,2,2,1] => [7,6,2,1,1] => 10011 => 00111 => 3
[5,3,2,2,2,1,1,1] => [8,5,2,1,1] => 01011 => 01011 => 3
[5,3,2,2,1,1,1,1,1] => [9,4,2,1,1] => 10011 => 00111 => 3
[5,3,2,1,1,1,1,1,1,1] => [10,3,2,1,1] => 01011 => 01011 => 3
[5,3,1,1,1,1,1,1,1,1,1] => [11,2,2,1,1] => 10011 => 00111 => 3
[5,2,2,2,2,2,2] => [7,7,1,1,1] => 11111 => 11111 => 5
[5,2,2,2,2,2,1,1] => [8,6,1,1,1] => 00111 => 00111 => 3
[5,2,2,2,2,1,1,1,1] => [9,5,1,1,1] => 11111 => 11111 => 5
[5,2,2,2,1,1,1,1,1,1] => [10,4,1,1,1] => 00111 => 00111 => 3
[5,2,2,1,1,1,1,1,1,1,1] => [11,3,1,1,1] => 11111 => 11111 => 5
[5,2,1,1,1,1,1,1,1,1,1,1] => [12,2,1,1,1] => 00111 => 00111 => 3
[5,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1,1,1] => 11111 => 11111 => 5
[4,4,4,4,1] => [5,4,4,4] => 1000 => 0001 => 1
[4,4,4,3,2] => [5,5,4,3] => 1101 => 0111 => 3
[4,4,4,3,1,1] => [6,4,4,3] => 0001 => 0001 => 1
[4,4,4,2,2,1] => [6,5,3,3] => 0111 => 0111 => 3
[4,4,4,2,1,1,1] => [7,4,3,3] => 1011 => 0111 => 3
[4,4,4,1,1,1,1,1] => [8,3,3,3] => 0111 => 0111 => 3
[4,4,3,3,3] => [5,5,5,2] => 1110 => 0111 => 3
[4,4,3,3,2,1] => [6,5,4,2] => 0100 => 0001 => 1
[4,4,3,3,1,1,1] => [7,4,4,2] => 1000 => 0001 => 1
[4,4,3,2,2,2] => [6,6,3,2] => 0010 => 0001 => 1
[4,4,3,2,2,1,1] => [7,5,3,2] => 1110 => 0111 => 3
[4,4,3,2,1,1,1,1] => [8,4,3,2] => 0010 => 0001 => 1
[4,4,3,1,1,1,1,1,1] => [9,3,3,2] => 1110 => 0111 => 3
[4,4,2,2,2,2,1] => [7,6,2,2] => 1000 => 0001 => 1
[4,4,2,2,2,1,1,1] => [8,5,2,2] => 0100 => 0001 => 1
[4,4,2,2,1,1,1,1,1] => [9,4,2,2] => 1000 => 0001 => 1
[4,4,2,1,1,1,1,1,1,1] => [10,3,2,2] => 0100 => 0001 => 1
[4,4,1,1,1,1,1,1,1,1,1] => [11,2,2,2] => 1000 => 0001 => 1
[4,3,3,3,3,1] => [6,5,5,1] => 0111 => 0111 => 3
[4,3,3,3,2,2] => [6,6,4,1] => 0001 => 0001 => 1
[4,3,3,3,2,1,1] => [7,5,4,1] => 1101 => 0111 => 3
[4,3,3,3,1,1,1,1] => [8,4,4,1] => 0001 => 0001 => 1
[4,3,3,2,2,2,1] => [7,6,3,1] => 1011 => 0111 => 3
[4,3,3,2,2,1,1,1] => [8,5,3,1] => 0111 => 0111 => 3
[4,3,3,2,1,1,1,1,1] => [9,4,3,1] => 1011 => 0111 => 3
[4,3,3,1,1,1,1,1,1,1] => [10,3,3,1] => 0111 => 0111 => 3
[4,3,2,2,2,2,2] => [7,7,2,1] => 1101 => 0111 => 3
[4,3,2,2,2,2,1,1] => [8,6,2,1] => 0001 => 0001 => 1
[4,3,2,2,2,1,1,1,1] => [9,5,2,1] => 1101 => 0111 => 3
[4,3,2,2,1,1,1,1,1,1] => [10,4,2,1] => 0001 => 0001 => 1
[4,3,2,1,1,1,1,1,1,1,1] => [11,3,2,1] => 1101 => 0111 => 3
[4,3,1,1,1,1,1,1,1,1,1,1] => [12,2,2,1] => 0001 => 0001 => 1
[4,2,2,2,2,2,2,1] => [8,7,1,1] => 0111 => 0111 => 3
[4,2,2,2,2,2,1,1,1] => [9,6,1,1] => 1011 => 0111 => 3
[4,2,2,2,2,1,1,1,1,1] => [10,5,1,1] => 0111 => 0111 => 3
[4,2,2,2,1,1,1,1,1,1,1] => [11,4,1,1] => 1011 => 0111 => 3
[4,2,2,1,1,1,1,1,1,1,1,1] => [12,3,1,1] => 0111 => 0111 => 3
[4,2,1,1,1,1,1,1,1,1,1,1,1] => [13,2,1,1] => 1011 => 0111 => 3
[4,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1,1,1] => 0111 => 0111 => 3
[3,3,3,3,3,2] => [6,6,5] => 001 => 001 => 1
[3,3,3,3,3,1,1] => [7,5,5] => 111 => 111 => 3
[3,3,3,3,2,2,1] => [7,6,4] => 100 => 001 => 1
[3,3,3,3,2,1,1,1] => [8,5,4] => 010 => 001 => 1
[3,3,3,3,1,1,1,1,1] => [9,4,4] => 100 => 001 => 1
[3,3,3,2,2,2,2] => [7,7,3] => 111 => 111 => 3
[3,3,3,2,2,2,1,1] => [8,6,3] => 001 => 001 => 1
[3,3,3,2,2,1,1,1,1] => [9,5,3] => 111 => 111 => 3
[3,3,3,2,1,1,1,1,1,1] => [10,4,3] => 001 => 001 => 1
[3,3,3,1,1,1,1,1,1,1,1] => [11,3,3] => 111 => 111 => 3
[3,3,2,2,2,2,2,1] => [8,7,2] => 010 => 001 => 1
[3,3,2,2,2,2,1,1,1] => [9,6,2] => 100 => 001 => 1
[3,3,2,2,2,1,1,1,1,1] => [10,5,2] => 010 => 001 => 1
[3,3,2,2,1,1,1,1,1,1,1] => [11,4,2] => 100 => 001 => 1
[3,3,2,1,1,1,1,1,1,1,1,1] => [12,3,2] => 010 => 001 => 1
[3,3,1,1,1,1,1,1,1,1,1,1,1] => [13,2,2] => 100 => 001 => 1
[3,2,2,2,2,2,2,2] => [8,8,1] => 001 => 001 => 1
[3,2,2,2,2,2,2,1,1] => [9,7,1] => 111 => 111 => 3
[3,2,2,2,2,2,1,1,1,1] => [10,6,1] => 001 => 001 => 1
[3,2,2,2,2,1,1,1,1,1,1] => [11,5,1] => 111 => 111 => 3
[3,2,2,2,1,1,1,1,1,1,1,1] => [12,4,1] => 001 => 001 => 1
[3,2,2,1,1,1,1,1,1,1,1,1,1] => [13,3,1] => 111 => 111 => 3
[3,2,1,1,1,1,1,1,1,1,1,1,1,1] => [14,2,1] => 001 => 001 => 1
[3,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,1,1] => 111 => 111 => 3
[2,2,2,2,2,2,2,2,1] => [9,8] => 10 => 01 => 1
[2,2,2,2,2,2,2,1,1,1] => [10,7] => 01 => 01 => 1
[2,2,2,2,2,2,1,1,1,1,1] => [11,6] => 10 => 01 => 1
[2,2,2,2,2,1,1,1,1,1,1,1] => [12,5] => 01 => 01 => 1
[2,2,2,2,1,1,1,1,1,1,1,1,1] => [13,4] => 10 => 01 => 1
[2,2,2,1,1,1,1,1,1,1,1,1,1,1] => [14,3] => 01 => 01 => 1
[2,2,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,2] => 10 => 01 => 1
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [16,1] => 01 => 01 => 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [17] => 1 => 1 => 1
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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:
1,0,1 2,0,1 2,0,2,0,1 4,0,2,0,1 3,0,5,0,2,0,1 7,0,5,0,2,0,1 5,0,9,0,5,0,2,0,1 12,0,10,0,5,0,2,0,1
F1=q
F2=1+q2
F3=2 q+q3
F4=2+2 q2+q4
F5=4 q+2 q3+q5
F6=3+5 q2+2 q4+q6
F7=7 q+5 q3+2 q5+q7
F8=5+9 q2+5 q4+2 q6+q8
F9=12 q+10 q3+5 q5+2 q7+q9
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
This is also known as the Hamming weight of the word.
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition λ of n is the partition λ∗ whose Ferrers diagram is obtained from the diagram of λ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
The conjugate partition of the partition λ of n is the partition λ∗ whose Ferrers diagram is obtained from the diagram of λ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
Map
odd parts
Description
Return the binary word indicating which parts of the partition are odd.
Map
runsort
Description
The word obtained by sorting the weakly increasing runs lexicographically.
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