Identifier
Values
0 => [2] => 10 => 1
1 => [1,1] => 11 => 0
00 => [3] => 100 => 1
01 => [2,1] => 101 => 1
10 => [1,2] => 110 => 1
11 => [1,1,1] => 111 => 0
000 => [4] => 1000 => 1
001 => [3,1] => 1001 => 1
010 => [2,2] => 1010 => 2
011 => [2,1,1] => 1011 => 1
100 => [1,3] => 1100 => 1
101 => [1,2,1] => 1101 => 1
110 => [1,1,2] => 1110 => 1
111 => [1,1,1,1] => 1111 => 0
0000 => [5] => 10000 => 1
0001 => [4,1] => 10001 => 1
0010 => [3,2] => 10010 => 2
0011 => [3,1,1] => 10011 => 1
0100 => [2,3] => 10100 => 2
0101 => [2,2,1] => 10101 => 2
0110 => [2,1,2] => 10110 => 2
0111 => [2,1,1,1] => 10111 => 1
1000 => [1,4] => 11000 => 1
1001 => [1,3,1] => 11001 => 1
1010 => [1,2,2] => 11010 => 2
1011 => [1,2,1,1] => 11011 => 1
1100 => [1,1,3] => 11100 => 1
1101 => [1,1,2,1] => 11101 => 1
1110 => [1,1,1,2] => 11110 => 1
1111 => [1,1,1,1,1] => 11111 => 0
00000 => [6] => 100000 => 1
00001 => [5,1] => 100001 => 1
00010 => [4,2] => 100010 => 2
00011 => [4,1,1] => 100011 => 1
00100 => [3,3] => 100100 => 2
00101 => [3,2,1] => 100101 => 2
00110 => [3,1,2] => 100110 => 2
00111 => [3,1,1,1] => 100111 => 1
01000 => [2,4] => 101000 => 2
01001 => [2,3,1] => 101001 => 2
01010 => [2,2,2] => 101010 => 3
01011 => [2,2,1,1] => 101011 => 2
01100 => [2,1,3] => 101100 => 2
01101 => [2,1,2,1] => 101101 => 2
01110 => [2,1,1,2] => 101110 => 2
01111 => [2,1,1,1,1] => 101111 => 1
10000 => [1,5] => 110000 => 1
10001 => [1,4,1] => 110001 => 1
10010 => [1,3,2] => 110010 => 2
10011 => [1,3,1,1] => 110011 => 1
10100 => [1,2,3] => 110100 => 2
10101 => [1,2,2,1] => 110101 => 2
10110 => [1,2,1,2] => 110110 => 2
10111 => [1,2,1,1,1] => 110111 => 1
11000 => [1,1,4] => 111000 => 1
11001 => [1,1,3,1] => 111001 => 1
11010 => [1,1,2,2] => 111010 => 2
11011 => [1,1,2,1,1] => 111011 => 1
11100 => [1,1,1,3] => 111100 => 1
11101 => [1,1,1,2,1] => 111101 => 1
11110 => [1,1,1,1,2] => 111110 => 1
11111 => [1,1,1,1,1,1] => 111111 => 0
000000 => [7] => 1000000 => 1
000001 => [6,1] => 1000001 => 1
000010 => [5,2] => 1000010 => 2
000011 => [5,1,1] => 1000011 => 1
000100 => [4,3] => 1000100 => 2
000101 => [4,2,1] => 1000101 => 2
000110 => [4,1,2] => 1000110 => 2
000111 => [4,1,1,1] => 1000111 => 1
001000 => [3,4] => 1001000 => 2
001001 => [3,3,1] => 1001001 => 2
001010 => [3,2,2] => 1001010 => 3
001011 => [3,2,1,1] => 1001011 => 2
001100 => [3,1,3] => 1001100 => 2
001101 => [3,1,2,1] => 1001101 => 2
001110 => [3,1,1,2] => 1001110 => 2
001111 => [3,1,1,1,1] => 1001111 => 1
010000 => [2,5] => 1010000 => 2
010001 => [2,4,1] => 1010001 => 2
010010 => [2,3,2] => 1010010 => 3
010011 => [2,3,1,1] => 1010011 => 2
010100 => [2,2,3] => 1010100 => 3
010101 => [2,2,2,1] => 1010101 => 3
010110 => [2,2,1,2] => 1010110 => 3
010111 => [2,2,1,1,1] => 1010111 => 2
011000 => [2,1,4] => 1011000 => 2
011001 => [2,1,3,1] => 1011001 => 2
011010 => [2,1,2,2] => 1011010 => 3
011011 => [2,1,2,1,1] => 1011011 => 2
011100 => [2,1,1,3] => 1011100 => 2
011101 => [2,1,1,2,1] => 1011101 => 2
011110 => [2,1,1,1,2] => 1011110 => 2
011111 => [2,1,1,1,1,1] => 1011111 => 1
100000 => [1,6] => 1100000 => 1
100001 => [1,5,1] => 1100001 => 1
100010 => [1,4,2] => 1100010 => 2
100011 => [1,4,1,1] => 1100011 => 1
100100 => [1,3,3] => 1100100 => 2
100101 => [1,3,2,1] => 1100101 => 2
100110 => [1,3,1,2] => 1100110 => 2
>>> Load all 660 entries. <<<
100111 => [1,3,1,1,1] => 1100111 => 1
101000 => [1,2,4] => 1101000 => 2
101001 => [1,2,3,1] => 1101001 => 2
101010 => [1,2,2,2] => 1101010 => 3
101011 => [1,2,2,1,1] => 1101011 => 2
101100 => [1,2,1,3] => 1101100 => 2
101101 => [1,2,1,2,1] => 1101101 => 2
101110 => [1,2,1,1,2] => 1101110 => 2
101111 => [1,2,1,1,1,1] => 1101111 => 1
110000 => [1,1,5] => 1110000 => 1
110001 => [1,1,4,1] => 1110001 => 1
110010 => [1,1,3,2] => 1110010 => 2
110011 => [1,1,3,1,1] => 1110011 => 1
110100 => [1,1,2,3] => 1110100 => 2
110101 => [1,1,2,2,1] => 1110101 => 2
110110 => [1,1,2,1,2] => 1110110 => 2
110111 => [1,1,2,1,1,1] => 1110111 => 1
111000 => [1,1,1,4] => 1111000 => 1
111001 => [1,1,1,3,1] => 1111001 => 1
111010 => [1,1,1,2,2] => 1111010 => 2
111011 => [1,1,1,2,1,1] => 1111011 => 1
111100 => [1,1,1,1,3] => 1111100 => 1
111101 => [1,1,1,1,2,1] => 1111101 => 1
111110 => [1,1,1,1,1,2] => 1111110 => 1
111111 => [1,1,1,1,1,1,1] => 1111111 => 0
0000000 => [8] => 10000000 => 1
0000001 => [7,1] => 10000001 => 1
0000010 => [6,2] => 10000010 => 2
0000011 => [6,1,1] => 10000011 => 1
0000100 => [5,3] => 10000100 => 2
0000101 => [5,2,1] => 10000101 => 2
0000110 => [5,1,2] => 10000110 => 2
0000111 => [5,1,1,1] => 10000111 => 1
0001000 => [4,4] => 10001000 => 2
0001001 => [4,3,1] => 10001001 => 2
0001010 => [4,2,2] => 10001010 => 3
0001011 => [4,2,1,1] => 10001011 => 2
0001100 => [4,1,3] => 10001100 => 2
0001101 => [4,1,2,1] => 10001101 => 2
0001110 => [4,1,1,2] => 10001110 => 2
0001111 => [4,1,1,1,1] => 10001111 => 1
0010000 => [3,5] => 10010000 => 2
0010001 => [3,4,1] => 10010001 => 2
0010010 => [3,3,2] => 10010010 => 3
0010011 => [3,3,1,1] => 10010011 => 2
0010100 => [3,2,3] => 10010100 => 3
0010101 => [3,2,2,1] => 10010101 => 3
0010110 => [3,2,1,2] => 10010110 => 3
0010111 => [3,2,1,1,1] => 10010111 => 2
0011000 => [3,1,4] => 10011000 => 2
0011001 => [3,1,3,1] => 10011001 => 2
0011010 => [3,1,2,2] => 10011010 => 3
0011011 => [3,1,2,1,1] => 10011011 => 2
0011100 => [3,1,1,3] => 10011100 => 2
0011101 => [3,1,1,2,1] => 10011101 => 2
0011110 => [3,1,1,1,2] => 10011110 => 2
0011111 => [3,1,1,1,1,1] => 10011111 => 1
0100000 => [2,6] => 10100000 => 2
0100001 => [2,5,1] => 10100001 => 2
0100010 => [2,4,2] => 10100010 => 3
0100011 => [2,4,1,1] => 10100011 => 2
0100100 => [2,3,3] => 10100100 => 3
0100101 => [2,3,2,1] => 10100101 => 3
0100110 => [2,3,1,2] => 10100110 => 3
0100111 => [2,3,1,1,1] => 10100111 => 2
0101000 => [2,2,4] => 10101000 => 3
0101001 => [2,2,3,1] => 10101001 => 3
0101010 => [2,2,2,2] => 10101010 => 4
0101011 => [2,2,2,1,1] => 10101011 => 3
0101100 => [2,2,1,3] => 10101100 => 3
0101101 => [2,2,1,2,1] => 10101101 => 3
0101110 => [2,2,1,1,2] => 10101110 => 3
0101111 => [2,2,1,1,1,1] => 10101111 => 2
0110000 => [2,1,5] => 10110000 => 2
0110001 => [2,1,4,1] => 10110001 => 2
0110010 => [2,1,3,2] => 10110010 => 3
0110011 => [2,1,3,1,1] => 10110011 => 2
0110100 => [2,1,2,3] => 10110100 => 3
0110101 => [2,1,2,2,1] => 10110101 => 3
0110110 => [2,1,2,1,2] => 10110110 => 3
0110111 => [2,1,2,1,1,1] => 10110111 => 2
0111000 => [2,1,1,4] => 10111000 => 2
0111001 => [2,1,1,3,1] => 10111001 => 2
0111010 => [2,1,1,2,2] => 10111010 => 3
0111011 => [2,1,1,2,1,1] => 10111011 => 2
0111100 => [2,1,1,1,3] => 10111100 => 2
0111101 => [2,1,1,1,2,1] => 10111101 => 2
0111110 => [2,1,1,1,1,2] => 10111110 => 2
0111111 => [2,1,1,1,1,1,1] => 10111111 => 1
1000000 => [1,7] => 11000000 => 1
1000001 => [1,6,1] => 11000001 => 1
1000010 => [1,5,2] => 11000010 => 2
1000011 => [1,5,1,1] => 11000011 => 1
1000100 => [1,4,3] => 11000100 => 2
1000101 => [1,4,2,1] => 11000101 => 2
1000110 => [1,4,1,2] => 11000110 => 2
1000111 => [1,4,1,1,1] => 11000111 => 1
1001000 => [1,3,4] => 11001000 => 2
1001001 => [1,3,3,1] => 11001001 => 2
1001010 => [1,3,2,2] => 11001010 => 3
1001011 => [1,3,2,1,1] => 11001011 => 2
1001100 => [1,3,1,3] => 11001100 => 2
1001101 => [1,3,1,2,1] => 11001101 => 2
1001110 => [1,3,1,1,2] => 11001110 => 2
1001111 => [1,3,1,1,1,1] => 11001111 => 1
1010000 => [1,2,5] => 11010000 => 2
1010001 => [1,2,4,1] => 11010001 => 2
1010010 => [1,2,3,2] => 11010010 => 3
1010011 => [1,2,3,1,1] => 11010011 => 2
1010100 => [1,2,2,3] => 11010100 => 3
1010101 => [1,2,2,2,1] => 11010101 => 3
1010110 => [1,2,2,1,2] => 11010110 => 3
1010111 => [1,2,2,1,1,1] => 11010111 => 2
1011000 => [1,2,1,4] => 11011000 => 2
1011001 => [1,2,1,3,1] => 11011001 => 2
1011010 => [1,2,1,2,2] => 11011010 => 3
1011011 => [1,2,1,2,1,1] => 11011011 => 2
1011100 => [1,2,1,1,3] => 11011100 => 2
1011101 => [1,2,1,1,2,1] => 11011101 => 2
1011110 => [1,2,1,1,1,2] => 11011110 => 2
1011111 => [1,2,1,1,1,1,1] => 11011111 => 1
1100000 => [1,1,6] => 11100000 => 1
1100001 => [1,1,5,1] => 11100001 => 1
1100010 => [1,1,4,2] => 11100010 => 2
1100011 => [1,1,4,1,1] => 11100011 => 1
1100100 => [1,1,3,3] => 11100100 => 2
1100101 => [1,1,3,2,1] => 11100101 => 2
1100110 => [1,1,3,1,2] => 11100110 => 2
1100111 => [1,1,3,1,1,1] => 11100111 => 1
1101000 => [1,1,2,4] => 11101000 => 2
1101001 => [1,1,2,3,1] => 11101001 => 2
1101010 => [1,1,2,2,2] => 11101010 => 3
1101011 => [1,1,2,2,1,1] => 11101011 => 2
1101100 => [1,1,2,1,3] => 11101100 => 2
1101101 => [1,1,2,1,2,1] => 11101101 => 2
1101110 => [1,1,2,1,1,2] => 11101110 => 2
1101111 => [1,1,2,1,1,1,1] => 11101111 => 1
1110000 => [1,1,1,5] => 11110000 => 1
1110001 => [1,1,1,4,1] => 11110001 => 1
1110010 => [1,1,1,3,2] => 11110010 => 2
1110011 => [1,1,1,3,1,1] => 11110011 => 1
1110100 => [1,1,1,2,3] => 11110100 => 2
1110101 => [1,1,1,2,2,1] => 11110101 => 2
1110110 => [1,1,1,2,1,2] => 11110110 => 2
1110111 => [1,1,1,2,1,1,1] => 11110111 => 1
1111000 => [1,1,1,1,4] => 11111000 => 1
1111001 => [1,1,1,1,3,1] => 11111001 => 1
1111010 => [1,1,1,1,2,2] => 11111010 => 2
1111011 => [1,1,1,1,2,1,1] => 11111011 => 1
1111100 => [1,1,1,1,1,3] => 11111100 => 1
1111101 => [1,1,1,1,1,2,1] => 11111101 => 1
1111110 => [1,1,1,1,1,1,2] => 11111110 => 1
1111111 => [1,1,1,1,1,1,1,1] => 11111111 => 0
00000000 => [9] => 100000000 => 1
00000001 => [8,1] => 100000001 => 1
00000010 => [7,2] => 100000010 => 2
00000011 => [7,1,1] => 100000011 => 1
00000100 => [6,3] => 100000100 => 2
00000101 => [6,2,1] => 100000101 => 2
00000110 => [6,1,2] => 100000110 => 2
00000111 => [6,1,1,1] => 100000111 => 1
00001000 => [5,4] => 100001000 => 2
00001001 => [5,3,1] => 100001001 => 2
00001010 => [5,2,2] => 100001010 => 3
00001011 => [5,2,1,1] => 100001011 => 2
00001100 => [5,1,3] => 100001100 => 2
00001101 => [5,1,2,1] => 100001101 => 2
00001110 => [5,1,1,2] => 100001110 => 2
00001111 => [5,1,1,1,1] => 100001111 => 1
00010000 => [4,5] => 100010000 => 2
00010001 => [4,4,1] => 100010001 => 2
00010010 => [4,3,2] => 100010010 => 3
00010011 => [4,3,1,1] => 100010011 => 2
00010100 => [4,2,3] => 100010100 => 3
00010101 => [4,2,2,1] => 100010101 => 3
00010110 => [4,2,1,2] => 100010110 => 3
00010111 => [4,2,1,1,1] => 100010111 => 2
00011000 => [4,1,4] => 100011000 => 2
00011001 => [4,1,3,1] => 100011001 => 2
00011010 => [4,1,2,2] => 100011010 => 3
00011011 => [4,1,2,1,1] => 100011011 => 2
00011100 => [4,1,1,3] => 100011100 => 2
00011101 => [4,1,1,2,1] => 100011101 => 2
00011110 => [4,1,1,1,2] => 100011110 => 2
00011111 => [4,1,1,1,1,1] => 100011111 => 1
00100000 => [3,6] => 100100000 => 2
00100001 => [3,5,1] => 100100001 => 2
00100010 => [3,4,2] => 100100010 => 3
00100011 => [3,4,1,1] => 100100011 => 2
00100100 => [3,3,3] => 100100100 => 3
00100101 => [3,3,2,1] => 100100101 => 3
00100110 => [3,3,1,2] => 100100110 => 3
00100111 => [3,3,1,1,1] => 100100111 => 2
00101000 => [3,2,4] => 100101000 => 3
00101001 => [3,2,3,1] => 100101001 => 3
00101010 => [3,2,2,2] => 100101010 => 4
00101011 => [3,2,2,1,1] => 100101011 => 3
00101100 => [3,2,1,3] => 100101100 => 3
00101101 => [3,2,1,2,1] => 100101101 => 3
00101110 => [3,2,1,1,2] => 100101110 => 3
00101111 => [3,2,1,1,1,1] => 100101111 => 2
00110000 => [3,1,5] => 100110000 => 2
00110001 => [3,1,4,1] => 100110001 => 2
00110010 => [3,1,3,2] => 100110010 => 3
00110011 => [3,1,3,1,1] => 100110011 => 2
00110100 => [3,1,2,3] => 100110100 => 3
00110101 => [3,1,2,2,1] => 100110101 => 3
00110110 => [3,1,2,1,2] => 100110110 => 3
00110111 => [3,1,2,1,1,1] => 100110111 => 2
00111000 => [3,1,1,4] => 100111000 => 2
00111001 => [3,1,1,3,1] => 100111001 => 2
00111010 => [3,1,1,2,2] => 100111010 => 3
00111011 => [3,1,1,2,1,1] => 100111011 => 2
00111100 => [3,1,1,1,3] => 100111100 => 2
00111101 => [3,1,1,1,2,1] => 100111101 => 2
00111110 => [3,1,1,1,1,2] => 100111110 => 2
00111111 => [3,1,1,1,1,1,1] => 100111111 => 1
01000000 => [2,7] => 101000000 => 2
01000001 => [2,6,1] => 101000001 => 2
01000010 => [2,5,2] => 101000010 => 3
01000011 => [2,5,1,1] => 101000011 => 2
01000100 => [2,4,3] => 101000100 => 3
01000101 => [2,4,2,1] => 101000101 => 3
01000110 => [2,4,1,2] => 101000110 => 3
01000111 => [2,4,1,1,1] => 101000111 => 2
01001000 => [2,3,4] => 101001000 => 3
01001001 => [2,3,3,1] => 101001001 => 3
01001010 => [2,3,2,2] => 101001010 => 4
01001011 => [2,3,2,1,1] => 101001011 => 3
01001100 => [2,3,1,3] => 101001100 => 3
01001101 => [2,3,1,2,1] => 101001101 => 3
01001110 => [2,3,1,1,2] => 101001110 => 3
01001111 => [2,3,1,1,1,1] => 101001111 => 2
01010000 => [2,2,5] => 101010000 => 3
01010001 => [2,2,4,1] => 101010001 => 3
01010010 => [2,2,3,2] => 101010010 => 4
01010011 => [2,2,3,1,1] => 101010011 => 3
01010100 => [2,2,2,3] => 101010100 => 4
01010101 => [2,2,2,2,1] => 101010101 => 4
01010110 => [2,2,2,1,2] => 101010110 => 4
01010111 => [2,2,2,1,1,1] => 101010111 => 3
01011000 => [2,2,1,4] => 101011000 => 3
01011001 => [2,2,1,3,1] => 101011001 => 3
01011010 => [2,2,1,2,2] => 101011010 => 4
01011011 => [2,2,1,2,1,1] => 101011011 => 3
01011100 => [2,2,1,1,3] => 101011100 => 3
01011101 => [2,2,1,1,2,1] => 101011101 => 3
01011110 => [2,2,1,1,1,2] => 101011110 => 3
01011111 => [2,2,1,1,1,1,1] => 101011111 => 2
01100000 => [2,1,6] => 101100000 => 2
01100001 => [2,1,5,1] => 101100001 => 2
01100010 => [2,1,4,2] => 101100010 => 3
01100011 => [2,1,4,1,1] => 101100011 => 2
01100100 => [2,1,3,3] => 101100100 => 3
01100101 => [2,1,3,2,1] => 101100101 => 3
01100110 => [2,1,3,1,2] => 101100110 => 3
01100111 => [2,1,3,1,1,1] => 101100111 => 2
01101000 => [2,1,2,4] => 101101000 => 3
01101001 => [2,1,2,3,1] => 101101001 => 3
01101010 => [2,1,2,2,2] => 101101010 => 4
01101011 => [2,1,2,2,1,1] => 101101011 => 3
01101100 => [2,1,2,1,3] => 101101100 => 3
01101101 => [2,1,2,1,2,1] => 101101101 => 3
01101110 => [2,1,2,1,1,2] => 101101110 => 3
01101111 => [2,1,2,1,1,1,1] => 101101111 => 2
01110000 => [2,1,1,5] => 101110000 => 2
01110001 => [2,1,1,4,1] => 101110001 => 2
01110010 => [2,1,1,3,2] => 101110010 => 3
01110011 => [2,1,1,3,1,1] => 101110011 => 2
01110100 => [2,1,1,2,3] => 101110100 => 3
01110101 => [2,1,1,2,2,1] => 101110101 => 3
01110110 => [2,1,1,2,1,2] => 101110110 => 3
01110111 => [2,1,1,2,1,1,1] => 101110111 => 2
01111000 => [2,1,1,1,4] => 101111000 => 2
01111001 => [2,1,1,1,3,1] => 101111001 => 2
01111010 => [2,1,1,1,2,2] => 101111010 => 3
01111011 => [2,1,1,1,2,1,1] => 101111011 => 2
01111100 => [2,1,1,1,1,3] => 101111100 => 2
01111101 => [2,1,1,1,1,2,1] => 101111101 => 2
01111110 => [2,1,1,1,1,1,2] => 101111110 => 2
01111111 => [2,1,1,1,1,1,1,1] => 101111111 => 1
10000000 => [1,8] => 110000000 => 1
10000001 => [1,7,1] => 110000001 => 1
10000010 => [1,6,2] => 110000010 => 2
10000011 => [1,6,1,1] => 110000011 => 1
10000100 => [1,5,3] => 110000100 => 2
10000101 => [1,5,2,1] => 110000101 => 2
10000110 => [1,5,1,2] => 110000110 => 2
10000111 => [1,5,1,1,1] => 110000111 => 1
10001000 => [1,4,4] => 110001000 => 2
10001001 => [1,4,3,1] => 110001001 => 2
10001010 => [1,4,2,2] => 110001010 => 3
10001011 => [1,4,2,1,1] => 110001011 => 2
10001100 => [1,4,1,3] => 110001100 => 2
10001101 => [1,4,1,2,1] => 110001101 => 2
10001110 => [1,4,1,1,2] => 110001110 => 2
10001111 => [1,4,1,1,1,1] => 110001111 => 1
10010000 => [1,3,5] => 110010000 => 2
10010001 => [1,3,4,1] => 110010001 => 2
10010010 => [1,3,3,2] => 110010010 => 3
10010011 => [1,3,3,1,1] => 110010011 => 2
10010100 => [1,3,2,3] => 110010100 => 3
10010101 => [1,3,2,2,1] => 110010101 => 3
10010110 => [1,3,2,1,2] => 110010110 => 3
10010111 => [1,3,2,1,1,1] => 110010111 => 2
10011000 => [1,3,1,4] => 110011000 => 2
10011001 => [1,3,1,3,1] => 110011001 => 2
10011010 => [1,3,1,2,2] => 110011010 => 3
10011011 => [1,3,1,2,1,1] => 110011011 => 2
10011100 => [1,3,1,1,3] => 110011100 => 2
10011101 => [1,3,1,1,2,1] => 110011101 => 2
10011110 => [1,3,1,1,1,2] => 110011110 => 2
10011111 => [1,3,1,1,1,1,1] => 110011111 => 1
10100000 => [1,2,6] => 110100000 => 2
10100001 => [1,2,5,1] => 110100001 => 2
10100010 => [1,2,4,2] => 110100010 => 3
10100011 => [1,2,4,1,1] => 110100011 => 2
10100100 => [1,2,3,3] => 110100100 => 3
10100101 => [1,2,3,2,1] => 110100101 => 3
10100110 => [1,2,3,1,2] => 110100110 => 3
10100111 => [1,2,3,1,1,1] => 110100111 => 2
10101000 => [1,2,2,4] => 110101000 => 3
10101001 => [1,2,2,3,1] => 110101001 => 3
10101010 => [1,2,2,2,2] => 110101010 => 4
10101011 => [1,2,2,2,1,1] => 110101011 => 3
10101100 => [1,2,2,1,3] => 110101100 => 3
10101101 => [1,2,2,1,2,1] => 110101101 => 3
10101110 => [1,2,2,1,1,2] => 110101110 => 3
10101111 => [1,2,2,1,1,1,1] => 110101111 => 2
10110000 => [1,2,1,5] => 110110000 => 2
10110001 => [1,2,1,4,1] => 110110001 => 2
10110010 => [1,2,1,3,2] => 110110010 => 3
10110011 => [1,2,1,3,1,1] => 110110011 => 2
10110100 => [1,2,1,2,3] => 110110100 => 3
10110101 => [1,2,1,2,2,1] => 110110101 => 3
10110110 => [1,2,1,2,1,2] => 110110110 => 3
10110111 => [1,2,1,2,1,1,1] => 110110111 => 2
10111000 => [1,2,1,1,4] => 110111000 => 2
10111001 => [1,2,1,1,3,1] => 110111001 => 2
10111010 => [1,2,1,1,2,2] => 110111010 => 3
10111011 => [1,2,1,1,2,1,1] => 110111011 => 2
10111100 => [1,2,1,1,1,3] => 110111100 => 2
10111101 => [1,2,1,1,1,2,1] => 110111101 => 2
10111110 => [1,2,1,1,1,1,2] => 110111110 => 2
10111111 => [1,2,1,1,1,1,1,1] => 110111111 => 1
11000000 => [1,1,7] => 111000000 => 1
11000001 => [1,1,6,1] => 111000001 => 1
11000010 => [1,1,5,2] => 111000010 => 2
11000011 => [1,1,5,1,1] => 111000011 => 1
11000100 => [1,1,4,3] => 111000100 => 2
11000101 => [1,1,4,2,1] => 111000101 => 2
11000110 => [1,1,4,1,2] => 111000110 => 2
11000111 => [1,1,4,1,1,1] => 111000111 => 1
11001000 => [1,1,3,4] => 111001000 => 2
11001001 => [1,1,3,3,1] => 111001001 => 2
11001010 => [1,1,3,2,2] => 111001010 => 3
11001011 => [1,1,3,2,1,1] => 111001011 => 2
11001100 => [1,1,3,1,3] => 111001100 => 2
11001101 => [1,1,3,1,2,1] => 111001101 => 2
11001110 => [1,1,3,1,1,2] => 111001110 => 2
11001111 => [1,1,3,1,1,1,1] => 111001111 => 1
11010000 => [1,1,2,5] => 111010000 => 2
11010001 => [1,1,2,4,1] => 111010001 => 2
11010010 => [1,1,2,3,2] => 111010010 => 3
11010011 => [1,1,2,3,1,1] => 111010011 => 2
11010100 => [1,1,2,2,3] => 111010100 => 3
11010101 => [1,1,2,2,2,1] => 111010101 => 3
11010110 => [1,1,2,2,1,2] => 111010110 => 3
11010111 => [1,1,2,2,1,1,1] => 111010111 => 2
11011000 => [1,1,2,1,4] => 111011000 => 2
11011001 => [1,1,2,1,3,1] => 111011001 => 2
11011010 => [1,1,2,1,2,2] => 111011010 => 3
11011011 => [1,1,2,1,2,1,1] => 111011011 => 2
11011100 => [1,1,2,1,1,3] => 111011100 => 2
11011101 => [1,1,2,1,1,2,1] => 111011101 => 2
11011110 => [1,1,2,1,1,1,2] => 111011110 => 2
11011111 => [1,1,2,1,1,1,1,1] => 111011111 => 1
11100000 => [1,1,1,6] => 111100000 => 1
11100001 => [1,1,1,5,1] => 111100001 => 1
11100010 => [1,1,1,4,2] => 111100010 => 2
11100011 => [1,1,1,4,1,1] => 111100011 => 1
11100100 => [1,1,1,3,3] => 111100100 => 2
11100101 => [1,1,1,3,2,1] => 111100101 => 2
11100110 => [1,1,1,3,1,2] => 111100110 => 2
11100111 => [1,1,1,3,1,1,1] => 111100111 => 1
11101000 => [1,1,1,2,4] => 111101000 => 2
11101001 => [1,1,1,2,3,1] => 111101001 => 2
11101010 => [1,1,1,2,2,2] => 111101010 => 3
11101011 => [1,1,1,2,2,1,1] => 111101011 => 2
11101100 => [1,1,1,2,1,3] => 111101100 => 2
11101101 => [1,1,1,2,1,2,1] => 111101101 => 2
11101110 => [1,1,1,2,1,1,2] => 111101110 => 2
11101111 => [1,1,1,2,1,1,1,1] => 111101111 => 1
11110000 => [1,1,1,1,5] => 111110000 => 1
11110001 => [1,1,1,1,4,1] => 111110001 => 1
11110010 => [1,1,1,1,3,2] => 111110010 => 2
11110011 => [1,1,1,1,3,1,1] => 111110011 => 1
11110100 => [1,1,1,1,2,3] => 111110100 => 2
11110101 => [1,1,1,1,2,2,1] => 111110101 => 2
11110110 => [1,1,1,1,2,1,2] => 111110110 => 2
11110111 => [1,1,1,1,2,1,1,1] => 111110111 => 1
11111000 => [1,1,1,1,1,4] => 111111000 => 1
11111001 => [1,1,1,1,1,3,1] => 111111001 => 1
11111010 => [1,1,1,1,1,2,2] => 111111010 => 2
11111011 => [1,1,1,1,1,2,1,1] => 111111011 => 1
11111100 => [1,1,1,1,1,1,3] => 111111100 => 1
11111101 => [1,1,1,1,1,1,2,1] => 111111101 => 1
11111110 => [1,1,1,1,1,1,1,2] => 111111110 => 1
11111111 => [1,1,1,1,1,1,1,1,1] => 111111111 => 0
000000000 => [10] => 1000000000 => 1
000000010 => [8,2] => 1000000010 => 2
000000100 => [7,3] => 1000000100 => 2
000000110 => [7,1,2] => 1000000110 => 2
000001010 => [6,2,2] => 1000001010 => 3
000001110 => [6,1,1,2] => 1000001110 => 2
000010110 => [5,2,1,2] => 1000010110 => 3
000011110 => [5,1,1,1,2] => 1000011110 => 2
000101110 => [4,2,1,1,2] => 1000101110 => 3
000110110 => [4,1,2,1,2] => 1000110110 => 3
000111010 => [4,1,1,2,2] => 1000111010 => 3
000111100 => [4,1,1,1,3] => 1000111100 => 2
000111110 => [4,1,1,1,1,2] => 1000111110 => 2
001001110 => [3,3,1,1,2] => 1001001110 => 3
001010101 => [3,2,2,2,1] => 1001010101 => 4
001010110 => [3,2,2,1,2] => 1001010110 => 4
001011001 => [3,2,1,3,1] => 1001011001 => 3
001011010 => [3,2,1,2,2] => 1001011010 => 4
001011110 => [3,2,1,1,1,2] => 1001011110 => 3
001100101 => [3,1,3,2,1] => 1001100101 => 3
001100110 => [3,1,3,1,2] => 1001100110 => 3
001101001 => [3,1,2,3,1] => 1001101001 => 3
001101010 => [3,1,2,2,2] => 1001101010 => 4
001101100 => [3,1,2,1,3] => 1001101100 => 3
001110001 => [3,1,1,4,1] => 1001110001 => 2
001110100 => [3,1,1,2,3] => 1001110100 => 3
001111000 => [3,1,1,1,4] => 1001111000 => 2
001111110 => [3,1,1,1,1,1,2] => 1001111110 => 2
010001110 => [2,4,1,1,2] => 1010001110 => 3
010010101 => [2,3,2,2,1] => 1010010101 => 4
010010110 => [2,3,2,1,2] => 1010010110 => 4
010011001 => [2,3,1,3,1] => 1010011001 => 3
010011010 => [2,3,1,2,2] => 1010011010 => 4
010100101 => [2,2,3,2,1] => 1010100101 => 4
010100110 => [2,2,3,1,2] => 1010100110 => 4
010101001 => [2,2,2,3,1] => 1010101001 => 4
010101010 => [2,2,2,2,2] => 1010101010 => 5
010101100 => [2,2,2,1,3] => 1010101100 => 4
010110001 => [2,2,1,4,1] => 1010110001 => 3
010110010 => [2,2,1,3,2] => 1010110010 => 4
010110100 => [2,2,1,2,3] => 1010110100 => 4
010111000 => [2,2,1,1,4] => 1010111000 => 3
010111110 => [2,2,1,1,1,1,2] => 1010111110 => 3
011000101 => [2,1,4,2,1] => 1011000101 => 3
011001001 => [2,1,3,3,1] => 1011001001 => 3
011001010 => [2,1,3,2,2] => 1011001010 => 4
011001100 => [2,1,3,1,3] => 1011001100 => 3
011010001 => [2,1,2,4,1] => 1011010001 => 3
011010010 => [2,1,2,3,2] => 1011010010 => 4
011010100 => [2,1,2,2,3] => 1011010100 => 4
011011000 => [2,1,2,1,4] => 1011011000 => 3
011100001 => [2,1,1,5,1] => 1011100001 => 2
011100010 => [2,1,1,4,2] => 1011100010 => 3
011100100 => [2,1,1,3,3] => 1011100100 => 3
011101000 => [2,1,1,2,4] => 1011101000 => 3
011110000 => [2,1,1,1,5] => 1011110000 => 2
011111101 => [2,1,1,1,1,1,2,1] => 1011111101 => 2
011111110 => [2,1,1,1,1,1,1,2] => 1011111110 => 2
100001000 => [1,5,4] => 1100001000 => 2
100010010 => [1,4,3,2] => 1100010010 => 3
100010101 => [1,4,2,2,1] => 1100010101 => 3
100011001 => [1,4,1,3,1] => 1100011001 => 2
100100101 => [1,3,3,2,1] => 1100100101 => 3
100100110 => [1,3,3,1,2] => 1100100110 => 3
100101001 => [1,3,2,3,1] => 1100101001 => 3
100101010 => [1,3,2,2,2] => 1100101010 => 4
100101100 => [1,3,2,1,3] => 1100101100 => 3
100110001 => [1,3,1,4,1] => 1100110001 => 2
100110010 => [1,3,1,3,2] => 1100110010 => 3
100110100 => [1,3,1,2,3] => 1100110100 => 3
100111000 => [1,3,1,1,4] => 1100111000 => 2
101000101 => [1,2,4,2,1] => 1101000101 => 3
101001001 => [1,2,3,3,1] => 1101001001 => 3
101001010 => [1,2,3,2,2] => 1101001010 => 4
101001100 => [1,2,3,1,3] => 1101001100 => 3
101001110 => [1,2,3,1,1,2] => 1101001110 => 3
101010001 => [1,2,2,4,1] => 1101010001 => 3
101010010 => [1,2,2,3,2] => 1101010010 => 4
101010100 => [1,2,2,2,3] => 1101010100 => 4
101011000 => [1,2,2,1,4] => 1101011000 => 3
101100001 => [1,2,1,5,1] => 1101100001 => 2
101100010 => [1,2,1,4,2] => 1101100010 => 3
101100100 => [1,2,1,3,3] => 1101100100 => 3
101101000 => [1,2,1,2,4] => 1101101000 => 3
101110000 => [1,2,1,1,5] => 1101110000 => 2
101111110 => [1,2,1,1,1,1,1,2] => 1101111110 => 2
110000101 => [1,1,5,2,1] => 1110000101 => 2
110001001 => [1,1,4,3,1] => 1110001001 => 2
110001010 => [1,1,4,2,2] => 1110001010 => 3
110001100 => [1,1,4,1,3] => 1110001100 => 2
110010001 => [1,1,3,4,1] => 1110010001 => 2
110010010 => [1,1,3,3,2] => 1110010010 => 3
110010100 => [1,1,3,2,3] => 1110010100 => 3
110011000 => [1,1,3,1,4] => 1110011000 => 2
110011110 => [1,1,3,1,1,1,2] => 1110011110 => 2
110100001 => [1,1,2,5,1] => 1110100001 => 2
110100010 => [1,1,2,4,2] => 1110100010 => 3
110100100 => [1,1,2,3,3] => 1110100100 => 3
110101000 => [1,1,2,2,4] => 1110101000 => 3
110110000 => [1,1,2,1,5] => 1110110000 => 2
110111110 => [1,1,2,1,1,1,1,2] => 1110111110 => 2
111000001 => [1,1,1,6,1] => 1111000001 => 1
111000010 => [1,1,1,5,2] => 1111000010 => 2
111000100 => [1,1,1,4,3] => 1111000100 => 2
111001000 => [1,1,1,3,4] => 1111001000 => 2
111010000 => [1,1,1,2,5] => 1111010000 => 2
111100000 => [1,1,1,1,6] => 1111100000 => 1
111111101 => [1,1,1,1,1,1,1,2,1] => 1111111101 => 1
111111110 => [1,1,1,1,1,1,1,1,2] => 1111111110 => 1
1111111110 => [1,1,1,1,1,1,1,1,1,2] => 11111111110 => 1
1111111101 => [1,1,1,1,1,1,1,1,2,1] => 11111111101 => 1
1011111110 => [1,2,1,1,1,1,1,1,2] => 11011111110 => 2
10010101010 => [1,3,2,2,2,2] => 110010101010 => 5
10100101010 => [1,2,3,2,2,2] => 110100101010 => 5
=> [1] => 1 => 0
01111000001 => [2,1,1,1,6,1] => 101111000001 => 2
0101010101 => [2,2,2,2,2,1] => 10101010101 => 5
11010101000 => [1,1,2,2,2,4] => 111010101000 => 4
11010100100 => [1,1,2,2,3,3] => 111010100100 => 4
11010010100 => [1,1,2,3,2,3] => 111010010100 => 4
11001010100 => [1,1,3,2,2,3] => 111001010100 => 4
10101010100 => [1,2,2,2,2,3] => 110101010100 => 5
0000000000 => [11] => 10000000000 => 1
00101010101 => [3,2,2,2,2,1] => 100101010101 => 5
0000000010 => [9,2] => 10000000010 => 2
0000111110 => [5,1,1,1,1,2] => 10000111110 => 2
0011111110 => [3,1,1,1,1,1,1,2] => 10011111110 => 2
0101111110 => [2,2,1,1,1,1,1,2] => 10101111110 => 3
10101010010 => [1,2,2,2,3,2] => 110101010010 => 5
10101001001 => [1,2,2,3,3,1] => 110101001001 => 4
10101001010 => [1,2,2,3,2,2] => 110101001010 => 5
10100100101 => [1,2,3,3,2,1] => 110100100101 => 4
10100101001 => [1,2,3,2,3,1] => 110100101001 => 4
10010010101 => [1,3,3,2,2,1] => 110010010101 => 4
10010100101 => [1,3,2,3,2,1] => 110010100101 => 4
10010101001 => [1,3,2,2,3,1] => 110010101001 => 4
01001010101 => [2,3,2,2,2,1] => 101001010101 => 5
01010010101 => [2,2,3,2,2,1] => 101010010101 => 5
01010100101 => [2,2,2,3,2,1] => 101010100101 => 5
01010101001 => [2,2,2,2,3,1] => 101010101001 => 5
01010101010 => [2,2,2,2,2,2] => 101010101010 => 6
10001010101 => [1,4,2,2,2,1] => 110001010101 => 4
0001000100 => [4,4,3] => 10001000100 => 3
10101010001 => [1,2,2,2,4,1] => 110101010001 => 4
01101010100 => [2,1,2,2,2,3] => 101101010100 => 5
01101001010 => [2,1,2,3,2,2] => 101101001010 => 5
01010110100 => [2,2,2,1,2,3] => 101010110100 => 5
11110001000 => [1,1,1,1,4,4] => 111110001000 => 2
11000111000 => [1,1,4,1,1,4] => 111000111000 => 2
00011111000 => [4,1,1,1,1,4] => 100011111000 => 2
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Description
The number of descents of a binary word.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
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 $0$.