Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001730: Binary words ⟶ ℤ
Values
[1,0,1,0] => [1] => 10 => 00 => 0
[1,0,1,0,1,0] => [2,1] => 1010 => 0010 => 0
[1,0,1,1,0,0] => [1,1] => 110 => 010 => 0
[1,1,0,0,1,0] => [2] => 100 => 000 => 0
[1,1,0,1,0,0] => [1] => 10 => 00 => 0
[1,0,1,0,1,0,1,0] => [3,2,1] => 101010 => 001010 => 0
[1,0,1,0,1,1,0,0] => [2,2,1] => 11010 => 01010 => 0
[1,0,1,1,0,0,1,0] => [3,1,1] => 100110 => 000110 => 0
[1,0,1,1,0,1,0,0] => [2,1,1] => 10110 => 00110 => 0
[1,0,1,1,1,0,0,0] => [1,1,1] => 1110 => 0110 => 1
[1,1,0,0,1,0,1,0] => [3,2] => 10100 => 00100 => 0
[1,1,0,0,1,1,0,0] => [2,2] => 1100 => 0100 => 0
[1,1,0,1,0,0,1,0] => [3,1] => 10010 => 00010 => 0
[1,1,0,1,0,1,0,0] => [2,1] => 1010 => 0010 => 0
[1,1,0,1,1,0,0,0] => [1,1] => 110 => 010 => 0
[1,1,1,0,0,0,1,0] => [3] => 1000 => 0000 => 0
[1,1,1,0,0,1,0,0] => [2] => 100 => 000 => 0
[1,1,1,0,1,0,0,0] => [1] => 10 => 00 => 0
[1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => 10101010 => 00101010 => 0
[1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 1101010 => 0101010 => 0
[1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 10011010 => 00011010 => 0
[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => 1011010 => 0011010 => 0
[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 111010 => 011010 => 1
[1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 10100110 => 00100110 => 0
[1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 1100110 => 0100110 => 0
[1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => 10010110 => 00010110 => 0
[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => 1010110 => 0010110 => 0
[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => 110110 => 010110 => 1
[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 10001110 => 00001110 => 0
[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => 1001110 => 0001110 => 0
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => 101110 => 001110 => 1
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 11110 => 01110 => 1
[1,1,0,0,1,0,1,0,1,0] => [4,3,2] => 1010100 => 0010100 => 0
[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 110100 => 010100 => 0
[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 1001100 => 0001100 => 0
[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => 101100 => 001100 => 0
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 11100 => 01100 => 2
[1,1,0,1,0,0,1,0,1,0] => [4,3,1] => 1010010 => 0010010 => 0
[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => 110010 => 010010 => 0
[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => 1001010 => 0001010 => 0
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => 101010 => 001010 => 0
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => 11010 => 01010 => 0
[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => 1000110 => 0000110 => 0
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => 100110 => 000110 => 0
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => 10110 => 00110 => 0
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => 1110 => 0110 => 1
[1,1,1,0,0,0,1,0,1,0] => [4,3] => 101000 => 001000 => 0
[1,1,1,0,0,0,1,1,0,0] => [3,3] => 11000 => 01000 => 0
[1,1,1,0,0,1,0,0,1,0] => [4,2] => 100100 => 000100 => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2] => 10100 => 00100 => 0
[1,1,1,0,0,1,1,0,0,0] => [2,2] => 1100 => 0100 => 0
[1,1,1,0,1,0,0,0,1,0] => [4,1] => 100010 => 000010 => 0
[1,1,1,0,1,0,0,1,0,0] => [3,1] => 10010 => 00010 => 0
[1,1,1,0,1,0,1,0,0,0] => [2,1] => 1010 => 0010 => 0
[1,1,1,0,1,1,0,0,0,0] => [1,1] => 110 => 010 => 0
[1,1,1,1,0,0,0,0,1,0] => [4] => 10000 => 00000 => 0
[1,1,1,1,0,0,0,1,0,0] => [3] => 1000 => 0000 => 0
[1,1,1,1,0,0,1,0,0,0] => [2] => 100 => 000 => 0
[1,1,1,1,0,1,0,0,0,0] => [1] => 10 => 00 => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [4,4,3,2,1] => 110101010 => 010101010 => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [4,3,3,2,1] => 101101010 => 001101010 => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 11101010 => 01101010 => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => 110011010 => 010011010 => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [4,3,2,2,1] => 101011010 => 001011010 => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [3,3,2,2,1] => 11011010 => 01011010 => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => 100111010 => 000111010 => 0
[1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => 10111010 => 00111010 => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => 1111010 => 0111010 => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [4,4,3,1,1] => 110100110 => 010100110 => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [4,3,3,1,1] => 101100110 => 001100110 => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => 11100110 => 01100110 => 3
[1,0,1,1,0,1,0,0,1,1,0,0] => [4,4,2,1,1] => 110010110 => 010010110 => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,1] => 101010110 => 001010110 => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [3,3,2,1,1] => 11010110 => 01010110 => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [4,2,2,1,1] => 100110110 => 000110110 => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [3,2,2,1,1] => 10110110 => 00110110 => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => 1110110 => 0110110 => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 110001110 => 010001110 => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [4,3,1,1,1] => 101001110 => 001001110 => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,3,1,1,1] => 11001110 => 01001110 => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [4,2,1,1,1] => 100101110 => 000101110 => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [3,2,1,1,1] => 10101110 => 00101110 => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => 1101110 => 0101110 => 1
[1,0,1,1,1,1,0,0,0,1,0,0] => [4,1,1,1,1] => 100011110 => 000011110 => 0
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => 10011110 => 00011110 => 1
[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => 1011110 => 0011110 => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => 111110 => 011110 => 1
[1,1,0,0,1,0,1,0,1,0,1,0] => [5,4,3,2] => 101010100 => 001010100 => 0
[1,1,0,0,1,0,1,0,1,1,0,0] => [4,4,3,2] => 11010100 => 01010100 => 0
[1,1,0,0,1,0,1,1,0,0,1,0] => [5,3,3,2] => 100110100 => 000110100 => 0
[1,1,0,0,1,0,1,1,0,1,0,0] => [4,3,3,2] => 10110100 => 00110100 => 0
[1,1,0,0,1,0,1,1,1,0,0,0] => [3,3,3,2] => 1110100 => 0110100 => 2
[1,1,0,0,1,1,0,0,1,0,1,0] => [5,4,2,2] => 101001100 => 001001100 => 0
[1,1,0,0,1,1,0,0,1,1,0,0] => [4,4,2,2] => 11001100 => 01001100 => 0
[1,1,0,0,1,1,0,1,0,0,1,0] => [5,3,2,2] => 100101100 => 000101100 => 0
[1,1,0,0,1,1,0,1,0,1,0,0] => [4,3,2,2] => 10101100 => 00101100 => 0
[1,1,0,0,1,1,0,1,1,0,0,0] => [3,3,2,2] => 1101100 => 0101100 => 2
[1,1,0,0,1,1,1,0,0,0,1,0] => [5,2,2,2] => 100011100 => 000011100 => 0
[1,1,0,0,1,1,1,0,0,1,0,0] => [4,2,2,2] => 10011100 => 00011100 => 0
[1,1,0,0,1,1,1,0,1,0,0,0] => [3,2,2,2] => 1011100 => 0011100 => 2
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => 111100 => 011100 => 1
>>> Load all 860 entries. <<<[1,1,0,1,0,0,1,0,1,0,1,0] => [5,4,3,1] => 101010010 => 001010010 => 0
[1,1,0,1,0,0,1,0,1,1,0,0] => [4,4,3,1] => 11010010 => 01010010 => 0
[1,1,0,1,0,0,1,1,0,0,1,0] => [5,3,3,1] => 100110010 => 000110010 => 0
[1,1,0,1,0,0,1,1,0,1,0,0] => [4,3,3,1] => 10110010 => 00110010 => 0
[1,1,0,1,0,0,1,1,1,0,0,0] => [3,3,3,1] => 1110010 => 0110010 => 2
[1,1,0,1,0,1,0,0,1,0,1,0] => [5,4,2,1] => 101001010 => 001001010 => 0
[1,1,0,1,0,1,0,0,1,1,0,0] => [4,4,2,1] => 11001010 => 01001010 => 0
[1,1,0,1,0,1,0,1,0,0,1,0] => [5,3,2,1] => 100101010 => 000101010 => 0
[1,1,0,1,0,1,0,1,0,1,0,0] => [4,3,2,1] => 10101010 => 00101010 => 0
[1,1,0,1,0,1,0,1,1,0,0,0] => [3,3,2,1] => 1101010 => 0101010 => 0
[1,1,0,1,0,1,1,0,0,0,1,0] => [5,2,2,1] => 100011010 => 000011010 => 0
[1,1,0,1,0,1,1,0,0,1,0,0] => [4,2,2,1] => 10011010 => 00011010 => 0
[1,1,0,1,0,1,1,0,1,0,0,0] => [3,2,2,1] => 1011010 => 0011010 => 0
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,2,2,1] => 111010 => 011010 => 1
[1,1,0,1,1,0,0,0,1,0,1,0] => [5,4,1,1] => 101000110 => 001000110 => 0
[1,1,0,1,1,0,0,0,1,1,0,0] => [4,4,1,1] => 11000110 => 01000110 => 0
[1,1,0,1,1,0,0,1,0,0,1,0] => [5,3,1,1] => 100100110 => 000100110 => 0
[1,1,0,1,1,0,0,1,0,1,0,0] => [4,3,1,1] => 10100110 => 00100110 => 0
[1,1,0,1,1,0,0,1,1,0,0,0] => [3,3,1,1] => 1100110 => 0100110 => 0
[1,1,0,1,1,0,1,0,0,0,1,0] => [5,2,1,1] => 100010110 => 000010110 => 0
[1,1,0,1,1,0,1,0,0,1,0,0] => [4,2,1,1] => 10010110 => 00010110 => 0
[1,1,0,1,1,0,1,0,1,0,0,0] => [3,2,1,1] => 1010110 => 0010110 => 0
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => 110110 => 010110 => 1
[1,1,0,1,1,1,0,0,0,0,1,0] => [5,1,1,1] => 100001110 => 000001110 => 0
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => 10001110 => 00001110 => 0
[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => 1001110 => 0001110 => 0
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => 101110 => 001110 => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => 11110 => 01110 => 1
[1,1,1,0,0,0,1,0,1,0,1,0] => [5,4,3] => 10101000 => 00101000 => 0
[1,1,1,0,0,0,1,0,1,1,0,0] => [4,4,3] => 1101000 => 0101000 => 0
[1,1,1,0,0,0,1,1,0,0,1,0] => [5,3,3] => 10011000 => 00011000 => 0
[1,1,1,0,0,0,1,1,0,1,0,0] => [4,3,3] => 1011000 => 0011000 => 0
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => 111000 => 011000 => 2
[1,1,1,0,0,1,0,0,1,0,1,0] => [5,4,2] => 10100100 => 00100100 => 0
[1,1,1,0,0,1,0,0,1,1,0,0] => [4,4,2] => 1100100 => 0100100 => 0
[1,1,1,0,0,1,0,1,0,0,1,0] => [5,3,2] => 10010100 => 00010100 => 0
[1,1,1,0,0,1,0,1,0,1,0,0] => [4,3,2] => 1010100 => 0010100 => 0
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,3,2] => 110100 => 010100 => 0
[1,1,1,0,0,1,1,0,0,0,1,0] => [5,2,2] => 10001100 => 00001100 => 0
[1,1,1,0,0,1,1,0,0,1,0,0] => [4,2,2] => 1001100 => 0001100 => 0
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => 101100 => 001100 => 0
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => 11100 => 01100 => 2
[1,1,1,0,1,0,0,0,1,0,1,0] => [5,4,1] => 10100010 => 00100010 => 0
[1,1,1,0,1,0,0,0,1,1,0,0] => [4,4,1] => 1100010 => 0100010 => 0
[1,1,1,0,1,0,0,1,0,0,1,0] => [5,3,1] => 10010010 => 00010010 => 0
[1,1,1,0,1,0,0,1,0,1,0,0] => [4,3,1] => 1010010 => 0010010 => 0
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => 110010 => 010010 => 0
[1,1,1,0,1,0,1,0,0,0,1,0] => [5,2,1] => 10001010 => 00001010 => 0
[1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => 1001010 => 0001010 => 0
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => 101010 => 001010 => 0
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => 11010 => 01010 => 0
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => 10000110 => 00000110 => 0
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => 1000110 => 0000110 => 0
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => 100110 => 000110 => 0
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => 10110 => 00110 => 0
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => 1110 => 0110 => 1
[1,1,1,1,0,0,0,0,1,0,1,0] => [5,4] => 1010000 => 0010000 => 0
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => 110000 => 010000 => 0
[1,1,1,1,0,0,0,1,0,0,1,0] => [5,3] => 1001000 => 0001000 => 0
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => 101000 => 001000 => 0
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => 11000 => 01000 => 0
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => 1000100 => 0000100 => 0
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => 100100 => 000100 => 0
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => 10100 => 00100 => 0
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => 1100 => 0100 => 0
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => 1000010 => 0000010 => 0
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => 100010 => 000010 => 0
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => 10010 => 00010 => 0
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => 1010 => 0010 => 0
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => 110 => 010 => 0
[1,1,1,1,1,0,0,0,0,0,1,0] => [5] => 100000 => 000000 => 0
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => 10000 => 00000 => 0
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => 1000 => 0000 => 0
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => 100 => 000 => 0
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => 10 => 00 => 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [3,3,3,3,2,1] => 111101010 => 011101010 => 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0] => [3,3,3,2,2,1] => 111011010 => 011011010 => 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [3,3,2,2,2,1] => 110111010 => 010111010 => 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,2,2,2,1] => 101111010 => 001111010 => 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2,1] => 11111010 => 01111010 => 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [3,3,3,3,1,1] => 111100110 => 011100110 => 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0] => [3,3,3,2,1,1] => 111010110 => 011010110 => 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0] => [3,3,2,2,1,1] => 110110110 => 010110110 => 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0] => [3,2,2,2,1,1] => 101110110 => 001110110 => 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [2,2,2,2,1,1] => 11110110 => 01110110 => 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0] => [3,3,3,1,1,1] => 111001110 => 011001110 => 3
[1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [3,3,2,1,1,1] => 110101110 => 010101110 => 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1,1,1] => 101101110 => 001101110 => 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1,1,1] => 11101110 => 01101110 => 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [3,3,1,1,1,1] => 110011110 => 010011110 => 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1,1,1] => 101011110 => 001011110 => 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1,1,1] => 11011110 => 01011110 => 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1,1,1] => 100111110 => 000111110 => 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1,1] => 10111110 => 00111110 => 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => 1111110 => 0111110 => 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [4,4,4,3,2] => 111010100 => 011010100 => 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0] => [4,4,3,3,2] => 110110100 => 010110100 => 2
[1,1,0,0,1,0,1,1,1,0,1,0,0,0] => [4,3,3,3,2] => 101110100 => 001110100 => 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [3,3,3,3,2] => 11110100 => 01110100 => 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [4,4,4,2,2] => 111001100 => 011001100 => 4
[1,1,0,0,1,1,0,1,0,1,1,0,0,0] => [4,4,3,2,2] => 110101100 => 010101100 => 2
[1,1,0,0,1,1,0,1,1,0,1,0,0,0] => [4,3,3,2,2] => 101101100 => 001101100 => 2
[1,1,0,0,1,1,0,1,1,1,0,0,0,0] => [3,3,3,2,2] => 11101100 => 01101100 => 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0] => [4,4,2,2,2] => 110011100 => 010011100 => 2
[1,1,0,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,2,2] => 101011100 => 001011100 => 2
[1,1,0,0,1,1,1,0,1,1,0,0,0,0] => [3,3,2,2,2] => 11011100 => 01011100 => 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0] => [4,2,2,2,2] => 100111100 => 000111100 => 2
[1,1,0,0,1,1,1,1,0,1,0,0,0,0] => [3,2,2,2,2] => 10111100 => 00111100 => 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2] => 1111100 => 0111100 => 1
[1,1,0,1,0,0,1,0,1,1,1,0,0,0] => [4,4,4,3,1] => 111010010 => 011010010 => 2
[1,1,0,1,0,0,1,1,0,1,1,0,0,0] => [4,4,3,3,1] => 110110010 => 010110010 => 2
[1,1,0,1,0,0,1,1,1,0,1,0,0,0] => [4,3,3,3,1] => 101110010 => 001110010 => 2
[1,1,0,1,0,0,1,1,1,1,0,0,0,0] => [3,3,3,3,1] => 11110010 => 01110010 => 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0] => [4,4,4,2,1] => 111001010 => 011001010 => 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [4,4,3,2,1] => 110101010 => 010101010 => 0
[1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [4,3,3,2,1] => 101101010 => 001101010 => 0
[1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [3,3,3,2,1] => 11101010 => 01101010 => 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [4,4,2,2,1] => 110011010 => 010011010 => 0
[1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [4,3,2,2,1] => 101011010 => 001011010 => 0
[1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [3,3,2,2,1] => 11011010 => 01011010 => 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [4,2,2,2,1] => 100111010 => 000111010 => 0
[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [3,2,2,2,1] => 10111010 => 00111010 => 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [2,2,2,2,1] => 1111010 => 0111010 => 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,0] => [4,4,4,1,1] => 111000110 => 011000110 => 2
[1,1,0,1,1,0,0,1,0,1,1,0,0,0] => [4,4,3,1,1] => 110100110 => 010100110 => 0
[1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [4,3,3,1,1] => 101100110 => 001100110 => 0
[1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [3,3,3,1,1] => 11100110 => 01100110 => 3
[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [4,4,2,1,1] => 110010110 => 010010110 => 0
[1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1,1] => 101010110 => 001010110 => 0
[1,1,0,1,1,0,1,0,1,1,0,0,0,0] => [3,3,2,1,1] => 11010110 => 01010110 => 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [4,2,2,1,1] => 100110110 => 000110110 => 0
[1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1,1] => 10110110 => 00110110 => 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1,1] => 1110110 => 0110110 => 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [4,4,1,1,1] => 110001110 => 010001110 => 0
[1,1,0,1,1,1,0,0,1,0,1,0,0,0] => [4,3,1,1,1] => 101001110 => 001001110 => 0
[1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [3,3,1,1,1] => 11001110 => 01001110 => 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [4,2,1,1,1] => 100101110 => 000101110 => 0
[1,1,0,1,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1,1] => 10101110 => 00101110 => 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1,1] => 1101110 => 0101110 => 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1,1] => 100011110 => 000011110 => 0
[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1,1] => 10011110 => 00011110 => 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1] => 1011110 => 0011110 => 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1] => 111110 => 011110 => 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [5,5,4,3] => 110101000 => 010101000 => 0
[1,1,1,0,0,0,1,0,1,1,0,1,0,0] => [5,4,4,3] => 101101000 => 001101000 => 0
[1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [4,4,4,3] => 11101000 => 01101000 => 2
[1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [5,5,3,3] => 110011000 => 010011000 => 0
[1,1,1,0,0,0,1,1,0,1,0,1,0,0] => [5,4,3,3] => 101011000 => 001011000 => 0
[1,1,1,0,0,0,1,1,0,1,1,0,0,0] => [4,4,3,3] => 11011000 => 01011000 => 2
[1,1,1,0,0,0,1,1,1,0,0,1,0,0] => [5,3,3,3] => 100111000 => 000111000 => 0
[1,1,1,0,0,0,1,1,1,0,1,0,0,0] => [4,3,3,3] => 10111000 => 00111000 => 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [3,3,3,3] => 1111000 => 0111000 => 2
[1,1,1,0,0,1,0,0,1,0,1,1,0,0] => [5,5,4,2] => 110100100 => 010100100 => 0
[1,1,1,0,0,1,0,0,1,1,0,1,0,0] => [5,4,4,2] => 101100100 => 001100100 => 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,0] => [4,4,4,2] => 11100100 => 01100100 => 2
[1,1,1,0,0,1,0,1,0,0,1,1,0,0] => [5,5,3,2] => 110010100 => 010010100 => 0
[1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [5,4,3,2] => 101010100 => 001010100 => 0
[1,1,1,0,0,1,0,1,0,1,1,0,0,0] => [4,4,3,2] => 11010100 => 01010100 => 0
[1,1,1,0,0,1,0,1,1,0,0,1,0,0] => [5,3,3,2] => 100110100 => 000110100 => 0
[1,1,1,0,0,1,0,1,1,0,1,0,0,0] => [4,3,3,2] => 10110100 => 00110100 => 0
[1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [3,3,3,2] => 1110100 => 0110100 => 2
[1,1,1,0,0,1,1,0,0,0,1,1,0,0] => [5,5,2,2] => 110001100 => 010001100 => 0
[1,1,1,0,0,1,1,0,0,1,0,1,0,0] => [5,4,2,2] => 101001100 => 001001100 => 0
[1,1,1,0,0,1,1,0,0,1,1,0,0,0] => [4,4,2,2] => 11001100 => 01001100 => 0
[1,1,1,0,0,1,1,0,1,0,0,1,0,0] => [5,3,2,2] => 100101100 => 000101100 => 0
[1,1,1,0,0,1,1,0,1,0,1,0,0,0] => [4,3,2,2] => 10101100 => 00101100 => 0
[1,1,1,0,0,1,1,0,1,1,0,0,0,0] => [3,3,2,2] => 1101100 => 0101100 => 2
[1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [5,2,2,2] => 100011100 => 000011100 => 0
[1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [4,2,2,2] => 10011100 => 00011100 => 0
[1,1,1,0,0,1,1,1,0,1,0,0,0,0] => [3,2,2,2] => 1011100 => 0011100 => 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [2,2,2,2] => 111100 => 011100 => 1
[1,1,1,0,1,0,0,0,1,0,1,1,0,0] => [5,5,4,1] => 110100010 => 010100010 => 0
[1,1,1,0,1,0,0,0,1,1,0,1,0,0] => [5,4,4,1] => 101100010 => 001100010 => 0
[1,1,1,0,1,0,0,0,1,1,1,0,0,0] => [4,4,4,1] => 11100010 => 01100010 => 2
[1,1,1,0,1,0,0,1,0,0,1,1,0,0] => [5,5,3,1] => 110010010 => 010010010 => 0
[1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [5,4,3,1] => 101010010 => 001010010 => 0
[1,1,1,0,1,0,0,1,0,1,1,0,0,0] => [4,4,3,1] => 11010010 => 01010010 => 0
[1,1,1,0,1,0,0,1,1,0,0,1,0,0] => [5,3,3,1] => 100110010 => 000110010 => 0
[1,1,1,0,1,0,0,1,1,0,1,0,0,0] => [4,3,3,1] => 10110010 => 00110010 => 0
[1,1,1,0,1,0,0,1,1,1,0,0,0,0] => [3,3,3,1] => 1110010 => 0110010 => 2
[1,1,1,0,1,0,1,0,0,0,1,1,0,0] => [5,5,2,1] => 110001010 => 010001010 => 0
[1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [5,4,2,1] => 101001010 => 001001010 => 0
[1,1,1,0,1,0,1,0,0,1,1,0,0,0] => [4,4,2,1] => 11001010 => 01001010 => 0
[1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [5,3,2,1] => 100101010 => 000101010 => 0
[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1] => 10101010 => 00101010 => 0
[1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [3,3,2,1] => 1101010 => 0101010 => 0
[1,1,1,0,1,0,1,1,0,0,0,1,0,0] => [5,2,2,1] => 100011010 => 000011010 => 0
[1,1,1,0,1,0,1,1,0,0,1,0,0,0] => [4,2,2,1] => 10011010 => 00011010 => 0
[1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1] => 1011010 => 0011010 => 0
[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1] => 111010 => 011010 => 1
[1,1,1,0,1,1,0,0,0,0,1,1,0,0] => [5,5,1,1] => 110000110 => 010000110 => 0
[1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [5,4,1,1] => 101000110 => 001000110 => 0
[1,1,1,0,1,1,0,0,0,1,1,0,0,0] => [4,4,1,1] => 11000110 => 01000110 => 0
[1,1,1,0,1,1,0,0,1,0,0,1,0,0] => [5,3,1,1] => 100100110 => 000100110 => 0
[1,1,1,0,1,1,0,0,1,0,1,0,0,0] => [4,3,1,1] => 10100110 => 00100110 => 0
[1,1,1,0,1,1,0,0,1,1,0,0,0,0] => [3,3,1,1] => 1100110 => 0100110 => 0
[1,1,1,0,1,1,0,1,0,0,0,1,0,0] => [5,2,1,1] => 100010110 => 000010110 => 0
[1,1,1,0,1,1,0,1,0,0,1,0,0,0] => [4,2,1,1] => 10010110 => 00010110 => 0
[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1] => 1010110 => 0010110 => 0
[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1] => 110110 => 010110 => 1
[1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [5,1,1,1] => 100001110 => 000001110 => 0
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1] => 10001110 => 00001110 => 0
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1] => 1001110 => 0001110 => 0
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1] => 101110 => 001110 => 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => 11110 => 01110 => 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [6,5,4] => 101010000 => 001010000 => 0
[1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [5,5,4] => 11010000 => 01010000 => 0
[1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [6,4,4] => 100110000 => 000110000 => 0
[1,1,1,1,0,0,0,0,1,1,0,1,0,0] => [5,4,4] => 10110000 => 00110000 => 0
[1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [4,4,4] => 1110000 => 0110000 => 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0] => [6,5,3] => 101001000 => 001001000 => 0
[1,1,1,1,0,0,0,1,0,0,1,1,0,0] => [5,5,3] => 11001000 => 01001000 => 0
[1,1,1,1,0,0,0,1,0,1,0,0,1,0] => [6,4,3] => 100101000 => 000101000 => 0
[1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [5,4,3] => 10101000 => 00101000 => 0
[1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [4,4,3] => 1101000 => 0101000 => 0
[1,1,1,1,0,0,0,1,1,0,0,0,1,0] => [6,3,3] => 100011000 => 000011000 => 0
[1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [5,3,3] => 10011000 => 00011000 => 0
[1,1,1,1,0,0,0,1,1,0,1,0,0,0] => [4,3,3] => 1011000 => 0011000 => 0
[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [3,3,3] => 111000 => 011000 => 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0] => [6,5,2] => 101000100 => 001000100 => 0
[1,1,1,1,0,0,1,0,0,0,1,1,0,0] => [5,5,2] => 11000100 => 01000100 => 0
[1,1,1,1,0,0,1,0,0,1,0,0,1,0] => [6,4,2] => 100100100 => 000100100 => 0
[1,1,1,1,0,0,1,0,0,1,0,1,0,0] => [5,4,2] => 10100100 => 00100100 => 0
[1,1,1,1,0,0,1,0,0,1,1,0,0,0] => [4,4,2] => 1100100 => 0100100 => 0
[1,1,1,1,0,0,1,0,1,0,0,0,1,0] => [6,3,2] => 100010100 => 000010100 => 0
[1,1,1,1,0,0,1,0,1,0,0,1,0,0] => [5,3,2] => 10010100 => 00010100 => 0
[1,1,1,1,0,0,1,0,1,0,1,0,0,0] => [4,3,2] => 1010100 => 0010100 => 0
[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [3,3,2] => 110100 => 010100 => 0
[1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [6,2,2] => 100001100 => 000001100 => 0
[1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [5,2,2] => 10001100 => 00001100 => 0
[1,1,1,1,0,0,1,1,0,0,1,0,0,0] => [4,2,2] => 1001100 => 0001100 => 0
[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [3,2,2] => 101100 => 001100 => 0
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => 11100 => 01100 => 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0] => [6,5,1] => 101000010 => 001000010 => 0
[1,1,1,1,0,1,0,0,0,0,1,1,0,0] => [5,5,1] => 11000010 => 01000010 => 0
[1,1,1,1,0,1,0,0,0,1,0,0,1,0] => [6,4,1] => 100100010 => 000100010 => 0
[1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [5,4,1] => 10100010 => 00100010 => 0
[1,1,1,1,0,1,0,0,0,1,1,0,0,0] => [4,4,1] => 1100010 => 0100010 => 0
[1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [6,3,1] => 100010010 => 000010010 => 0
[1,1,1,1,0,1,0,0,1,0,0,1,0,0] => [5,3,1] => 10010010 => 00010010 => 0
[1,1,1,1,0,1,0,0,1,0,1,0,0,0] => [4,3,1] => 1010010 => 0010010 => 0
[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [3,3,1] => 110010 => 010010 => 0
[1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [6,2,1] => 100001010 => 000001010 => 0
[1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [5,2,1] => 10001010 => 00001010 => 0
[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [4,2,1] => 1001010 => 0001010 => 0
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [3,2,1] => 101010 => 001010 => 0
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => 11010 => 01010 => 0
[1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [6,1,1] => 100000110 => 000000110 => 0
[1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [5,1,1] => 10000110 => 00000110 => 0
[1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [4,1,1] => 1000110 => 0000110 => 0
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [3,1,1] => 100110 => 000110 => 0
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [2,1,1] => 10110 => 00110 => 0
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => 1110 => 0110 => 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [6,5] => 10100000 => 00100000 => 0
[1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [5,5] => 1100000 => 0100000 => 0
[1,1,1,1,1,0,0,0,0,1,0,0,1,0] => [6,4] => 10010000 => 00010000 => 0
[1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [5,4] => 1010000 => 0010000 => 0
[1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [4,4] => 110000 => 010000 => 0
[1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [6,3] => 10001000 => 00001000 => 0
[1,1,1,1,1,0,0,0,1,0,0,1,0,0] => [5,3] => 1001000 => 0001000 => 0
[1,1,1,1,1,0,0,0,1,0,1,0,0,0] => [4,3] => 101000 => 001000 => 0
[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [3,3] => 11000 => 01000 => 0
[1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [6,2] => 10000100 => 00000100 => 0
[1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [5,2] => 1000100 => 0000100 => 0
[1,1,1,1,1,0,0,1,0,0,1,0,0,0] => [4,2] => 100100 => 000100 => 0
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [3,2] => 10100 => 00100 => 0
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => 1100 => 0100 => 0
[1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [6,1] => 10000010 => 00000010 => 0
[1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [5,1] => 1000010 => 0000010 => 0
[1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [4,1] => 100010 => 000010 => 0
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => 10010 => 00010 => 0
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => 1010 => 0010 => 0
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => 110 => 010 => 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [6] => 1000000 => 0000000 => 0
[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [5] => 100000 => 000000 => 0
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [4] => 10000 => 00000 => 0
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => 1000 => 0000 => 0
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => 100 => 000 => 0
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1] => 10 => 00 => 0
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2,2,1] => 111111010 => 011111010 => 1
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2,1,1] => 111110110 => 011110110 => 1
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,1,1,1] => 111101110 => 011101110 => 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1,1,1,1] => 111011110 => 011011110 => 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1,1,1,1] => 110111110 => 010111110 => 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1,1,1] => 101111110 => 001111110 => 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => 11111110 => 01111110 => 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0] => [3,3,3,3,3,2] => 111110100 => 011110100 => 1
[1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0] => [3,3,3,3,2,2] => 111101100 => 011101100 => 1
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0] => [3,3,3,2,2,2] => 111011100 => 011011100 => 1
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2,2,2] => 110111100 => 010111100 => 1
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,2,2] => 101111100 => 001111100 => 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2,2] => 11111100 => 01111100 => 1
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0] => [3,3,3,3,3,1] => 111110010 => 011110010 => 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0] => [3,3,3,3,2,1] => 111101010 => 011101010 => 1
[1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0] => [3,3,3,2,2,1] => 111011010 => 011011010 => 1
[1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2,2,1] => 110111010 => 010111010 => 1
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,2,1] => 101111010 => 001111010 => 1
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2,1] => 11111010 => 01111010 => 1
[1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0] => [3,3,3,3,1,1] => 111100110 => 011100110 => 1
[1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0] => [3,3,3,2,1,1] => 111010110 => 011010110 => 1
[1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2,1,1] => 110110110 => 010110110 => 1
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,1,1] => 101110110 => 001110110 => 1
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,1,1] => 11110110 => 01110110 => 1
[1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0] => [3,3,3,1,1,1] => 111001110 => 011001110 => 3
[1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0] => [3,3,2,1,1,1] => 110101110 => 010101110 => 1
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0] => [3,2,2,1,1,1] => 101101110 => 001101110 => 1
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1,1,1] => 11101110 => 01101110 => 1
[1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0] => [3,3,1,1,1,1] => 110011110 => 010011110 => 1
[1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1,1,1] => 101011110 => 001011110 => 1
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1,1,1] => 11011110 => 01011110 => 1
[1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1,1,1] => 100111110 => 000111110 => 1
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1,1] => 10111110 => 00111110 => 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1] => 1111110 => 0111110 => 1
[1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0] => [4,4,3,3,3] => 110111000 => 010111000 => 2
[1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0] => [4,3,3,3,3] => 101111000 => 001111000 => 2
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0] => [3,3,3,3,3] => 11111000 => 01111000 => 1
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0] => [4,4,4,4,2] => 111100100 => 011100100 => 2
[1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0] => [4,4,4,3,2] => 111010100 => 011010100 => 2
[1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0] => [4,4,3,3,2] => 110110100 => 010110100 => 2
[1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0] => [4,3,3,3,2] => 101110100 => 001110100 => 2
[1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0] => [3,3,3,3,2] => 11110100 => 01110100 => 1
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0] => [4,4,4,2,2] => 111001100 => 011001100 => 4
[1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0] => [4,4,3,2,2] => 110101100 => 010101100 => 2
[1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0] => [4,3,3,2,2] => 101101100 => 001101100 => 2
[1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0] => [3,3,3,2,2] => 11101100 => 01101100 => 1
[1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0] => [4,4,2,2,2] => 110011100 => 010011100 => 2
[1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0] => [4,3,2,2,2] => 101011100 => 001011100 => 2
[1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2,2] => 11011100 => 01011100 => 1
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0] => [4,2,2,2,2] => 100111100 => 000111100 => 2
[1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,2] => 10111100 => 00111100 => 1
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2] => 1111100 => 0111100 => 1
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0] => [4,4,4,4,1] => 111100010 => 011100010 => 2
[1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0] => [4,4,4,3,1] => 111010010 => 011010010 => 2
[1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0] => [4,4,3,3,1] => 110110010 => 010110010 => 2
[1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0] => [4,3,3,3,1] => 101110010 => 001110010 => 2
[1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0] => [3,3,3,3,1] => 11110010 => 01110010 => 1
[1,1,1,0,1,0,1,0,0,1,1,1,0,0,0,0] => [4,4,4,2,1] => 111001010 => 011001010 => 2
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0] => [4,4,3,2,1] => 110101010 => 010101010 => 0
[1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0] => [4,3,3,2,1] => 101101010 => 001101010 => 0
[1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0] => [3,3,3,2,1] => 11101010 => 01101010 => 1
[1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0] => [4,4,2,2,1] => 110011010 => 010011010 => 0
[1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0] => [4,3,2,2,1] => 101011010 => 001011010 => 0
[1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2,1] => 11011010 => 01011010 => 1
[1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0] => [4,2,2,2,1] => 100111010 => 000111010 => 0
[1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,1] => 10111010 => 00111010 => 1
[1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,1] => 1111010 => 0111010 => 1
[1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0] => [4,4,4,1,1] => 111000110 => 011000110 => 2
[1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0] => [4,4,3,1,1] => 110100110 => 010100110 => 0
[1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0] => [4,3,3,1,1] => 101100110 => 001100110 => 0
[1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0] => [3,3,3,1,1] => 11100110 => 01100110 => 3
[1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0] => [4,4,2,1,1] => 110010110 => 010010110 => 0
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0] => [4,3,2,1,1] => 101010110 => 001010110 => 0
[1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0] => [3,3,2,1,1] => 11010110 => 01010110 => 1
[1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0] => [4,2,2,1,1] => 100110110 => 000110110 => 0
[1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0] => [3,2,2,1,1] => 10110110 => 00110110 => 1
[1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1,1] => 1110110 => 0110110 => 1
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0] => [4,4,1,1,1] => 110001110 => 010001110 => 0
[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0] => [4,3,1,1,1] => 101001110 => 001001110 => 0
[1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0] => [3,3,1,1,1] => 11001110 => 01001110 => 1
[1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0] => [4,2,1,1,1] => 100101110 => 000101110 => 0
[1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1,1] => 10101110 => 00101110 => 1
[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1,1] => 1101110 => 0101110 => 1
[1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0] => [4,1,1,1,1] => 100011110 => 000011110 => 0
[1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1,1] => 10011110 => 00011110 => 1
[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1] => 1011110 => 0011110 => 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1] => 111110 => 011110 => 1
[1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0] => [5,4,4,4] => 101110000 => 001110000 => 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [4,4,4,4] => 11110000 => 01110000 => 2
[1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0] => [5,5,4,3] => 110101000 => 010101000 => 0
[1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0] => [5,4,4,3] => 101101000 => 001101000 => 0
[1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0] => [4,4,4,3] => 11101000 => 01101000 => 2
[1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0] => [5,5,3,3] => 110011000 => 010011000 => 0
[1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0] => [5,4,3,3] => 101011000 => 001011000 => 0
[1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0] => [4,4,3,3] => 11011000 => 01011000 => 2
[1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0] => [5,3,3,3] => 100111000 => 000111000 => 0
[1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0] => [4,3,3,3] => 10111000 => 00111000 => 2
[1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0] => [3,3,3,3] => 1111000 => 0111000 => 2
[1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0] => [5,5,5,2] => 111000100 => 011000100 => 2
[1,1,1,1,0,0,1,0,0,1,0,1,1,0,0,0] => [5,5,4,2] => 110100100 => 010100100 => 0
[1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0] => [5,4,4,2] => 101100100 => 001100100 => 0
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0] => [4,4,4,2] => 11100100 => 01100100 => 2
[1,1,1,1,0,0,1,0,1,0,0,1,1,0,0,0] => [5,5,3,2] => 110010100 => 010010100 => 0
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0] => [5,4,3,2] => 101010100 => 001010100 => 0
[1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0] => [4,4,3,2] => 11010100 => 01010100 => 0
[1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,0] => [5,3,3,2] => 100110100 => 000110100 => 0
[1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0] => [4,3,3,2] => 10110100 => 00110100 => 0
[1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0] => [3,3,3,2] => 1110100 => 0110100 => 2
[1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0] => [5,5,2,2] => 110001100 => 010001100 => 0
[1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0] => [5,4,2,2] => 101001100 => 001001100 => 0
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0] => [4,4,2,2] => 11001100 => 01001100 => 0
[1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0] => [5,3,2,2] => 100101100 => 000101100 => 0
[1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0] => [4,3,2,2] => 10101100 => 00101100 => 0
[1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0] => [3,3,2,2] => 1101100 => 0101100 => 2
[1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0] => [5,2,2,2] => 100011100 => 000011100 => 0
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0] => [4,2,2,2] => 10011100 => 00011100 => 0
[1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2] => 1011100 => 0011100 => 2
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2] => 111100 => 011100 => 1
[1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0] => [5,5,5,1] => 111000010 => 011000010 => 2
[1,1,1,1,0,1,0,0,0,1,0,1,1,0,0,0] => [5,5,4,1] => 110100010 => 010100010 => 0
[1,1,1,1,0,1,0,0,0,1,1,0,1,0,0,0] => [5,4,4,1] => 101100010 => 001100010 => 0
[1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0] => [4,4,4,1] => 11100010 => 01100010 => 2
[1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0] => [5,5,3,1] => 110010010 => 010010010 => 0
[1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0] => [5,4,3,1] => 101010010 => 001010010 => 0
[1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0] => [4,4,3,1] => 11010010 => 01010010 => 0
[1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0] => [5,3,3,1] => 100110010 => 000110010 => 0
[1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0] => [4,3,3,1] => 10110010 => 00110010 => 0
[1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0] => [3,3,3,1] => 1110010 => 0110010 => 2
[1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0] => [5,5,2,1] => 110001010 => 010001010 => 0
[1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0] => [5,4,2,1] => 101001010 => 001001010 => 0
[1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0] => [4,4,2,1] => 11001010 => 01001010 => 0
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0] => [5,3,2,1] => 100101010 => 000101010 => 0
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0] => [4,3,2,1] => 10101010 => 00101010 => 0
[1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0] => [3,3,2,1] => 1101010 => 0101010 => 0
[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0] => [5,2,2,1] => 100011010 => 000011010 => 0
[1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0] => [4,2,2,1] => 10011010 => 00011010 => 0
[1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0] => [3,2,2,1] => 1011010 => 0011010 => 0
[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1] => 111010 => 011010 => 1
[1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0] => [5,5,1,1] => 110000110 => 010000110 => 0
[1,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0] => [5,4,1,1] => 101000110 => 001000110 => 0
[1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0] => [4,4,1,1] => 11000110 => 01000110 => 0
[1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0] => [5,3,1,1] => 100100110 => 000100110 => 0
[1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0] => [4,3,1,1] => 10100110 => 00100110 => 0
[1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0] => [3,3,1,1] => 1100110 => 0100110 => 0
[1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0] => [5,2,1,1] => 100010110 => 000010110 => 0
[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0] => [4,2,1,1] => 10010110 => 00010110 => 0
[1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1] => 1010110 => 0010110 => 0
[1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1] => 110110 => 010110 => 1
[1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0] => [5,1,1,1] => 100001110 => 000001110 => 0
[1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0] => [4,1,1,1] => 10001110 => 00001110 => 0
[1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1] => 1001110 => 0001110 => 0
[1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1] => 101110 => 001110 => 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1] => 11110 => 01110 => 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0] => [6,6,5] => 110100000 => 010100000 => 0
[1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0] => [6,5,5] => 101100000 => 001100000 => 0
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0] => [5,5,5] => 11100000 => 01100000 => 2
[1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0] => [6,6,4] => 110010000 => 010010000 => 0
[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0] => [6,5,4] => 101010000 => 001010000 => 0
[1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0] => [5,5,4] => 11010000 => 01010000 => 0
[1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0] => [6,4,4] => 100110000 => 000110000 => 0
[1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0] => [5,4,4] => 10110000 => 00110000 => 0
[1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0] => [4,4,4] => 1110000 => 0110000 => 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0] => [6,6,3] => 110001000 => 010001000 => 0
[1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0] => [6,5,3] => 101001000 => 001001000 => 0
[1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0] => [5,5,3] => 11001000 => 01001000 => 0
[1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0] => [6,4,3] => 100101000 => 000101000 => 0
[1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0] => [5,4,3] => 10101000 => 00101000 => 0
[1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0] => [4,4,3] => 1101000 => 0101000 => 0
[1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0] => [6,3,3] => 100011000 => 000011000 => 0
[1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0] => [5,3,3] => 10011000 => 00011000 => 0
[1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0] => [4,3,3] => 1011000 => 0011000 => 0
[1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0] => [3,3,3] => 111000 => 011000 => 2
[1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0] => [6,6,2] => 110000100 => 010000100 => 0
[1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0] => [6,5,2] => 101000100 => 001000100 => 0
[1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0] => [5,5,2] => 11000100 => 01000100 => 0
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0] => [6,4,2] => 100100100 => 000100100 => 0
[1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0] => [5,4,2] => 10100100 => 00100100 => 0
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0] => [4,4,2] => 1100100 => 0100100 => 0
[1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0] => [6,3,2] => 100010100 => 000010100 => 0
[1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0] => [5,3,2] => 10010100 => 00010100 => 0
[1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0] => [4,3,2] => 1010100 => 0010100 => 0
[1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0] => [3,3,2] => 110100 => 010100 => 0
[1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0] => [6,2,2] => 100001100 => 000001100 => 0
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0] => [5,2,2] => 10001100 => 00001100 => 0
[1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0] => [4,2,2] => 1001100 => 0001100 => 0
[1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0] => [3,2,2] => 101100 => 001100 => 0
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0] => [2,2,2] => 11100 => 01100 => 2
[1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0] => [6,6,1] => 110000010 => 010000010 => 0
[1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0] => [6,5,1] => 101000010 => 001000010 => 0
[1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0] => [5,5,1] => 11000010 => 01000010 => 0
[1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0] => [6,4,1] => 100100010 => 000100010 => 0
[1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0] => [5,4,1] => 10100010 => 00100010 => 0
[1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0] => [4,4,1] => 1100010 => 0100010 => 0
[1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0] => [6,3,1] => 100010010 => 000010010 => 0
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0] => [5,3,1] => 10010010 => 00010010 => 0
[1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0] => [4,3,1] => 1010010 => 0010010 => 0
[1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0] => [3,3,1] => 110010 => 010010 => 0
[1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0] => [6,2,1] => 100001010 => 000001010 => 0
[1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0] => [5,2,1] => 10001010 => 00001010 => 0
[1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0] => [4,2,1] => 1001010 => 0001010 => 0
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [3,2,1] => 101010 => 001010 => 0
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [2,2,1] => 11010 => 01010 => 0
[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0] => [6,1,1] => 100000110 => 000000110 => 0
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0] => [5,1,1] => 10000110 => 00000110 => 0
[1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [4,1,1] => 1000110 => 0000110 => 0
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [3,1,1] => 100110 => 000110 => 0
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [2,1,1] => 10110 => 00110 => 0
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [1,1,1] => 1110 => 0110 => 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => [7,6] => 101000000 => 001000000 => 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [6,6] => 11000000 => 01000000 => 0
[1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0] => [7,5] => 100100000 => 000100000 => 0
[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0] => [6,5] => 10100000 => 00100000 => 0
[1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0] => [5,5] => 1100000 => 0100000 => 0
[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0] => [7,4] => 100010000 => 000010000 => 0
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0] => [6,4] => 10010000 => 00010000 => 0
[1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0] => [5,4] => 1010000 => 0010000 => 0
[1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0] => [4,4] => 110000 => 010000 => 0
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => [7,3] => 100001000 => 000001000 => 0
[1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0] => [6,3] => 10001000 => 00001000 => 0
[1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0] => [5,3] => 1001000 => 0001000 => 0
[1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0] => [4,3] => 101000 => 001000 => 0
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0] => [3,3] => 11000 => 01000 => 0
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0] => [7,2] => 100000100 => 000000100 => 0
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0] => [6,2] => 10000100 => 00000100 => 0
[1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0] => [5,2] => 1000100 => 0000100 => 0
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0] => [4,2] => 100100 => 000100 => 0
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [3,2] => 10100 => 00100 => 0
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [2,2] => 1100 => 0100 => 0
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => [7,1] => 100000010 => 000000010 => 0
[1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0] => [6,1] => 10000010 => 00000010 => 0
[1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0] => [5,1] => 1000010 => 0000010 => 0
[1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [4,1] => 100010 => 000010 => 0
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [3,1] => 10010 => 00010 => 0
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => 1010 => 0010 => 0
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => 110 => 010 => 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [7] => 10000000 => 00000000 => 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [6] => 1000000 => 0000000 => 0
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [5] => 100000 => 000000 => 0
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [4] => 10000 => 00000 => 0
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [3] => 1000 => 0000 => 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => 100 => 000 => 0
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1] => 10 => 00 => 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [8] => 100000000 => 000000000 => 0
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1] => 111111110 => 011111110 => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0] => [7,7] => 110000000 => 010000000 => 0
[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,2,2,2] => 111111100 => 011111100 => 1
[1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,2,2,1] => 111111010 => 011111010 => 1
[1,1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,2,1,1] => 111110110 => 011110110 => 1
[1,1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2,1,1,1,1] => 111011110 => 011011110 => 1
[1,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1,1,1,1] => 110111110 => 010111110 => 1
[1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1,1,1,1] => 101111110 => 001111110 => 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => 11111110 => 01111110 => 1
[1,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,0] => [3,3,3,3,3,2] => 111110100 => 011110100 => 1
[1,1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0,0] => [3,3,2,2,2,2] => 110111100 => 010111100 => 1
[1,1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0,0] => [3,2,2,2,2,2] => 101111100 => 001111100 => 1
[1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,2,2] => 11111100 => 01111100 => 1
[1,1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0,0] => [3,3,3,3,3,1] => 111110010 => 011110010 => 1
[1,1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0,0] => [3,3,3,3,2,1] => 111101010 => 011101010 => 1
[1,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,2,1] => 11111010 => 01111010 => 1
[1,1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,1,1] => 11110110 => 01110110 => 1
[1,1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [3,3,2,1,1,1] => 110101110 => 010101110 => 1
[1,1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2,1,1,1] => 11101110 => 01101110 => 1
[1,1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1,1,1] => 101011110 => 001011110 => 1
[1,1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1,1,1] => 11011110 => 01011110 => 1
[1,1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1,1,1,1] => 100111110 => 000111110 => 1
[1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1,1,1] => 10111110 => 00111110 => 1
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1] => 1111110 => 0111110 => 1
[1,1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0,0] => [4,3,3,3,3] => 101111000 => 001111000 => 2
[1,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,0] => [3,3,3,3,3] => 11111000 => 01111000 => 1
[1,1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0,0] => [4,4,4,3,2] => 111010100 => 011010100 => 2
[1,1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0,0] => [4,4,3,3,2] => 110110100 => 010110100 => 2
[1,1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0,0] => [4,3,2,2,2] => 101011100 => 001011100 => 2
[1,1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0,0] => [4,2,2,2,2] => 100111100 => 000111100 => 2
[1,1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0,0] => [3,2,2,2,2] => 10111100 => 00111100 => 1
[1,1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,2] => 1111100 => 0111100 => 1
[1,1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0,0] => [4,4,4,4,1] => 111100010 => 011100010 => 2
[1,1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0,0] => [3,3,3,3,1] => 11110010 => 01110010 => 1
[1,1,1,1,0,1,0,1,0,0,1,1,1,0,0,0,0,0] => [4,4,4,2,1] => 111001010 => 011001010 => 2
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0] => [4,4,3,2,1] => 110101010 => 010101010 => 0
[1,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0] => [4,3,3,2,1] => 101101010 => 001101010 => 0
[1,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0] => [3,3,3,2,1] => 11101010 => 01101010 => 1
[1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0] => [4,3,2,2,1] => 101011010 => 001011010 => 0
[1,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0,0] => [3,2,2,2,1] => 10111010 => 00111010 => 1
[1,1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,1] => 1111010 => 0111010 => 1
[1,1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0] => [4,4,3,1,1] => 110100110 => 010100110 => 0
[1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0] => [4,3,2,1,1] => 101010110 => 001010110 => 0
[1,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0] => [3,2,2,1,1] => 10110110 => 00110110 => 1
[1,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2,1,1] => 1110110 => 0110110 => 1
[1,1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0,0] => [4,4,1,1,1] => 110001110 => 010001110 => 0
[1,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0,0] => [4,3,1,1,1] => 101001110 => 001001110 => 0
[1,1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1,1,1] => 11001110 => 01001110 => 1
[1,1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1,1] => 10101110 => 00101110 => 1
[1,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1,1] => 1101110 => 0101110 => 1
[1,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1,1] => 1011110 => 0011110 => 1
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1] => 111110 => 011110 => 1
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0] => [4,4,4,4] => 11110000 => 01110000 => 2
[1,1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0,0] => [5,5,4,3] => 110101000 => 010101000 => 0
[1,1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0,0] => [4,4,4,3] => 11101000 => 01101000 => 2
[1,1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0,0] => [5,5,3,3] => 110011000 => 010011000 => 0
[1,1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0,0] => [4,3,3,3] => 10111000 => 00111000 => 2
[1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0] => [5,4,3,2] => 101010100 => 001010100 => 0
[1,1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0,0] => [4,4,3,2] => 11010100 => 01010100 => 0
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0] => [4,4,2,2] => 11001100 => 01001100 => 0
[1,1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0,0] => [3,3,2,2] => 1101100 => 0101100 => 2
[1,1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0,0] => [5,2,2,2] => 100011100 => 000011100 => 0
[1,1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0,0] => [5,5,5,1] => 111000010 => 011000010 => 2
[1,1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0,0] => [5,4,3,1] => 101010010 => 001010010 => 0
[1,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,0] => [4,3,3,1] => 10110010 => 00110010 => 0
[1,1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0,0] => [3,3,3,1] => 1110010 => 0110010 => 2
[1,1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0,0] => [5,4,2,1] => 101001010 => 001001010 => 0
[1,1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0,0] => [4,4,2,1] => 11001010 => 01001010 => 0
[1,1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0,0] => [5,3,2,1] => 100101010 => 000101010 => 0
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0] => [4,3,2,1] => 10101010 => 00101010 => 0
[1,1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0] => [3,3,2,1] => 1101010 => 0101010 => 0
[1,1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0,0] => [4,2,2,1] => 10011010 => 00011010 => 0
[1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0,0] => [3,2,2,1] => 1011010 => 0011010 => 0
[1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2,1] => 111010 => 011010 => 1
[1,1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0,0] => [4,3,1,1] => 10100110 => 00100110 => 0
[1,1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1,1] => 1100110 => 0100110 => 0
[1,1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0,0] => [4,2,1,1] => 10010110 => 00010110 => 0
[1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1] => 1010110 => 0010110 => 0
[1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1,1] => 110110 => 010110 => 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0,0] => [5,1,1,1] => 100001110 => 000001110 => 0
[1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0] => [4,1,1,1] => 10001110 => 00001110 => 0
[1,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1,1] => 1001110 => 0001110 => 0
[1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1] => 101110 => 001110 => 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1] => 11110 => 01110 => 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0,0] => [6,6,5] => 110100000 => 010100000 => 0
[1,1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0,0] => [6,5,5] => 101100000 => 001100000 => 0
[1,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,0] => [5,5,5] => 11100000 => 01100000 => 2
[1,1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0,0] => [5,4,4] => 10110000 => 00110000 => 0
[1,1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0] => [4,4,4] => 1110000 => 0110000 => 2
[1,1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0,0] => [5,4,3] => 10101000 => 00101000 => 0
[1,1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0,0] => [4,4,3] => 1101000 => 0101000 => 0
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0,0] => [6,3,3] => 100011000 => 000011000 => 0
[1,1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0,0] => [4,3,3] => 1011000 => 0011000 => 0
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0] => [3,3,3] => 111000 => 011000 => 2
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,0] => [6,4,2] => 100100100 => 000100100 => 0
[1,1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0,0] => [5,4,2] => 10100100 => 00100100 => 0
[1,1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0,0] => [4,4,2] => 1100100 => 0100100 => 0
[1,1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0,0] => [5,3,2] => 10010100 => 00010100 => 0
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0] => [4,3,2] => 1010100 => 0010100 => 0
[1,1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0] => [3,3,2] => 110100 => 010100 => 0
[1,1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0,0] => [4,2,2] => 1001100 => 0001100 => 0
[1,1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0,0] => [3,2,2] => 101100 => 001100 => 0
[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2] => 11100 => 01100 => 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0,0] => [6,6,1] => 110000010 => 010000010 => 0
[1,1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0,0] => [6,5,1] => 101000010 => 001000010 => 0
[1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0] => [5,4,1] => 10100010 => 00100010 => 0
[1,1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0,0] => [4,4,1] => 1100010 => 0100010 => 0
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0] => [6,3,1] => 100010010 => 000010010 => 0
[1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0] => [5,3,1] => 10010010 => 00010010 => 0
[1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0] => [4,3,1] => 1010010 => 0010010 => 0
[1,1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1] => 110010 => 010010 => 0
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0,0] => [6,2,1] => 100001010 => 000001010 => 0
[1,1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0,0] => [5,2,1] => 10001010 => 00001010 => 0
[1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0] => [4,2,1] => 1001010 => 0001010 => 0
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1] => 101010 => 001010 => 0
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1] => 11010 => 01010 => 0
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0,0] => [6,1,1] => 100000110 => 000000110 => 0
[1,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0,0] => [4,1,1] => 1000110 => 0000110 => 0
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1] => 100110 => 000110 => 0
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1] => 10110 => 00110 => 0
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1] => 1110 => 0110 => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0] => [7,6] => 101000000 => 001000000 => 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0] => [6,6] => 11000000 => 01000000 => 0
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0] => [7,5] => 100100000 => 000100000 => 0
[1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0] => [5,5] => 1100000 => 0100000 => 0
[1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0] => [4,4] => 110000 => 010000 => 0
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0] => [7,3] => 100001000 => 000001000 => 0
[1,1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,0] => [4,3] => 101000 => 001000 => 0
[1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0] => [3,3] => 11000 => 01000 => 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0] => [7,2] => 100000100 => 000000100 => 0
[1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0] => [4,2] => 100100 => 000100 => 0
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0] => [3,2] => 10100 => 00100 => 0
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0] => [2,2] => 1100 => 0100 => 0
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0] => [7,1] => 100000010 => 000000010 => 0
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0] => [6,1] => 10000010 => 00000010 => 0
[1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0] => [5,1] => 1000010 => 0000010 => 0
[1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0] => [4,1] => 100010 => 000010 => 0
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0] => [3,1] => 10010 => 00010 => 0
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => 1010 => 0010 => 0
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => 110 => 010 => 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0] => [7] => 10000000 => 00000000 => 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0] => [6] => 1000000 => 0000000 => 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0] => [5] => 100000 => 000000 => 0
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [4] => 10000 => 00000 => 0
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [3] => 1000 => 0000 => 0
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [2] => 100 => 000 => 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [1] => 10 => 00 => 0
[1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [3,3,3,3,3,3] => 111111000 => 011111000 => 1
[1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0] => [4,4,4,4,4] => 111110000 => 011110000 => 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0] => [5,5,5,5] => 111100000 => 011100000 => 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0] => [6,6,6] => 111000000 => 011000000 => 2
[1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0] => [4,4,3,2,1] => 110101010 => 010101010 => 0
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0,0] => [5,4,3,2] => 101010100 => 001010100 => 0
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [1,1] => 110 => 010 => 0
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [2] => 100 => 000 => 0
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1] => 110 => 010 => 0
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1] => 11010 => 01010 => 0
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0] => [3,2] => 10100 => 00100 => 0
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0] => [2] => 100 => 000 => 0
[1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1] => 111111110 => 011111110 => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0] => [8] => 100000000 => 000000000 => 0
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [1] => 10 => 00 => 0
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1] => 1110 => 0110 => 1
[1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1] => 11110 => 01110 => 1
[1,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1] => 111110 => 011110 => 1
[1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1] => 1111110 => 0111110 => 1
[1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => 11111110 => 01111110 => 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0] => [1] => 10 => 00 => 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0] => [7] => 10000000 => 00000000 => 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0] => [6] => 1000000 => 0000000 => 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0] => [5] => 100000 => 000000 => 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0] => [4] => 10000 => 00000 => 0
[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0] => [3] => 1000 => 0000 => 0
[1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0] => [2,2] => 1100 => 0100 => 0
[1,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0] => [3,3] => 11000 => 01000 => 0
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0] => [3,3,3] => 111000 => 011000 => 2
[1,1,1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0,0] => [4,4,4] => 1110000 => 0110000 => 2
[1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0] => [4,4,4,4] => 11110000 => 01110000 => 2
[1,1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,0] => [5,5,5,5] => 111100000 => 011100000 => 2
[1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1] => 10110 => 00110 => 0
[1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0,0] => [4,3,2] => 1010100 => 0010100 => 0
[1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0] => [3,1] => 10010 => 00010 => 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,0] => [6,6,6] => 111000000 => 011000000 => 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0,0] => [6,6] => 11000000 => 01000000 => 0
[1,1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1,1,1,1] => 10111110 => 00111110 => 1
[1,1,1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [3,1,1,1,1,1] => 100111110 => 000111110 => 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [2,1] => 1010 => 0010 => 0
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1] => 1010 => 0010 => 0
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [4,3,2,1] => 10101010 => 00101010 => 0
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1] => 101010 => 001010 => 0
[1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0] => [3] => 1000 => 0000 => 0
[1,1,1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0] => [3,3,3,3,2,1] => 111101010 => 011101010 => 1
[1,1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2,2,2,2,1] => 111111010 => 011111010 => 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0] => [7,7] => 110000000 => 010000000 => 0
[1,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0] => [4,4,4,4,4] => 111110000 => 011110000 => 2
[1,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0] => [3,3,3,3,3,3] => 111111000 => 011111000 => 1
[1,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2,2,2,2,2] => 111111100 => 011111100 => 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0,0,0] => [5,3,2,1] => 100101010 => 000101010 => 0
[1,1,1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0,0,0] => [5,4,2,1] => 101001010 => 001001010 => 0
[1,1,1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0,0,0] => [5,4,3,1] => 101010010 => 001010010 => 0
[1,1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0] => [3,1] => 10010 => 00010 => 0
[1,1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0,0] => [4,2] => 100100 => 000100 => 0
[1,1,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2,2,2,1] => 11111010 => 01111010 => 1
[1,1,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0,0] => [4,3,3,2,1] => 101101010 => 001101010 => 0
[1,1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0] => [4,3,2,2,1] => 101011010 => 001011010 => 0
[1,1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0,0] => [4,3,2,1,1] => 101010110 => 001010110 => 0
[1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0,0,0] => [6,5,1] => 101000010 => 001000010 => 0
[1,1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0,0] => [5,4,1] => 10100010 => 00100010 => 0
[1,1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0,0] => [4,3,1] => 1010010 => 0010010 => 0
[1,1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2] => 11100 => 01100 => 2
[1,1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2,2,2,2] => 11111100 => 01111100 => 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0] => [7,1] => 100000010 => 000000010 => 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0] => [6,1] => 10000010 => 00000010 => 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,0] => [5,1] => 1000010 => 0000010 => 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0] => [4,1] => 100010 => 000010 => 0
[1,1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1,1] => 101110 => 001110 => 1
[1,1,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1,1,1] => 1011110 => 0011110 => 1
[1,1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1,1,1,1,1] => 101111110 => 001111110 => 1
[1,1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1,1] => 10110 => 00110 => 0
[1,1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0,0,0] => [3,2,2,1] => 1011010 => 0011010 => 0
[1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1,1] => 1110 => 0110 => 1
[1,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,0] => [3,1,1] => 100110 => 000110 => 0
[1,1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1,1] => 110110 => 010110 => 1
[1,1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1,1] => 1010110 => 0010110 => 0
[1,1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0] => [2,2] => 1100 => 0100 => 0
[1,1,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0,0] => [3,3] => 11000 => 01000 => 0
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0] => [3] => 1000 => 0000 => 0
[1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0] => [1,1] => 110 => 010 => 0
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0,0] => [7,1] => 100000010 => 000000010 => 0
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0,0] => [6,1] => 10000010 => 00000010 => 0
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,0,0] => [5,1] => 1000010 => 0000010 => 0
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0] => [4,1] => 100010 => 000010 => 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0,0] => [5,5] => 1100000 => 0100000 => 0
[1,1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0] => [4,4] => 110000 => 010000 => 0
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0,0] => [5,3,1] => 10010010 => 00010010 => 0
[1,1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2,1] => 111010 => 011010 => 1
[1,1,1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0] => [3,3,2] => 110100 => 010100 => 0
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Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each 0 as a step (1,−1) and 1 as a step (1,1). Then this statistic counts the number of times the path crosses the x-axis.
Interpret each 0 as a step (1,−1) and 1 as a step (1,1). Then this statistic counts the number of times the path crosses the x-axis.
Map
twist
Description
Return the binary word with first letter inverted.
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition λ associated to a Dyck path is defined to be the complementary partition inside the staircase partition (n−1,…,2,1) when cutting out D considered as a path from (0,0) to (n,n).
In other words, λi is the number of down-steps before the (n+1−i)-th up-step of D.
This map is a bijection between Dyck paths of size n and partitions inside the staircase partition (n−1,…,2,1).
The partition λ associated to a Dyck path is defined to be the complementary partition inside the staircase partition (n−1,…,2,1) when cutting out D considered as a path from (0,0) to (n,n).
In other words, λi is the number of down-steps before the (n+1−i)-th up-step of D.
This map is a bijection between Dyck paths of size n and partitions inside the staircase partition (n−1,…,2,1).
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.
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