Identifier
-
Mp00097:
Binary words
—delta morphism⟶
Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000290: Binary words ⟶ ℤ
Values
0 => [1] => 1 => 0
1 => [1] => 1 => 0
00 => [2] => 10 => 1
01 => [1,1] => 11 => 0
10 => [1,1] => 11 => 0
11 => [2] => 10 => 1
000 => [3] => 100 => 1
001 => [2,1] => 101 => 1
010 => [1,1,1] => 111 => 0
011 => [1,2] => 110 => 2
100 => [1,2] => 110 => 2
101 => [1,1,1] => 111 => 0
110 => [2,1] => 101 => 1
111 => [3] => 100 => 1
0000 => [4] => 1000 => 1
0001 => [3,1] => 1001 => 1
0010 => [2,1,1] => 1011 => 1
0011 => [2,2] => 1010 => 4
0100 => [1,1,2] => 1110 => 3
0101 => [1,1,1,1] => 1111 => 0
0110 => [1,2,1] => 1101 => 2
0111 => [1,3] => 1100 => 2
1000 => [1,3] => 1100 => 2
1001 => [1,2,1] => 1101 => 2
1010 => [1,1,1,1] => 1111 => 0
1011 => [1,1,2] => 1110 => 3
1100 => [2,2] => 1010 => 4
1101 => [2,1,1] => 1011 => 1
1110 => [3,1] => 1001 => 1
1111 => [4] => 1000 => 1
00000 => [5] => 10000 => 1
00001 => [4,1] => 10001 => 1
00010 => [3,1,1] => 10011 => 1
00011 => [3,2] => 10010 => 5
00100 => [2,1,2] => 10110 => 5
00101 => [2,1,1,1] => 10111 => 1
00110 => [2,2,1] => 10101 => 4
00111 => [2,3] => 10100 => 4
01000 => [1,1,3] => 11100 => 3
01001 => [1,1,2,1] => 11101 => 3
01010 => [1,1,1,1,1] => 11111 => 0
01011 => [1,1,1,2] => 11110 => 4
01100 => [1,2,2] => 11010 => 6
01101 => [1,2,1,1] => 11011 => 2
01110 => [1,3,1] => 11001 => 2
01111 => [1,4] => 11000 => 2
10000 => [1,4] => 11000 => 2
10001 => [1,3,1] => 11001 => 2
10010 => [1,2,1,1] => 11011 => 2
10011 => [1,2,2] => 11010 => 6
10100 => [1,1,1,2] => 11110 => 4
10101 => [1,1,1,1,1] => 11111 => 0
10110 => [1,1,2,1] => 11101 => 3
10111 => [1,1,3] => 11100 => 3
11000 => [2,3] => 10100 => 4
11001 => [2,2,1] => 10101 => 4
11010 => [2,1,1,1] => 10111 => 1
11011 => [2,1,2] => 10110 => 5
11100 => [3,2] => 10010 => 5
11101 => [3,1,1] => 10011 => 1
11110 => [4,1] => 10001 => 1
11111 => [5] => 10000 => 1
000000 => [6] => 100000 => 1
000001 => [5,1] => 100001 => 1
000010 => [4,1,1] => 100011 => 1
000011 => [4,2] => 100010 => 6
000100 => [3,1,2] => 100110 => 6
000101 => [3,1,1,1] => 100111 => 1
000110 => [3,2,1] => 100101 => 5
000111 => [3,3] => 100100 => 5
001000 => [2,1,3] => 101100 => 5
001001 => [2,1,2,1] => 101101 => 5
001010 => [2,1,1,1,1] => 101111 => 1
001011 => [2,1,1,2] => 101110 => 6
001100 => [2,2,2] => 101010 => 9
001101 => [2,2,1,1] => 101011 => 4
001110 => [2,3,1] => 101001 => 4
001111 => [2,4] => 101000 => 4
010000 => [1,1,4] => 111000 => 3
010001 => [1,1,3,1] => 111001 => 3
010010 => [1,1,2,1,1] => 111011 => 3
010011 => [1,1,2,2] => 111010 => 8
010100 => [1,1,1,1,2] => 111110 => 5
010101 => [1,1,1,1,1,1] => 111111 => 0
010110 => [1,1,1,2,1] => 111101 => 4
010111 => [1,1,1,3] => 111100 => 4
011000 => [1,2,3] => 110100 => 6
011001 => [1,2,2,1] => 110101 => 6
011010 => [1,2,1,1,1] => 110111 => 2
011011 => [1,2,1,2] => 110110 => 7
011100 => [1,3,2] => 110010 => 7
011101 => [1,3,1,1] => 110011 => 2
011110 => [1,4,1] => 110001 => 2
011111 => [1,5] => 110000 => 2
100000 => [1,5] => 110000 => 2
100001 => [1,4,1] => 110001 => 2
100010 => [1,3,1,1] => 110011 => 2
100011 => [1,3,2] => 110010 => 7
100100 => [1,2,1,2] => 110110 => 7
100101 => [1,2,1,1,1] => 110111 => 2
100110 => [1,2,2,1] => 110101 => 6
>>> Load all 1208 entries. <<<100111 => [1,2,3] => 110100 => 6
101000 => [1,1,1,3] => 111100 => 4
101001 => [1,1,1,2,1] => 111101 => 4
101010 => [1,1,1,1,1,1] => 111111 => 0
101011 => [1,1,1,1,2] => 111110 => 5
101100 => [1,1,2,2] => 111010 => 8
101101 => [1,1,2,1,1] => 111011 => 3
101110 => [1,1,3,1] => 111001 => 3
101111 => [1,1,4] => 111000 => 3
110000 => [2,4] => 101000 => 4
110001 => [2,3,1] => 101001 => 4
110010 => [2,2,1,1] => 101011 => 4
110011 => [2,2,2] => 101010 => 9
110100 => [2,1,1,2] => 101110 => 6
110101 => [2,1,1,1,1] => 101111 => 1
110110 => [2,1,2,1] => 101101 => 5
110111 => [2,1,3] => 101100 => 5
111000 => [3,3] => 100100 => 5
111001 => [3,2,1] => 100101 => 5
111010 => [3,1,1,1] => 100111 => 1
111011 => [3,1,2] => 100110 => 6
111100 => [4,2] => 100010 => 6
111101 => [4,1,1] => 100011 => 1
111110 => [5,1] => 100001 => 1
111111 => [6] => 100000 => 1
0000000 => [7] => 1000000 => 1
0000001 => [6,1] => 1000001 => 1
0000010 => [5,1,1] => 1000011 => 1
0000011 => [5,2] => 1000010 => 7
0000100 => [4,1,2] => 1000110 => 7
0000101 => [4,1,1,1] => 1000111 => 1
0000110 => [4,2,1] => 1000101 => 6
0000111 => [4,3] => 1000100 => 6
0001000 => [3,1,3] => 1001100 => 6
0001001 => [3,1,2,1] => 1001101 => 6
0001010 => [3,1,1,1,1] => 1001111 => 1
0001011 => [3,1,1,2] => 1001110 => 7
0001100 => [3,2,2] => 1001010 => 11
0001101 => [3,2,1,1] => 1001011 => 5
0001110 => [3,3,1] => 1001001 => 5
0001111 => [3,4] => 1001000 => 5
0010000 => [2,1,4] => 1011000 => 5
0010001 => [2,1,3,1] => 1011001 => 5
0010010 => [2,1,2,1,1] => 1011011 => 5
0010011 => [2,1,2,2] => 1011010 => 11
0010100 => [2,1,1,1,2] => 1011110 => 7
0010101 => [2,1,1,1,1,1] => 1011111 => 1
0010110 => [2,1,1,2,1] => 1011101 => 6
0010111 => [2,1,1,3] => 1011100 => 6
0011000 => [2,2,3] => 1010100 => 9
0011001 => [2,2,2,1] => 1010101 => 9
0011010 => [2,2,1,1,1] => 1010111 => 4
0011011 => [2,2,1,2] => 1010110 => 10
0011100 => [2,3,2] => 1010010 => 10
0011101 => [2,3,1,1] => 1010011 => 4
0011110 => [2,4,1] => 1010001 => 4
0011111 => [2,5] => 1010000 => 4
0100000 => [1,1,5] => 1110000 => 3
0100001 => [1,1,4,1] => 1110001 => 3
0100010 => [1,1,3,1,1] => 1110011 => 3
0100011 => [1,1,3,2] => 1110010 => 9
0100100 => [1,1,2,1,2] => 1110110 => 9
0100101 => [1,1,2,1,1,1] => 1110111 => 3
0100110 => [1,1,2,2,1] => 1110101 => 8
0100111 => [1,1,2,3] => 1110100 => 8
0101000 => [1,1,1,1,3] => 1111100 => 5
0101001 => [1,1,1,1,2,1] => 1111101 => 5
0101010 => [1,1,1,1,1,1,1] => 1111111 => 0
0101011 => [1,1,1,1,1,2] => 1111110 => 6
0101100 => [1,1,1,2,2] => 1111010 => 10
0101101 => [1,1,1,2,1,1] => 1111011 => 4
0101110 => [1,1,1,3,1] => 1111001 => 4
0101111 => [1,1,1,4] => 1111000 => 4
0110000 => [1,2,4] => 1101000 => 6
0110001 => [1,2,3,1] => 1101001 => 6
0110010 => [1,2,2,1,1] => 1101011 => 6
0110011 => [1,2,2,2] => 1101010 => 12
0110100 => [1,2,1,1,2] => 1101110 => 8
0110101 => [1,2,1,1,1,1] => 1101111 => 2
0110110 => [1,2,1,2,1] => 1101101 => 7
0110111 => [1,2,1,3] => 1101100 => 7
0111000 => [1,3,3] => 1100100 => 7
0111001 => [1,3,2,1] => 1100101 => 7
0111010 => [1,3,1,1,1] => 1100111 => 2
0111011 => [1,3,1,2] => 1100110 => 8
0111100 => [1,4,2] => 1100010 => 8
0111101 => [1,4,1,1] => 1100011 => 2
0111110 => [1,5,1] => 1100001 => 2
0111111 => [1,6] => 1100000 => 2
1000000 => [1,6] => 1100000 => 2
1000001 => [1,5,1] => 1100001 => 2
1000010 => [1,4,1,1] => 1100011 => 2
1000011 => [1,4,2] => 1100010 => 8
1000100 => [1,3,1,2] => 1100110 => 8
1000101 => [1,3,1,1,1] => 1100111 => 2
1000110 => [1,3,2,1] => 1100101 => 7
1000111 => [1,3,3] => 1100100 => 7
1001000 => [1,2,1,3] => 1101100 => 7
1001001 => [1,2,1,2,1] => 1101101 => 7
1001010 => [1,2,1,1,1,1] => 1101111 => 2
1001011 => [1,2,1,1,2] => 1101110 => 8
1001100 => [1,2,2,2] => 1101010 => 12
1001101 => [1,2,2,1,1] => 1101011 => 6
1001110 => [1,2,3,1] => 1101001 => 6
1001111 => [1,2,4] => 1101000 => 6
1010000 => [1,1,1,4] => 1111000 => 4
1010001 => [1,1,1,3,1] => 1111001 => 4
1010010 => [1,1,1,2,1,1] => 1111011 => 4
1010011 => [1,1,1,2,2] => 1111010 => 10
1010100 => [1,1,1,1,1,2] => 1111110 => 6
1010101 => [1,1,1,1,1,1,1] => 1111111 => 0
1010110 => [1,1,1,1,2,1] => 1111101 => 5
1010111 => [1,1,1,1,3] => 1111100 => 5
1011000 => [1,1,2,3] => 1110100 => 8
1011001 => [1,1,2,2,1] => 1110101 => 8
1011010 => [1,1,2,1,1,1] => 1110111 => 3
1011011 => [1,1,2,1,2] => 1110110 => 9
1011100 => [1,1,3,2] => 1110010 => 9
1011101 => [1,1,3,1,1] => 1110011 => 3
1011110 => [1,1,4,1] => 1110001 => 3
1011111 => [1,1,5] => 1110000 => 3
1100000 => [2,5] => 1010000 => 4
1100001 => [2,4,1] => 1010001 => 4
1100010 => [2,3,1,1] => 1010011 => 4
1100011 => [2,3,2] => 1010010 => 10
1100100 => [2,2,1,2] => 1010110 => 10
1100101 => [2,2,1,1,1] => 1010111 => 4
1100110 => [2,2,2,1] => 1010101 => 9
1100111 => [2,2,3] => 1010100 => 9
1101000 => [2,1,1,3] => 1011100 => 6
1101001 => [2,1,1,2,1] => 1011101 => 6
1101010 => [2,1,1,1,1,1] => 1011111 => 1
1101011 => [2,1,1,1,2] => 1011110 => 7
1101100 => [2,1,2,2] => 1011010 => 11
1101101 => [2,1,2,1,1] => 1011011 => 5
1101110 => [2,1,3,1] => 1011001 => 5
1101111 => [2,1,4] => 1011000 => 5
1110000 => [3,4] => 1001000 => 5
1110001 => [3,3,1] => 1001001 => 5
1110010 => [3,2,1,1] => 1001011 => 5
1110011 => [3,2,2] => 1001010 => 11
1110100 => [3,1,1,2] => 1001110 => 7
1110101 => [3,1,1,1,1] => 1001111 => 1
1110110 => [3,1,2,1] => 1001101 => 6
1110111 => [3,1,3] => 1001100 => 6
1111000 => [4,3] => 1000100 => 6
1111001 => [4,2,1] => 1000101 => 6
1111010 => [4,1,1,1] => 1000111 => 1
1111011 => [4,1,2] => 1000110 => 7
1111100 => [5,2] => 1000010 => 7
1111101 => [5,1,1] => 1000011 => 1
1111110 => [6,1] => 1000001 => 1
1111111 => [7] => 1000000 => 1
00000000 => [8] => 10000000 => 1
00000001 => [7,1] => 10000001 => 1
00000010 => [6,1,1] => 10000011 => 1
00000011 => [6,2] => 10000010 => 8
00000100 => [5,1,2] => 10000110 => 8
00000101 => [5,1,1,1] => 10000111 => 1
00000110 => [5,2,1] => 10000101 => 7
00000111 => [5,3] => 10000100 => 7
00001000 => [4,1,3] => 10001100 => 7
00001001 => [4,1,2,1] => 10001101 => 7
00001010 => [4,1,1,1,1] => 10001111 => 1
00001011 => [4,1,1,2] => 10001110 => 8
00001100 => [4,2,2] => 10001010 => 13
00001101 => [4,2,1,1] => 10001011 => 6
00001110 => [4,3,1] => 10001001 => 6
00001111 => [4,4] => 10001000 => 6
00010000 => [3,1,4] => 10011000 => 6
00010001 => [3,1,3,1] => 10011001 => 6
00010010 => [3,1,2,1,1] => 10011011 => 6
00010011 => [3,1,2,2] => 10011010 => 13
00010100 => [3,1,1,1,2] => 10011110 => 8
00010101 => [3,1,1,1,1,1] => 10011111 => 1
00010110 => [3,1,1,2,1] => 10011101 => 7
00010111 => [3,1,1,3] => 10011100 => 7
00011000 => [3,2,3] => 10010100 => 11
00011001 => [3,2,2,1] => 10010101 => 11
00011010 => [3,2,1,1,1] => 10010111 => 5
00011011 => [3,2,1,2] => 10010110 => 12
00011100 => [3,3,2] => 10010010 => 12
00011101 => [3,3,1,1] => 10010011 => 5
00011110 => [3,4,1] => 10010001 => 5
00011111 => [3,5] => 10010000 => 5
00100000 => [2,1,5] => 10110000 => 5
00100001 => [2,1,4,1] => 10110001 => 5
00100010 => [2,1,3,1,1] => 10110011 => 5
00100011 => [2,1,3,2] => 10110010 => 12
00100100 => [2,1,2,1,2] => 10110110 => 12
00100101 => [2,1,2,1,1,1] => 10110111 => 5
00100110 => [2,1,2,2,1] => 10110101 => 11
00100111 => [2,1,2,3] => 10110100 => 11
00101000 => [2,1,1,1,3] => 10111100 => 7
00101001 => [2,1,1,1,2,1] => 10111101 => 7
00101010 => [2,1,1,1,1,1,1] => 10111111 => 1
00101011 => [2,1,1,1,1,2] => 10111110 => 8
00101100 => [2,1,1,2,2] => 10111010 => 13
00101101 => [2,1,1,2,1,1] => 10111011 => 6
00101110 => [2,1,1,3,1] => 10111001 => 6
00101111 => [2,1,1,4] => 10111000 => 6
00110000 => [2,2,4] => 10101000 => 9
00110001 => [2,2,3,1] => 10101001 => 9
00110010 => [2,2,2,1,1] => 10101011 => 9
00110011 => [2,2,2,2] => 10101010 => 16
00110100 => [2,2,1,1,2] => 10101110 => 11
00110101 => [2,2,1,1,1,1] => 10101111 => 4
00110110 => [2,2,1,2,1] => 10101101 => 10
00110111 => [2,2,1,3] => 10101100 => 10
00111000 => [2,3,3] => 10100100 => 10
00111001 => [2,3,2,1] => 10100101 => 10
00111010 => [2,3,1,1,1] => 10100111 => 4
00111011 => [2,3,1,2] => 10100110 => 11
00111100 => [2,4,2] => 10100010 => 11
00111101 => [2,4,1,1] => 10100011 => 4
00111110 => [2,5,1] => 10100001 => 4
00111111 => [2,6] => 10100000 => 4
01000000 => [1,1,6] => 11100000 => 3
01000001 => [1,1,5,1] => 11100001 => 3
01000010 => [1,1,4,1,1] => 11100011 => 3
01000011 => [1,1,4,2] => 11100010 => 10
01000100 => [1,1,3,1,2] => 11100110 => 10
01000101 => [1,1,3,1,1,1] => 11100111 => 3
01000110 => [1,1,3,2,1] => 11100101 => 9
01000111 => [1,1,3,3] => 11100100 => 9
01001000 => [1,1,2,1,3] => 11101100 => 9
01001001 => [1,1,2,1,2,1] => 11101101 => 9
01001010 => [1,1,2,1,1,1,1] => 11101111 => 3
01001011 => [1,1,2,1,1,2] => 11101110 => 10
01001100 => [1,1,2,2,2] => 11101010 => 15
01001101 => [1,1,2,2,1,1] => 11101011 => 8
01001110 => [1,1,2,3,1] => 11101001 => 8
01001111 => [1,1,2,4] => 11101000 => 8
01010000 => [1,1,1,1,4] => 11111000 => 5
01010001 => [1,1,1,1,3,1] => 11111001 => 5
01010010 => [1,1,1,1,2,1,1] => 11111011 => 5
01010011 => [1,1,1,1,2,2] => 11111010 => 12
01010100 => [1,1,1,1,1,1,2] => 11111110 => 7
01010101 => [1,1,1,1,1,1,1,1] => 11111111 => 0
01010110 => [1,1,1,1,1,2,1] => 11111101 => 6
01010111 => [1,1,1,1,1,3] => 11111100 => 6
01011000 => [1,1,1,2,3] => 11110100 => 10
01011001 => [1,1,1,2,2,1] => 11110101 => 10
01011010 => [1,1,1,2,1,1,1] => 11110111 => 4
01011011 => [1,1,1,2,1,2] => 11110110 => 11
01011100 => [1,1,1,3,2] => 11110010 => 11
01011101 => [1,1,1,3,1,1] => 11110011 => 4
01011110 => [1,1,1,4,1] => 11110001 => 4
01011111 => [1,1,1,5] => 11110000 => 4
01100000 => [1,2,5] => 11010000 => 6
01100001 => [1,2,4,1] => 11010001 => 6
01100010 => [1,2,3,1,1] => 11010011 => 6
01100011 => [1,2,3,2] => 11010010 => 13
01100100 => [1,2,2,1,2] => 11010110 => 13
01100101 => [1,2,2,1,1,1] => 11010111 => 6
01100110 => [1,2,2,2,1] => 11010101 => 12
01100111 => [1,2,2,3] => 11010100 => 12
01101000 => [1,2,1,1,3] => 11011100 => 8
01101001 => [1,2,1,1,2,1] => 11011101 => 8
01101010 => [1,2,1,1,1,1,1] => 11011111 => 2
01101011 => [1,2,1,1,1,2] => 11011110 => 9
01101100 => [1,2,1,2,2] => 11011010 => 14
01101101 => [1,2,1,2,1,1] => 11011011 => 7
01101110 => [1,2,1,3,1] => 11011001 => 7
01101111 => [1,2,1,4] => 11011000 => 7
01110000 => [1,3,4] => 11001000 => 7
01110001 => [1,3,3,1] => 11001001 => 7
01110010 => [1,3,2,1,1] => 11001011 => 7
01110011 => [1,3,2,2] => 11001010 => 14
01110100 => [1,3,1,1,2] => 11001110 => 9
01110101 => [1,3,1,1,1,1] => 11001111 => 2
01110110 => [1,3,1,2,1] => 11001101 => 8
01110111 => [1,3,1,3] => 11001100 => 8
01111000 => [1,4,3] => 11000100 => 8
01111001 => [1,4,2,1] => 11000101 => 8
01111010 => [1,4,1,1,1] => 11000111 => 2
01111011 => [1,4,1,2] => 11000110 => 9
01111100 => [1,5,2] => 11000010 => 9
01111101 => [1,5,1,1] => 11000011 => 2
01111110 => [1,6,1] => 11000001 => 2
01111111 => [1,7] => 11000000 => 2
10000000 => [1,7] => 11000000 => 2
10000001 => [1,6,1] => 11000001 => 2
10000010 => [1,5,1,1] => 11000011 => 2
10000011 => [1,5,2] => 11000010 => 9
10000100 => [1,4,1,2] => 11000110 => 9
10000101 => [1,4,1,1,1] => 11000111 => 2
10000110 => [1,4,2,1] => 11000101 => 8
10000111 => [1,4,3] => 11000100 => 8
10001000 => [1,3,1,3] => 11001100 => 8
10001001 => [1,3,1,2,1] => 11001101 => 8
10001010 => [1,3,1,1,1,1] => 11001111 => 2
10001011 => [1,3,1,1,2] => 11001110 => 9
10001100 => [1,3,2,2] => 11001010 => 14
10001101 => [1,3,2,1,1] => 11001011 => 7
10001110 => [1,3,3,1] => 11001001 => 7
10001111 => [1,3,4] => 11001000 => 7
10010000 => [1,2,1,4] => 11011000 => 7
10010001 => [1,2,1,3,1] => 11011001 => 7
10010010 => [1,2,1,2,1,1] => 11011011 => 7
10010011 => [1,2,1,2,2] => 11011010 => 14
10010100 => [1,2,1,1,1,2] => 11011110 => 9
10010101 => [1,2,1,1,1,1,1] => 11011111 => 2
10010110 => [1,2,1,1,2,1] => 11011101 => 8
10010111 => [1,2,1,1,3] => 11011100 => 8
10011000 => [1,2,2,3] => 11010100 => 12
10011001 => [1,2,2,2,1] => 11010101 => 12
10011010 => [1,2,2,1,1,1] => 11010111 => 6
10011011 => [1,2,2,1,2] => 11010110 => 13
10011100 => [1,2,3,2] => 11010010 => 13
10011101 => [1,2,3,1,1] => 11010011 => 6
10011110 => [1,2,4,1] => 11010001 => 6
10011111 => [1,2,5] => 11010000 => 6
10100000 => [1,1,1,5] => 11110000 => 4
10100001 => [1,1,1,4,1] => 11110001 => 4
10100010 => [1,1,1,3,1,1] => 11110011 => 4
10100011 => [1,1,1,3,2] => 11110010 => 11
10100100 => [1,1,1,2,1,2] => 11110110 => 11
10100101 => [1,1,1,2,1,1,1] => 11110111 => 4
10100110 => [1,1,1,2,2,1] => 11110101 => 10
10100111 => [1,1,1,2,3] => 11110100 => 10
10101000 => [1,1,1,1,1,3] => 11111100 => 6
10101001 => [1,1,1,1,1,2,1] => 11111101 => 6
10101010 => [1,1,1,1,1,1,1,1] => 11111111 => 0
10101011 => [1,1,1,1,1,1,2] => 11111110 => 7
10101100 => [1,1,1,1,2,2] => 11111010 => 12
10101101 => [1,1,1,1,2,1,1] => 11111011 => 5
10101110 => [1,1,1,1,3,1] => 11111001 => 5
10101111 => [1,1,1,1,4] => 11111000 => 5
10110000 => [1,1,2,4] => 11101000 => 8
10110001 => [1,1,2,3,1] => 11101001 => 8
10110010 => [1,1,2,2,1,1] => 11101011 => 8
10110011 => [1,1,2,2,2] => 11101010 => 15
10110100 => [1,1,2,1,1,2] => 11101110 => 10
10110101 => [1,1,2,1,1,1,1] => 11101111 => 3
10110110 => [1,1,2,1,2,1] => 11101101 => 9
10110111 => [1,1,2,1,3] => 11101100 => 9
10111000 => [1,1,3,3] => 11100100 => 9
10111001 => [1,1,3,2,1] => 11100101 => 9
10111010 => [1,1,3,1,1,1] => 11100111 => 3
10111011 => [1,1,3,1,2] => 11100110 => 10
10111100 => [1,1,4,2] => 11100010 => 10
10111101 => [1,1,4,1,1] => 11100011 => 3
10111110 => [1,1,5,1] => 11100001 => 3
10111111 => [1,1,6] => 11100000 => 3
11000000 => [2,6] => 10100000 => 4
11000001 => [2,5,1] => 10100001 => 4
11000010 => [2,4,1,1] => 10100011 => 4
11000011 => [2,4,2] => 10100010 => 11
11000100 => [2,3,1,2] => 10100110 => 11
11000101 => [2,3,1,1,1] => 10100111 => 4
11000110 => [2,3,2,1] => 10100101 => 10
11000111 => [2,3,3] => 10100100 => 10
11001000 => [2,2,1,3] => 10101100 => 10
11001001 => [2,2,1,2,1] => 10101101 => 10
11001010 => [2,2,1,1,1,1] => 10101111 => 4
11001011 => [2,2,1,1,2] => 10101110 => 11
11001100 => [2,2,2,2] => 10101010 => 16
11001101 => [2,2,2,1,1] => 10101011 => 9
11001110 => [2,2,3,1] => 10101001 => 9
11001111 => [2,2,4] => 10101000 => 9
11010000 => [2,1,1,4] => 10111000 => 6
11010001 => [2,1,1,3,1] => 10111001 => 6
11010010 => [2,1,1,2,1,1] => 10111011 => 6
11010011 => [2,1,1,2,2] => 10111010 => 13
11010100 => [2,1,1,1,1,2] => 10111110 => 8
11010101 => [2,1,1,1,1,1,1] => 10111111 => 1
11010110 => [2,1,1,1,2,1] => 10111101 => 7
11010111 => [2,1,1,1,3] => 10111100 => 7
11011000 => [2,1,2,3] => 10110100 => 11
11011001 => [2,1,2,2,1] => 10110101 => 11
11011010 => [2,1,2,1,1,1] => 10110111 => 5
11011011 => [2,1,2,1,2] => 10110110 => 12
11011100 => [2,1,3,2] => 10110010 => 12
11011101 => [2,1,3,1,1] => 10110011 => 5
11011110 => [2,1,4,1] => 10110001 => 5
11011111 => [2,1,5] => 10110000 => 5
11100000 => [3,5] => 10010000 => 5
11100001 => [3,4,1] => 10010001 => 5
11100010 => [3,3,1,1] => 10010011 => 5
11100011 => [3,3,2] => 10010010 => 12
11100100 => [3,2,1,2] => 10010110 => 12
11100101 => [3,2,1,1,1] => 10010111 => 5
11100110 => [3,2,2,1] => 10010101 => 11
11100111 => [3,2,3] => 10010100 => 11
11101000 => [3,1,1,3] => 10011100 => 7
11101001 => [3,1,1,2,1] => 10011101 => 7
11101010 => [3,1,1,1,1,1] => 10011111 => 1
11101011 => [3,1,1,1,2] => 10011110 => 8
11101100 => [3,1,2,2] => 10011010 => 13
11101101 => [3,1,2,1,1] => 10011011 => 6
11101110 => [3,1,3,1] => 10011001 => 6
11101111 => [3,1,4] => 10011000 => 6
11110000 => [4,4] => 10001000 => 6
11110001 => [4,3,1] => 10001001 => 6
11110010 => [4,2,1,1] => 10001011 => 6
11110011 => [4,2,2] => 10001010 => 13
11110100 => [4,1,1,2] => 10001110 => 8
11110101 => [4,1,1,1,1] => 10001111 => 1
11110110 => [4,1,2,1] => 10001101 => 7
11110111 => [4,1,3] => 10001100 => 7
11111000 => [5,3] => 10000100 => 7
11111001 => [5,2,1] => 10000101 => 7
11111010 => [5,1,1,1] => 10000111 => 1
11111011 => [5,1,2] => 10000110 => 8
11111100 => [6,2] => 10000010 => 8
11111101 => [6,1,1] => 10000011 => 1
11111110 => [7,1] => 10000001 => 1
11111111 => [8] => 10000000 => 1
000000000 => [9] => 100000000 => 1
000000001 => [8,1] => 100000001 => 1
000000010 => [7,1,1] => 100000011 => 1
000000011 => [7,2] => 100000010 => 9
000000100 => [6,1,2] => 100000110 => 9
000000101 => [6,1,1,1] => 100000111 => 1
000000110 => [6,2,1] => 100000101 => 8
000000111 => [6,3] => 100000100 => 8
000001000 => [5,1,3] => 100001100 => 8
000001001 => [5,1,2,1] => 100001101 => 8
000001010 => [5,1,1,1,1] => 100001111 => 1
000001011 => [5,1,1,2] => 100001110 => 9
000001100 => [5,2,2] => 100001010 => 15
000001101 => [5,2,1,1] => 100001011 => 7
000001110 => [5,3,1] => 100001001 => 7
000001111 => [5,4] => 100001000 => 7
000010000 => [4,1,4] => 100011000 => 7
000010001 => [4,1,3,1] => 100011001 => 7
000010010 => [4,1,2,1,1] => 100011011 => 7
000010011 => [4,1,2,2] => 100011010 => 15
000010100 => [4,1,1,1,2] => 100011110 => 9
000010101 => [4,1,1,1,1,1] => 100011111 => 1
000010110 => [4,1,1,2,1] => 100011101 => 8
000010111 => [4,1,1,3] => 100011100 => 8
000011000 => [4,2,3] => 100010100 => 13
000011001 => [4,2,2,1] => 100010101 => 13
000011010 => [4,2,1,1,1] => 100010111 => 6
000011011 => [4,2,1,2] => 100010110 => 14
000011100 => [4,3,2] => 100010010 => 14
000011101 => [4,3,1,1] => 100010011 => 6
000011110 => [4,4,1] => 100010001 => 6
000011111 => [4,5] => 100010000 => 6
000100000 => [3,1,5] => 100110000 => 6
000100001 => [3,1,4,1] => 100110001 => 6
000100010 => [3,1,3,1,1] => 100110011 => 6
000100011 => [3,1,3,2] => 100110010 => 14
000100100 => [3,1,2,1,2] => 100110110 => 14
000100101 => [3,1,2,1,1,1] => 100110111 => 6
000100110 => [3,1,2,2,1] => 100110101 => 13
000100111 => [3,1,2,3] => 100110100 => 13
000101000 => [3,1,1,1,3] => 100111100 => 8
000101001 => [3,1,1,1,2,1] => 100111101 => 8
000101010 => [3,1,1,1,1,1,1] => 100111111 => 1
000101011 => [3,1,1,1,1,2] => 100111110 => 9
000101100 => [3,1,1,2,2] => 100111010 => 15
000101101 => [3,1,1,2,1,1] => 100111011 => 7
000101110 => [3,1,1,3,1] => 100111001 => 7
000101111 => [3,1,1,4] => 100111000 => 7
000110000 => [3,2,4] => 100101000 => 11
000110001 => [3,2,3,1] => 100101001 => 11
000110010 => [3,2,2,1,1] => 100101011 => 11
000110011 => [3,2,2,2] => 100101010 => 19
000110100 => [3,2,1,1,2] => 100101110 => 13
000110101 => [3,2,1,1,1,1] => 100101111 => 5
000110110 => [3,2,1,2,1] => 100101101 => 12
000110111 => [3,2,1,3] => 100101100 => 12
000111000 => [3,3,3] => 100100100 => 12
000111001 => [3,3,2,1] => 100100101 => 12
000111010 => [3,3,1,1,1] => 100100111 => 5
000111011 => [3,3,1,2] => 100100110 => 13
000111100 => [3,4,2] => 100100010 => 13
000111101 => [3,4,1,1] => 100100011 => 5
000111110 => [3,5,1] => 100100001 => 5
000111111 => [3,6] => 100100000 => 5
001000000 => [2,1,6] => 101100000 => 5
001000001 => [2,1,5,1] => 101100001 => 5
001000010 => [2,1,4,1,1] => 101100011 => 5
001000011 => [2,1,4,2] => 101100010 => 13
001000100 => [2,1,3,1,2] => 101100110 => 13
001000101 => [2,1,3,1,1,1] => 101100111 => 5
001000110 => [2,1,3,2,1] => 101100101 => 12
001000111 => [2,1,3,3] => 101100100 => 12
001001000 => [2,1,2,1,3] => 101101100 => 12
001001001 => [2,1,2,1,2,1] => 101101101 => 12
001001010 => [2,1,2,1,1,1,1] => 101101111 => 5
001001011 => [2,1,2,1,1,2] => 101101110 => 13
001001100 => [2,1,2,2,2] => 101101010 => 19
001001101 => [2,1,2,2,1,1] => 101101011 => 11
001001110 => [2,1,2,3,1] => 101101001 => 11
001001111 => [2,1,2,4] => 101101000 => 11
001010000 => [2,1,1,1,4] => 101111000 => 7
001010001 => [2,1,1,1,3,1] => 101111001 => 7
001010010 => [2,1,1,1,2,1,1] => 101111011 => 7
001010011 => [2,1,1,1,2,2] => 101111010 => 15
001010100 => [2,1,1,1,1,1,2] => 101111110 => 9
001010101 => [2,1,1,1,1,1,1,1] => 101111111 => 1
001010110 => [2,1,1,1,1,2,1] => 101111101 => 8
001010111 => [2,1,1,1,1,3] => 101111100 => 8
001011000 => [2,1,1,2,3] => 101110100 => 13
001011001 => [2,1,1,2,2,1] => 101110101 => 13
001011010 => [2,1,1,2,1,1,1] => 101110111 => 6
001011011 => [2,1,1,2,1,2] => 101110110 => 14
001011100 => [2,1,1,3,2] => 101110010 => 14
001011101 => [2,1,1,3,1,1] => 101110011 => 6
001011110 => [2,1,1,4,1] => 101110001 => 6
001011111 => [2,1,1,5] => 101110000 => 6
001100000 => [2,2,5] => 101010000 => 9
001100001 => [2,2,4,1] => 101010001 => 9
001100010 => [2,2,3,1,1] => 101010011 => 9
001100011 => [2,2,3,2] => 101010010 => 17
001100100 => [2,2,2,1,2] => 101010110 => 17
001100101 => [2,2,2,1,1,1] => 101010111 => 9
001100110 => [2,2,2,2,1] => 101010101 => 16
001100111 => [2,2,2,3] => 101010100 => 16
001101000 => [2,2,1,1,3] => 101011100 => 11
001101001 => [2,2,1,1,2,1] => 101011101 => 11
001101010 => [2,2,1,1,1,1,1] => 101011111 => 4
001101011 => [2,2,1,1,1,2] => 101011110 => 12
001101100 => [2,2,1,2,2] => 101011010 => 18
001101101 => [2,2,1,2,1,1] => 101011011 => 10
001101110 => [2,2,1,3,1] => 101011001 => 10
001101111 => [2,2,1,4] => 101011000 => 10
001110000 => [2,3,4] => 101001000 => 10
001110001 => [2,3,3,1] => 101001001 => 10
001110010 => [2,3,2,1,1] => 101001011 => 10
001110011 => [2,3,2,2] => 101001010 => 18
001110100 => [2,3,1,1,2] => 101001110 => 12
001110101 => [2,3,1,1,1,1] => 101001111 => 4
001110110 => [2,3,1,2,1] => 101001101 => 11
001110111 => [2,3,1,3] => 101001100 => 11
001111000 => [2,4,3] => 101000100 => 11
001111001 => [2,4,2,1] => 101000101 => 11
001111010 => [2,4,1,1,1] => 101000111 => 4
001111011 => [2,4,1,2] => 101000110 => 12
001111100 => [2,5,2] => 101000010 => 12
001111101 => [2,5,1,1] => 101000011 => 4
001111110 => [2,6,1] => 101000001 => 4
001111111 => [2,7] => 101000000 => 4
010000000 => [1,1,7] => 111000000 => 3
010000001 => [1,1,6,1] => 111000001 => 3
010000010 => [1,1,5,1,1] => 111000011 => 3
010000011 => [1,1,5,2] => 111000010 => 11
010000100 => [1,1,4,1,2] => 111000110 => 11
010000101 => [1,1,4,1,1,1] => 111000111 => 3
010000110 => [1,1,4,2,1] => 111000101 => 10
010000111 => [1,1,4,3] => 111000100 => 10
010001000 => [1,1,3,1,3] => 111001100 => 10
010001001 => [1,1,3,1,2,1] => 111001101 => 10
010001010 => [1,1,3,1,1,1,1] => 111001111 => 3
010001011 => [1,1,3,1,1,2] => 111001110 => 11
010001100 => [1,1,3,2,2] => 111001010 => 17
010001101 => [1,1,3,2,1,1] => 111001011 => 9
010001110 => [1,1,3,3,1] => 111001001 => 9
010001111 => [1,1,3,4] => 111001000 => 9
010010000 => [1,1,2,1,4] => 111011000 => 9
010010001 => [1,1,2,1,3,1] => 111011001 => 9
010010010 => [1,1,2,1,2,1,1] => 111011011 => 9
010010011 => [1,1,2,1,2,2] => 111011010 => 17
010010100 => [1,1,2,1,1,1,2] => 111011110 => 11
010010101 => [1,1,2,1,1,1,1,1] => 111011111 => 3
010010110 => [1,1,2,1,1,2,1] => 111011101 => 10
010010111 => [1,1,2,1,1,3] => 111011100 => 10
010011000 => [1,1,2,2,3] => 111010100 => 15
010011001 => [1,1,2,2,2,1] => 111010101 => 15
010011010 => [1,1,2,2,1,1,1] => 111010111 => 8
010011011 => [1,1,2,2,1,2] => 111010110 => 16
010011100 => [1,1,2,3,2] => 111010010 => 16
010011101 => [1,1,2,3,1,1] => 111010011 => 8
010011110 => [1,1,2,4,1] => 111010001 => 8
010011111 => [1,1,2,5] => 111010000 => 8
010100000 => [1,1,1,1,5] => 111110000 => 5
010100001 => [1,1,1,1,4,1] => 111110001 => 5
010100010 => [1,1,1,1,3,1,1] => 111110011 => 5
010100011 => [1,1,1,1,3,2] => 111110010 => 13
010100100 => [1,1,1,1,2,1,2] => 111110110 => 13
010100101 => [1,1,1,1,2,1,1,1] => 111110111 => 5
010100110 => [1,1,1,1,2,2,1] => 111110101 => 12
010100111 => [1,1,1,1,2,3] => 111110100 => 12
010101000 => [1,1,1,1,1,1,3] => 111111100 => 7
010101001 => [1,1,1,1,1,1,2,1] => 111111101 => 7
010101010 => [1,1,1,1,1,1,1,1,1] => 111111111 => 0
010101011 => [1,1,1,1,1,1,1,2] => 111111110 => 8
010101100 => [1,1,1,1,1,2,2] => 111111010 => 14
010101101 => [1,1,1,1,1,2,1,1] => 111111011 => 6
010101110 => [1,1,1,1,1,3,1] => 111111001 => 6
010101111 => [1,1,1,1,1,4] => 111111000 => 6
010110000 => [1,1,1,2,4] => 111101000 => 10
010110001 => [1,1,1,2,3,1] => 111101001 => 10
010110010 => [1,1,1,2,2,1,1] => 111101011 => 10
010110011 => [1,1,1,2,2,2] => 111101010 => 18
010110100 => [1,1,1,2,1,1,2] => 111101110 => 12
010110101 => [1,1,1,2,1,1,1,1] => 111101111 => 4
010110110 => [1,1,1,2,1,2,1] => 111101101 => 11
010110111 => [1,1,1,2,1,3] => 111101100 => 11
010111000 => [1,1,1,3,3] => 111100100 => 11
010111001 => [1,1,1,3,2,1] => 111100101 => 11
010111010 => [1,1,1,3,1,1,1] => 111100111 => 4
010111011 => [1,1,1,3,1,2] => 111100110 => 12
010111100 => [1,1,1,4,2] => 111100010 => 12
010111101 => [1,1,1,4,1,1] => 111100011 => 4
010111110 => [1,1,1,5,1] => 111100001 => 4
010111111 => [1,1,1,6] => 111100000 => 4
011000000 => [1,2,6] => 110100000 => 6
011000001 => [1,2,5,1] => 110100001 => 6
011000010 => [1,2,4,1,1] => 110100011 => 6
011000011 => [1,2,4,2] => 110100010 => 14
011000100 => [1,2,3,1,2] => 110100110 => 14
011000101 => [1,2,3,1,1,1] => 110100111 => 6
011000110 => [1,2,3,2,1] => 110100101 => 13
011000111 => [1,2,3,3] => 110100100 => 13
011001000 => [1,2,2,1,3] => 110101100 => 13
011001001 => [1,2,2,1,2,1] => 110101101 => 13
011001010 => [1,2,2,1,1,1,1] => 110101111 => 6
011001011 => [1,2,2,1,1,2] => 110101110 => 14
011001100 => [1,2,2,2,2] => 110101010 => 20
011001101 => [1,2,2,2,1,1] => 110101011 => 12
011001110 => [1,2,2,3,1] => 110101001 => 12
011001111 => [1,2,2,4] => 110101000 => 12
011010000 => [1,2,1,1,4] => 110111000 => 8
011010001 => [1,2,1,1,3,1] => 110111001 => 8
011010010 => [1,2,1,1,2,1,1] => 110111011 => 8
011010011 => [1,2,1,1,2,2] => 110111010 => 16
011010100 => [1,2,1,1,1,1,2] => 110111110 => 10
011010101 => [1,2,1,1,1,1,1,1] => 110111111 => 2
011010110 => [1,2,1,1,1,2,1] => 110111101 => 9
011010111 => [1,2,1,1,1,3] => 110111100 => 9
011011000 => [1,2,1,2,3] => 110110100 => 14
011011001 => [1,2,1,2,2,1] => 110110101 => 14
011011010 => [1,2,1,2,1,1,1] => 110110111 => 7
011011011 => [1,2,1,2,1,2] => 110110110 => 15
011011100 => [1,2,1,3,2] => 110110010 => 15
011011101 => [1,2,1,3,1,1] => 110110011 => 7
011011110 => [1,2,1,4,1] => 110110001 => 7
011011111 => [1,2,1,5] => 110110000 => 7
011100000 => [1,3,5] => 110010000 => 7
011100001 => [1,3,4,1] => 110010001 => 7
011100010 => [1,3,3,1,1] => 110010011 => 7
011100011 => [1,3,3,2] => 110010010 => 15
011100100 => [1,3,2,1,2] => 110010110 => 15
011100101 => [1,3,2,1,1,1] => 110010111 => 7
011100110 => [1,3,2,2,1] => 110010101 => 14
011100111 => [1,3,2,3] => 110010100 => 14
011101000 => [1,3,1,1,3] => 110011100 => 9
011101001 => [1,3,1,1,2,1] => 110011101 => 9
011101010 => [1,3,1,1,1,1,1] => 110011111 => 2
011101011 => [1,3,1,1,1,2] => 110011110 => 10
011101100 => [1,3,1,2,2] => 110011010 => 16
011101101 => [1,3,1,2,1,1] => 110011011 => 8
011101110 => [1,3,1,3,1] => 110011001 => 8
011101111 => [1,3,1,4] => 110011000 => 8
011110000 => [1,4,4] => 110001000 => 8
011110001 => [1,4,3,1] => 110001001 => 8
011110010 => [1,4,2,1,1] => 110001011 => 8
011110011 => [1,4,2,2] => 110001010 => 16
011110100 => [1,4,1,1,2] => 110001110 => 10
011110101 => [1,4,1,1,1,1] => 110001111 => 2
011110110 => [1,4,1,2,1] => 110001101 => 9
011110111 => [1,4,1,3] => 110001100 => 9
011111000 => [1,5,3] => 110000100 => 9
011111001 => [1,5,2,1] => 110000101 => 9
011111010 => [1,5,1,1,1] => 110000111 => 2
011111011 => [1,5,1,2] => 110000110 => 10
011111100 => [1,6,2] => 110000010 => 10
011111101 => [1,6,1,1] => 110000011 => 2
011111110 => [1,7,1] => 110000001 => 2
011111111 => [1,8] => 110000000 => 2
100000000 => [1,8] => 110000000 => 2
100000001 => [1,7,1] => 110000001 => 2
100000010 => [1,6,1,1] => 110000011 => 2
100000011 => [1,6,2] => 110000010 => 10
100000100 => [1,5,1,2] => 110000110 => 10
100000101 => [1,5,1,1,1] => 110000111 => 2
100000110 => [1,5,2,1] => 110000101 => 9
100000111 => [1,5,3] => 110000100 => 9
100001000 => [1,4,1,3] => 110001100 => 9
100001001 => [1,4,1,2,1] => 110001101 => 9
100001010 => [1,4,1,1,1,1] => 110001111 => 2
100001011 => [1,4,1,1,2] => 110001110 => 10
100001100 => [1,4,2,2] => 110001010 => 16
100001101 => [1,4,2,1,1] => 110001011 => 8
100001110 => [1,4,3,1] => 110001001 => 8
100001111 => [1,4,4] => 110001000 => 8
100010000 => [1,3,1,4] => 110011000 => 8
100010001 => [1,3,1,3,1] => 110011001 => 8
100010010 => [1,3,1,2,1,1] => 110011011 => 8
100010011 => [1,3,1,2,2] => 110011010 => 16
100010100 => [1,3,1,1,1,2] => 110011110 => 10
100010101 => [1,3,1,1,1,1,1] => 110011111 => 2
100010110 => [1,3,1,1,2,1] => 110011101 => 9
100010111 => [1,3,1,1,3] => 110011100 => 9
100011000 => [1,3,2,3] => 110010100 => 14
100011001 => [1,3,2,2,1] => 110010101 => 14
100011010 => [1,3,2,1,1,1] => 110010111 => 7
100011011 => [1,3,2,1,2] => 110010110 => 15
100011100 => [1,3,3,2] => 110010010 => 15
100011101 => [1,3,3,1,1] => 110010011 => 7
100011110 => [1,3,4,1] => 110010001 => 7
100011111 => [1,3,5] => 110010000 => 7
100100000 => [1,2,1,5] => 110110000 => 7
100100001 => [1,2,1,4,1] => 110110001 => 7
100100010 => [1,2,1,3,1,1] => 110110011 => 7
100100011 => [1,2,1,3,2] => 110110010 => 15
100100100 => [1,2,1,2,1,2] => 110110110 => 15
100100101 => [1,2,1,2,1,1,1] => 110110111 => 7
100100110 => [1,2,1,2,2,1] => 110110101 => 14
100100111 => [1,2,1,2,3] => 110110100 => 14
100101000 => [1,2,1,1,1,3] => 110111100 => 9
100101001 => [1,2,1,1,1,2,1] => 110111101 => 9
100101010 => [1,2,1,1,1,1,1,1] => 110111111 => 2
100101011 => [1,2,1,1,1,1,2] => 110111110 => 10
100101100 => [1,2,1,1,2,2] => 110111010 => 16
100101101 => [1,2,1,1,2,1,1] => 110111011 => 8
100101110 => [1,2,1,1,3,1] => 110111001 => 8
100101111 => [1,2,1,1,4] => 110111000 => 8
100110000 => [1,2,2,4] => 110101000 => 12
100110001 => [1,2,2,3,1] => 110101001 => 12
100110010 => [1,2,2,2,1,1] => 110101011 => 12
100110011 => [1,2,2,2,2] => 110101010 => 20
100110100 => [1,2,2,1,1,2] => 110101110 => 14
100110101 => [1,2,2,1,1,1,1] => 110101111 => 6
100110110 => [1,2,2,1,2,1] => 110101101 => 13
100110111 => [1,2,2,1,3] => 110101100 => 13
100111000 => [1,2,3,3] => 110100100 => 13
100111001 => [1,2,3,2,1] => 110100101 => 13
100111010 => [1,2,3,1,1,1] => 110100111 => 6
100111011 => [1,2,3,1,2] => 110100110 => 14
100111100 => [1,2,4,2] => 110100010 => 14
100111101 => [1,2,4,1,1] => 110100011 => 6
100111110 => [1,2,5,1] => 110100001 => 6
100111111 => [1,2,6] => 110100000 => 6
101000000 => [1,1,1,6] => 111100000 => 4
101000001 => [1,1,1,5,1] => 111100001 => 4
101000010 => [1,1,1,4,1,1] => 111100011 => 4
101000011 => [1,1,1,4,2] => 111100010 => 12
101000100 => [1,1,1,3,1,2] => 111100110 => 12
101000101 => [1,1,1,3,1,1,1] => 111100111 => 4
101000110 => [1,1,1,3,2,1] => 111100101 => 11
101000111 => [1,1,1,3,3] => 111100100 => 11
101001000 => [1,1,1,2,1,3] => 111101100 => 11
101001001 => [1,1,1,2,1,2,1] => 111101101 => 11
101001010 => [1,1,1,2,1,1,1,1] => 111101111 => 4
101001011 => [1,1,1,2,1,1,2] => 111101110 => 12
101001100 => [1,1,1,2,2,2] => 111101010 => 18
101001101 => [1,1,1,2,2,1,1] => 111101011 => 10
101001110 => [1,1,1,2,3,1] => 111101001 => 10
101001111 => [1,1,1,2,4] => 111101000 => 10
101010000 => [1,1,1,1,1,4] => 111111000 => 6
101010001 => [1,1,1,1,1,3,1] => 111111001 => 6
101010010 => [1,1,1,1,1,2,1,1] => 111111011 => 6
101010011 => [1,1,1,1,1,2,2] => 111111010 => 14
101010100 => [1,1,1,1,1,1,1,2] => 111111110 => 8
101010101 => [1,1,1,1,1,1,1,1,1] => 111111111 => 0
101010110 => [1,1,1,1,1,1,2,1] => 111111101 => 7
101010111 => [1,1,1,1,1,1,3] => 111111100 => 7
101011000 => [1,1,1,1,2,3] => 111110100 => 12
101011001 => [1,1,1,1,2,2,1] => 111110101 => 12
101011010 => [1,1,1,1,2,1,1,1] => 111110111 => 5
101011011 => [1,1,1,1,2,1,2] => 111110110 => 13
101011100 => [1,1,1,1,3,2] => 111110010 => 13
101011101 => [1,1,1,1,3,1,1] => 111110011 => 5
101011110 => [1,1,1,1,4,1] => 111110001 => 5
101011111 => [1,1,1,1,5] => 111110000 => 5
101100000 => [1,1,2,5] => 111010000 => 8
101100001 => [1,1,2,4,1] => 111010001 => 8
101100010 => [1,1,2,3,1,1] => 111010011 => 8
101100011 => [1,1,2,3,2] => 111010010 => 16
101100100 => [1,1,2,2,1,2] => 111010110 => 16
101100101 => [1,1,2,2,1,1,1] => 111010111 => 8
101100110 => [1,1,2,2,2,1] => 111010101 => 15
101100111 => [1,1,2,2,3] => 111010100 => 15
101101000 => [1,1,2,1,1,3] => 111011100 => 10
101101001 => [1,1,2,1,1,2,1] => 111011101 => 10
101101010 => [1,1,2,1,1,1,1,1] => 111011111 => 3
101101011 => [1,1,2,1,1,1,2] => 111011110 => 11
101101100 => [1,1,2,1,2,2] => 111011010 => 17
101101101 => [1,1,2,1,2,1,1] => 111011011 => 9
101101110 => [1,1,2,1,3,1] => 111011001 => 9
101101111 => [1,1,2,1,4] => 111011000 => 9
101110000 => [1,1,3,4] => 111001000 => 9
101110001 => [1,1,3,3,1] => 111001001 => 9
101110010 => [1,1,3,2,1,1] => 111001011 => 9
101110011 => [1,1,3,2,2] => 111001010 => 17
101110100 => [1,1,3,1,1,2] => 111001110 => 11
101110101 => [1,1,3,1,1,1,1] => 111001111 => 3
101110110 => [1,1,3,1,2,1] => 111001101 => 10
101110111 => [1,1,3,1,3] => 111001100 => 10
101111000 => [1,1,4,3] => 111000100 => 10
101111001 => [1,1,4,2,1] => 111000101 => 10
101111010 => [1,1,4,1,1,1] => 111000111 => 3
101111011 => [1,1,4,1,2] => 111000110 => 11
101111100 => [1,1,5,2] => 111000010 => 11
101111101 => [1,1,5,1,1] => 111000011 => 3
101111110 => [1,1,6,1] => 111000001 => 3
101111111 => [1,1,7] => 111000000 => 3
110000000 => [2,7] => 101000000 => 4
110000001 => [2,6,1] => 101000001 => 4
110000010 => [2,5,1,1] => 101000011 => 4
110000011 => [2,5,2] => 101000010 => 12
110000100 => [2,4,1,2] => 101000110 => 12
110000101 => [2,4,1,1,1] => 101000111 => 4
110000110 => [2,4,2,1] => 101000101 => 11
110000111 => [2,4,3] => 101000100 => 11
110001000 => [2,3,1,3] => 101001100 => 11
110001001 => [2,3,1,2,1] => 101001101 => 11
110001010 => [2,3,1,1,1,1] => 101001111 => 4
110001011 => [2,3,1,1,2] => 101001110 => 12
110001100 => [2,3,2,2] => 101001010 => 18
110001101 => [2,3,2,1,1] => 101001011 => 10
110001110 => [2,3,3,1] => 101001001 => 10
110001111 => [2,3,4] => 101001000 => 10
110010000 => [2,2,1,4] => 101011000 => 10
110010001 => [2,2,1,3,1] => 101011001 => 10
110010010 => [2,2,1,2,1,1] => 101011011 => 10
110010011 => [2,2,1,2,2] => 101011010 => 18
110010100 => [2,2,1,1,1,2] => 101011110 => 12
110010101 => [2,2,1,1,1,1,1] => 101011111 => 4
110010110 => [2,2,1,1,2,1] => 101011101 => 11
110010111 => [2,2,1,1,3] => 101011100 => 11
110011000 => [2,2,2,3] => 101010100 => 16
110011001 => [2,2,2,2,1] => 101010101 => 16
110011010 => [2,2,2,1,1,1] => 101010111 => 9
110011011 => [2,2,2,1,2] => 101010110 => 17
110011100 => [2,2,3,2] => 101010010 => 17
110011101 => [2,2,3,1,1] => 101010011 => 9
110011110 => [2,2,4,1] => 101010001 => 9
110011111 => [2,2,5] => 101010000 => 9
110100000 => [2,1,1,5] => 101110000 => 6
110100001 => [2,1,1,4,1] => 101110001 => 6
110100010 => [2,1,1,3,1,1] => 101110011 => 6
110100011 => [2,1,1,3,2] => 101110010 => 14
110100100 => [2,1,1,2,1,2] => 101110110 => 14
110100101 => [2,1,1,2,1,1,1] => 101110111 => 6
110100110 => [2,1,1,2,2,1] => 101110101 => 13
110100111 => [2,1,1,2,3] => 101110100 => 13
110101000 => [2,1,1,1,1,3] => 101111100 => 8
110101001 => [2,1,1,1,1,2,1] => 101111101 => 8
110101010 => [2,1,1,1,1,1,1,1] => 101111111 => 1
110101011 => [2,1,1,1,1,1,2] => 101111110 => 9
110101100 => [2,1,1,1,2,2] => 101111010 => 15
110101101 => [2,1,1,1,2,1,1] => 101111011 => 7
110101110 => [2,1,1,1,3,1] => 101111001 => 7
110101111 => [2,1,1,1,4] => 101111000 => 7
110110000 => [2,1,2,4] => 101101000 => 11
110110001 => [2,1,2,3,1] => 101101001 => 11
110110010 => [2,1,2,2,1,1] => 101101011 => 11
110110011 => [2,1,2,2,2] => 101101010 => 19
110110100 => [2,1,2,1,1,2] => 101101110 => 13
110110101 => [2,1,2,1,1,1,1] => 101101111 => 5
110110110 => [2,1,2,1,2,1] => 101101101 => 12
110110111 => [2,1,2,1,3] => 101101100 => 12
110111000 => [2,1,3,3] => 101100100 => 12
110111001 => [2,1,3,2,1] => 101100101 => 12
110111010 => [2,1,3,1,1,1] => 101100111 => 5
110111011 => [2,1,3,1,2] => 101100110 => 13
110111100 => [2,1,4,2] => 101100010 => 13
110111101 => [2,1,4,1,1] => 101100011 => 5
110111110 => [2,1,5,1] => 101100001 => 5
110111111 => [2,1,6] => 101100000 => 5
111000000 => [3,6] => 100100000 => 5
111000001 => [3,5,1] => 100100001 => 5
111000010 => [3,4,1,1] => 100100011 => 5
111000011 => [3,4,2] => 100100010 => 13
111000100 => [3,3,1,2] => 100100110 => 13
111000101 => [3,3,1,1,1] => 100100111 => 5
111000110 => [3,3,2,1] => 100100101 => 12
111000111 => [3,3,3] => 100100100 => 12
111001000 => [3,2,1,3] => 100101100 => 12
111001001 => [3,2,1,2,1] => 100101101 => 12
111001010 => [3,2,1,1,1,1] => 100101111 => 5
111001011 => [3,2,1,1,2] => 100101110 => 13
111001100 => [3,2,2,2] => 100101010 => 19
111001101 => [3,2,2,1,1] => 100101011 => 11
111001110 => [3,2,3,1] => 100101001 => 11
111001111 => [3,2,4] => 100101000 => 11
111010000 => [3,1,1,4] => 100111000 => 7
111010001 => [3,1,1,3,1] => 100111001 => 7
111010010 => [3,1,1,2,1,1] => 100111011 => 7
111010011 => [3,1,1,2,2] => 100111010 => 15
111010100 => [3,1,1,1,1,2] => 100111110 => 9
111010101 => [3,1,1,1,1,1,1] => 100111111 => 1
111010110 => [3,1,1,1,2,1] => 100111101 => 8
111010111 => [3,1,1,1,3] => 100111100 => 8
111011000 => [3,1,2,3] => 100110100 => 13
111011001 => [3,1,2,2,1] => 100110101 => 13
111011010 => [3,1,2,1,1,1] => 100110111 => 6
111011011 => [3,1,2,1,2] => 100110110 => 14
111011100 => [3,1,3,2] => 100110010 => 14
111011101 => [3,1,3,1,1] => 100110011 => 6
111011110 => [3,1,4,1] => 100110001 => 6
111011111 => [3,1,5] => 100110000 => 6
111100000 => [4,5] => 100010000 => 6
111100001 => [4,4,1] => 100010001 => 6
111100010 => [4,3,1,1] => 100010011 => 6
111100011 => [4,3,2] => 100010010 => 14
111100100 => [4,2,1,2] => 100010110 => 14
111100101 => [4,2,1,1,1] => 100010111 => 6
111100110 => [4,2,2,1] => 100010101 => 13
111100111 => [4,2,3] => 100010100 => 13
111101000 => [4,1,1,3] => 100011100 => 8
111101001 => [4,1,1,2,1] => 100011101 => 8
111101010 => [4,1,1,1,1,1] => 100011111 => 1
111101011 => [4,1,1,1,2] => 100011110 => 9
111101100 => [4,1,2,2] => 100011010 => 15
111101101 => [4,1,2,1,1] => 100011011 => 7
111101110 => [4,1,3,1] => 100011001 => 7
111101111 => [4,1,4] => 100011000 => 7
111110000 => [5,4] => 100001000 => 7
111110001 => [5,3,1] => 100001001 => 7
111110010 => [5,2,1,1] => 100001011 => 7
111110011 => [5,2,2] => 100001010 => 15
111110100 => [5,1,1,2] => 100001110 => 9
111110101 => [5,1,1,1,1] => 100001111 => 1
111110110 => [5,1,2,1] => 100001101 => 8
111110111 => [5,1,3] => 100001100 => 8
111111000 => [6,3] => 100000100 => 8
111111001 => [6,2,1] => 100000101 => 8
111111010 => [6,1,1,1] => 100000111 => 1
111111011 => [6,1,2] => 100000110 => 9
111111100 => [7,2] => 100000010 => 9
111111101 => [7,1,1] => 100000011 => 1
111111110 => [8,1] => 100000001 => 1
111111111 => [9] => 100000000 => 1
1011010100 => [1,1,2,1,1,1,1,2] => 1110111110 => 12
1101010100 => [2,1,1,1,1,1,1,2] => 1011111110 => 10
1110010100 => [3,2,1,1,1,2] => 1001011110 => 14
101011110000 => [1,1,1,1,4,4] => 111110001000 => 14
101100111000 => [1,1,2,2,3,3] => 111010100100 => 25
101101110000 => [1,1,2,1,3,4] => 111011001000 => 18
101110001100 => [1,1,3,3,2,2] => 111001001010 => 29
101110011000 => [1,1,3,2,2,3] => 111001010100 => 27
101110110000 => [1,1,3,1,2,4] => 111001101000 => 19
101111001000 => [1,1,4,2,1,3] => 111000101100 => 20
101111010000 => [1,1,4,1,1,4] => 111000111000 => 12
110010111000 => [2,2,1,1,3,3] => 101011100100 => 21
110011001100 => [2,2,2,2,2,2] => 101010101010 => 36
110011011000 => [2,2,2,1,2,3] => 101010110100 => 27
110100111000 => [2,1,1,2,3,3] => 101110100100 => 23
110101110000 => [2,1,1,1,3,4] => 101111001000 => 16
110110001100 => [2,1,2,3,2,2] => 101101001010 => 31
110110011000 => [2,1,2,2,2,3] => 101101010100 => 29
110110110000 => [2,1,2,1,2,4] => 101101101000 => 21
110111001000 => [2,1,3,2,1,3] => 101100101100 => 22
110111010000 => [2,1,3,1,1,4] => 101100111000 => 14
111001011000 => [3,2,1,1,2,3] => 100101110100 => 23
111010011000 => [3,1,1,2,2,3] => 100111010100 => 25
111010110000 => [3,1,1,1,2,4] => 100111101000 => 17
111101001000 => [4,1,1,2,1,3] => 100011101100 => 18
111101010000 => [4,1,1,1,1,4] => 100011111000 => 10
1111111111 => [10] => 1000000000 => 1
1111111100 => [8,2] => 1000000010 => 10
1111111011 => [7,1,2] => 1000000110 => 10
1111110011 => [6,2,2] => 1000001010 => 17
1111100100 => [5,2,1,2] => 1000010110 => 16
1111011011 => [4,1,2,1,2] => 1000110110 => 16
1111010111 => [4,1,1,1,3] => 1000111100 => 9
1111010100 => [4,1,1,1,1,2] => 1000111110 => 10
1111010011 => [4,1,1,2,2] => 1000111010 => 17
1111001011 => [4,2,1,1,2] => 1000101110 => 15
1110111011 => [3,1,3,1,2] => 1001100110 => 15
1110110001 => [3,1,2,3,1] => 1001101001 => 13
1110101111 => [3,1,1,1,4] => 1001111000 => 8
1110101011 => [3,1,1,1,1,1,2] => 1001111110 => 10
1110100001 => [3,1,1,4,1] => 1001110001 => 7
1110011001 => [3,2,2,2,1] => 1001010101 => 19
1110010011 => [3,2,1,2,2] => 1001011010 => 21
1110010001 => [3,2,1,3,1] => 1001011001 => 12
1101111001 => [2,1,4,2,1] => 1011000101 => 13
1101110111 => [2,1,3,1,3] => 1011001100 => 13
1101110001 => [2,1,3,3,1] => 1011001001 => 12
1101101111 => [2,1,2,1,4] => 1011011000 => 12
1101100011 => [2,1,2,3,2] => 1011010010 => 20
1101100001 => [2,1,2,4,1] => 1011010001 => 11
1101011111 => [2,1,1,1,5] => 1011110000 => 7
1101001111 => [2,1,1,2,4] => 1011101000 => 13
1101000111 => [2,1,1,3,3] => 1011100100 => 14
1101000011 => [2,1,1,4,2] => 1011100010 => 15
1101000001 => [2,1,1,5,1] => 1011100001 => 6
1100111001 => [2,2,3,2,1] => 1010100101 => 17
1100110011 => [2,2,2,2,2] => 1010101010 => 25
1100110001 => [2,2,2,3,1] => 1010101001 => 16
1100101011 => [2,2,1,1,1,1,2] => 1010111110 => 13
1100100111 => [2,2,1,2,3] => 1010110100 => 18
1100100011 => [2,2,1,3,2] => 1010110010 => 19
1100011011 => [2,3,2,1,2] => 1010010110 => 19
1100011001 => [2,3,2,2,1] => 1010010101 => 18
1100010011 => [2,3,1,2,2] => 1010011010 => 20
1100001011 => [2,4,1,1,2] => 1010001110 => 13
1011110111 => [1,1,4,1,3] => 1110001100 => 11
1011110001 => [1,1,4,3,1] => 1110001001 => 10
1011100011 => [1,1,3,3,2] => 1110010010 => 18
1011100001 => [1,1,3,4,1] => 1110010001 => 9
1011000111 => [1,1,2,3,3] => 1110100100 => 16
1011000011 => [1,1,2,4,2] => 1110100010 => 17
1010111111 => [1,1,1,1,6] => 1111100000 => 5
1010101011 => [1,1,1,1,1,1,1,1,2] => 1111111110 => 9
1010101001 => [1,1,1,1,1,1,1,2,1] => 1111111101 => 8
1010001111 => [1,1,1,3,4] => 1111001000 => 11
1010000111 => [1,1,1,4,3] => 1111000100 => 12
1010000001 => [1,1,1,6,1] => 1111000001 => 4
1001110100 => [1,2,3,1,1,2] => 1101001110 => 15
1001110011 => [1,2,3,2,2] => 1101001010 => 22
1001110001 => [1,2,3,3,1] => 1101001001 => 13
1001100111 => [1,2,2,2,3] => 1101010100 => 20
1001100011 => [1,2,2,3,2] => 1101010010 => 21
1001011111 => [1,2,1,1,5] => 1101110000 => 8
1001010100 => [1,2,1,1,1,1,1,2] => 1101111110 => 11
1001001111 => [1,2,1,2,4] => 1101101000 => 14
1001000111 => [1,2,1,3,3] => 1101100100 => 15
1000111011 => [1,3,3,1,2] => 1100100110 => 16
1000111001 => [1,3,3,2,1] => 1100100101 => 15
1000110111 => [1,3,2,1,3] => 1100101100 => 15
1000110011 => [1,3,2,2,2] => 1100101010 => 23
1000110001 => [1,3,2,3,1] => 1100101001 => 14
1000101111 => [1,3,1,1,4] => 1100111000 => 9
1000100111 => [1,3,1,2,3] => 1100110100 => 16
1000001111 => [1,5,4] => 1100001000 => 9
100011001110 => [1,3,2,2,3,1] => 110010101001 => 23
100110001110 => [1,2,2,3,3,1] => 110101001001 => 21
100011100110 => [1,3,3,2,2,1] => 110010010101 => 25
0010101011 => [2,1,1,1,1,1,1,2] => 1011111110 => 10
0001101011 => [3,2,1,1,1,2] => 1001011110 => 14
010100001111 => [1,1,1,1,4,4] => 111110001000 => 14
010011000111 => [1,1,2,2,3,3] => 111010100100 => 25
010010001111 => [1,1,2,1,3,4] => 111011001000 => 18
010000110111 => [1,1,4,2,1,3] => 111000101100 => 20
010000101111 => [1,1,4,1,1,4] => 111000111000 => 12
001100110011 => [2,2,2,2,2,2] => 101010101010 => 36
001100100111 => [2,2,2,1,2,3] => 101010110100 => 27
001011000111 => [2,1,1,2,3,3] => 101110100100 => 23
001010001111 => [2,1,1,1,3,4] => 101111001000 => 16
001000110111 => [2,1,3,2,1,3] => 101100101100 => 22
001000101111 => [2,1,3,1,1,4] => 101100111000 => 14
000010110111 => [4,1,1,2,1,3] => 100011101100 => 18
000010101111 => [4,1,1,1,1,4] => 100011111000 => 10
0000000000 => [10] => 1000000000 => 1
0110011000 => [1,2,2,2,3] => 1101010100 => 20
0111001000 => [1,3,2,1,3] => 1100101100 => 15
0100111000 => [1,1,2,3,3] => 1110100100 => 16
0100111110 => [1,1,2,5,1] => 1110100001 => 8
0011010100 => [2,2,1,1,1,1,2] => 1010111110 => 13
0101010100 => [1,1,1,1,1,1,1,1,2] => 1111111110 => 9
0010111100 => [2,1,1,4,2] => 1011100010 => 15
0011110100 => [2,4,1,1,2] => 1010001110 => 13
0000010100 => [5,1,1,1,2] => 1000011110 => 10
00000000000 => [11] => 10000000000 => 1
0000000100 => [7,1,2] => 1000000110 => 10
0001011110 => [3,1,1,4,1] => 1001110001 => 7
0010111110 => [2,1,1,5,1] => 1011100001 => 6
0101111110 => [1,1,1,6,1] => 1111000001 => 4
0010111000 => [2,1,1,3,3] => 1011100100 => 14
0011001100 => [2,2,2,2,2] => 1010101010 => 25
0011011000 => [2,2,1,2,3] => 1010110100 => 18
0011100100 => [2,3,2,1,2] => 1010010110 => 19
0101110000 => [1,1,1,3,4] => 1111001000 => 11
0110001100 => [1,2,3,2,2] => 1101001010 => 22
0110110000 => [1,2,1,2,4] => 1101101000 => 14
0111000100 => [1,3,3,1,2] => 1100100110 => 16
0111010000 => [1,3,1,1,4] => 1100111000 => 9
01010101001 => [1,1,1,1,1,1,1,1,2,1] => 11111111101 => 9
0101010110 => [1,1,1,1,1,1,1,2,1] => 1111111101 => 8
0101111000 => [1,1,1,4,3] => 1111000100 => 12
0110011100 => [1,2,2,3,2] => 1101010010 => 21
0110111000 => [1,2,1,3,3] => 1101100100 => 15
0111000110 => [1,3,3,2,1] => 1100100101 => 15
0111001100 => [1,3,2,2,2] => 1100101010 => 23
0111011000 => [1,3,1,2,3] => 1100110100 => 16
0111110000 => [1,5,4] => 1100001000 => 9
011001110001 => [1,2,2,3,3,1] => 110101001001 => 21
011100011001 => [1,3,3,2,2,1] => 110010010101 => 25
011100110001 => [1,3,2,2,3,1] => 110010101001 => 23
0011100110 => [2,3,2,2,1] => 1010010101 => 18
0011101100 => [2,3,1,2,2] => 1010011010 => 20
0001101110 => [3,2,1,3,1] => 1001011001 => 12
0100111100 => [1,1,2,4,2] => 1110100010 => 17
100111001100 => [1,2,3,2,2,2] => 110100101010 => 33
0111100011 => [1,4,3,2] => 1100010010 => 17
011100110011 => [1,3,2,2,2,2] => 110010101010 => 34
011100111001 => [1,3,2,3,2,1] => 110010100101 => 24
011110011001 => [1,4,2,2,2,1] => 110001010101 => 26
0001101100 => [3,2,1,2,2] => 1001011010 => 21
0011000110 => [2,2,3,2,1] => 1010100101 => 17
0010001110 => [2,1,3,3,1] => 1011001001 => 12
11111111111 => [11] => 10000000000 => 1
0000001100 => [6,2,2] => 1000001010 => 17
0000110100 => [4,2,1,1,2] => 1000101110 => 15
0000000011 => [8,2] => 1000000010 => 10
0000011011 => [5,2,1,2] => 1000010110 => 16
00000000011 => [9,2] => 10000000010 => 11
0110001011 => [1,2,3,1,1,2] => 1101001110 => 15
011000110011 => [1,2,3,2,2,2] => 110100101010 => 33
0110001110 => [1,2,3,3,1] => 1101001001 => 13
0011001110 => [2,2,2,3,1] => 1010101001 => 16
0100011110 => [1,1,3,4,1] => 1110010001 => 9
0010011110 => [2,1,2,4,1] => 1011010001 => 11
0101000000 => [1,1,1,1,6] => 1111100000 => 5
0001010000 => [3,1,1,1,4] => 1001111000 => 8
0100010000 => [1,1,3,1,4] => 1110011000 => 10
0001000100 => [3,1,3,1,2] => 1001100110 => 15
0111011100 => [1,3,1,3,2] => 1100110010 => 17
0110101011 => [1,2,1,1,1,1,1,2] => 1101111110 => 11
0000000111 => [7,3] => 1000000100 => 9
0000001011 => [6,1,1,2] => 1000001110 => 10
0011011110 => [2,2,1,4,1] => 1010110001 => 10
0011101110 => [2,3,1,3,1] => 1010011001 => 11
01101010100 => [1,2,1,1,1,1,1,1,2] => 11011111110 => 12
110011000110 => [2,2,2,3,2,1] => 101010100101 => 26
110001100110 => [2,3,2,2,2,1] => 101001010101 => 28
011001100011 => [1,2,2,2,3,2] => 110101010010 => 31
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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:
2,2 2,4,2 2,6,4,2,2 2,8,6,4,6,4,2 2,10,8,6,10,10,10,4,2,2 2,12,10,8,14,16,20,16,10,8,6,4,2 2,14,12,10,18,22,30,30,28,22,18,18,14,10,4,2,2 2,16,14,12,22,28,40,44,48,48,40,40,40,36,28,20,14,8,6,4,2
$F_{1} = 2$
$F_{2} = 2 + 2\ q$
$F_{3} = 2 + 4\ q + 2\ q^{2}$
$F_{4} = 2 + 6\ q + 4\ q^{2} + 2\ q^{3} + 2\ q^{4}$
$F_{5} = 2 + 8\ q + 6\ q^{2} + 4\ q^{3} + 6\ q^{4} + 4\ q^{5} + 2\ q^{6}$
$F_{6} = 2 + 10\ q + 8\ q^{2} + 6\ q^{3} + 10\ q^{4} + 10\ q^{5} + 10\ q^{6} + 4\ q^{7} + 2\ q^{8} + 2\ q^{9}$
$F_{7} = 2 + 12\ q + 10\ q^{2} + 8\ q^{3} + 14\ q^{4} + 16\ q^{5} + 20\ q^{6} + 16\ q^{7} + 10\ q^{8} + 8\ q^{9} + 6\ q^{10} + 4\ q^{11} + 2\ q^{12}$
$F_{8} = 2 + 14\ q + 12\ q^{2} + 10\ q^{3} + 18\ q^{4} + 22\ q^{5} + 30\ q^{6} + 30\ q^{7} + 28\ q^{8} + 22\ q^{9} + 18\ q^{10} + 18\ q^{11} + 14\ q^{12} + 10\ q^{13} + 4\ q^{14} + 2\ q^{15} + 2\ q^{16}$
$F_{9} = 2 + 16\ q + 14\ q^{2} + 12\ q^{3} + 22\ q^{4} + 28\ q^{5} + 40\ q^{6} + 44\ q^{7} + 48\ q^{8} + 48\ q^{9} + 40\ q^{10} + 40\ q^{11} + 40\ q^{12} + 36\ q^{13} + 28\ q^{14} + 20\ q^{15} + 14\ q^{16} + 8\ q^{17} + 6\ q^{18} + 4\ q^{19} + 2\ q^{20}$
Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Map
to binary word
Description
Return the composition as a binary word, treating ones as separators.
Encoding a positive integer $i$ as the word $10\dots 0$ consisting of a one followed by $i-1$ zeros, the binary word of a composition $(i_1,\dots,i_k)$ is the concatenation of of words for $i_1,\dots,i_k$.
The image of this map contains precisely the words which do not begin with a $0$.
Encoding a positive integer $i$ as the word $10\dots 0$ consisting of a one followed by $i-1$ zeros, the binary word of a composition $(i_1,\dots,i_k)$ is the concatenation of of words for $i_1,\dots,i_k$.
The image of this map contains precisely the words which do not begin with a $0$.
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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