Identifier
-
Mp00040:
Integer compositions
—to partition⟶
Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤ
Values
[1] => [1] => 10 => [1,1] => 1
[1,1] => [1,1] => 110 => [2,1] => 1
[2] => [2] => 100 => [1,2] => 2
[1,1,1] => [1,1,1] => 1110 => [3,1] => 1
[1,2] => [2,1] => 1010 => [1,1,1,1] => 1
[2,1] => [2,1] => 1010 => [1,1,1,1] => 1
[3] => [3] => 1000 => [1,3] => 3
[1,1,1,1] => [1,1,1,1] => 11110 => [4,1] => 1
[1,1,2] => [2,1,1] => 10110 => [1,1,2,1] => 1
[1,2,1] => [2,1,1] => 10110 => [1,1,2,1] => 1
[1,3] => [3,1] => 10010 => [1,2,1,1] => 1
[2,1,1] => [2,1,1] => 10110 => [1,1,2,1] => 1
[2,2] => [2,2] => 1100 => [2,2] => 2
[3,1] => [3,1] => 10010 => [1,2,1,1] => 1
[4] => [4] => 10000 => [1,4] => 4
[1,1,1,1,1] => [1,1,1,1,1] => 111110 => [5,1] => 1
[1,1,1,2] => [2,1,1,1] => 101110 => [1,1,3,1] => 1
[1,1,2,1] => [2,1,1,1] => 101110 => [1,1,3,1] => 1
[1,1,3] => [3,1,1] => 100110 => [1,2,2,1] => 1
[1,2,1,1] => [2,1,1,1] => 101110 => [1,1,3,1] => 1
[1,2,2] => [2,2,1] => 11010 => [2,1,1,1] => 1
[1,3,1] => [3,1,1] => 100110 => [1,2,2,1] => 1
[1,4] => [4,1] => 100010 => [1,3,1,1] => 1
[2,1,1,1] => [2,1,1,1] => 101110 => [1,1,3,1] => 1
[2,1,2] => [2,2,1] => 11010 => [2,1,1,1] => 1
[2,2,1] => [2,2,1] => 11010 => [2,1,1,1] => 1
[2,3] => [3,2] => 10100 => [1,1,1,2] => 2
[3,1,1] => [3,1,1] => 100110 => [1,2,2,1] => 1
[3,2] => [3,2] => 10100 => [1,1,1,2] => 2
[4,1] => [4,1] => 100010 => [1,3,1,1] => 1
[5] => [5] => 100000 => [1,5] => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 1111110 => [6,1] => 1
[1,1,1,1,2] => [2,1,1,1,1] => 1011110 => [1,1,4,1] => 1
[1,1,1,2,1] => [2,1,1,1,1] => 1011110 => [1,1,4,1] => 1
[1,1,1,3] => [3,1,1,1] => 1001110 => [1,2,3,1] => 1
[1,1,2,1,1] => [2,1,1,1,1] => 1011110 => [1,1,4,1] => 1
[1,1,2,2] => [2,2,1,1] => 110110 => [2,1,2,1] => 1
[1,1,3,1] => [3,1,1,1] => 1001110 => [1,2,3,1] => 1
[1,1,4] => [4,1,1] => 1000110 => [1,3,2,1] => 1
[1,2,1,1,1] => [2,1,1,1,1] => 1011110 => [1,1,4,1] => 1
[1,2,1,2] => [2,2,1,1] => 110110 => [2,1,2,1] => 1
[1,2,2,1] => [2,2,1,1] => 110110 => [2,1,2,1] => 1
[1,2,3] => [3,2,1] => 101010 => [1,1,1,1,1,1] => 1
[1,3,1,1] => [3,1,1,1] => 1001110 => [1,2,3,1] => 1
[1,3,2] => [3,2,1] => 101010 => [1,1,1,1,1,1] => 1
[1,4,1] => [4,1,1] => 1000110 => [1,3,2,1] => 1
[1,5] => [5,1] => 1000010 => [1,4,1,1] => 1
[2,1,1,1,1] => [2,1,1,1,1] => 1011110 => [1,1,4,1] => 1
[2,1,1,2] => [2,2,1,1] => 110110 => [2,1,2,1] => 1
[2,1,2,1] => [2,2,1,1] => 110110 => [2,1,2,1] => 1
[2,1,3] => [3,2,1] => 101010 => [1,1,1,1,1,1] => 1
[2,2,1,1] => [2,2,1,1] => 110110 => [2,1,2,1] => 1
[2,2,2] => [2,2,2] => 11100 => [3,2] => 2
[2,3,1] => [3,2,1] => 101010 => [1,1,1,1,1,1] => 1
[2,4] => [4,2] => 100100 => [1,2,1,2] => 2
[3,1,1,1] => [3,1,1,1] => 1001110 => [1,2,3,1] => 1
[3,1,2] => [3,2,1] => 101010 => [1,1,1,1,1,1] => 1
[3,2,1] => [3,2,1] => 101010 => [1,1,1,1,1,1] => 1
[3,3] => [3,3] => 11000 => [2,3] => 3
[4,1,1] => [4,1,1] => 1000110 => [1,3,2,1] => 1
[4,2] => [4,2] => 100100 => [1,2,1,2] => 2
[5,1] => [5,1] => 1000010 => [1,4,1,1] => 1
[6] => [6] => 1000000 => [1,6] => 6
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 11111110 => [7,1] => 1
[1,1,1,1,1,2] => [2,1,1,1,1,1] => 10111110 => [1,1,5,1] => 1
[1,1,1,1,2,1] => [2,1,1,1,1,1] => 10111110 => [1,1,5,1] => 1
[1,1,1,1,3] => [3,1,1,1,1] => 10011110 => [1,2,4,1] => 1
[1,1,1,2,1,1] => [2,1,1,1,1,1] => 10111110 => [1,1,5,1] => 1
[1,1,1,2,2] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 1
[1,1,1,3,1] => [3,1,1,1,1] => 10011110 => [1,2,4,1] => 1
[1,1,1,4] => [4,1,1,1] => 10001110 => [1,3,3,1] => 1
[1,1,2,1,1,1] => [2,1,1,1,1,1] => 10111110 => [1,1,5,1] => 1
[1,1,2,1,2] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 1
[1,1,2,2,1] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 1
[1,1,2,3] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1
[1,1,3,1,1] => [3,1,1,1,1] => 10011110 => [1,2,4,1] => 1
[1,1,3,2] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1
[1,1,4,1] => [4,1,1,1] => 10001110 => [1,3,3,1] => 1
[1,1,5] => [5,1,1] => 10000110 => [1,4,2,1] => 1
[1,2,1,1,1,1] => [2,1,1,1,1,1] => 10111110 => [1,1,5,1] => 1
[1,2,1,1,2] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 1
[1,2,1,2,1] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 1
[1,2,1,3] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1
[1,2,2,1,1] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 1
[1,2,2,2] => [2,2,2,1] => 111010 => [3,1,1,1] => 1
[1,2,3,1] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1
[1,2,4] => [4,2,1] => 1001010 => [1,2,1,1,1,1] => 1
[1,3,1,1,1] => [3,1,1,1,1] => 10011110 => [1,2,4,1] => 1
[1,3,1,2] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1
[1,3,2,1] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1
[1,3,3] => [3,3,1] => 110010 => [2,2,1,1] => 1
[1,4,1,1] => [4,1,1,1] => 10001110 => [1,3,3,1] => 1
[1,4,2] => [4,2,1] => 1001010 => [1,2,1,1,1,1] => 1
[1,5,1] => [5,1,1] => 10000110 => [1,4,2,1] => 1
[1,6] => [6,1] => 10000010 => [1,5,1,1] => 1
[2,1,1,1,1,1] => [2,1,1,1,1,1] => 10111110 => [1,1,5,1] => 1
[2,1,1,1,2] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 1
[2,1,1,2,1] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 1
[2,1,1,3] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1
[2,1,2,1,1] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 1
[2,1,2,2] => [2,2,2,1] => 111010 => [3,1,1,1] => 1
>>> Load all 1064 entries. <<<[2,1,3,1] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1
[2,1,4] => [4,2,1] => 1001010 => [1,2,1,1,1,1] => 1
[2,2,1,1,1] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 1
[2,2,1,2] => [2,2,2,1] => 111010 => [3,1,1,1] => 1
[2,2,2,1] => [2,2,2,1] => 111010 => [3,1,1,1] => 1
[2,2,3] => [3,2,2] => 101100 => [1,1,2,2] => 2
[2,3,1,1] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1
[2,3,2] => [3,2,2] => 101100 => [1,1,2,2] => 2
[2,4,1] => [4,2,1] => 1001010 => [1,2,1,1,1,1] => 1
[2,5] => [5,2] => 1000100 => [1,3,1,2] => 2
[3,1,1,1,1] => [3,1,1,1,1] => 10011110 => [1,2,4,1] => 1
[3,1,1,2] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1
[3,1,2,1] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1
[3,1,3] => [3,3,1] => 110010 => [2,2,1,1] => 1
[3,2,1,1] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1
[3,2,2] => [3,2,2] => 101100 => [1,1,2,2] => 2
[3,3,1] => [3,3,1] => 110010 => [2,2,1,1] => 1
[3,4] => [4,3] => 101000 => [1,1,1,3] => 3
[4,1,1,1] => [4,1,1,1] => 10001110 => [1,3,3,1] => 1
[4,1,2] => [4,2,1] => 1001010 => [1,2,1,1,1,1] => 1
[4,2,1] => [4,2,1] => 1001010 => [1,2,1,1,1,1] => 1
[4,3] => [4,3] => 101000 => [1,1,1,3] => 3
[5,1,1] => [5,1,1] => 10000110 => [1,4,2,1] => 1
[5,2] => [5,2] => 1000100 => [1,3,1,2] => 2
[6,1] => [6,1] => 10000010 => [1,5,1,1] => 1
[7] => [7] => 10000000 => [1,7] => 7
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 111111110 => [8,1] => 1
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => 101111110 => [1,1,6,1] => 1
[1,1,1,1,1,2,1] => [2,1,1,1,1,1,1] => 101111110 => [1,1,6,1] => 1
[1,1,1,1,1,3] => [3,1,1,1,1,1] => 100111110 => [1,2,5,1] => 1
[1,1,1,1,2,1,1] => [2,1,1,1,1,1,1] => 101111110 => [1,1,6,1] => 1
[1,1,1,1,2,2] => [2,2,1,1,1,1] => 11011110 => [2,1,4,1] => 1
[1,1,1,1,3,1] => [3,1,1,1,1,1] => 100111110 => [1,2,5,1] => 1
[1,1,1,1,4] => [4,1,1,1,1] => 100011110 => [1,3,4,1] => 1
[1,1,1,2,1,1,1] => [2,1,1,1,1,1,1] => 101111110 => [1,1,6,1] => 1
[1,1,1,2,1,2] => [2,2,1,1,1,1] => 11011110 => [2,1,4,1] => 1
[1,1,1,2,2,1] => [2,2,1,1,1,1] => 11011110 => [2,1,4,1] => 1
[1,1,1,2,3] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[1,1,1,3,1,1] => [3,1,1,1,1,1] => 100111110 => [1,2,5,1] => 1
[1,1,1,3,2] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[1,1,1,4,1] => [4,1,1,1,1] => 100011110 => [1,3,4,1] => 1
[1,1,1,5] => [5,1,1,1] => 100001110 => [1,4,3,1] => 1
[1,1,2,1,1,1,1] => [2,1,1,1,1,1,1] => 101111110 => [1,1,6,1] => 1
[1,1,2,1,1,2] => [2,2,1,1,1,1] => 11011110 => [2,1,4,1] => 1
[1,1,2,1,2,1] => [2,2,1,1,1,1] => 11011110 => [2,1,4,1] => 1
[1,1,2,1,3] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[1,1,2,2,1,1] => [2,2,1,1,1,1] => 11011110 => [2,1,4,1] => 1
[1,1,2,2,2] => [2,2,2,1,1] => 1110110 => [3,1,2,1] => 1
[1,1,2,3,1] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[1,1,2,4] => [4,2,1,1] => 10010110 => [1,2,1,1,2,1] => 1
[1,1,3,1,1,1] => [3,1,1,1,1,1] => 100111110 => [1,2,5,1] => 1
[1,1,3,1,2] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[1,1,3,2,1] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[1,1,3,3] => [3,3,1,1] => 1100110 => [2,2,2,1] => 1
[1,1,4,1,1] => [4,1,1,1,1] => 100011110 => [1,3,4,1] => 1
[1,1,4,2] => [4,2,1,1] => 10010110 => [1,2,1,1,2,1] => 1
[1,1,5,1] => [5,1,1,1] => 100001110 => [1,4,3,1] => 1
[1,1,6] => [6,1,1] => 100000110 => [1,5,2,1] => 1
[1,2,1,1,1,1,1] => [2,1,1,1,1,1,1] => 101111110 => [1,1,6,1] => 1
[1,2,1,1,1,2] => [2,2,1,1,1,1] => 11011110 => [2,1,4,1] => 1
[1,2,1,1,2,1] => [2,2,1,1,1,1] => 11011110 => [2,1,4,1] => 1
[1,2,1,1,3] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[1,2,1,2,1,1] => [2,2,1,1,1,1] => 11011110 => [2,1,4,1] => 1
[1,2,1,2,2] => [2,2,2,1,1] => 1110110 => [3,1,2,1] => 1
[1,2,1,3,1] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[1,2,1,4] => [4,2,1,1] => 10010110 => [1,2,1,1,2,1] => 1
[1,2,2,1,1,1] => [2,2,1,1,1,1] => 11011110 => [2,1,4,1] => 1
[1,2,2,1,2] => [2,2,2,1,1] => 1110110 => [3,1,2,1] => 1
[1,2,2,2,1] => [2,2,2,1,1] => 1110110 => [3,1,2,1] => 1
[1,2,2,3] => [3,2,2,1] => 1011010 => [1,1,2,1,1,1] => 1
[1,2,3,1,1] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[1,2,3,2] => [3,2,2,1] => 1011010 => [1,1,2,1,1,1] => 1
[1,2,4,1] => [4,2,1,1] => 10010110 => [1,2,1,1,2,1] => 1
[1,2,5] => [5,2,1] => 10001010 => [1,3,1,1,1,1] => 1
[1,3,1,1,1,1] => [3,1,1,1,1,1] => 100111110 => [1,2,5,1] => 1
[1,3,1,1,2] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[1,3,1,2,1] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[1,3,1,3] => [3,3,1,1] => 1100110 => [2,2,2,1] => 1
[1,3,2,1,1] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[1,3,2,2] => [3,2,2,1] => 1011010 => [1,1,2,1,1,1] => 1
[1,3,3,1] => [3,3,1,1] => 1100110 => [2,2,2,1] => 1
[1,3,4] => [4,3,1] => 1010010 => [1,1,1,2,1,1] => 1
[1,4,1,1,1] => [4,1,1,1,1] => 100011110 => [1,3,4,1] => 1
[1,4,1,2] => [4,2,1,1] => 10010110 => [1,2,1,1,2,1] => 1
[1,4,2,1] => [4,2,1,1] => 10010110 => [1,2,1,1,2,1] => 1
[1,4,3] => [4,3,1] => 1010010 => [1,1,1,2,1,1] => 1
[1,5,1,1] => [5,1,1,1] => 100001110 => [1,4,3,1] => 1
[1,5,2] => [5,2,1] => 10001010 => [1,3,1,1,1,1] => 1
[1,6,1] => [6,1,1] => 100000110 => [1,5,2,1] => 1
[1,7] => [7,1] => 100000010 => [1,6,1,1] => 1
[2,1,1,1,1,1,1] => [2,1,1,1,1,1,1] => 101111110 => [1,1,6,1] => 1
[2,1,1,1,1,2] => [2,2,1,1,1,1] => 11011110 => [2,1,4,1] => 1
[2,1,1,1,2,1] => [2,2,1,1,1,1] => 11011110 => [2,1,4,1] => 1
[2,1,1,1,3] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[2,1,1,2,1,1] => [2,2,1,1,1,1] => 11011110 => [2,1,4,1] => 1
[2,1,1,2,2] => [2,2,2,1,1] => 1110110 => [3,1,2,1] => 1
[2,1,1,3,1] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[2,1,1,4] => [4,2,1,1] => 10010110 => [1,2,1,1,2,1] => 1
[2,1,2,1,1,1] => [2,2,1,1,1,1] => 11011110 => [2,1,4,1] => 1
[2,1,2,1,2] => [2,2,2,1,1] => 1110110 => [3,1,2,1] => 1
[2,1,2,2,1] => [2,2,2,1,1] => 1110110 => [3,1,2,1] => 1
[2,1,2,3] => [3,2,2,1] => 1011010 => [1,1,2,1,1,1] => 1
[2,1,3,1,1] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[2,1,3,2] => [3,2,2,1] => 1011010 => [1,1,2,1,1,1] => 1
[2,1,4,1] => [4,2,1,1] => 10010110 => [1,2,1,1,2,1] => 1
[2,1,5] => [5,2,1] => 10001010 => [1,3,1,1,1,1] => 1
[2,2,1,1,1,1] => [2,2,1,1,1,1] => 11011110 => [2,1,4,1] => 1
[2,2,1,1,2] => [2,2,2,1,1] => 1110110 => [3,1,2,1] => 1
[2,2,1,2,1] => [2,2,2,1,1] => 1110110 => [3,1,2,1] => 1
[2,2,1,3] => [3,2,2,1] => 1011010 => [1,1,2,1,1,1] => 1
[2,2,2,1,1] => [2,2,2,1,1] => 1110110 => [3,1,2,1] => 1
[2,2,2,2] => [2,2,2,2] => 111100 => [4,2] => 2
[2,2,3,1] => [3,2,2,1] => 1011010 => [1,1,2,1,1,1] => 1
[2,2,4] => [4,2,2] => 1001100 => [1,2,2,2] => 2
[2,3,1,1,1] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[2,3,1,2] => [3,2,2,1] => 1011010 => [1,1,2,1,1,1] => 1
[2,3,2,1] => [3,2,2,1] => 1011010 => [1,1,2,1,1,1] => 1
[2,3,3] => [3,3,2] => 110100 => [2,1,1,2] => 2
[2,4,1,1] => [4,2,1,1] => 10010110 => [1,2,1,1,2,1] => 1
[2,4,2] => [4,2,2] => 1001100 => [1,2,2,2] => 2
[2,5,1] => [5,2,1] => 10001010 => [1,3,1,1,1,1] => 1
[2,6] => [6,2] => 10000100 => [1,4,1,2] => 2
[3,1,1,1,1,1] => [3,1,1,1,1,1] => 100111110 => [1,2,5,1] => 1
[3,1,1,1,2] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[3,1,1,2,1] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[3,1,1,3] => [3,3,1,1] => 1100110 => [2,2,2,1] => 1
[3,1,2,1,1] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[3,1,2,2] => [3,2,2,1] => 1011010 => [1,1,2,1,1,1] => 1
[3,1,3,1] => [3,3,1,1] => 1100110 => [2,2,2,1] => 1
[3,1,4] => [4,3,1] => 1010010 => [1,1,1,2,1,1] => 1
[3,2,1,1,1] => [3,2,1,1,1] => 10101110 => [1,1,1,1,3,1] => 1
[3,2,1,2] => [3,2,2,1] => 1011010 => [1,1,2,1,1,1] => 1
[3,2,2,1] => [3,2,2,1] => 1011010 => [1,1,2,1,1,1] => 1
[3,2,3] => [3,3,2] => 110100 => [2,1,1,2] => 2
[3,3,1,1] => [3,3,1,1] => 1100110 => [2,2,2,1] => 1
[3,3,2] => [3,3,2] => 110100 => [2,1,1,2] => 2
[3,4,1] => [4,3,1] => 1010010 => [1,1,1,2,1,1] => 1
[3,5] => [5,3] => 1001000 => [1,2,1,3] => 3
[4,1,1,1,1] => [4,1,1,1,1] => 100011110 => [1,3,4,1] => 1
[4,1,1,2] => [4,2,1,1] => 10010110 => [1,2,1,1,2,1] => 1
[4,1,2,1] => [4,2,1,1] => 10010110 => [1,2,1,1,2,1] => 1
[4,1,3] => [4,3,1] => 1010010 => [1,1,1,2,1,1] => 1
[4,2,1,1] => [4,2,1,1] => 10010110 => [1,2,1,1,2,1] => 1
[4,2,2] => [4,2,2] => 1001100 => [1,2,2,2] => 2
[4,3,1] => [4,3,1] => 1010010 => [1,1,1,2,1,1] => 1
[4,4] => [4,4] => 110000 => [2,4] => 4
[5,1,1,1] => [5,1,1,1] => 100001110 => [1,4,3,1] => 1
[5,1,2] => [5,2,1] => 10001010 => [1,3,1,1,1,1] => 1
[5,2,1] => [5,2,1] => 10001010 => [1,3,1,1,1,1] => 1
[5,3] => [5,3] => 1001000 => [1,2,1,3] => 3
[6,1,1] => [6,1,1] => 100000110 => [1,5,2,1] => 1
[6,2] => [6,2] => 10000100 => [1,4,1,2] => 2
[7,1] => [7,1] => 100000010 => [1,6,1,1] => 1
[8] => [8] => 100000000 => [1,8] => 8
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 1111111110 => [9,1] => 1
[1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1] => 1011111110 => [1,1,7,1] => 1
[1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1] => 1011111110 => [1,1,7,1] => 1
[1,1,1,1,1,1,3] => [3,1,1,1,1,1,1] => 1001111110 => [1,2,6,1] => 1
[1,1,1,1,1,2,1,1] => [2,1,1,1,1,1,1,1] => 1011111110 => [1,1,7,1] => 1
[1,1,1,1,1,2,2] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[1,1,1,1,1,3,1] => [3,1,1,1,1,1,1] => 1001111110 => [1,2,6,1] => 1
[1,1,1,1,1,4] => [4,1,1,1,1,1] => 1000111110 => [1,3,5,1] => 1
[1,1,1,1,2,1,1,1] => [2,1,1,1,1,1,1,1] => 1011111110 => [1,1,7,1] => 1
[1,1,1,1,2,1,2] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[1,1,1,1,2,2,1] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[1,1,1,1,2,3] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,1,1,1,3,1,1] => [3,1,1,1,1,1,1] => 1001111110 => [1,2,6,1] => 1
[1,1,1,1,3,2] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,1,1,1,4,1] => [4,1,1,1,1,1] => 1000111110 => [1,3,5,1] => 1
[1,1,1,1,5] => [5,1,1,1,1] => 1000011110 => [1,4,4,1] => 1
[1,1,1,2,1,1,1,1] => [2,1,1,1,1,1,1,1] => 1011111110 => [1,1,7,1] => 1
[1,1,1,2,1,1,2] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[1,1,1,2,1,2,1] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[1,1,1,2,1,3] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,1,1,2,2,1,1] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[1,1,1,2,2,2] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[1,1,1,2,3,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,1,1,2,4] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[1,1,1,3,1,1,1] => [3,1,1,1,1,1,1] => 1001111110 => [1,2,6,1] => 1
[1,1,1,3,1,2] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,1,1,3,2,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,1,1,3,3] => [3,3,1,1,1] => 11001110 => [2,2,3,1] => 1
[1,1,1,4,1,1] => [4,1,1,1,1,1] => 1000111110 => [1,3,5,1] => 1
[1,1,1,4,2] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[1,1,1,5,1] => [5,1,1,1,1] => 1000011110 => [1,4,4,1] => 1
[1,1,1,6] => [6,1,1,1] => 1000001110 => [1,5,3,1] => 1
[1,1,2,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 1011111110 => [1,1,7,1] => 1
[1,1,2,1,1,1,2] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[1,1,2,1,1,2,1] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[1,1,2,1,1,3] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,1,2,1,2,1,1] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[1,1,2,1,2,2] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[1,1,2,1,3,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,1,2,1,4] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[1,1,2,2,1,1,1] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[1,1,2,2,1,2] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[1,1,2,2,2,1] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[1,1,2,2,3] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[1,1,2,3,1,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,1,2,3,2] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[1,1,2,4,1] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[1,1,2,5] => [5,2,1,1] => 100010110 => [1,3,1,1,2,1] => 1
[1,1,3,1,1,1,1] => [3,1,1,1,1,1,1] => 1001111110 => [1,2,6,1] => 1
[1,1,3,1,1,2] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,1,3,1,2,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,1,3,1,3] => [3,3,1,1,1] => 11001110 => [2,2,3,1] => 1
[1,1,3,2,1,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,1,3,2,2] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[1,1,3,3,1] => [3,3,1,1,1] => 11001110 => [2,2,3,1] => 1
[1,1,3,4] => [4,3,1,1] => 10100110 => [1,1,1,2,2,1] => 1
[1,1,4,1,1,1] => [4,1,1,1,1,1] => 1000111110 => [1,3,5,1] => 1
[1,1,4,1,2] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[1,1,4,2,1] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[1,1,4,3] => [4,3,1,1] => 10100110 => [1,1,1,2,2,1] => 1
[1,1,5,1,1] => [5,1,1,1,1] => 1000011110 => [1,4,4,1] => 1
[1,1,5,2] => [5,2,1,1] => 100010110 => [1,3,1,1,2,1] => 1
[1,1,6,1] => [6,1,1,1] => 1000001110 => [1,5,3,1] => 1
[1,1,7] => [7,1,1] => 1000000110 => [1,6,2,1] => 1
[1,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 1011111110 => [1,1,7,1] => 1
[1,2,1,1,1,1,2] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[1,2,1,1,1,2,1] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[1,2,1,1,1,3] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,2,1,1,2,1,1] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[1,2,1,1,2,2] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[1,2,1,1,3,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,2,1,1,4] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[1,2,1,2,1,1,1] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[1,2,1,2,1,2] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[1,2,1,2,2,1] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[1,2,1,2,3] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[1,2,1,3,1,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,2,1,3,2] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[1,2,1,4,1] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[1,2,1,5] => [5,2,1,1] => 100010110 => [1,3,1,1,2,1] => 1
[1,2,2,1,1,1,1] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[1,2,2,1,1,2] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[1,2,2,1,2,1] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[1,2,2,1,3] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[1,2,2,2,1,1] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[1,2,2,2,2] => [2,2,2,2,1] => 1111010 => [4,1,1,1] => 1
[1,2,2,3,1] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[1,2,2,4] => [4,2,2,1] => 10011010 => [1,2,2,1,1,1] => 1
[1,2,3,1,1,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,2,3,1,2] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[1,2,3,2,1] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[1,2,3,3] => [3,3,2,1] => 1101010 => [2,1,1,1,1,1] => 1
[1,2,4,1,1] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[1,2,4,2] => [4,2,2,1] => 10011010 => [1,2,2,1,1,1] => 1
[1,2,5,1] => [5,2,1,1] => 100010110 => [1,3,1,1,2,1] => 1
[1,2,6] => [6,2,1] => 100001010 => [1,4,1,1,1,1] => 1
[1,3,1,1,1,1,1] => [3,1,1,1,1,1,1] => 1001111110 => [1,2,6,1] => 1
[1,3,1,1,1,2] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,3,1,1,2,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,3,1,1,3] => [3,3,1,1,1] => 11001110 => [2,2,3,1] => 1
[1,3,1,2,1,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,3,1,2,2] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[1,3,1,3,1] => [3,3,1,1,1] => 11001110 => [2,2,3,1] => 1
[1,3,1,4] => [4,3,1,1] => 10100110 => [1,1,1,2,2,1] => 1
[1,3,2,1,1,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[1,3,2,1,2] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[1,3,2,2,1] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[1,3,2,3] => [3,3,2,1] => 1101010 => [2,1,1,1,1,1] => 1
[1,3,3,1,1] => [3,3,1,1,1] => 11001110 => [2,2,3,1] => 1
[1,3,3,2] => [3,3,2,1] => 1101010 => [2,1,1,1,1,1] => 1
[1,3,4,1] => [4,3,1,1] => 10100110 => [1,1,1,2,2,1] => 1
[1,3,5] => [5,3,1] => 10010010 => [1,2,1,2,1,1] => 1
[1,4,1,1,1,1] => [4,1,1,1,1,1] => 1000111110 => [1,3,5,1] => 1
[1,4,1,1,2] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[1,4,1,2,1] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[1,4,1,3] => [4,3,1,1] => 10100110 => [1,1,1,2,2,1] => 1
[1,4,2,1,1] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[1,4,2,2] => [4,2,2,1] => 10011010 => [1,2,2,1,1,1] => 1
[1,4,3,1] => [4,3,1,1] => 10100110 => [1,1,1,2,2,1] => 1
[1,4,4] => [4,4,1] => 1100010 => [2,3,1,1] => 1
[1,5,1,1,1] => [5,1,1,1,1] => 1000011110 => [1,4,4,1] => 1
[1,5,1,2] => [5,2,1,1] => 100010110 => [1,3,1,1,2,1] => 1
[1,5,2,1] => [5,2,1,1] => 100010110 => [1,3,1,1,2,1] => 1
[1,5,3] => [5,3,1] => 10010010 => [1,2,1,2,1,1] => 1
[1,6,1,1] => [6,1,1,1] => 1000001110 => [1,5,3,1] => 1
[1,6,2] => [6,2,1] => 100001010 => [1,4,1,1,1,1] => 1
[1,7,1] => [7,1,1] => 1000000110 => [1,6,2,1] => 1
[1,8] => [8,1] => 1000000010 => [1,7,1,1] => 1
[2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 1011111110 => [1,1,7,1] => 1
[2,1,1,1,1,1,2] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[2,1,1,1,1,2,1] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[2,1,1,1,1,3] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[2,1,1,1,2,1,1] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[2,1,1,1,2,2] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[2,1,1,1,3,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[2,1,1,1,4] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[2,1,1,2,1,1,1] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[2,1,1,2,1,2] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[2,1,1,2,2,1] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[2,1,1,2,3] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[2,1,1,3,1,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[2,1,1,3,2] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[2,1,1,4,1] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[2,1,1,5] => [5,2,1,1] => 100010110 => [1,3,1,1,2,1] => 1
[2,1,2,1,1,1,1] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[2,1,2,1,1,2] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[2,1,2,1,2,1] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[2,1,2,1,3] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[2,1,2,2,1,1] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[2,1,2,2,2] => [2,2,2,2,1] => 1111010 => [4,1,1,1] => 1
[2,1,2,3,1] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[2,1,2,4] => [4,2,2,1] => 10011010 => [1,2,2,1,1,1] => 1
[2,1,3,1,1,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[2,1,3,1,2] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[2,1,3,2,1] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[2,1,3,3] => [3,3,2,1] => 1101010 => [2,1,1,1,1,1] => 1
[2,1,4,1,1] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[2,1,4,2] => [4,2,2,1] => 10011010 => [1,2,2,1,1,1] => 1
[2,1,5,1] => [5,2,1,1] => 100010110 => [1,3,1,1,2,1] => 1
[2,1,6] => [6,2,1] => 100001010 => [1,4,1,1,1,1] => 1
[2,2,1,1,1,1,1] => [2,2,1,1,1,1,1] => 110111110 => [2,1,5,1] => 1
[2,2,1,1,1,2] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[2,2,1,1,2,1] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[2,2,1,1,3] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[2,2,1,2,1,1] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[2,2,1,2,2] => [2,2,2,2,1] => 1111010 => [4,1,1,1] => 1
[2,2,1,3,1] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[2,2,1,4] => [4,2,2,1] => 10011010 => [1,2,2,1,1,1] => 1
[2,2,2,1,1,1] => [2,2,2,1,1,1] => 11101110 => [3,1,3,1] => 1
[2,2,2,1,2] => [2,2,2,2,1] => 1111010 => [4,1,1,1] => 1
[2,2,2,2,1] => [2,2,2,2,1] => 1111010 => [4,1,1,1] => 1
[2,2,2,3] => [3,2,2,2] => 1011100 => [1,1,3,2] => 2
[2,2,3,1,1] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[2,2,3,2] => [3,2,2,2] => 1011100 => [1,1,3,2] => 2
[2,2,4,1] => [4,2,2,1] => 10011010 => [1,2,2,1,1,1] => 1
[2,2,5] => [5,2,2] => 10001100 => [1,3,2,2] => 2
[2,3,1,1,1,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[2,3,1,1,2] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[2,3,1,2,1] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[2,3,1,3] => [3,3,2,1] => 1101010 => [2,1,1,1,1,1] => 1
[2,3,2,1,1] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[2,3,2,2] => [3,2,2,2] => 1011100 => [1,1,3,2] => 2
[2,3,3,1] => [3,3,2,1] => 1101010 => [2,1,1,1,1,1] => 1
[2,3,4] => [4,3,2] => 1010100 => [1,1,1,1,1,2] => 2
[2,4,1,1,1] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[2,4,1,2] => [4,2,2,1] => 10011010 => [1,2,2,1,1,1] => 1
[2,4,2,1] => [4,2,2,1] => 10011010 => [1,2,2,1,1,1] => 1
[2,4,3] => [4,3,2] => 1010100 => [1,1,1,1,1,2] => 2
[2,5,1,1] => [5,2,1,1] => 100010110 => [1,3,1,1,2,1] => 1
[2,5,2] => [5,2,2] => 10001100 => [1,3,2,2] => 2
[2,6,1] => [6,2,1] => 100001010 => [1,4,1,1,1,1] => 1
[2,7] => [7,2] => 100000100 => [1,5,1,2] => 2
[3,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 1001111110 => [1,2,6,1] => 1
[3,1,1,1,1,2] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[3,1,1,1,2,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[3,1,1,1,3] => [3,3,1,1,1] => 11001110 => [2,2,3,1] => 1
[3,1,1,2,1,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[3,1,1,2,2] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[3,1,1,3,1] => [3,3,1,1,1] => 11001110 => [2,2,3,1] => 1
[3,1,1,4] => [4,3,1,1] => 10100110 => [1,1,1,2,2,1] => 1
[3,1,2,1,1,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[3,1,2,1,2] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[3,1,2,2,1] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[3,1,2,3] => [3,3,2,1] => 1101010 => [2,1,1,1,1,1] => 1
[3,1,3,1,1] => [3,3,1,1,1] => 11001110 => [2,2,3,1] => 1
[3,1,3,2] => [3,3,2,1] => 1101010 => [2,1,1,1,1,1] => 1
[3,1,4,1] => [4,3,1,1] => 10100110 => [1,1,1,2,2,1] => 1
[3,1,5] => [5,3,1] => 10010010 => [1,2,1,2,1,1] => 1
[3,2,1,1,1,1] => [3,2,1,1,1,1] => 101011110 => [1,1,1,1,4,1] => 1
[3,2,1,1,2] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[3,2,1,2,1] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[3,2,1,3] => [3,3,2,1] => 1101010 => [2,1,1,1,1,1] => 1
[3,2,2,1,1] => [3,2,2,1,1] => 10110110 => [1,1,2,1,2,1] => 1
[3,2,2,2] => [3,2,2,2] => 1011100 => [1,1,3,2] => 2
[3,2,3,1] => [3,3,2,1] => 1101010 => [2,1,1,1,1,1] => 1
[3,2,4] => [4,3,2] => 1010100 => [1,1,1,1,1,2] => 2
[3,3,1,1,1] => [3,3,1,1,1] => 11001110 => [2,2,3,1] => 1
[3,3,1,2] => [3,3,2,1] => 1101010 => [2,1,1,1,1,1] => 1
[3,3,2,1] => [3,3,2,1] => 1101010 => [2,1,1,1,1,1] => 1
[3,3,3] => [3,3,3] => 111000 => [3,3] => 3
[3,4,1,1] => [4,3,1,1] => 10100110 => [1,1,1,2,2,1] => 1
[3,4,2] => [4,3,2] => 1010100 => [1,1,1,1,1,2] => 2
[3,5,1] => [5,3,1] => 10010010 => [1,2,1,2,1,1] => 1
[3,6] => [6,3] => 10001000 => [1,3,1,3] => 3
[4,1,1,1,1,1] => [4,1,1,1,1,1] => 1000111110 => [1,3,5,1] => 1
[4,1,1,1,2] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[4,1,1,2,1] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[4,1,1,3] => [4,3,1,1] => 10100110 => [1,1,1,2,2,1] => 1
[4,1,2,1,1] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[4,1,2,2] => [4,2,2,1] => 10011010 => [1,2,2,1,1,1] => 1
[4,1,3,1] => [4,3,1,1] => 10100110 => [1,1,1,2,2,1] => 1
[4,1,4] => [4,4,1] => 1100010 => [2,3,1,1] => 1
[4,2,1,1,1] => [4,2,1,1,1] => 100101110 => [1,2,1,1,3,1] => 1
[4,2,1,2] => [4,2,2,1] => 10011010 => [1,2,2,1,1,1] => 1
[4,2,2,1] => [4,2,2,1] => 10011010 => [1,2,2,1,1,1] => 1
[4,2,3] => [4,3,2] => 1010100 => [1,1,1,1,1,2] => 2
[4,3,1,1] => [4,3,1,1] => 10100110 => [1,1,1,2,2,1] => 1
[4,3,2] => [4,3,2] => 1010100 => [1,1,1,1,1,2] => 2
[4,4,1] => [4,4,1] => 1100010 => [2,3,1,1] => 1
[4,5] => [5,4] => 1010000 => [1,1,1,4] => 4
[5,1,1,1,1] => [5,1,1,1,1] => 1000011110 => [1,4,4,1] => 1
[5,1,1,2] => [5,2,1,1] => 100010110 => [1,3,1,1,2,1] => 1
[5,1,2,1] => [5,2,1,1] => 100010110 => [1,3,1,1,2,1] => 1
[5,1,3] => [5,3,1] => 10010010 => [1,2,1,2,1,1] => 1
[5,2,1,1] => [5,2,1,1] => 100010110 => [1,3,1,1,2,1] => 1
[5,2,2] => [5,2,2] => 10001100 => [1,3,2,2] => 2
[5,3,1] => [5,3,1] => 10010010 => [1,2,1,2,1,1] => 1
[5,4] => [5,4] => 1010000 => [1,1,1,4] => 4
[6,1,1,1] => [6,1,1,1] => 1000001110 => [1,5,3,1] => 1
[6,1,2] => [6,2,1] => 100001010 => [1,4,1,1,1,1] => 1
[6,2,1] => [6,2,1] => 100001010 => [1,4,1,1,1,1] => 1
[6,3] => [6,3] => 10001000 => [1,3,1,3] => 3
[7,1,1] => [7,1,1] => 1000000110 => [1,6,2,1] => 1
[7,2] => [7,2] => 100000100 => [1,5,1,2] => 2
[8,1] => [8,1] => 1000000010 => [1,7,1,1] => 1
[9] => [9] => 1000000000 => [1,9] => 9
[1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 11111111110 => [10,1] => 1
[1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1] => 10111111110 => [1,1,8,1] => 1
[1,1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1,1] => 10111111110 => [1,1,8,1] => 1
[1,1,1,1,1,1,1,3] => [3,1,1,1,1,1,1,1] => 10011111110 => [1,2,7,1] => 1
[1,1,1,1,1,1,2,1,1] => [2,1,1,1,1,1,1,1,1] => 10111111110 => [1,1,8,1] => 1
[1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,1,1,1,1,1,3,1] => [3,1,1,1,1,1,1,1] => 10011111110 => [1,2,7,1] => 1
[1,1,1,1,1,1,4] => [4,1,1,1,1,1,1] => 10001111110 => [1,3,6,1] => 1
[1,1,1,1,1,2,1,1,1] => [2,1,1,1,1,1,1,1,1] => 10111111110 => [1,1,8,1] => 1
[1,1,1,1,1,2,1,2] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,1,1,1,1,2,2,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,1,1,1,1,2,3] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,1,1,1,3,1,1] => [3,1,1,1,1,1,1,1] => 10011111110 => [1,2,7,1] => 1
[1,1,1,1,1,3,2] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,1,1,1,4,1] => [4,1,1,1,1,1,1] => 10001111110 => [1,3,6,1] => 1
[1,1,1,1,1,5] => [5,1,1,1,1,1] => 10000111110 => [1,4,5,1] => 1
[1,1,1,1,2,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 10111111110 => [1,1,8,1] => 1
[1,1,1,1,2,1,1,2] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,1,1,1,2,1,2,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,1,1,1,2,1,3] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,1,1,2,2,1,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,1,1,1,2,2,2] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,1,1,1,2,3,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,1,1,2,4] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,1,1,1,3,1,1,1] => [3,1,1,1,1,1,1,1] => 10011111110 => [1,2,7,1] => 1
[1,1,1,1,3,1,2] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,1,1,3,2,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,1,1,3,3] => [3,3,1,1,1,1] => 110011110 => [2,2,4,1] => 1
[1,1,1,1,4,1,1] => [4,1,1,1,1,1,1] => 10001111110 => [1,3,6,1] => 1
[1,1,1,1,4,2] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,1,1,1,5,1] => [5,1,1,1,1,1] => 10000111110 => [1,4,5,1] => 1
[1,1,1,1,6] => [6,1,1,1,1] => 10000011110 => [1,5,4,1] => 1
[1,1,1,2,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 10111111110 => [1,1,8,1] => 1
[1,1,1,2,1,1,1,2] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,1,1,2,1,1,2,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,1,1,2,1,1,3] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,1,2,1,2,1,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,1,1,2,1,2,2] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,1,1,2,1,3,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,1,2,1,4] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,1,1,2,2,1,1,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,1,1,2,2,1,2] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,1,1,2,2,2,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,1,1,2,2,3] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,1,1,2,3,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,1,2,3,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,1,1,2,4,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,1,1,2,5] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[1,1,1,3,1,1,1,1] => [3,1,1,1,1,1,1,1] => 10011111110 => [1,2,7,1] => 1
[1,1,1,3,1,1,2] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,1,3,1,2,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,1,3,1,3] => [3,3,1,1,1,1] => 110011110 => [2,2,4,1] => 1
[1,1,1,3,2,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,1,3,2,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,1,1,3,3,1] => [3,3,1,1,1,1] => 110011110 => [2,2,4,1] => 1
[1,1,1,3,4] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[1,1,1,4,1,1,1] => [4,1,1,1,1,1,1] => 10001111110 => [1,3,6,1] => 1
[1,1,1,4,1,2] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,1,1,4,2,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,1,1,4,3] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[1,1,1,5,1,1] => [5,1,1,1,1,1] => 10000111110 => [1,4,5,1] => 1
[1,1,1,5,2] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[1,1,1,6,1] => [6,1,1,1,1] => 10000011110 => [1,5,4,1] => 1
[1,1,1,7] => [7,1,1,1] => 10000001110 => [1,6,3,1] => 1
[1,1,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 10111111110 => [1,1,8,1] => 1
[1,1,2,1,1,1,1,2] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,1,2,1,1,1,2,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,1,2,1,1,1,3] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,2,1,1,2,1,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,1,2,1,1,2,2] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,1,2,1,1,3,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,2,1,1,4] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,1,2,1,2,1,1,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,1,2,1,2,1,2] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,1,2,1,2,2,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,1,2,1,2,3] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,1,2,1,3,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,2,1,3,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,1,2,1,4,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,1,2,1,5] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[1,1,2,2,1,1,1,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,1,2,2,1,1,2] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,1,2,2,1,2,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,1,2,2,1,3] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,1,2,2,2,1,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,1,2,2,2,2] => [2,2,2,2,1,1] => 11110110 => [4,1,2,1] => 1
[1,1,2,2,3,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,1,2,2,4] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[1,1,2,3,1,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,2,3,1,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,1,2,3,2,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,1,2,3,3] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[1,1,2,4,1,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,1,2,4,2] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[1,1,2,5,1] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[1,1,2,6] => [6,2,1,1] => 1000010110 => [1,4,1,1,2,1] => 1
[1,1,3,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 10011111110 => [1,2,7,1] => 1
[1,1,3,1,1,1,2] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,3,1,1,2,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,3,1,1,3] => [3,3,1,1,1,1] => 110011110 => [2,2,4,1] => 1
[1,1,3,1,2,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,3,1,2,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,1,3,1,3,1] => [3,3,1,1,1,1] => 110011110 => [2,2,4,1] => 1
[1,1,3,1,4] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[1,1,3,2,1,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,1,3,2,1,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,1,3,2,2,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,1,3,2,3] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[1,1,3,3,1,1] => [3,3,1,1,1,1] => 110011110 => [2,2,4,1] => 1
[1,1,3,3,2] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[1,1,3,4,1] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[1,1,3,5] => [5,3,1,1] => 100100110 => [1,2,1,2,2,1] => 1
[1,1,4,1,1,1,1] => [4,1,1,1,1,1,1] => 10001111110 => [1,3,6,1] => 1
[1,1,4,1,1,2] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,1,4,1,2,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,1,4,1,3] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[1,1,4,2,1,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,1,4,2,2] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[1,1,4,3,1] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[1,1,4,4] => [4,4,1,1] => 11000110 => [2,3,2,1] => 1
[1,1,5,1,1,1] => [5,1,1,1,1,1] => 10000111110 => [1,4,5,1] => 1
[1,1,5,1,2] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[1,1,5,2,1] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[1,1,5,3] => [5,3,1,1] => 100100110 => [1,2,1,2,2,1] => 1
[1,1,6,1,1] => [6,1,1,1,1] => 10000011110 => [1,5,4,1] => 1
[1,1,6,2] => [6,2,1,1] => 1000010110 => [1,4,1,1,2,1] => 1
[1,1,7,1] => [7,1,1,1] => 10000001110 => [1,6,3,1] => 1
[1,1,8] => [8,1,1] => 10000000110 => [1,7,2,1] => 1
[1,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 10111111110 => [1,1,8,1] => 1
[1,2,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,2,1,1,1,1,2,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,2,1,1,1,1,3] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,2,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,2,1,1,1,2,2] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,2,1,1,1,3,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,2,1,1,1,4] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,2,1,1,2,1,1,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,2,1,1,2,1,2] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,2,1,1,2,2,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,2,1,1,2,3] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,2,1,1,3,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,2,1,1,3,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,2,1,1,4,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,2,1,1,5] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[1,2,1,2,1,1,1,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,2,1,2,1,1,2] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,2,1,2,1,2,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,2,1,2,1,3] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,2,1,2,2,1,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,2,1,2,2,2] => [2,2,2,2,1,1] => 11110110 => [4,1,2,1] => 1
[1,2,1,2,3,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,2,1,2,4] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[1,2,1,3,1,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,2,1,3,1,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,2,1,3,2,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,2,1,3,3] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[1,2,1,4,1,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,2,1,4,2] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[1,2,1,5,1] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[1,2,1,6] => [6,2,1,1] => 1000010110 => [1,4,1,1,2,1] => 1
[1,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[1,2,2,1,1,1,2] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,2,2,1,1,2,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,2,2,1,1,3] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,2,2,1,2,1,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,2,2,1,2,2] => [2,2,2,2,1,1] => 11110110 => [4,1,2,1] => 1
[1,2,2,1,3,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,2,2,1,4] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[1,2,2,2,1,1,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[1,2,2,2,1,2] => [2,2,2,2,1,1] => 11110110 => [4,1,2,1] => 1
[1,2,2,2,2,1] => [2,2,2,2,1,1] => 11110110 => [4,1,2,1] => 1
[1,2,2,2,3] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[1,2,2,3,1,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,2,2,3,2] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[1,2,2,4,1] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[1,2,2,5] => [5,2,2,1] => 100011010 => [1,3,2,1,1,1] => 1
[1,2,3,1,1,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,2,3,1,1,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,2,3,1,2,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,2,3,1,3] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[1,2,3,2,1,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,2,3,2,2] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[1,2,3,3,1] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[1,2,3,4] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[1,2,4,1,1,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,2,4,1,2] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[1,2,4,2,1] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[1,2,4,3] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[1,2,5,1,1] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[1,2,5,2] => [5,2,2,1] => 100011010 => [1,3,2,1,1,1] => 1
[1,2,6,1] => [6,2,1,1] => 1000010110 => [1,4,1,1,2,1] => 1
[1,2,7] => [7,2,1] => 1000001010 => [1,5,1,1,1,1] => 1
[1,3,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 10011111110 => [1,2,7,1] => 1
[1,3,1,1,1,1,2] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,3,1,1,1,2,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,3,1,1,1,3] => [3,3,1,1,1,1] => 110011110 => [2,2,4,1] => 1
[1,3,1,1,2,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,3,1,1,2,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,3,1,1,3,1] => [3,3,1,1,1,1] => 110011110 => [2,2,4,1] => 1
[1,3,1,1,4] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[1,3,1,2,1,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,3,1,2,1,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,3,1,2,2,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,3,1,2,3] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[1,3,1,3,1,1] => [3,3,1,1,1,1] => 110011110 => [2,2,4,1] => 1
[1,3,1,3,2] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[1,3,1,4,1] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[1,3,1,5] => [5,3,1,1] => 100100110 => [1,2,1,2,2,1] => 1
[1,3,2,1,1,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[1,3,2,1,1,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,3,2,1,2,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,3,2,1,3] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[1,3,2,2,1,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[1,3,2,2,2] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[1,3,2,3,1] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[1,3,2,4] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[1,3,3,1,1,1] => [3,3,1,1,1,1] => 110011110 => [2,2,4,1] => 1
[1,3,3,1,2] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[1,3,3,2,1] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[1,3,3,3] => [3,3,3,1] => 1110010 => [3,2,1,1] => 1
[1,3,4,1,1] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[1,3,4,2] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[1,3,5,1] => [5,3,1,1] => 100100110 => [1,2,1,2,2,1] => 1
[1,3,6] => [6,3,1] => 100010010 => [1,3,1,2,1,1] => 1
[1,4,1,1,1,1,1] => [4,1,1,1,1,1,1] => 10001111110 => [1,3,6,1] => 1
[1,4,1,1,1,2] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,4,1,1,2,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,4,1,1,3] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[1,4,1,2,1,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,4,1,2,2] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[1,4,1,3,1] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[1,4,1,4] => [4,4,1,1] => 11000110 => [2,3,2,1] => 1
[1,4,2,1,1,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[1,4,2,1,2] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[1,4,2,2,1] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[1,4,2,3] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[1,4,3,1,1] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[1,4,3,2] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[1,4,4,1] => [4,4,1,1] => 11000110 => [2,3,2,1] => 1
[1,4,5] => [5,4,1] => 10100010 => [1,1,1,3,1,1] => 1
[1,5,1,1,1,1] => [5,1,1,1,1,1] => 10000111110 => [1,4,5,1] => 1
[1,5,1,1,2] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[1,5,1,2,1] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[1,5,1,3] => [5,3,1,1] => 100100110 => [1,2,1,2,2,1] => 1
[1,5,2,1,1] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[1,5,2,2] => [5,2,2,1] => 100011010 => [1,3,2,1,1,1] => 1
[1,5,3,1] => [5,3,1,1] => 100100110 => [1,2,1,2,2,1] => 1
[1,5,4] => [5,4,1] => 10100010 => [1,1,1,3,1,1] => 1
[1,6,1,1,1] => [6,1,1,1,1] => 10000011110 => [1,5,4,1] => 1
[1,6,1,2] => [6,2,1,1] => 1000010110 => [1,4,1,1,2,1] => 1
[1,6,2,1] => [6,2,1,1] => 1000010110 => [1,4,1,1,2,1] => 1
[1,6,3] => [6,3,1] => 100010010 => [1,3,1,2,1,1] => 1
[1,7,1,1] => [7,1,1,1] => 10000001110 => [1,6,3,1] => 1
[1,7,2] => [7,2,1] => 1000001010 => [1,5,1,1,1,1] => 1
[1,8,1] => [8,1,1] => 10000000110 => [1,7,2,1] => 1
[1,9] => [9,1] => 10000000010 => [1,8,1,1] => 1
[2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 10111111110 => [1,1,8,1] => 1
[2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[2,1,1,1,1,1,2,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[2,1,1,1,1,1,3] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[2,1,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[2,1,1,1,1,2,2] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[2,1,1,1,1,3,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[2,1,1,1,1,4] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[2,1,1,1,2,1,1,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[2,1,1,1,2,1,2] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[2,1,1,1,2,2,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[2,1,1,1,2,3] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,1,1,1,3,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[2,1,1,1,3,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,1,1,1,4,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[2,1,1,1,5] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[2,1,1,2,1,1,1,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[2,1,1,2,1,1,2] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[2,1,1,2,1,2,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[2,1,1,2,1,3] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,1,1,2,2,1,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[2,1,1,2,2,2] => [2,2,2,2,1,1] => 11110110 => [4,1,2,1] => 1
[2,1,1,2,3,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,1,1,2,4] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[2,1,1,3,1,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[2,1,1,3,1,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,1,1,3,2,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,1,1,3,3] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[2,1,1,4,1,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[2,1,1,4,2] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[2,1,1,5,1] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[2,1,1,6] => [6,2,1,1] => 1000010110 => [1,4,1,1,2,1] => 1
[2,1,2,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[2,1,2,1,1,1,2] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[2,1,2,1,1,2,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[2,1,2,1,1,3] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,1,2,1,2,1,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[2,1,2,1,2,2] => [2,2,2,2,1,1] => 11110110 => [4,1,2,1] => 1
[2,1,2,1,3,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,1,2,1,4] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[2,1,2,2,1,1,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[2,1,2,2,1,2] => [2,2,2,2,1,1] => 11110110 => [4,1,2,1] => 1
[2,1,2,2,2,1] => [2,2,2,2,1,1] => 11110110 => [4,1,2,1] => 1
[2,1,2,2,3] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[2,1,2,3,1,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,1,2,3,2] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[2,1,2,4,1] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[2,1,2,5] => [5,2,2,1] => 100011010 => [1,3,2,1,1,1] => 1
[2,1,3,1,1,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[2,1,3,1,1,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,1,3,1,2,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,1,3,1,3] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[2,1,3,2,1,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,1,3,2,2] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[2,1,3,3,1] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[2,1,3,4] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[2,1,4,1,1,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[2,1,4,1,2] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[2,1,4,2,1] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[2,1,4,3] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[2,1,5,1,1] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[2,1,5,2] => [5,2,2,1] => 100011010 => [1,3,2,1,1,1] => 1
[2,1,6,1] => [6,2,1,1] => 1000010110 => [1,4,1,1,2,1] => 1
[2,1,7] => [7,2,1] => 1000001010 => [1,5,1,1,1,1] => 1
[2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 1101111110 => [2,1,6,1] => 1
[2,2,1,1,1,1,2] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[2,2,1,1,1,2,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[2,2,1,1,1,3] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,2,1,1,2,1,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[2,2,1,1,2,2] => [2,2,2,2,1,1] => 11110110 => [4,1,2,1] => 1
[2,2,1,1,3,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,2,1,1,4] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[2,2,1,2,1,1,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[2,2,1,2,1,2] => [2,2,2,2,1,1] => 11110110 => [4,1,2,1] => 1
[2,2,1,2,2,1] => [2,2,2,2,1,1] => 11110110 => [4,1,2,1] => 1
[2,2,1,2,3] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[2,2,1,3,1,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,2,1,3,2] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[2,2,1,4,1] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[2,2,1,5] => [5,2,2,1] => 100011010 => [1,3,2,1,1,1] => 1
[2,2,2,1,1,1,1] => [2,2,2,1,1,1,1] => 111011110 => [3,1,4,1] => 1
[2,2,2,1,1,2] => [2,2,2,2,1,1] => 11110110 => [4,1,2,1] => 1
[2,2,2,1,2,1] => [2,2,2,2,1,1] => 11110110 => [4,1,2,1] => 1
[2,2,2,1,3] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[2,2,2,2,1,1] => [2,2,2,2,1,1] => 11110110 => [4,1,2,1] => 1
[2,2,2,2,2] => [2,2,2,2,2] => 1111100 => [5,2] => 2
[2,2,2,3,1] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[2,2,2,4] => [4,2,2,2] => 10011100 => [1,2,3,2] => 2
[2,2,3,1,1,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,2,3,1,2] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[2,2,3,2,1] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[2,2,3,3] => [3,3,2,2] => 1101100 => [2,1,2,2] => 2
[2,2,4,1,1] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[2,2,4,2] => [4,2,2,2] => 10011100 => [1,2,3,2] => 2
[2,2,5,1] => [5,2,2,1] => 100011010 => [1,3,2,1,1,1] => 1
[2,2,6] => [6,2,2] => 100001100 => [1,4,2,2] => 2
[2,3,1,1,1,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[2,3,1,1,1,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,3,1,1,2,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,3,1,1,3] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[2,3,1,2,1,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,3,1,2,2] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[2,3,1,3,1] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[2,3,1,4] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[2,3,2,1,1,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[2,3,2,1,2] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[2,3,2,2,1] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[2,3,2,3] => [3,3,2,2] => 1101100 => [2,1,2,2] => 2
[2,3,3,1,1] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[2,3,3,2] => [3,3,2,2] => 1101100 => [2,1,2,2] => 2
[2,3,4,1] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[2,3,5] => [5,3,2] => 10010100 => [1,2,1,1,1,2] => 2
[2,4,1,1,1,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[2,4,1,1,2] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[2,4,1,2,1] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[2,4,1,3] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[2,4,2,1,1] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[2,4,2,2] => [4,2,2,2] => 10011100 => [1,2,3,2] => 2
[2,4,3,1] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[2,4,4] => [4,4,2] => 1100100 => [2,2,1,2] => 2
[2,5,1,1,1] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[2,5,1,2] => [5,2,2,1] => 100011010 => [1,3,2,1,1,1] => 1
[2,5,2,1] => [5,2,2,1] => 100011010 => [1,3,2,1,1,1] => 1
[2,5,3] => [5,3,2] => 10010100 => [1,2,1,1,1,2] => 2
[2,6,1,1] => [6,2,1,1] => 1000010110 => [1,4,1,1,2,1] => 1
[2,6,2] => [6,2,2] => 100001100 => [1,4,2,2] => 2
[2,7,1] => [7,2,1] => 1000001010 => [1,5,1,1,1,1] => 1
[2,8] => [8,2] => 1000000100 => [1,6,1,2] => 2
[3,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 10011111110 => [1,2,7,1] => 1
[3,1,1,1,1,1,2] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[3,1,1,1,1,2,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[3,1,1,1,1,3] => [3,3,1,1,1,1] => 110011110 => [2,2,4,1] => 1
[3,1,1,1,2,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[3,1,1,1,2,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[3,1,1,1,3,1] => [3,3,1,1,1,1] => 110011110 => [2,2,4,1] => 1
[3,1,1,1,4] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[3,1,1,2,1,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[3,1,1,2,1,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[3,1,1,2,2,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[3,1,1,2,3] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[3,1,1,3,1,1] => [3,3,1,1,1,1] => 110011110 => [2,2,4,1] => 1
[3,1,1,3,2] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[3,1,1,4,1] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[3,1,1,5] => [5,3,1,1] => 100100110 => [1,2,1,2,2,1] => 1
[3,1,2,1,1,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[3,1,2,1,1,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[3,1,2,1,2,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[3,1,2,1,3] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[3,1,2,2,1,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[3,1,2,2,2] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[3,1,2,3,1] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[3,1,2,4] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[3,1,3,1,1,1] => [3,3,1,1,1,1] => 110011110 => [2,2,4,1] => 1
[3,1,3,1,2] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[3,1,3,2,1] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[3,1,3,3] => [3,3,3,1] => 1110010 => [3,2,1,1] => 1
[3,1,4,1,1] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[3,1,4,2] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[3,1,5,1] => [5,3,1,1] => 100100110 => [1,2,1,2,2,1] => 1
[3,1,6] => [6,3,1] => 100010010 => [1,3,1,2,1,1] => 1
[3,2,1,1,1,1,1] => [3,2,1,1,1,1,1] => 1010111110 => [1,1,1,1,5,1] => 1
[3,2,1,1,1,2] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[3,2,1,1,2,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[3,2,1,1,3] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[3,2,1,2,1,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[3,2,1,2,2] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[3,2,1,3,1] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[3,2,1,4] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[3,2,2,1,1,1] => [3,2,2,1,1,1] => 101101110 => [1,1,2,1,3,1] => 1
[3,2,2,1,2] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[3,2,2,2,1] => [3,2,2,2,1] => 10111010 => [1,1,3,1,1,1] => 1
[3,2,2,3] => [3,3,2,2] => 1101100 => [2,1,2,2] => 2
[3,2,3,1,1] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[3,2,3,2] => [3,3,2,2] => 1101100 => [2,1,2,2] => 2
[3,2,4,1] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[3,2,5] => [5,3,2] => 10010100 => [1,2,1,1,1,2] => 2
[3,3,1,1,1,1] => [3,3,1,1,1,1] => 110011110 => [2,2,4,1] => 1
[3,3,1,1,2] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[3,3,1,2,1] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[3,3,1,3] => [3,3,3,1] => 1110010 => [3,2,1,1] => 1
[3,3,2,1,1] => [3,3,2,1,1] => 11010110 => [2,1,1,1,2,1] => 1
[3,3,2,2] => [3,3,2,2] => 1101100 => [2,1,2,2] => 2
[3,3,3,1] => [3,3,3,1] => 1110010 => [3,2,1,1] => 1
[3,3,4] => [4,3,3] => 1011000 => [1,1,2,3] => 3
[3,4,1,1,1] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[3,4,1,2] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[3,4,2,1] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[3,4,3] => [4,3,3] => 1011000 => [1,1,2,3] => 3
[3,5,1,1] => [5,3,1,1] => 100100110 => [1,2,1,2,2,1] => 1
[3,5,2] => [5,3,2] => 10010100 => [1,2,1,1,1,2] => 2
[3,6,1] => [6,3,1] => 100010010 => [1,3,1,2,1,1] => 1
[3,7] => [7,3] => 100001000 => [1,4,1,3] => 3
[4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => 10001111110 => [1,3,6,1] => 1
[4,1,1,1,1,2] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[4,1,1,1,2,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[4,1,1,1,3] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[4,1,1,2,1,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[4,1,1,2,2] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[4,1,1,3,1] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[4,1,1,4] => [4,4,1,1] => 11000110 => [2,3,2,1] => 1
[4,1,2,1,1,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[4,1,2,1,2] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[4,1,2,2,1] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[4,1,2,3] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[4,1,3,1,1] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[4,1,3,2] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[4,1,4,1] => [4,4,1,1] => 11000110 => [2,3,2,1] => 1
[4,1,5] => [5,4,1] => 10100010 => [1,1,1,3,1,1] => 1
[4,2,1,1,1,1] => [4,2,1,1,1,1] => 1001011110 => [1,2,1,1,4,1] => 1
[4,2,1,1,2] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[4,2,1,2,1] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[4,2,1,3] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[4,2,2,1,1] => [4,2,2,1,1] => 100110110 => [1,2,2,1,2,1] => 1
[4,2,2,2] => [4,2,2,2] => 10011100 => [1,2,3,2] => 2
[4,2,3,1] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[4,2,4] => [4,4,2] => 1100100 => [2,2,1,2] => 2
[4,3,1,1,1] => [4,3,1,1,1] => 101001110 => [1,1,1,2,3,1] => 1
[4,3,1,2] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[4,3,2,1] => [4,3,2,1] => 10101010 => [1,1,1,1,1,1,1,1] => 1
[4,3,3] => [4,3,3] => 1011000 => [1,1,2,3] => 3
[4,4,1,1] => [4,4,1,1] => 11000110 => [2,3,2,1] => 1
[4,4,2] => [4,4,2] => 1100100 => [2,2,1,2] => 2
[4,5,1] => [5,4,1] => 10100010 => [1,1,1,3,1,1] => 1
[4,6] => [6,4] => 10010000 => [1,2,1,4] => 4
[5,1,1,1,1,1] => [5,1,1,1,1,1] => 10000111110 => [1,4,5,1] => 1
[5,1,1,1,2] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[5,1,1,2,1] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[5,1,1,3] => [5,3,1,1] => 100100110 => [1,2,1,2,2,1] => 1
[5,1,2,1,1] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[5,1,2,2] => [5,2,2,1] => 100011010 => [1,3,2,1,1,1] => 1
[5,1,3,1] => [5,3,1,1] => 100100110 => [1,2,1,2,2,1] => 1
[5,1,4] => [5,4,1] => 10100010 => [1,1,1,3,1,1] => 1
[5,2,1,1,1] => [5,2,1,1,1] => 1000101110 => [1,3,1,1,3,1] => 1
[5,2,1,2] => [5,2,2,1] => 100011010 => [1,3,2,1,1,1] => 1
[5,2,2,1] => [5,2,2,1] => 100011010 => [1,3,2,1,1,1] => 1
[5,2,3] => [5,3,2] => 10010100 => [1,2,1,1,1,2] => 2
[5,3,1,1] => [5,3,1,1] => 100100110 => [1,2,1,2,2,1] => 1
[5,3,2] => [5,3,2] => 10010100 => [1,2,1,1,1,2] => 2
[5,4,1] => [5,4,1] => 10100010 => [1,1,1,3,1,1] => 1
[5,5] => [5,5] => 1100000 => [2,5] => 5
[6,1,1,1,1] => [6,1,1,1,1] => 10000011110 => [1,5,4,1] => 1
[6,1,1,2] => [6,2,1,1] => 1000010110 => [1,4,1,1,2,1] => 1
[6,1,2,1] => [6,2,1,1] => 1000010110 => [1,4,1,1,2,1] => 1
[6,1,3] => [6,3,1] => 100010010 => [1,3,1,2,1,1] => 1
[6,2,1,1] => [6,2,1,1] => 1000010110 => [1,4,1,1,2,1] => 1
[6,2,2] => [6,2,2] => 100001100 => [1,4,2,2] => 2
[6,3,1] => [6,3,1] => 100010010 => [1,3,1,2,1,1] => 1
[6,4] => [6,4] => 10010000 => [1,2,1,4] => 4
[7,1,1,1] => [7,1,1,1] => 10000001110 => [1,6,3,1] => 1
[7,1,2] => [7,2,1] => 1000001010 => [1,5,1,1,1,1] => 1
[7,2,1] => [7,2,1] => 1000001010 => [1,5,1,1,1,1] => 1
[7,3] => [7,3] => 100001000 => [1,4,1,3] => 3
[8,1,1] => [8,1,1] => 10000000110 => [1,7,2,1] => 1
[8,2] => [8,2] => 1000000100 => [1,6,1,2] => 2
[9,1] => [9,1] => 10000000010 => [1,8,1,1] => 1
[10] => [10] => 10000000000 => [1,10] => 10
[2,2,4,4] => [4,4,2,2] => 11001100 => [2,2,2,2] => 2
[3,3,3,3] => [3,3,3,3] => 1111000 => [4,3] => 3
[3,2,3,4] => [4,3,3,2] => 10110100 => [1,1,2,1,1,2] => 2
[4,4,2,2] => [4,4,2,2] => 11001100 => [2,2,2,2] => 2
[4,3,2,3] => [4,3,3,2] => 10110100 => [1,1,2,1,1,2] => 2
[4,2,2,4] => [4,4,2,2] => 11001100 => [2,2,2,2] => 2
[1,3,4,4] => [4,4,3,1] => 11010010 => [2,1,1,2,1,1] => 1
[1,4,4,3] => [4,4,3,1] => 11010010 => [2,1,1,2,1,1] => 1
[1,4,3,4] => [4,4,3,1] => 11010010 => [2,1,1,2,1,1] => 1
[2,3,2,3,2] => [3,3,2,2,2] => 11011100 => [2,1,3,2] => 2
[2,3,3,2,2] => [3,3,2,2,2] => 11011100 => [2,1,3,2] => 2
[2,3,4,3] => [4,3,3,2] => 10110100 => [1,1,2,1,1,2] => 2
[2,4,4,2] => [4,4,2,2] => 11001100 => [2,2,2,2] => 2
[2,4,3,3] => [4,3,3,2] => 10110100 => [1,1,2,1,1,2] => 2
[2,2,3,3,2] => [3,3,2,2,2] => 11011100 => [2,1,3,2] => 2
[2,2,2,3,3] => [3,3,2,2,2] => 11011100 => [2,1,3,2] => 2
[2,2,3,2,3] => [3,3,2,2,2] => 11011100 => [2,1,3,2] => 2
[2,3,2,2,3] => [3,3,2,2,2] => 11011100 => [2,1,3,2] => 2
[2,3,3,4] => [4,3,3,2] => 10110100 => [1,1,2,1,1,2] => 2
[2,4,2,4] => [4,4,2,2] => 11001100 => [2,2,2,2] => 2
[2,5,5] => [5,5,2] => 11000100 => [2,3,1,2] => 2
[3,4,3,2] => [4,3,3,2] => 10110100 => [1,1,2,1,1,2] => 2
[3,3,4,2] => [4,3,3,2] => 10110100 => [1,1,2,1,1,2] => 2
[3,3,2,2,2] => [3,3,2,2,2] => 11011100 => [2,1,3,2] => 2
[3,4,4,1] => [4,4,3,1] => 11010010 => [2,1,1,2,1,1] => 1
[3,4,2,3] => [4,3,3,2] => 10110100 => [1,1,2,1,1,2] => 2
[3,2,3,2,2] => [3,3,2,2,2] => 11011100 => [2,1,3,2] => 2
[3,2,4,3] => [4,3,3,2] => 10110100 => [1,1,2,1,1,2] => 2
[3,2,2,3,2] => [3,3,2,2,2] => 11011100 => [2,1,3,2] => 2
[3,2,2,2,3] => [3,3,2,2,2] => 11011100 => [2,1,3,2] => 2
[3,3,2,4] => [4,3,3,2] => 10110100 => [1,1,2,1,1,2] => 2
[4,4,3,1] => [4,4,3,1] => 11010010 => [2,1,1,2,1,1] => 1
[4,3,4,1] => [4,4,3,1] => 11010010 => [2,1,1,2,1,1] => 1
[4,3,3,2] => [4,3,3,2] => 10110100 => [1,1,2,1,1,2] => 2
[4,4,4] => [4,4,4] => 1110000 => [3,4] => 4
[4,2,4,2] => [4,4,2,2] => 11001100 => [2,2,2,2] => 2
[4,2,3,3] => [4,3,3,2] => 10110100 => [1,1,2,1,1,2] => 2
[5,5,2] => [5,5,2] => 11000100 => [2,3,1,2] => 2
[5,2,5] => [5,5,2] => 11000100 => [2,3,1,2] => 2
[1,2,2,2,2,2] => [2,2,2,2,2,1] => 11111010 => [5,1,1,1] => 1
[2,2,2,2,2,1] => [2,2,2,2,2,1] => 11111010 => [5,1,1,1] => 1
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Generating function
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Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
1,1 3,0,1 6,1,0,1 13,2,0,0,1 27,3,1,0,0,1 56,5,2,0,0,0,1 115,9,2,1,0,0,0,1 235,15,3,2,0,0,0,0,1 478,25,5,2,1,0,0,0,0,1
$F_{1} = q$
$F_{2} = q + q^{2}$
$F_{3} = 3\ q + q^{3}$
$F_{4} = 6\ q + q^{2} + q^{4}$
$F_{5} = 13\ q + 2\ q^{2} + q^{5}$
$F_{6} = 27\ q + 3\ q^{2} + q^{3} + q^{6}$
$F_{7} = 56\ q + 5\ q^{2} + 2\ q^{3} + q^{7}$
$F_{8} = 115\ q + 9\ q^{2} + 2\ q^{3} + q^{4} + q^{8}$
$F_{9} = 235\ q + 15\ q^{2} + 3\ q^{3} + 2\ q^{4} + q^{9}$
$F_{10} = 478\ q + 25\ q^{2} + 5\ q^{3} + 2\ q^{4} + q^{5} + q^{10}$
Description
The last part of an integer composition.
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.
Map
to partition
Description
Sends a composition to the partition obtained by sorting the entries.
Map
delta morphism
Description
Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
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