Identifier
-
Mp00308:
Integer partitions
—Bulgarian solitaire⟶
Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000629: Binary words ⟶ ℤ
Values
[1] => [1] => 1 => 0
[2] => [1,1] => 11 => 0
[1,1] => [2] => 0 => 0
[3] => [2,1] => 01 => 0
[2,1] => [2,1] => 01 => 0
[1,1,1] => [3] => 1 => 0
[4] => [3,1] => 11 => 0
[3,1] => [2,2] => 00 => 0
[2,2] => [2,1,1] => 011 => 0
[2,1,1] => [3,1] => 11 => 0
[1,1,1,1] => [4] => 0 => 0
[5] => [4,1] => 01 => 0
[4,1] => [3,2] => 10 => 0
[3,2] => [2,2,1] => 001 => 0
[3,1,1] => [3,2] => 10 => 0
[2,2,1] => [3,1,1] => 111 => 0
[2,1,1,1] => [4,1] => 01 => 0
[1,1,1,1,1] => [5] => 1 => 0
[6] => [5,1] => 11 => 0
[5,1] => [4,2] => 00 => 0
[4,2] => [3,2,1] => 101 => 0
[4,1,1] => [3,3] => 11 => 0
[3,3] => [2,2,2] => 000 => 0
[3,2,1] => [3,2,1] => 101 => 0
[3,1,1,1] => [4,2] => 00 => 0
[2,2,2] => [3,1,1,1] => 1111 => 0
[2,2,1,1] => [4,1,1] => 011 => 0
[2,1,1,1,1] => [5,1] => 11 => 0
[1,1,1,1,1,1] => [6] => 0 => 0
[7] => [6,1] => 01 => 0
[6,1] => [5,2] => 10 => 0
[5,2] => [4,2,1] => 001 => 0
[5,1,1] => [4,3] => 01 => 0
[4,3] => [3,2,2] => 100 => 0
[4,2,1] => [3,3,1] => 111 => 0
[4,1,1,1] => [4,3] => 01 => 0
[3,3,1] => [3,2,2] => 100 => 0
[3,2,2] => [3,2,1,1] => 1011 => 0
[3,2,1,1] => [4,2,1] => 001 => 0
[3,1,1,1,1] => [5,2] => 10 => 0
[2,2,2,1] => [4,1,1,1] => 0111 => 0
[2,2,1,1,1] => [5,1,1] => 111 => 0
[2,1,1,1,1,1] => [6,1] => 01 => 0
[1,1,1,1,1,1,1] => [7] => 1 => 0
[8] => [7,1] => 11 => 0
[7,1] => [6,2] => 00 => 0
[6,2] => [5,2,1] => 101 => 0
[6,1,1] => [5,3] => 11 => 0
[5,3] => [4,2,2] => 000 => 0
[5,2,1] => [4,3,1] => 011 => 0
[5,1,1,1] => [4,4] => 00 => 0
[4,4] => [3,3,2] => 110 => 0
[4,3,1] => [3,3,2] => 110 => 0
[4,2,2] => [3,3,1,1] => 1111 => 0
[4,2,1,1] => [4,3,1] => 011 => 0
[4,1,1,1,1] => [5,3] => 11 => 0
[3,3,2] => [3,2,2,1] => 1001 => 0
[3,3,1,1] => [4,2,2] => 000 => 0
[3,2,2,1] => [4,2,1,1] => 0011 => 0
[3,2,1,1,1] => [5,2,1] => 101 => 0
[3,1,1,1,1,1] => [6,2] => 00 => 0
[2,2,2,2] => [4,1,1,1,1] => 01111 => 0
[2,2,2,1,1] => [5,1,1,1] => 1111 => 0
[2,2,1,1,1,1] => [6,1,1] => 011 => 0
[2,1,1,1,1,1,1] => [7,1] => 11 => 0
[1,1,1,1,1,1,1,1] => [8] => 0 => 0
[9] => [8,1] => 01 => 0
[8,1] => [7,2] => 10 => 0
[7,2] => [6,2,1] => 001 => 0
[7,1,1] => [6,3] => 01 => 0
[6,3] => [5,2,2] => 100 => 0
[6,2,1] => [5,3,1] => 111 => 0
[6,1,1,1] => [5,4] => 10 => 0
[5,4] => [4,3,2] => 010 => 0
[5,3,1] => [4,3,2] => 010 => 0
[5,2,2] => [4,3,1,1] => 0111 => 0
[5,2,1,1] => [4,4,1] => 001 => 0
[5,1,1,1,1] => [5,4] => 10 => 0
[4,4,1] => [3,3,3] => 111 => 0
[4,3,2] => [3,3,2,1] => 1101 => 0
[4,3,1,1] => [4,3,2] => 010 => 0
[4,2,2,1] => [4,3,1,1] => 0111 => 0
[4,2,1,1,1] => [5,3,1] => 111 => 0
[4,1,1,1,1,1] => [6,3] => 01 => 0
[3,3,3] => [3,2,2,2] => 1000 => 0
[3,3,2,1] => [4,2,2,1] => 0001 => 0
[3,3,1,1,1] => [5,2,2] => 100 => 0
[3,2,2,2] => [4,2,1,1,1] => 00111 => 0
[3,2,2,1,1] => [5,2,1,1] => 1011 => 0
[3,2,1,1,1,1] => [6,2,1] => 001 => 0
[3,1,1,1,1,1,1] => [7,2] => 10 => 0
[2,2,2,2,1] => [5,1,1,1,1] => 11111 => 0
[2,2,2,1,1,1] => [6,1,1,1] => 0111 => 0
[2,2,1,1,1,1,1] => [7,1,1] => 111 => 0
[2,1,1,1,1,1,1,1] => [8,1] => 01 => 0
[1,1,1,1,1,1,1,1,1] => [9] => 1 => 0
[10] => [9,1] => 11 => 0
[9,1] => [8,2] => 00 => 0
[8,2] => [7,2,1] => 101 => 0
[8,1,1] => [7,3] => 11 => 0
[7,3] => [6,2,2] => 000 => 0
>>> Load all 1211 entries. <<<[7,2,1] => [6,3,1] => 011 => 0
[7,1,1,1] => [6,4] => 00 => 0
[6,4] => [5,3,2] => 110 => 0
[6,3,1] => [5,3,2] => 110 => 0
[6,2,2] => [5,3,1,1] => 1111 => 0
[6,2,1,1] => [5,4,1] => 101 => 0
[6,1,1,1,1] => [5,5] => 11 => 0
[5,5] => [4,4,2] => 000 => 0
[5,4,1] => [4,3,3] => 011 => 0
[5,3,2] => [4,3,2,1] => 0101 => 0
[5,3,1,1] => [4,4,2] => 000 => 0
[5,2,2,1] => [4,4,1,1] => 0011 => 0
[5,2,1,1,1] => [5,4,1] => 101 => 0
[5,1,1,1,1,1] => [6,4] => 00 => 0
[4,4,2] => [3,3,3,1] => 1111 => 0
[4,4,1,1] => [4,3,3] => 011 => 0
[4,3,3] => [3,3,2,2] => 1100 => 0
[4,3,2,1] => [4,3,2,1] => 0101 => 0
[4,3,1,1,1] => [5,3,2] => 110 => 0
[4,2,2,2] => [4,3,1,1,1] => 01111 => 0
[4,2,2,1,1] => [5,3,1,1] => 1111 => 0
[4,2,1,1,1,1] => [6,3,1] => 011 => 0
[4,1,1,1,1,1,1] => [7,3] => 11 => 0
[3,3,3,1] => [4,2,2,2] => 0000 => 0
[3,3,2,2] => [4,2,2,1,1] => 00011 => 0
[3,3,2,1,1] => [5,2,2,1] => 1001 => 0
[3,3,1,1,1,1] => [6,2,2] => 000 => 0
[3,2,2,2,1] => [5,2,1,1,1] => 10111 => 0
[3,2,2,1,1,1] => [6,2,1,1] => 0011 => 0
[3,2,1,1,1,1,1] => [7,2,1] => 101 => 0
[3,1,1,1,1,1,1,1] => [8,2] => 00 => 0
[2,2,2,2,2] => [5,1,1,1,1,1] => 111111 => 0
[2,2,2,2,1,1] => [6,1,1,1,1] => 01111 => 0
[2,2,2,1,1,1,1] => [7,1,1,1] => 1111 => 0
[2,2,1,1,1,1,1,1] => [8,1,1] => 011 => 0
[2,1,1,1,1,1,1,1,1] => [9,1] => 11 => 0
[1,1,1,1,1,1,1,1,1,1] => [10] => 0 => 0
[11] => [10,1] => 01 => 0
[10,1] => [9,2] => 10 => 0
[9,2] => [8,2,1] => 001 => 0
[9,1,1] => [8,3] => 01 => 0
[8,3] => [7,2,2] => 100 => 0
[8,2,1] => [7,3,1] => 111 => 0
[8,1,1,1] => [7,4] => 10 => 0
[7,4] => [6,3,2] => 010 => 0
[7,3,1] => [6,3,2] => 010 => 0
[7,2,2] => [6,3,1,1] => 0111 => 0
[7,2,1,1] => [6,4,1] => 001 => 0
[7,1,1,1,1] => [6,5] => 01 => 0
[6,5] => [5,4,2] => 100 => 0
[6,4,1] => [5,3,3] => 111 => 0
[6,3,2] => [5,3,2,1] => 1101 => 0
[6,3,1,1] => [5,4,2] => 100 => 0
[6,2,2,1] => [5,4,1,1] => 1011 => 0
[6,2,1,1,1] => [5,5,1] => 111 => 0
[6,1,1,1,1,1] => [6,5] => 01 => 0
[5,5,1] => [4,4,3] => 001 => 0
[5,4,2] => [4,3,3,1] => 0111 => 0
[5,4,1,1] => [4,4,3] => 001 => 0
[5,3,3] => [4,3,2,2] => 0100 => 0
[5,3,2,1] => [4,4,2,1] => 0001 => 0
[5,3,1,1,1] => [5,4,2] => 100 => 0
[5,2,2,2] => [4,4,1,1,1] => 00111 => 0
[5,2,2,1,1] => [5,4,1,1] => 1011 => 0
[5,2,1,1,1,1] => [6,4,1] => 001 => 0
[5,1,1,1,1,1,1] => [7,4] => 10 => 0
[4,4,3] => [3,3,3,2] => 1110 => 0
[4,4,2,1] => [4,3,3,1] => 0111 => 0
[4,4,1,1,1] => [5,3,3] => 111 => 0
[4,3,3,1] => [4,3,2,2] => 0100 => 0
[4,3,2,2] => [4,3,2,1,1] => 01011 => 0
[4,3,2,1,1] => [5,3,2,1] => 1101 => 0
[4,3,1,1,1,1] => [6,3,2] => 010 => 0
[4,2,2,2,1] => [5,3,1,1,1] => 11111 => 0
[4,2,2,1,1,1] => [6,3,1,1] => 0111 => 0
[4,2,1,1,1,1,1] => [7,3,1] => 111 => 0
[4,1,1,1,1,1,1,1] => [8,3] => 01 => 0
[3,3,3,2] => [4,2,2,2,1] => 00001 => 0
[3,3,3,1,1] => [5,2,2,2] => 1000 => 0
[3,3,2,2,1] => [5,2,2,1,1] => 10011 => 0
[3,3,2,1,1,1] => [6,2,2,1] => 0001 => 0
[3,3,1,1,1,1,1] => [7,2,2] => 100 => 0
[3,2,2,2,2] => [5,2,1,1,1,1] => 101111 => 0
[3,2,2,2,1,1] => [6,2,1,1,1] => 00111 => 0
[3,2,2,1,1,1,1] => [7,2,1,1] => 1011 => 0
[3,2,1,1,1,1,1,1] => [8,2,1] => 001 => 0
[3,1,1,1,1,1,1,1,1] => [9,2] => 10 => 0
[2,2,2,2,2,1] => [6,1,1,1,1,1] => 011111 => 0
[2,2,2,2,1,1,1] => [7,1,1,1,1] => 11111 => 0
[2,2,2,1,1,1,1,1] => [8,1,1,1] => 0111 => 0
[2,2,1,1,1,1,1,1,1] => [9,1,1] => 111 => 0
[2,1,1,1,1,1,1,1,1,1] => [10,1] => 01 => 0
[1,1,1,1,1,1,1,1,1,1,1] => [11] => 1 => 0
[12] => [11,1] => 11 => 0
[11,1] => [10,2] => 00 => 0
[10,2] => [9,2,1] => 101 => 0
[10,1,1] => [9,3] => 11 => 0
[9,3] => [8,2,2] => 000 => 0
[9,2,1] => [8,3,1] => 011 => 0
[9,1,1,1] => [8,4] => 00 => 0
[8,4] => [7,3,2] => 110 => 0
[8,3,1] => [7,3,2] => 110 => 0
[8,2,2] => [7,3,1,1] => 1111 => 0
[8,2,1,1] => [7,4,1] => 101 => 0
[8,1,1,1,1] => [7,5] => 11 => 0
[7,5] => [6,4,2] => 000 => 0
[7,4,1] => [6,3,3] => 011 => 0
[7,3,2] => [6,3,2,1] => 0101 => 0
[7,3,1,1] => [6,4,2] => 000 => 0
[7,2,2,1] => [6,4,1,1] => 0011 => 0
[7,2,1,1,1] => [6,5,1] => 011 => 0
[7,1,1,1,1,1] => [6,6] => 00 => 0
[6,6] => [5,5,2] => 110 => 0
[6,5,1] => [5,4,3] => 101 => 0
[6,4,2] => [5,3,3,1] => 1111 => 0
[6,4,1,1] => [5,4,3] => 101 => 0
[6,3,3] => [5,3,2,2] => 1100 => 0
[6,3,2,1] => [5,4,2,1] => 1001 => 0
[6,3,1,1,1] => [5,5,2] => 110 => 0
[6,2,2,2] => [5,4,1,1,1] => 10111 => 0
[6,2,2,1,1] => [5,5,1,1] => 1111 => 0
[6,2,1,1,1,1] => [6,5,1] => 011 => 0
[6,1,1,1,1,1,1] => [7,5] => 11 => 0
[5,5,2] => [4,4,3,1] => 0011 => 0
[5,5,1,1] => [4,4,4] => 000 => 0
[5,4,3] => [4,3,3,2] => 0110 => 0
[5,4,2,1] => [4,4,3,1] => 0011 => 0
[5,4,1,1,1] => [5,4,3] => 101 => 0
[5,3,3,1] => [4,4,2,2] => 0000 => 0
[5,3,2,2] => [4,4,2,1,1] => 00011 => 0
[5,3,2,1,1] => [5,4,2,1] => 1001 => 0
[5,3,1,1,1,1] => [6,4,2] => 000 => 0
[5,2,2,2,1] => [5,4,1,1,1] => 10111 => 0
[5,2,2,1,1,1] => [6,4,1,1] => 0011 => 0
[5,2,1,1,1,1,1] => [7,4,1] => 101 => 0
[5,1,1,1,1,1,1,1] => [8,4] => 00 => 0
[4,4,4] => [3,3,3,3] => 1111 => 0
[4,4,3,1] => [4,3,3,2] => 0110 => 0
[4,4,2,2] => [4,3,3,1,1] => 01111 => 0
[4,4,2,1,1] => [5,3,3,1] => 1111 => 0
[4,4,1,1,1,1] => [6,3,3] => 011 => 0
[4,3,3,2] => [4,3,2,2,1] => 01001 => 0
[4,3,3,1,1] => [5,3,2,2] => 1100 => 0
[4,3,2,2,1] => [5,3,2,1,1] => 11011 => 0
[4,3,2,1,1,1] => [6,3,2,1] => 0101 => 0
[4,3,1,1,1,1,1] => [7,3,2] => 110 => 0
[4,2,2,2,2] => [5,3,1,1,1,1] => 111111 => 0
[4,2,2,2,1,1] => [6,3,1,1,1] => 01111 => 0
[4,2,2,1,1,1,1] => [7,3,1,1] => 1111 => 0
[4,2,1,1,1,1,1,1] => [8,3,1] => 011 => 0
[4,1,1,1,1,1,1,1,1] => [9,3] => 11 => 0
[3,3,3,3] => [4,2,2,2,2] => 00000 => 0
[3,3,3,2,1] => [5,2,2,2,1] => 10001 => 0
[3,3,3,1,1,1] => [6,2,2,2] => 0000 => 0
[3,3,2,2,2] => [5,2,2,1,1,1] => 100111 => 0
[3,3,2,2,1,1] => [6,2,2,1,1] => 00011 => 0
[3,3,2,1,1,1,1] => [7,2,2,1] => 1001 => 0
[3,3,1,1,1,1,1,1] => [8,2,2] => 000 => 0
[3,2,2,2,2,1] => [6,2,1,1,1,1] => 001111 => 0
[3,2,2,2,1,1,1] => [7,2,1,1,1] => 10111 => 0
[3,2,2,1,1,1,1,1] => [8,2,1,1] => 0011 => 0
[3,2,1,1,1,1,1,1,1] => [9,2,1] => 101 => 0
[3,1,1,1,1,1,1,1,1,1] => [10,2] => 00 => 0
[2,2,2,2,2,2] => [6,1,1,1,1,1,1] => 0111111 => 0
[2,2,2,2,2,1,1] => [7,1,1,1,1,1] => 111111 => 0
[2,2,2,2,1,1,1,1] => [8,1,1,1,1] => 01111 => 0
[2,2,2,1,1,1,1,1,1] => [9,1,1,1] => 1111 => 0
[2,2,1,1,1,1,1,1,1,1] => [10,1,1] => 011 => 0
[2,1,1,1,1,1,1,1,1,1,1] => [11,1] => 11 => 0
[1,1,1,1,1,1,1,1,1,1,1,1] => [12] => 0 => 0
[13] => [12,1] => 01 => 0
[12,1] => [11,2] => 10 => 0
[11,2] => [10,2,1] => 001 => 0
[11,1,1] => [10,3] => 01 => 0
[10,3] => [9,2,2] => 100 => 0
[10,2,1] => [9,3,1] => 111 => 0
[10,1,1,1] => [9,4] => 10 => 0
[9,4] => [8,3,2] => 010 => 0
[9,3,1] => [8,3,2] => 010 => 0
[9,2,2] => [8,3,1,1] => 0111 => 0
[9,2,1,1] => [8,4,1] => 001 => 0
[9,1,1,1,1] => [8,5] => 01 => 0
[8,5] => [7,4,2] => 100 => 0
[8,4,1] => [7,3,3] => 111 => 0
[8,3,2] => [7,3,2,1] => 1101 => 0
[8,3,1,1] => [7,4,2] => 100 => 0
[8,2,2,1] => [7,4,1,1] => 1011 => 0
[8,2,1,1,1] => [7,5,1] => 111 => 0
[8,1,1,1,1,1] => [7,6] => 10 => 0
[7,6] => [6,5,2] => 010 => 0
[7,5,1] => [6,4,3] => 001 => 0
[7,4,2] => [6,3,3,1] => 0111 => 0
[7,4,1,1] => [6,4,3] => 001 => 0
[7,3,3] => [6,3,2,2] => 0100 => 0
[7,3,2,1] => [6,4,2,1] => 0001 => 0
[7,3,1,1,1] => [6,5,2] => 010 => 0
[7,2,2,2] => [6,4,1,1,1] => 00111 => 0
[7,2,2,1,1] => [6,5,1,1] => 0111 => 0
[7,2,1,1,1,1] => [6,6,1] => 001 => 0
[7,1,1,1,1,1,1] => [7,6] => 10 => 0
[6,6,1] => [5,5,3] => 111 => 0
[6,5,2] => [5,4,3,1] => 1011 => 0
[6,5,1,1] => [5,4,4] => 100 => 0
[6,4,3] => [5,3,3,2] => 1110 => 0
[6,4,2,1] => [5,4,3,1] => 1011 => 0
[6,4,1,1,1] => [5,5,3] => 111 => 0
[6,3,3,1] => [5,4,2,2] => 1000 => 0
[6,3,2,2] => [5,4,2,1,1] => 10011 => 0
[6,3,2,1,1] => [5,5,2,1] => 1101 => 0
[6,3,1,1,1,1] => [6,5,2] => 010 => 0
[6,2,2,2,1] => [5,5,1,1,1] => 11111 => 0
[6,2,2,1,1,1] => [6,5,1,1] => 0111 => 0
[6,2,1,1,1,1,1] => [7,5,1] => 111 => 0
[6,1,1,1,1,1,1,1] => [8,5] => 01 => 0
[5,5,3] => [4,4,3,2] => 0010 => 0
[5,5,2,1] => [4,4,4,1] => 0001 => 0
[5,5,1,1,1] => [5,4,4] => 100 => 0
[5,4,4] => [4,3,3,3] => 0111 => 0
[5,4,3,1] => [4,4,3,2] => 0010 => 0
[5,4,2,2] => [4,4,3,1,1] => 00111 => 0
[5,4,2,1,1] => [5,4,3,1] => 1011 => 0
[5,4,1,1,1,1] => [6,4,3] => 001 => 0
[5,3,3,2] => [4,4,2,2,1] => 00001 => 0
[5,3,3,1,1] => [5,4,2,2] => 1000 => 0
[5,3,2,2,1] => [5,4,2,1,1] => 10011 => 0
[5,3,2,1,1,1] => [6,4,2,1] => 0001 => 0
[5,3,1,1,1,1,1] => [7,4,2] => 100 => 0
[5,2,2,2,2] => [5,4,1,1,1,1] => 101111 => 0
[5,2,2,2,1,1] => [6,4,1,1,1] => 00111 => 0
[5,2,2,1,1,1,1] => [7,4,1,1] => 1011 => 0
[5,2,1,1,1,1,1,1] => [8,4,1] => 001 => 0
[5,1,1,1,1,1,1,1,1] => [9,4] => 10 => 0
[4,4,4,1] => [4,3,3,3] => 0111 => 0
[4,4,3,2] => [4,3,3,2,1] => 01101 => 0
[4,4,3,1,1] => [5,3,3,2] => 1110 => 0
[4,4,2,2,1] => [5,3,3,1,1] => 11111 => 0
[4,4,2,1,1,1] => [6,3,3,1] => 0111 => 0
[4,4,1,1,1,1,1] => [7,3,3] => 111 => 0
[4,3,3,3] => [4,3,2,2,2] => 01000 => 0
[4,3,3,2,1] => [5,3,2,2,1] => 11001 => 0
[4,3,3,1,1,1] => [6,3,2,2] => 0100 => 0
[4,3,2,2,2] => [5,3,2,1,1,1] => 110111 => 0
[4,3,2,2,1,1] => [6,3,2,1,1] => 01011 => 0
[4,3,2,1,1,1,1] => [7,3,2,1] => 1101 => 0
[4,3,1,1,1,1,1,1] => [8,3,2] => 010 => 0
[4,2,2,2,2,1] => [6,3,1,1,1,1] => 011111 => 0
[4,2,2,2,1,1,1] => [7,3,1,1,1] => 11111 => 0
[4,2,2,1,1,1,1,1] => [8,3,1,1] => 0111 => 0
[4,2,1,1,1,1,1,1,1] => [9,3,1] => 111 => 0
[4,1,1,1,1,1,1,1,1,1] => [10,3] => 01 => 0
[3,3,3,3,1] => [5,2,2,2,2] => 10000 => 0
[3,3,3,2,2] => [5,2,2,2,1,1] => 100011 => 0
[3,3,3,2,1,1] => [6,2,2,2,1] => 00001 => 0
[3,3,3,1,1,1,1] => [7,2,2,2] => 1000 => 0
[3,3,2,2,2,1] => [6,2,2,1,1,1] => 000111 => 0
[3,3,2,2,1,1,1] => [7,2,2,1,1] => 10011 => 0
[3,3,2,1,1,1,1,1] => [8,2,2,1] => 0001 => 0
[3,3,1,1,1,1,1,1,1] => [9,2,2] => 100 => 0
[3,2,2,2,2,2] => [6,2,1,1,1,1,1] => 0011111 => 0
[3,2,2,2,2,1,1] => [7,2,1,1,1,1] => 101111 => 0
[3,2,2,2,1,1,1,1] => [8,2,1,1,1] => 00111 => 0
[3,2,2,1,1,1,1,1,1] => [9,2,1,1] => 1011 => 0
[3,2,1,1,1,1,1,1,1,1] => [10,2,1] => 001 => 0
[3,1,1,1,1,1,1,1,1,1,1] => [11,2] => 10 => 0
[2,2,2,2,2,2,1] => [7,1,1,1,1,1,1] => 1111111 => 0
[2,2,2,2,2,1,1,1] => [8,1,1,1,1,1] => 011111 => 0
[2,2,2,2,1,1,1,1,1] => [9,1,1,1,1] => 11111 => 0
[2,2,2,1,1,1,1,1,1,1] => [10,1,1,1] => 0111 => 0
[2,2,1,1,1,1,1,1,1,1,1] => [11,1,1] => 111 => 0
[2,1,1,1,1,1,1,1,1,1,1,1] => [12,1] => 01 => 0
[1,1,1,1,1,1,1,1,1,1,1,1,1] => [13] => 1 => 0
[14] => [13,1] => 11 => 0
[13,1] => [12,2] => 00 => 0
[12,2] => [11,2,1] => 101 => 0
[12,1,1] => [11,3] => 11 => 0
[11,3] => [10,2,2] => 000 => 0
[11,2,1] => [10,3,1] => 011 => 0
[11,1,1,1] => [10,4] => 00 => 0
[10,4] => [9,3,2] => 110 => 0
[10,3,1] => [9,3,2] => 110 => 0
[10,2,2] => [9,3,1,1] => 1111 => 0
[10,2,1,1] => [9,4,1] => 101 => 0
[10,1,1,1,1] => [9,5] => 11 => 0
[9,5] => [8,4,2] => 000 => 0
[9,4,1] => [8,3,3] => 011 => 0
[9,3,2] => [8,3,2,1] => 0101 => 0
[9,3,1,1] => [8,4,2] => 000 => 0
[9,2,2,1] => [8,4,1,1] => 0011 => 0
[9,2,1,1,1] => [8,5,1] => 011 => 0
[9,1,1,1,1,1] => [8,6] => 00 => 0
[8,6] => [7,5,2] => 110 => 0
[8,5,1] => [7,4,3] => 101 => 0
[8,4,2] => [7,3,3,1] => 1111 => 0
[8,4,1,1] => [7,4,3] => 101 => 0
[8,3,3] => [7,3,2,2] => 1100 => 0
[8,3,2,1] => [7,4,2,1] => 1001 => 0
[8,3,1,1,1] => [7,5,2] => 110 => 0
[8,2,2,2] => [7,4,1,1,1] => 10111 => 0
[8,2,2,1,1] => [7,5,1,1] => 1111 => 0
[8,2,1,1,1,1] => [7,6,1] => 101 => 0
[8,1,1,1,1,1,1] => [7,7] => 11 => 0
[7,7] => [6,6,2] => 000 => 0
[7,6,1] => [6,5,3] => 011 => 0
[7,5,2] => [6,4,3,1] => 0011 => 0
[7,5,1,1] => [6,4,4] => 000 => 0
[7,4,3] => [6,3,3,2] => 0110 => 0
[7,4,2,1] => [6,4,3,1] => 0011 => 0
[7,4,1,1,1] => [6,5,3] => 011 => 0
[7,3,3,1] => [6,4,2,2] => 0000 => 0
[7,3,2,2] => [6,4,2,1,1] => 00011 => 0
[7,3,2,1,1] => [6,5,2,1] => 0101 => 0
[7,3,1,1,1,1] => [6,6,2] => 000 => 0
[7,2,2,2,1] => [6,5,1,1,1] => 01111 => 0
[7,2,2,1,1,1] => [6,6,1,1] => 0011 => 0
[7,2,1,1,1,1,1] => [7,6,1] => 101 => 0
[7,1,1,1,1,1,1,1] => [8,6] => 00 => 0
[6,6,2] => [5,5,3,1] => 1111 => 0
[6,6,1,1] => [5,5,4] => 110 => 0
[6,5,3] => [5,4,3,2] => 1010 => 0
[6,5,2,1] => [5,4,4,1] => 1001 => 0
[6,5,1,1,1] => [5,5,4] => 110 => 0
[6,4,4] => [5,3,3,3] => 1111 => 0
[6,4,3,1] => [5,4,3,2] => 1010 => 0
[6,4,2,2] => [5,4,3,1,1] => 10111 => 0
[6,4,2,1,1] => [5,5,3,1] => 1111 => 0
[6,4,1,1,1,1] => [6,5,3] => 011 => 0
[6,3,3,2] => [5,4,2,2,1] => 10001 => 0
[6,3,3,1,1] => [5,5,2,2] => 1100 => 0
[6,3,2,2,1] => [5,5,2,1,1] => 11011 => 0
[6,3,2,1,1,1] => [6,5,2,1] => 0101 => 0
[6,3,1,1,1,1,1] => [7,5,2] => 110 => 0
[6,2,2,2,2] => [5,5,1,1,1,1] => 111111 => 0
[6,2,2,2,1,1] => [6,5,1,1,1] => 01111 => 0
[6,2,2,1,1,1,1] => [7,5,1,1] => 1111 => 0
[6,2,1,1,1,1,1,1] => [8,5,1] => 011 => 0
[6,1,1,1,1,1,1,1,1] => [9,5] => 11 => 0
[5,5,4] => [4,4,3,3] => 0011 => 0
[5,5,3,1] => [4,4,4,2] => 0000 => 0
[5,5,2,2] => [4,4,4,1,1] => 00011 => 0
[5,5,2,1,1] => [5,4,4,1] => 1001 => 0
[5,5,1,1,1,1] => [6,4,4] => 000 => 0
[5,4,4,1] => [4,4,3,3] => 0011 => 0
[5,4,3,2] => [4,4,3,2,1] => 00101 => 0
[5,4,3,1,1] => [5,4,3,2] => 1010 => 0
[5,4,2,2,1] => [5,4,3,1,1] => 10111 => 0
[5,4,2,1,1,1] => [6,4,3,1] => 0011 => 0
[5,4,1,1,1,1,1] => [7,4,3] => 101 => 0
[5,3,3,3] => [4,4,2,2,2] => 00000 => 0
[5,3,3,2,1] => [5,4,2,2,1] => 10001 => 0
[5,3,3,1,1,1] => [6,4,2,2] => 0000 => 0
[5,3,2,2,2] => [5,4,2,1,1,1] => 100111 => 0
[5,3,2,2,1,1] => [6,4,2,1,1] => 00011 => 0
[5,3,2,1,1,1,1] => [7,4,2,1] => 1001 => 0
[5,3,1,1,1,1,1,1] => [8,4,2] => 000 => 0
[5,2,2,2,2,1] => [6,4,1,1,1,1] => 001111 => 0
[5,2,2,2,1,1,1] => [7,4,1,1,1] => 10111 => 0
[5,2,2,1,1,1,1,1] => [8,4,1,1] => 0011 => 0
[5,2,1,1,1,1,1,1,1] => [9,4,1] => 101 => 0
[5,1,1,1,1,1,1,1,1,1] => [10,4] => 00 => 0
[4,4,4,2] => [4,3,3,3,1] => 01111 => 0
[4,4,4,1,1] => [5,3,3,3] => 1111 => 0
[4,4,3,3] => [4,3,3,2,2] => 01100 => 0
[4,4,3,2,1] => [5,3,3,2,1] => 11101 => 0
[4,4,3,1,1,1] => [6,3,3,2] => 0110 => 0
[4,4,2,2,2] => [5,3,3,1,1,1] => 111111 => 0
[4,4,2,2,1,1] => [6,3,3,1,1] => 01111 => 0
[4,4,2,1,1,1,1] => [7,3,3,1] => 1111 => 0
[4,4,1,1,1,1,1,1] => [8,3,3] => 011 => 0
[4,3,3,3,1] => [5,3,2,2,2] => 11000 => 0
[4,3,3,2,2] => [5,3,2,2,1,1] => 110011 => 0
[4,3,3,2,1,1] => [6,3,2,2,1] => 01001 => 0
[4,3,3,1,1,1,1] => [7,3,2,2] => 1100 => 0
[4,3,2,2,2,1] => [6,3,2,1,1,1] => 010111 => 0
[4,3,2,2,1,1,1] => [7,3,2,1,1] => 11011 => 0
[4,3,2,1,1,1,1,1] => [8,3,2,1] => 0101 => 0
[4,3,1,1,1,1,1,1,1] => [9,3,2] => 110 => 0
[4,2,2,2,2,2] => [6,3,1,1,1,1,1] => 0111111 => 0
[4,2,2,2,2,1,1] => [7,3,1,1,1,1] => 111111 => 0
[4,2,2,2,1,1,1,1] => [8,3,1,1,1] => 01111 => 0
[4,2,2,1,1,1,1,1,1] => [9,3,1,1] => 1111 => 0
[4,2,1,1,1,1,1,1,1,1] => [10,3,1] => 011 => 0
[4,1,1,1,1,1,1,1,1,1,1] => [11,3] => 11 => 0
[3,3,3,3,2] => [5,2,2,2,2,1] => 100001 => 0
[3,3,3,3,1,1] => [6,2,2,2,2] => 00000 => 0
[3,3,3,2,2,1] => [6,2,2,2,1,1] => 000011 => 0
[3,3,3,2,1,1,1] => [7,2,2,2,1] => 10001 => 0
[3,3,3,1,1,1,1,1] => [8,2,2,2] => 0000 => 0
[3,3,2,2,2,2] => [6,2,2,1,1,1,1] => 0001111 => 0
[3,3,2,2,2,1,1] => [7,2,2,1,1,1] => 100111 => 0
[3,3,2,2,1,1,1,1] => [8,2,2,1,1] => 00011 => 0
[3,3,2,1,1,1,1,1,1] => [9,2,2,1] => 1001 => 0
[3,3,1,1,1,1,1,1,1,1] => [10,2,2] => 000 => 0
[3,2,2,2,2,2,1] => [7,2,1,1,1,1,1] => 1011111 => 0
[3,2,2,2,2,1,1,1] => [8,2,1,1,1,1] => 001111 => 0
[3,2,2,2,1,1,1,1,1] => [9,2,1,1,1] => 10111 => 0
[3,2,2,1,1,1,1,1,1,1] => [10,2,1,1] => 0011 => 0
[3,2,1,1,1,1,1,1,1,1,1] => [11,2,1] => 101 => 0
[3,1,1,1,1,1,1,1,1,1,1,1] => [12,2] => 00 => 0
[2,2,2,2,2,2,2] => [7,1,1,1,1,1,1,1] => 11111111 => 0
[2,2,2,2,2,2,1,1] => [8,1,1,1,1,1,1] => 0111111 => 0
[2,2,2,2,2,1,1,1,1] => [9,1,1,1,1,1] => 111111 => 0
[2,2,2,2,1,1,1,1,1,1] => [10,1,1,1,1] => 01111 => 0
[2,2,2,1,1,1,1,1,1,1,1] => [11,1,1,1] => 1111 => 0
[2,2,1,1,1,1,1,1,1,1,1,1] => [12,1,1] => 011 => 0
[2,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1] => 11 => 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14] => 0 => 0
[15] => [14,1] => 01 => 0
[14,1] => [13,2] => 10 => 0
[13,2] => [12,2,1] => 001 => 0
[13,1,1] => [12,3] => 01 => 0
[12,3] => [11,2,2] => 100 => 0
[12,2,1] => [11,3,1] => 111 => 0
[12,1,1,1] => [11,4] => 10 => 0
[11,4] => [10,3,2] => 010 => 0
[11,3,1] => [10,3,2] => 010 => 0
[11,2,2] => [10,3,1,1] => 0111 => 0
[11,2,1,1] => [10,4,1] => 001 => 0
[11,1,1,1,1] => [10,5] => 01 => 0
[10,5] => [9,4,2] => 100 => 0
[10,4,1] => [9,3,3] => 111 => 0
[10,3,2] => [9,3,2,1] => 1101 => 0
[10,3,1,1] => [9,4,2] => 100 => 0
[10,2,2,1] => [9,4,1,1] => 1011 => 0
[10,2,1,1,1] => [9,5,1] => 111 => 0
[10,1,1,1,1,1] => [9,6] => 10 => 0
[9,6] => [8,5,2] => 010 => 0
[9,5,1] => [8,4,3] => 001 => 0
[9,4,2] => [8,3,3,1] => 0111 => 0
[9,4,1,1] => [8,4,3] => 001 => 0
[9,3,3] => [8,3,2,2] => 0100 => 0
[9,3,2,1] => [8,4,2,1] => 0001 => 0
[9,3,1,1,1] => [8,5,2] => 010 => 0
[9,2,2,2] => [8,4,1,1,1] => 00111 => 0
[9,2,2,1,1] => [8,5,1,1] => 0111 => 0
[9,2,1,1,1,1] => [8,6,1] => 001 => 0
[9,1,1,1,1,1,1] => [8,7] => 01 => 0
[8,7] => [7,6,2] => 100 => 0
[8,6,1] => [7,5,3] => 111 => 0
[8,5,2] => [7,4,3,1] => 1011 => 0
[8,5,1,1] => [7,4,4] => 100 => 0
[8,4,3] => [7,3,3,2] => 1110 => 0
[8,4,2,1] => [7,4,3,1] => 1011 => 0
[8,4,1,1,1] => [7,5,3] => 111 => 0
[8,3,3,1] => [7,4,2,2] => 1000 => 0
[8,3,2,2] => [7,4,2,1,1] => 10011 => 0
[8,3,2,1,1] => [7,5,2,1] => 1101 => 0
[8,3,1,1,1,1] => [7,6,2] => 100 => 0
[8,2,2,2,1] => [7,5,1,1,1] => 11111 => 0
[8,2,2,1,1,1] => [7,6,1,1] => 1011 => 0
[8,2,1,1,1,1,1] => [7,7,1] => 111 => 0
[8,1,1,1,1,1,1,1] => [8,7] => 01 => 0
[7,7,1] => [6,6,3] => 001 => 0
[7,6,2] => [6,5,3,1] => 0111 => 0
[7,6,1,1] => [6,5,4] => 010 => 0
[7,5,3] => [6,4,3,2] => 0010 => 0
[7,5,2,1] => [6,4,4,1] => 0001 => 0
[7,5,1,1,1] => [6,5,4] => 010 => 0
[7,4,4] => [6,3,3,3] => 0111 => 0
[7,4,3,1] => [6,4,3,2] => 0010 => 0
[7,4,2,2] => [6,4,3,1,1] => 00111 => 0
[7,4,2,1,1] => [6,5,3,1] => 0111 => 0
[7,4,1,1,1,1] => [6,6,3] => 001 => 0
[7,3,3,2] => [6,4,2,2,1] => 00001 => 0
[7,3,3,1,1] => [6,5,2,2] => 0100 => 0
[7,3,2,2,1] => [6,5,2,1,1] => 01011 => 0
[7,3,2,1,1,1] => [6,6,2,1] => 0001 => 0
[7,3,1,1,1,1,1] => [7,6,2] => 100 => 0
[7,2,2,2,2] => [6,5,1,1,1,1] => 011111 => 0
[7,2,2,2,1,1] => [6,6,1,1,1] => 00111 => 0
[7,2,2,1,1,1,1] => [7,6,1,1] => 1011 => 0
[7,2,1,1,1,1,1,1] => [8,6,1] => 001 => 0
[7,1,1,1,1,1,1,1,1] => [9,6] => 10 => 0
[6,6,3] => [5,5,3,2] => 1110 => 0
[6,6,2,1] => [5,5,4,1] => 1101 => 0
[6,6,1,1,1] => [5,5,5] => 111 => 0
[6,5,4] => [5,4,3,3] => 1011 => 0
[6,5,3,1] => [5,4,4,2] => 1000 => 0
[6,5,2,2] => [5,4,4,1,1] => 10011 => 0
[6,5,2,1,1] => [5,5,4,1] => 1101 => 0
[6,5,1,1,1,1] => [6,5,4] => 010 => 0
[6,4,4,1] => [5,4,3,3] => 1011 => 0
[6,4,3,2] => [5,4,3,2,1] => 10101 => 0
[6,4,3,1,1] => [5,5,3,2] => 1110 => 0
[6,4,2,2,1] => [5,5,3,1,1] => 11111 => 0
[6,4,2,1,1,1] => [6,5,3,1] => 0111 => 0
[6,4,1,1,1,1,1] => [7,5,3] => 111 => 0
[6,3,3,3] => [5,4,2,2,2] => 10000 => 0
[6,3,3,2,1] => [5,5,2,2,1] => 11001 => 0
[6,3,3,1,1,1] => [6,5,2,2] => 0100 => 0
[6,3,2,2,2] => [5,5,2,1,1,1] => 110111 => 0
[6,3,2,2,1,1] => [6,5,2,1,1] => 01011 => 0
[6,3,2,1,1,1,1] => [7,5,2,1] => 1101 => 0
[6,3,1,1,1,1,1,1] => [8,5,2] => 010 => 0
[6,2,2,2,2,1] => [6,5,1,1,1,1] => 011111 => 0
[6,2,2,2,1,1,1] => [7,5,1,1,1] => 11111 => 0
[6,2,2,1,1,1,1,1] => [8,5,1,1] => 0111 => 0
[6,2,1,1,1,1,1,1,1] => [9,5,1] => 111 => 0
[6,1,1,1,1,1,1,1,1,1] => [10,5] => 01 => 0
[5,5,5] => [4,4,4,3] => 0001 => 0
[5,5,4,1] => [4,4,4,3] => 0001 => 0
[5,5,3,2] => [4,4,4,2,1] => 00001 => 0
[5,5,3,1,1] => [5,4,4,2] => 1000 => 0
[5,5,2,2,1] => [5,4,4,1,1] => 10011 => 0
[5,5,2,1,1,1] => [6,4,4,1] => 0001 => 0
[5,5,1,1,1,1,1] => [7,4,4] => 100 => 0
[5,4,4,2] => [4,4,3,3,1] => 00111 => 0
[5,4,4,1,1] => [5,4,3,3] => 1011 => 0
[5,4,3,3] => [4,4,3,2,2] => 00100 => 0
[5,4,3,2,1] => [5,4,3,2,1] => 10101 => 0
[5,4,3,1,1,1] => [6,4,3,2] => 0010 => 0
[5,4,2,2,2] => [5,4,3,1,1,1] => 101111 => 0
[5,4,2,2,1,1] => [6,4,3,1,1] => 00111 => 0
[5,4,2,1,1,1,1] => [7,4,3,1] => 1011 => 0
[5,4,1,1,1,1,1,1] => [8,4,3] => 001 => 0
[5,3,3,3,1] => [5,4,2,2,2] => 10000 => 0
[5,3,3,2,2] => [5,4,2,2,1,1] => 100011 => 0
[5,3,3,2,1,1] => [6,4,2,2,1] => 00001 => 0
[5,3,3,1,1,1,1] => [7,4,2,2] => 1000 => 0
[5,3,2,2,2,1] => [6,4,2,1,1,1] => 000111 => 0
[5,3,2,2,1,1,1] => [7,4,2,1,1] => 10011 => 0
[5,3,2,1,1,1,1,1] => [8,4,2,1] => 0001 => 0
[5,3,1,1,1,1,1,1,1] => [9,4,2] => 100 => 0
[5,2,2,2,2,2] => [6,4,1,1,1,1,1] => 0011111 => 0
[5,2,2,2,2,1,1] => [7,4,1,1,1,1] => 101111 => 0
[5,2,2,2,1,1,1,1] => [8,4,1,1,1] => 00111 => 0
[5,2,2,1,1,1,1,1,1] => [9,4,1,1] => 1011 => 0
[5,2,1,1,1,1,1,1,1,1] => [10,4,1] => 001 => 0
[5,1,1,1,1,1,1,1,1,1,1] => [11,4] => 10 => 0
[4,4,4,3] => [4,3,3,3,2] => 01110 => 0
[4,4,4,2,1] => [5,3,3,3,1] => 11111 => 0
[4,4,4,1,1,1] => [6,3,3,3] => 0111 => 0
[4,4,3,3,1] => [5,3,3,2,2] => 11100 => 0
[4,4,3,2,2] => [5,3,3,2,1,1] => 111011 => 0
[4,4,3,2,1,1] => [6,3,3,2,1] => 01101 => 0
[4,4,3,1,1,1,1] => [7,3,3,2] => 1110 => 0
[4,4,2,2,2,1] => [6,3,3,1,1,1] => 011111 => 0
[4,4,2,2,1,1,1] => [7,3,3,1,1] => 11111 => 0
[4,4,2,1,1,1,1,1] => [8,3,3,1] => 0111 => 0
[4,4,1,1,1,1,1,1,1] => [9,3,3] => 111 => 0
[4,3,3,3,2] => [5,3,2,2,2,1] => 110001 => 0
[4,3,3,3,1,1] => [6,3,2,2,2] => 01000 => 0
[4,3,3,2,2,1] => [6,3,2,2,1,1] => 010011 => 0
[4,3,3,2,1,1,1] => [7,3,2,2,1] => 11001 => 0
[4,3,3,1,1,1,1,1] => [8,3,2,2] => 0100 => 0
[4,3,2,2,2,2] => [6,3,2,1,1,1,1] => 0101111 => 0
[4,3,2,2,2,1,1] => [7,3,2,1,1,1] => 110111 => 0
[4,3,2,2,1,1,1,1] => [8,3,2,1,1] => 01011 => 0
[4,3,2,1,1,1,1,1,1] => [9,3,2,1] => 1101 => 0
[4,3,1,1,1,1,1,1,1,1] => [10,3,2] => 010 => 0
[4,2,2,2,2,2,1] => [7,3,1,1,1,1,1] => 1111111 => 0
[4,2,2,2,2,1,1,1] => [8,3,1,1,1,1] => 011111 => 0
[4,2,2,2,1,1,1,1,1] => [9,3,1,1,1] => 11111 => 0
[4,2,2,1,1,1,1,1,1,1] => [10,3,1,1] => 0111 => 0
[4,2,1,1,1,1,1,1,1,1,1] => [11,3,1] => 111 => 0
[4,1,1,1,1,1,1,1,1,1,1,1] => [12,3] => 01 => 0
[3,3,3,3,3] => [5,2,2,2,2,2] => 100000 => 0
[3,3,3,3,2,1] => [6,2,2,2,2,1] => 000001 => 0
[3,3,3,3,1,1,1] => [7,2,2,2,2] => 10000 => 0
[3,3,3,2,2,2] => [6,2,2,2,1,1,1] => 0000111 => 0
[3,3,3,2,2,1,1] => [7,2,2,2,1,1] => 100011 => 0
[3,3,3,2,1,1,1,1] => [8,2,2,2,1] => 00001 => 0
[3,3,3,1,1,1,1,1,1] => [9,2,2,2] => 1000 => 0
[3,3,2,2,2,2,1] => [7,2,2,1,1,1,1] => 1001111 => 0
[3,3,2,2,2,1,1,1] => [8,2,2,1,1,1] => 000111 => 0
[3,3,2,2,1,1,1,1,1] => [9,2,2,1,1] => 10011 => 0
[3,3,2,1,1,1,1,1,1,1] => [10,2,2,1] => 0001 => 0
[3,3,1,1,1,1,1,1,1,1,1] => [11,2,2] => 100 => 0
[3,2,2,2,2,2,2] => [7,2,1,1,1,1,1,1] => 10111111 => 0
[3,2,2,2,2,2,1,1] => [8,2,1,1,1,1,1] => 0011111 => 0
[3,2,2,2,2,1,1,1,1] => [9,2,1,1,1,1] => 101111 => 0
[3,2,2,2,1,1,1,1,1,1] => [10,2,1,1,1] => 00111 => 0
[3,2,2,1,1,1,1,1,1,1,1] => [11,2,1,1] => 1011 => 0
[3,2,1,1,1,1,1,1,1,1,1,1] => [12,2,1] => 001 => 0
[3,1,1,1,1,1,1,1,1,1,1,1,1] => [13,2] => 10 => 0
[2,2,2,2,2,2,2,1] => [8,1,1,1,1,1,1,1] => 01111111 => 0
[2,2,2,2,2,2,1,1,1] => [9,1,1,1,1,1,1] => 1111111 => 0
[2,2,2,2,2,1,1,1,1,1] => [10,1,1,1,1,1] => 011111 => 0
[2,2,2,2,1,1,1,1,1,1,1] => [11,1,1,1,1] => 11111 => 0
[2,2,2,1,1,1,1,1,1,1,1,1] => [12,1,1,1] => 0111 => 0
[2,2,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1] => 111 => 0
[2,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1] => 01 => 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15] => 1 => 0
[16] => [15,1] => 11 => 0
[15,1] => [14,2] => 00 => 0
[14,2] => [13,2,1] => 101 => 0
[14,1,1] => [13,3] => 11 => 0
[13,3] => [12,2,2] => 000 => 0
[13,2,1] => [12,3,1] => 011 => 0
[13,1,1,1] => [12,4] => 00 => 0
[12,4] => [11,3,2] => 110 => 0
[12,3,1] => [11,3,2] => 110 => 0
[12,2,2] => [11,3,1,1] => 1111 => 0
[12,2,1,1] => [11,4,1] => 101 => 0
[12,1,1,1,1] => [11,5] => 11 => 0
[11,5] => [10,4,2] => 000 => 0
[11,4,1] => [10,3,3] => 011 => 0
[11,3,2] => [10,3,2,1] => 0101 => 0
[11,3,1,1] => [10,4,2] => 000 => 0
[11,2,2,1] => [10,4,1,1] => 0011 => 0
[11,2,1,1,1] => [10,5,1] => 011 => 0
[11,1,1,1,1,1] => [10,6] => 00 => 0
[10,6] => [9,5,2] => 110 => 0
[10,5,1] => [9,4,3] => 101 => 0
[10,4,2] => [9,3,3,1] => 1111 => 0
[10,4,1,1] => [9,4,3] => 101 => 0
[10,3,3] => [9,3,2,2] => 1100 => 0
[10,3,2,1] => [9,4,2,1] => 1001 => 0
[10,3,1,1,1] => [9,5,2] => 110 => 0
[10,2,2,2] => [9,4,1,1,1] => 10111 => 0
[10,2,2,1,1] => [9,5,1,1] => 1111 => 0
[10,2,1,1,1,1] => [9,6,1] => 101 => 0
[10,1,1,1,1,1,1] => [9,7] => 11 => 0
[9,7] => [8,6,2] => 000 => 0
[9,6,1] => [8,5,3] => 011 => 0
[9,5,2] => [8,4,3,1] => 0011 => 0
[9,5,1,1] => [8,4,4] => 000 => 0
[9,4,3] => [8,3,3,2] => 0110 => 0
[9,4,2,1] => [8,4,3,1] => 0011 => 0
[9,4,1,1,1] => [8,5,3] => 011 => 0
[9,3,3,1] => [8,4,2,2] => 0000 => 0
[9,3,2,2] => [8,4,2,1,1] => 00011 => 0
[9,3,2,1,1] => [8,5,2,1] => 0101 => 0
[9,3,1,1,1,1] => [8,6,2] => 000 => 0
[9,2,2,2,1] => [8,5,1,1,1] => 01111 => 0
[9,2,2,1,1,1] => [8,6,1,1] => 0011 => 0
[9,2,1,1,1,1,1] => [8,7,1] => 011 => 0
[9,1,1,1,1,1,1,1] => [8,8] => 00 => 0
[8,8] => [7,7,2] => 110 => 0
[8,7,1] => [7,6,3] => 101 => 0
[8,6,2] => [7,5,3,1] => 1111 => 0
[8,6,1,1] => [7,5,4] => 110 => 0
[8,5,3] => [7,4,3,2] => 1010 => 0
[8,5,2,1] => [7,4,4,1] => 1001 => 0
[8,5,1,1,1] => [7,5,4] => 110 => 0
[8,4,4] => [7,3,3,3] => 1111 => 0
[8,4,3,1] => [7,4,3,2] => 1010 => 0
[8,4,2,2] => [7,4,3,1,1] => 10111 => 0
[8,4,2,1,1] => [7,5,3,1] => 1111 => 0
[8,4,1,1,1,1] => [7,6,3] => 101 => 0
[8,3,3,2] => [7,4,2,2,1] => 10001 => 0
[8,3,3,1,1] => [7,5,2,2] => 1100 => 0
[8,3,2,2,1] => [7,5,2,1,1] => 11011 => 0
[8,3,2,1,1,1] => [7,6,2,1] => 1001 => 0
[8,3,1,1,1,1,1] => [7,7,2] => 110 => 0
[8,2,2,2,2] => [7,5,1,1,1,1] => 111111 => 0
[8,2,2,2,1,1] => [7,6,1,1,1] => 10111 => 0
[8,2,2,1,1,1,1] => [7,7,1,1] => 1111 => 0
[8,2,1,1,1,1,1,1] => [8,7,1] => 011 => 0
[8,1,1,1,1,1,1,1,1] => [9,7] => 11 => 0
[7,7,2] => [6,6,3,1] => 0011 => 0
[7,7,1,1] => [6,6,4] => 000 => 0
[7,6,3] => [6,5,3,2] => 0110 => 0
[7,6,2,1] => [6,5,4,1] => 0101 => 0
[7,6,1,1,1] => [6,5,5] => 011 => 0
[7,5,4] => [6,4,3,3] => 0011 => 0
[7,5,3,1] => [6,4,4,2] => 0000 => 0
[7,5,2,2] => [6,4,4,1,1] => 00011 => 0
[7,5,2,1,1] => [6,5,4,1] => 0101 => 0
[7,5,1,1,1,1] => [6,6,4] => 000 => 0
[7,4,4,1] => [6,4,3,3] => 0011 => 0
[7,4,3,2] => [6,4,3,2,1] => 00101 => 0
[7,4,3,1,1] => [6,5,3,2] => 0110 => 0
[7,4,2,2,1] => [6,5,3,1,1] => 01111 => 0
[7,4,2,1,1,1] => [6,6,3,1] => 0011 => 0
[7,4,1,1,1,1,1] => [7,6,3] => 101 => 0
[7,3,3,3] => [6,4,2,2,2] => 00000 => 0
[7,3,3,2,1] => [6,5,2,2,1] => 01001 => 0
[7,3,3,1,1,1] => [6,6,2,2] => 0000 => 0
[7,3,2,2,2] => [6,5,2,1,1,1] => 010111 => 0
[7,3,2,2,1,1] => [6,6,2,1,1] => 00011 => 0
[7,3,2,1,1,1,1] => [7,6,2,1] => 1001 => 0
[7,3,1,1,1,1,1,1] => [8,6,2] => 000 => 0
[7,2,2,2,2,1] => [6,6,1,1,1,1] => 001111 => 0
[7,2,2,2,1,1,1] => [7,6,1,1,1] => 10111 => 0
[7,2,2,1,1,1,1,1] => [8,6,1,1] => 0011 => 0
[7,2,1,1,1,1,1,1,1] => [9,6,1] => 101 => 0
[7,1,1,1,1,1,1,1,1,1] => [10,6] => 00 => 0
[6,6,4] => [5,5,3,3] => 1111 => 0
[6,6,3,1] => [5,5,4,2] => 1100 => 0
[6,6,2,2] => [5,5,4,1,1] => 11011 => 0
[6,6,2,1,1] => [5,5,5,1] => 1111 => 0
[6,6,1,1,1,1] => [6,5,5] => 011 => 0
[6,5,5] => [5,4,4,3] => 1001 => 0
[6,5,4,1] => [5,4,4,3] => 1001 => 0
[6,5,3,2] => [5,4,4,2,1] => 10001 => 0
[6,5,3,1,1] => [5,5,4,2] => 1100 => 0
[6,5,2,2,1] => [5,5,4,1,1] => 11011 => 0
[6,5,2,1,1,1] => [6,5,4,1] => 0101 => 0
[6,5,1,1,1,1,1] => [7,5,4] => 110 => 0
[6,4,4,2] => [5,4,3,3,1] => 10111 => 0
[6,4,4,1,1] => [5,5,3,3] => 1111 => 0
[6,4,3,3] => [5,4,3,2,2] => 10100 => 0
[6,4,3,2,1] => [5,5,3,2,1] => 11101 => 0
[6,4,3,1,1,1] => [6,5,3,2] => 0110 => 0
[6,4,2,2,2] => [5,5,3,1,1,1] => 111111 => 0
[6,4,2,2,1,1] => [6,5,3,1,1] => 01111 => 0
[6,4,2,1,1,1,1] => [7,5,3,1] => 1111 => 0
[6,4,1,1,1,1,1,1] => [8,5,3] => 011 => 0
[6,3,3,3,1] => [5,5,2,2,2] => 11000 => 0
[6,3,3,2,2] => [5,5,2,2,1,1] => 110011 => 0
[6,3,3,2,1,1] => [6,5,2,2,1] => 01001 => 0
[6,3,3,1,1,1,1] => [7,5,2,2] => 1100 => 0
[6,3,2,2,2,1] => [6,5,2,1,1,1] => 010111 => 0
[6,3,2,2,1,1,1] => [7,5,2,1,1] => 11011 => 0
[6,3,2,1,1,1,1,1] => [8,5,2,1] => 0101 => 0
[6,3,1,1,1,1,1,1,1] => [9,5,2] => 110 => 0
[6,2,2,2,2,2] => [6,5,1,1,1,1,1] => 0111111 => 0
[6,2,2,2,2,1,1] => [7,5,1,1,1,1] => 111111 => 0
[6,2,2,2,1,1,1,1] => [8,5,1,1,1] => 01111 => 0
[6,2,2,1,1,1,1,1,1] => [9,5,1,1] => 1111 => 0
[6,2,1,1,1,1,1,1,1,1] => [10,5,1] => 011 => 0
[6,1,1,1,1,1,1,1,1,1,1] => [11,5] => 11 => 0
[5,5,5,1] => [4,4,4,4] => 0000 => 0
[5,5,4,2] => [4,4,4,3,1] => 00011 => 0
[5,5,4,1,1] => [5,4,4,3] => 1001 => 0
[5,5,3,3] => [4,4,4,2,2] => 00000 => 0
[5,5,3,2,1] => [5,4,4,2,1] => 10001 => 0
[5,5,3,1,1,1] => [6,4,4,2] => 0000 => 0
[5,5,2,2,2] => [5,4,4,1,1,1] => 100111 => 0
[5,5,2,2,1,1] => [6,4,4,1,1] => 00011 => 0
[5,5,2,1,1,1,1] => [7,4,4,1] => 1001 => 0
[5,5,1,1,1,1,1,1] => [8,4,4] => 000 => 0
[5,4,4,3] => [4,4,3,3,2] => 00110 => 0
[5,4,4,2,1] => [5,4,3,3,1] => 10111 => 0
[5,4,4,1,1,1] => [6,4,3,3] => 0011 => 0
[5,4,3,3,1] => [5,4,3,2,2] => 10100 => 0
[5,4,3,2,2] => [5,4,3,2,1,1] => 101011 => 0
[5,4,3,2,1,1] => [6,4,3,2,1] => 00101 => 0
[5,4,3,1,1,1,1] => [7,4,3,2] => 1010 => 0
[5,4,2,2,2,1] => [6,4,3,1,1,1] => 001111 => 0
[5,4,2,2,1,1,1] => [7,4,3,1,1] => 10111 => 0
[5,4,2,1,1,1,1,1] => [8,4,3,1] => 0011 => 0
[5,4,1,1,1,1,1,1,1] => [9,4,3] => 101 => 0
[5,3,3,3,2] => [5,4,2,2,2,1] => 100001 => 0
[5,3,3,3,1,1] => [6,4,2,2,2] => 00000 => 0
[5,3,3,2,2,1] => [6,4,2,2,1,1] => 000011 => 0
[5,3,3,2,1,1,1] => [7,4,2,2,1] => 10001 => 0
[5,3,3,1,1,1,1,1] => [8,4,2,2] => 0000 => 0
[5,3,2,2,2,2] => [6,4,2,1,1,1,1] => 0001111 => 0
[5,3,2,2,2,1,1] => [7,4,2,1,1,1] => 100111 => 0
[5,3,2,2,1,1,1,1] => [8,4,2,1,1] => 00011 => 0
[5,3,2,1,1,1,1,1,1] => [9,4,2,1] => 1001 => 0
[5,3,1,1,1,1,1,1,1,1] => [10,4,2] => 000 => 0
[5,2,2,2,2,2,1] => [7,4,1,1,1,1,1] => 1011111 => 0
[5,2,2,2,2,1,1,1] => [8,4,1,1,1,1] => 001111 => 0
[5,2,2,2,1,1,1,1,1] => [9,4,1,1,1] => 10111 => 0
[5,2,2,1,1,1,1,1,1,1] => [10,4,1,1] => 0011 => 0
[5,2,1,1,1,1,1,1,1,1,1] => [11,4,1] => 101 => 0
[5,1,1,1,1,1,1,1,1,1,1,1] => [12,4] => 00 => 0
[4,4,4,4] => [4,3,3,3,3] => 01111 => 0
[4,4,4,3,1] => [5,3,3,3,2] => 11110 => 0
[4,4,4,2,2] => [5,3,3,3,1,1] => 111111 => 0
[4,4,4,2,1,1] => [6,3,3,3,1] => 01111 => 0
[4,4,4,1,1,1,1] => [7,3,3,3] => 1111 => 0
[4,4,3,3,2] => [5,3,3,2,2,1] => 111001 => 0
[4,4,3,3,1,1] => [6,3,3,2,2] => 01100 => 0
[4,4,3,2,2,1] => [6,3,3,2,1,1] => 011011 => 0
[4,4,3,2,1,1,1] => [7,3,3,2,1] => 11101 => 0
[4,4,3,1,1,1,1,1] => [8,3,3,2] => 0110 => 0
[4,4,2,2,2,2] => [6,3,3,1,1,1,1] => 0111111 => 0
[4,4,2,2,2,1,1] => [7,3,3,1,1,1] => 111111 => 0
[4,4,2,2,1,1,1,1] => [8,3,3,1,1] => 01111 => 0
[4,4,2,1,1,1,1,1,1] => [9,3,3,1] => 1111 => 0
[4,4,1,1,1,1,1,1,1,1] => [10,3,3] => 011 => 0
[4,3,3,3,3] => [5,3,2,2,2,2] => 110000 => 0
[4,3,3,3,2,1] => [6,3,2,2,2,1] => 010001 => 0
[4,3,3,3,1,1,1] => [7,3,2,2,2] => 11000 => 0
[4,3,3,2,2,2] => [6,3,2,2,1,1,1] => 0100111 => 0
[4,3,3,2,2,1,1] => [7,3,2,2,1,1] => 110011 => 0
[4,3,3,2,1,1,1,1] => [8,3,2,2,1] => 01001 => 0
[4,3,3,1,1,1,1,1,1] => [9,3,2,2] => 1100 => 0
[4,3,2,2,2,2,1] => [7,3,2,1,1,1,1] => 1101111 => 0
[4,3,2,2,2,1,1,1] => [8,3,2,1,1,1] => 010111 => 0
[4,3,2,2,1,1,1,1,1] => [9,3,2,1,1] => 11011 => 0
[4,3,2,1,1,1,1,1,1,1] => [10,3,2,1] => 0101 => 0
[4,3,1,1,1,1,1,1,1,1,1] => [11,3,2] => 110 => 0
[4,2,2,2,2,2,2] => [7,3,1,1,1,1,1,1] => 11111111 => 0
[4,2,2,2,2,2,1,1] => [8,3,1,1,1,1,1] => 0111111 => 0
[4,2,2,2,2,1,1,1,1] => [9,3,1,1,1,1] => 111111 => 0
[4,2,2,2,1,1,1,1,1,1] => [10,3,1,1,1] => 01111 => 0
[4,2,2,1,1,1,1,1,1,1,1] => [11,3,1,1] => 1111 => 0
[4,2,1,1,1,1,1,1,1,1,1,1] => [12,3,1] => 011 => 0
[4,1,1,1,1,1,1,1,1,1,1,1,1] => [13,3] => 11 => 0
[3,3,3,3,3,1] => [6,2,2,2,2,2] => 000000 => 0
[3,3,3,3,2,2] => [6,2,2,2,2,1,1] => 0000011 => 0
[3,3,3,3,2,1,1] => [7,2,2,2,2,1] => 100001 => 0
[3,3,3,3,1,1,1,1] => [8,2,2,2,2] => 00000 => 0
[3,3,3,2,2,2,1] => [7,2,2,2,1,1,1] => 1000111 => 0
[3,3,3,2,2,1,1,1] => [8,2,2,2,1,1] => 000011 => 0
[3,3,3,2,1,1,1,1,1] => [9,2,2,2,1] => 10001 => 0
[3,3,3,1,1,1,1,1,1,1] => [10,2,2,2] => 0000 => 0
[3,3,2,2,2,2,2] => [7,2,2,1,1,1,1,1] => 10011111 => 0
[3,3,2,2,2,2,1,1] => [8,2,2,1,1,1,1] => 0001111 => 0
[3,3,2,2,2,1,1,1,1] => [9,2,2,1,1,1] => 100111 => 0
[3,3,2,2,1,1,1,1,1,1] => [10,2,2,1,1] => 00011 => 0
[3,3,2,1,1,1,1,1,1,1,1] => [11,2,2,1] => 1001 => 0
[3,3,1,1,1,1,1,1,1,1,1,1] => [12,2,2] => 000 => 0
[3,2,2,2,2,2,2,1] => [8,2,1,1,1,1,1,1] => 00111111 => 0
[3,2,2,2,2,2,1,1,1] => [9,2,1,1,1,1,1] => 1011111 => 0
[3,2,2,2,2,1,1,1,1,1] => [10,2,1,1,1,1] => 001111 => 0
[3,2,2,2,1,1,1,1,1,1,1] => [11,2,1,1,1] => 10111 => 0
[3,2,2,1,1,1,1,1,1,1,1,1] => [12,2,1,1] => 0011 => 0
[3,2,1,1,1,1,1,1,1,1,1,1,1] => [13,2,1] => 101 => 0
[3,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,2] => 00 => 0
[2,2,2,2,2,2,2,2] => [8,1,1,1,1,1,1,1,1] => 011111111 => 0
[2,2,2,2,2,2,2,1,1] => [9,1,1,1,1,1,1,1] => 11111111 => 0
[2,2,2,2,2,2,1,1,1,1] => [10,1,1,1,1,1,1] => 0111111 => 0
[2,2,2,2,2,1,1,1,1,1,1] => [11,1,1,1,1,1] => 111111 => 0
[2,2,2,2,1,1,1,1,1,1,1,1] => [12,1,1,1,1] => 01111 => 0
[2,2,2,1,1,1,1,1,1,1,1,1,1] => [13,1,1,1] => 1111 => 0
[2,2,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1,1] => 011 => 0
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,1] => 11 => 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [16] => 0 => 0
[17] => [16,1] => 01 => 0
[16,1] => [15,2] => 10 => 0
[15,2] => [14,2,1] => 001 => 0
[15,1,1] => [14,3] => 01 => 0
[14,3] => [13,2,2] => 100 => 0
[14,2,1] => [13,3,1] => 111 => 0
[14,1,1,1] => [13,4] => 10 => 0
[13,4] => [12,3,2] => 010 => 0
[13,3,1] => [12,3,2] => 010 => 0
[13,2,2] => [12,3,1,1] => 0111 => 0
[13,2,1,1] => [12,4,1] => 001 => 0
[13,1,1,1,1] => [12,5] => 01 => 0
[12,5] => [11,4,2] => 100 => 0
[12,4,1] => [11,3,3] => 111 => 0
[12,3,2] => [11,3,2,1] => 1101 => 0
[12,3,1,1] => [11,4,2] => 100 => 0
[12,2,2,1] => [11,4,1,1] => 1011 => 0
[12,2,1,1,1] => [11,5,1] => 111 => 0
[12,1,1,1,1,1] => [11,6] => 10 => 0
[11,6] => [10,5,2] => 010 => 0
[11,5,1] => [10,4,3] => 001 => 0
[11,4,2] => [10,3,3,1] => 0111 => 0
[11,4,1,1] => [10,4,3] => 001 => 0
[11,3,3] => [10,3,2,2] => 0100 => 0
[11,3,2,1] => [10,4,2,1] => 0001 => 0
[11,3,1,1,1] => [10,5,2] => 010 => 0
[11,2,2,2] => [10,4,1,1,1] => 00111 => 0
[11,2,2,1,1] => [10,5,1,1] => 0111 => 0
[11,2,1,1,1,1] => [10,6,1] => 001 => 0
[11,1,1,1,1,1,1] => [10,7] => 01 => 0
[10,7] => [9,6,2] => 100 => 0
[10,6,1] => [9,5,3] => 111 => 0
[10,5,2] => [9,4,3,1] => 1011 => 0
[10,5,1,1] => [9,4,4] => 100 => 0
[10,4,3] => [9,3,3,2] => 1110 => 0
[10,4,2,1] => [9,4,3,1] => 1011 => 0
[10,4,1,1,1] => [9,5,3] => 111 => 0
[10,3,3,1] => [9,4,2,2] => 1000 => 0
[10,3,2,2] => [9,4,2,1,1] => 10011 => 0
[10,3,2,1,1] => [9,5,2,1] => 1101 => 0
[10,3,1,1,1,1] => [9,6,2] => 100 => 0
[10,2,2,2,1] => [9,5,1,1,1] => 11111 => 0
[10,2,2,1,1,1] => [9,6,1,1] => 1011 => 0
[10,2,1,1,1,1,1] => [9,7,1] => 111 => 0
[10,1,1,1,1,1,1,1] => [9,8] => 10 => 0
[9,8] => [8,7,2] => 010 => 0
[9,7,1] => [8,6,3] => 001 => 0
[9,6,2] => [8,5,3,1] => 0111 => 0
[9,6,1,1] => [8,5,4] => 010 => 0
[9,5,3] => [8,4,3,2] => 0010 => 0
[9,5,2,1] => [8,4,4,1] => 0001 => 0
[9,5,1,1,1] => [8,5,4] => 010 => 0
[9,4,4] => [8,3,3,3] => 0111 => 0
[9,4,3,1] => [8,4,3,2] => 0010 => 0
[9,4,2,2] => [8,4,3,1,1] => 00111 => 0
[9,4,2,1,1] => [8,5,3,1] => 0111 => 0
[9,4,1,1,1,1] => [8,6,3] => 001 => 0
[9,3,3,2] => [8,4,2,2,1] => 00001 => 0
[9,3,3,1,1] => [8,5,2,2] => 0100 => 0
[9,3,2,2,1] => [8,5,2,1,1] => 01011 => 0
[9,3,2,1,1,1] => [8,6,2,1] => 0001 => 0
[9,3,1,1,1,1,1] => [8,7,2] => 010 => 0
[9,2,2,2,2] => [8,5,1,1,1,1] => 011111 => 0
[9,2,2,2,1,1] => [8,6,1,1,1] => 00111 => 0
[9,2,2,1,1,1,1] => [8,7,1,1] => 0111 => 0
[9,2,1,1,1,1,1,1] => [8,8,1] => 001 => 0
[9,1,1,1,1,1,1,1,1] => [9,8] => 10 => 0
[8,8,1] => [7,7,3] => 111 => 0
[8,7,2] => [7,6,3,1] => 1011 => 0
[8,7,1,1] => [7,6,4] => 100 => 0
[8,6,3] => [7,5,3,2] => 1110 => 0
[8,6,2,1] => [7,5,4,1] => 1101 => 0
[8,6,1,1,1] => [7,5,5] => 111 => 0
[8,5,4] => [7,4,3,3] => 1011 => 0
[8,5,3,1] => [7,4,4,2] => 1000 => 0
[8,5,2,2] => [7,4,4,1,1] => 10011 => 0
[8,5,2,1,1] => [7,5,4,1] => 1101 => 0
[8,5,1,1,1,1] => [7,6,4] => 100 => 0
[8,4,4,1] => [7,4,3,3] => 1011 => 0
[8,4,3,2] => [7,4,3,2,1] => 10101 => 0
[8,4,3,1,1] => [7,5,3,2] => 1110 => 0
[8,4,2,2,1] => [7,5,3,1,1] => 11111 => 0
[8,4,2,1,1,1] => [7,6,3,1] => 1011 => 0
[8,4,1,1,1,1,1] => [7,7,3] => 111 => 0
[8,3,3,3] => [7,4,2,2,2] => 10000 => 0
[8,3,3,2,1] => [7,5,2,2,1] => 11001 => 0
[8,3,3,1,1,1] => [7,6,2,2] => 1000 => 0
[8,3,2,2,2] => [7,5,2,1,1,1] => 110111 => 0
[8,3,2,2,1,1] => [7,6,2,1,1] => 10011 => 0
[8,3,2,1,1,1,1] => [7,7,2,1] => 1101 => 0
[8,3,1,1,1,1,1,1] => [8,7,2] => 010 => 0
[8,2,2,2,2,1] => [7,6,1,1,1,1] => 101111 => 0
[8,2,2,2,1,1,1] => [7,7,1,1,1] => 11111 => 0
[8,2,2,1,1,1,1,1] => [8,7,1,1] => 0111 => 0
[8,2,1,1,1,1,1,1,1] => [9,7,1] => 111 => 0
[8,1,1,1,1,1,1,1,1,1] => [10,7] => 01 => 0
[7,7,3] => [6,6,3,2] => 0010 => 0
[7,7,2,1] => [6,6,4,1] => 0001 => 0
[7,7,1,1,1] => [6,6,5] => 001 => 0
[7,6,4] => [6,5,3,3] => 0111 => 0
[7,6,3,1] => [6,5,4,2] => 0100 => 0
[7,6,2,2] => [6,5,4,1,1] => 01011 => 0
[7,6,2,1,1] => [6,5,5,1] => 0111 => 0
[7,6,1,1,1,1] => [6,6,5] => 001 => 0
[7,5,5] => [6,4,4,3] => 0001 => 0
[7,5,4,1] => [6,4,4,3] => 0001 => 0
[7,5,3,2] => [6,4,4,2,1] => 00001 => 0
[7,5,3,1,1] => [6,5,4,2] => 0100 => 0
[7,5,2,2,1] => [6,5,4,1,1] => 01011 => 0
[7,5,2,1,1,1] => [6,6,4,1] => 0001 => 0
[7,5,1,1,1,1,1] => [7,6,4] => 100 => 0
[7,4,4,2] => [6,4,3,3,1] => 00111 => 0
[7,4,4,1,1] => [6,5,3,3] => 0111 => 0
[7,4,3,3] => [6,4,3,2,2] => 00100 => 0
[7,4,3,2,1] => [6,5,3,2,1] => 01101 => 0
[7,4,3,1,1,1] => [6,6,3,2] => 0010 => 0
[7,4,2,2,2] => [6,5,3,1,1,1] => 011111 => 0
[7,4,2,2,1,1] => [6,6,3,1,1] => 00111 => 0
[7,4,2,1,1,1,1] => [7,6,3,1] => 1011 => 0
[7,4,1,1,1,1,1,1] => [8,6,3] => 001 => 0
[7,3,3,3,1] => [6,5,2,2,2] => 01000 => 0
[7,3,3,2,2] => [6,5,2,2,1,1] => 010011 => 0
[7,3,3,2,1,1] => [6,6,2,2,1] => 00001 => 0
[7,3,3,1,1,1,1] => [7,6,2,2] => 1000 => 0
[7,3,2,2,2,1] => [6,6,2,1,1,1] => 000111 => 0
[7,3,2,2,1,1,1] => [7,6,2,1,1] => 10011 => 0
[7,3,2,1,1,1,1,1] => [8,6,2,1] => 0001 => 0
[7,3,1,1,1,1,1,1,1] => [9,6,2] => 100 => 0
[7,2,2,2,2,2] => [6,6,1,1,1,1,1] => 0011111 => 0
[7,2,2,2,2,1,1] => [7,6,1,1,1,1] => 101111 => 0
[7,2,2,2,1,1,1,1] => [8,6,1,1,1] => 00111 => 0
[7,2,2,1,1,1,1,1,1] => [9,6,1,1] => 1011 => 0
[7,2,1,1,1,1,1,1,1,1] => [10,6,1] => 001 => 0
[7,1,1,1,1,1,1,1,1,1,1] => [11,6] => 10 => 0
[6,6,5] => [5,5,4,3] => 1101 => 0
[6,6,4,1] => [5,5,4,3] => 1101 => 0
[6,6,3,2] => [5,5,4,2,1] => 11001 => 0
[6,6,3,1,1] => [5,5,5,2] => 1110 => 0
[6,6,2,2,1] => [5,5,5,1,1] => 11111 => 0
[6,6,2,1,1,1] => [6,5,5,1] => 0111 => 0
[6,6,1,1,1,1,1] => [7,5,5] => 111 => 0
[6,5,5,1] => [5,4,4,4] => 1000 => 0
[6,5,4,2] => [5,4,4,3,1] => 10011 => 0
[6,5,4,1,1] => [5,5,4,3] => 1101 => 0
[6,5,3,3] => [5,4,4,2,2] => 10000 => 0
[6,5,3,2,1] => [5,5,4,2,1] => 11001 => 0
[6,5,3,1,1,1] => [6,5,4,2] => 0100 => 0
[6,5,2,2,2] => [5,5,4,1,1,1] => 110111 => 0
[6,5,2,2,1,1] => [6,5,4,1,1] => 01011 => 0
[6,5,2,1,1,1,1] => [7,5,4,1] => 1101 => 0
[6,5,1,1,1,1,1,1] => [8,5,4] => 010 => 0
[6,4,4,3] => [5,4,3,3,2] => 10110 => 0
[6,4,4,2,1] => [5,5,3,3,1] => 11111 => 0
[6,4,4,1,1,1] => [6,5,3,3] => 0111 => 0
[6,4,3,3,1] => [5,5,3,2,2] => 11100 => 0
[6,4,3,2,2] => [5,5,3,2,1,1] => 111011 => 0
[6,4,3,2,1,1] => [6,5,3,2,1] => 01101 => 0
[6,4,3,1,1,1,1] => [7,5,3,2] => 1110 => 0
[6,4,2,2,2,1] => [6,5,3,1,1,1] => 011111 => 0
[6,4,2,2,1,1,1] => [7,5,3,1,1] => 11111 => 0
[6,4,2,1,1,1,1,1] => [8,5,3,1] => 0111 => 0
[6,4,1,1,1,1,1,1,1] => [9,5,3] => 111 => 0
[6,3,3,3,2] => [5,5,2,2,2,1] => 110001 => 0
[6,3,3,3,1,1] => [6,5,2,2,2] => 01000 => 0
[6,3,3,2,2,1] => [6,5,2,2,1,1] => 010011 => 0
[6,3,3,2,1,1,1] => [7,5,2,2,1] => 11001 => 0
[6,3,3,1,1,1,1,1] => [8,5,2,2] => 0100 => 0
[6,3,2,2,2,2] => [6,5,2,1,1,1,1] => 0101111 => 0
[6,3,2,2,2,1,1] => [7,5,2,1,1,1] => 110111 => 0
[6,3,2,2,1,1,1,1] => [8,5,2,1,1] => 01011 => 0
[6,3,2,1,1,1,1,1,1] => [9,5,2,1] => 1101 => 0
[6,3,1,1,1,1,1,1,1,1] => [10,5,2] => 010 => 0
[6,2,2,2,2,2,1] => [7,5,1,1,1,1,1] => 1111111 => 0
[6,2,2,2,2,1,1,1] => [8,5,1,1,1,1] => 011111 => 0
[6,2,2,2,1,1,1,1,1] => [9,5,1,1,1] => 11111 => 0
[6,2,2,1,1,1,1,1,1,1] => [10,5,1,1] => 0111 => 0
[6,2,1,1,1,1,1,1,1,1,1] => [11,5,1] => 111 => 0
[6,1,1,1,1,1,1,1,1,1,1,1] => [12,5] => 01 => 0
[5,5,5,2] => [4,4,4,4,1] => 00001 => 0
[5,5,5,1,1] => [5,4,4,4] => 1000 => 0
[5,5,4,3] => [4,4,4,3,2] => 00010 => 0
[5,5,4,2,1] => [5,4,4,3,1] => 10011 => 0
[5,5,4,1,1,1] => [6,4,4,3] => 0001 => 0
[5,5,3,3,1] => [5,4,4,2,2] => 10000 => 0
[5,5,3,2,2] => [5,4,4,2,1,1] => 100011 => 0
[5,5,3,2,1,1] => [6,4,4,2,1] => 00001 => 0
[5,5,3,1,1,1,1] => [7,4,4,2] => 1000 => 0
[5,5,2,2,2,1] => [6,4,4,1,1,1] => 000111 => 0
[5,5,2,2,1,1,1] => [7,4,4,1,1] => 10011 => 0
[5,5,2,1,1,1,1,1] => [8,4,4,1] => 0001 => 0
[5,5,1,1,1,1,1,1,1] => [9,4,4] => 100 => 0
[5,4,4,4] => [4,4,3,3,3] => 00111 => 0
[5,4,4,3,1] => [5,4,3,3,2] => 10110 => 0
[5,4,4,2,2] => [5,4,3,3,1,1] => 101111 => 0
[5,4,4,2,1,1] => [6,4,3,3,1] => 00111 => 0
[5,4,4,1,1,1,1] => [7,4,3,3] => 1011 => 0
[5,4,3,3,2] => [5,4,3,2,2,1] => 101001 => 0
[5,4,3,3,1,1] => [6,4,3,2,2] => 00100 => 0
[5,4,3,2,2,1] => [6,4,3,2,1,1] => 001011 => 0
[5,4,3,2,1,1,1] => [7,4,3,2,1] => 10101 => 0
[5,4,3,1,1,1,1,1] => [8,4,3,2] => 0010 => 0
[5,4,2,2,2,2] => [6,4,3,1,1,1,1] => 0011111 => 0
[5,4,2,2,2,1,1] => [7,4,3,1,1,1] => 101111 => 0
[5,4,2,2,1,1,1,1] => [8,4,3,1,1] => 00111 => 0
[5,4,2,1,1,1,1,1,1] => [9,4,3,1] => 1011 => 0
[5,4,1,1,1,1,1,1,1,1] => [10,4,3] => 001 => 0
[5,3,3,3,3] => [5,4,2,2,2,2] => 100000 => 0
[5,3,3,3,2,1] => [6,4,2,2,2,1] => 000001 => 0
[5,3,3,3,1,1,1] => [7,4,2,2,2] => 10000 => 0
[5,3,3,2,2,2] => [6,4,2,2,1,1,1] => 0000111 => 0
[5,3,3,2,2,1,1] => [7,4,2,2,1,1] => 100011 => 0
[5,3,3,2,1,1,1,1] => [8,4,2,2,1] => 00001 => 0
[5,3,3,1,1,1,1,1,1] => [9,4,2,2] => 1000 => 0
[5,3,2,2,2,2,1] => [7,4,2,1,1,1,1] => 1001111 => 0
[5,3,2,2,2,1,1,1] => [8,4,2,1,1,1] => 000111 => 0
[5,3,2,2,1,1,1,1,1] => [9,4,2,1,1] => 10011 => 0
[5,3,2,1,1,1,1,1,1,1] => [10,4,2,1] => 0001 => 0
[5,3,1,1,1,1,1,1,1,1,1] => [11,4,2] => 100 => 0
[5,2,2,2,2,2,2] => [7,4,1,1,1,1,1,1] => 10111111 => 0
[5,2,2,2,2,2,1,1] => [8,4,1,1,1,1,1] => 0011111 => 0
[5,2,2,2,2,1,1,1,1] => [9,4,1,1,1,1] => 101111 => 0
[5,2,2,2,1,1,1,1,1,1] => [10,4,1,1,1] => 00111 => 0
[5,2,2,1,1,1,1,1,1,1,1] => [11,4,1,1] => 1011 => 0
[5,2,1,1,1,1,1,1,1,1,1,1] => [12,4,1] => 001 => 0
[5,1,1,1,1,1,1,1,1,1,1,1,1] => [13,4] => 10 => 0
[4,4,4,4,1] => [5,3,3,3,3] => 11111 => 0
[4,4,4,3,2] => [5,3,3,3,2,1] => 111101 => 0
[4,4,4,3,1,1] => [6,3,3,3,2] => 01110 => 0
[4,4,4,2,2,1] => [6,3,3,3,1,1] => 011111 => 0
[4,4,4,2,1,1,1] => [7,3,3,3,1] => 11111 => 0
[4,4,4,1,1,1,1,1] => [8,3,3,3] => 0111 => 0
[4,4,3,3,3] => [5,3,3,2,2,2] => 111000 => 0
[4,4,3,3,2,1] => [6,3,3,2,2,1] => 011001 => 0
[4,4,3,3,1,1,1] => [7,3,3,2,2] => 11100 => 0
[4,4,3,2,2,2] => [6,3,3,2,1,1,1] => 0110111 => 0
[4,4,3,2,2,1,1] => [7,3,3,2,1,1] => 111011 => 0
[4,4,3,2,1,1,1,1] => [8,3,3,2,1] => 01101 => 0
[4,4,3,1,1,1,1,1,1] => [9,3,3,2] => 1110 => 0
[4,4,2,2,2,2,1] => [7,3,3,1,1,1,1] => 1111111 => 0
[4,4,2,2,2,1,1,1] => [8,3,3,1,1,1] => 011111 => 0
[4,4,2,2,1,1,1,1,1] => [9,3,3,1,1] => 11111 => 0
[4,4,2,1,1,1,1,1,1,1] => [10,3,3,1] => 0111 => 0
[4,4,1,1,1,1,1,1,1,1,1] => [11,3,3] => 111 => 0
[4,3,3,3,3,1] => [6,3,2,2,2,2] => 010000 => 0
[4,3,3,3,2,2] => [6,3,2,2,2,1,1] => 0100011 => 0
[4,3,3,3,2,1,1] => [7,3,2,2,2,1] => 110001 => 0
[4,3,3,3,1,1,1,1] => [8,3,2,2,2] => 01000 => 0
[4,3,3,2,2,2,1] => [7,3,2,2,1,1,1] => 1100111 => 0
[4,3,3,2,2,1,1,1] => [8,3,2,2,1,1] => 010011 => 0
[4,3,3,2,1,1,1,1,1] => [9,3,2,2,1] => 11001 => 0
[4,3,3,1,1,1,1,1,1,1] => [10,3,2,2] => 0100 => 0
[4,3,2,2,2,2,2] => [7,3,2,1,1,1,1,1] => 11011111 => 0
[4,3,2,2,2,2,1,1] => [8,3,2,1,1,1,1] => 0101111 => 0
[4,3,2,2,2,1,1,1,1] => [9,3,2,1,1,1] => 110111 => 0
[4,3,2,2,1,1,1,1,1,1] => [10,3,2,1,1] => 01011 => 0
[4,3,2,1,1,1,1,1,1,1,1] => [11,3,2,1] => 1101 => 0
[4,3,1,1,1,1,1,1,1,1,1,1] => [12,3,2] => 010 => 0
[4,2,2,2,2,2,2,1] => [8,3,1,1,1,1,1,1] => 01111111 => 0
[4,2,2,2,2,2,1,1,1] => [9,3,1,1,1,1,1] => 1111111 => 0
[4,2,2,2,2,1,1,1,1,1] => [10,3,1,1,1,1] => 011111 => 0
[4,2,2,2,1,1,1,1,1,1,1] => [11,3,1,1,1] => 11111 => 0
[4,2,2,1,1,1,1,1,1,1,1,1] => [12,3,1,1] => 0111 => 0
[4,2,1,1,1,1,1,1,1,1,1,1,1] => [13,3,1] => 111 => 0
[4,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,3] => 01 => 0
[3,3,3,3,3,2] => [6,2,2,2,2,2,1] => 0000001 => 0
[3,3,3,3,3,1,1] => [7,2,2,2,2,2] => 100000 => 0
[3,3,3,3,2,2,1] => [7,2,2,2,2,1,1] => 1000011 => 0
[3,3,3,3,2,1,1,1] => [8,2,2,2,2,1] => 000001 => 0
[3,3,3,3,1,1,1,1,1] => [9,2,2,2,2] => 10000 => 0
[3,3,3,2,2,2,2] => [7,2,2,2,1,1,1,1] => 10001111 => 0
[3,3,3,2,2,2,1,1] => [8,2,2,2,1,1,1] => 0000111 => 0
[3,3,3,2,2,1,1,1,1] => [9,2,2,2,1,1] => 100011 => 0
[3,3,3,2,1,1,1,1,1,1] => [10,2,2,2,1] => 00001 => 0
[3,3,3,1,1,1,1,1,1,1,1] => [11,2,2,2] => 1000 => 0
[3,3,2,2,2,2,2,1] => [8,2,2,1,1,1,1,1] => 00011111 => 0
[3,3,2,2,2,2,1,1,1] => [9,2,2,1,1,1,1] => 1001111 => 0
[3,3,2,2,2,1,1,1,1,1] => [10,2,2,1,1,1] => 000111 => 0
[3,3,2,2,1,1,1,1,1,1,1] => [11,2,2,1,1] => 10011 => 0
[3,3,2,1,1,1,1,1,1,1,1,1] => [12,2,2,1] => 0001 => 0
[3,3,1,1,1,1,1,1,1,1,1,1,1] => [13,2,2] => 100 => 0
[3,2,2,2,2,2,2,2] => [8,2,1,1,1,1,1,1,1] => 001111111 => 0
[3,2,2,2,2,2,2,1,1] => [9,2,1,1,1,1,1,1] => 10111111 => 0
[3,2,2,2,2,2,1,1,1,1] => [10,2,1,1,1,1,1] => 0011111 => 0
[3,2,2,2,2,1,1,1,1,1,1] => [11,2,1,1,1,1] => 101111 => 0
[3,2,2,2,1,1,1,1,1,1,1,1] => [12,2,1,1,1] => 00111 => 0
[3,2,2,1,1,1,1,1,1,1,1,1,1] => [13,2,1,1] => 1011 => 0
[3,2,1,1,1,1,1,1,1,1,1,1,1,1] => [14,2,1] => 001 => 0
[3,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,2] => 10 => 0
[2,2,2,2,2,2,2,2,1] => [9,1,1,1,1,1,1,1,1] => 111111111 => 0
[2,2,2,2,2,2,2,1,1,1] => [10,1,1,1,1,1,1,1] => 01111111 => 0
[2,2,2,2,2,2,1,1,1,1,1] => [11,1,1,1,1,1,1] => 1111111 => 0
[2,2,2,2,2,1,1,1,1,1,1,1] => [12,1,1,1,1,1] => 011111 => 0
[2,2,2,2,1,1,1,1,1,1,1,1,1] => [13,1,1,1,1] => 11111 => 0
[2,2,2,1,1,1,1,1,1,1,1,1,1,1] => [14,1,1,1] => 0111 => 0
[2,2,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,1,1] => 111 => 0
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [16,1] => 01 => 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [17] => 1 => 0
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searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Generating function
$F_{1} = 1$
$F_{2} = 2$
$F_{3} = 3$
$F_{4} = 5$
$F_{5} = 7$
$F_{6} = 11$
$F_{7} = 15$
$F_{8} = 22$
$F_{9} = 30$
$F_{10} = 42$
$F_{11} = 56$
$F_{12} = 77$
$F_{13} = 101$
$F_{14} = 135$
$F_{15} = 176$
$F_{16} = 231$
$F_{17} = 297$
Description
The defect of a binary word.
The defect of a finite word $w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in $w$. The maximum possible number of palindromic factors in a word $w$ is $|w|+1$.
The defect of a finite word $w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in $w$. The maximum possible number of palindromic factors in a word $w$ is $|w|+1$.
Map
odd parts
Description
Return the binary word indicating which parts of the partition are odd.
Map
Bulgarian solitaire
Description
A move in Bulgarian solitaire.
Remove the first column of the Ferrers diagram and insert it as a new row.
If the partition is empty, return the empty partition.
Remove the first column of the Ferrers diagram and insert it as a new row.
If the partition is empty, return the empty partition.
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