Identifier
Values
0 => 0 => 1 => [1,1] => 1
1 => 1 => 0 => [2] => 2
00 => 00 => 01 => [2,1] => 1
01 => 10 => 11 => [1,1,1] => 1
10 => 01 => 10 => [1,2] => 2
11 => 11 => 00 => [3] => 3
000 => 000 => 001 => [3,1] => 1
001 => 100 => 011 => [2,1,1] => 1
010 => 010 => 101 => [1,2,1] => 1
011 => 110 => 111 => [1,1,1,1] => 1
100 => 001 => 010 => [2,2] => 2
101 => 101 => 110 => [1,1,2] => 2
110 => 011 => 100 => [1,3] => 3
111 => 111 => 000 => [4] => 4
0000 => 0000 => 0001 => [4,1] => 1
0001 => 1000 => 0011 => [3,1,1] => 1
0010 => 0100 => 1001 => [1,3,1] => 1
0011 => 1100 => 0111 => [2,1,1,1] => 1
0100 => 0010 => 0101 => [2,2,1] => 1
0101 => 1010 => 1101 => [1,1,2,1] => 1
0110 => 0110 => 1011 => [1,2,1,1] => 1
0111 => 1110 => 1111 => [1,1,1,1,1] => 1
1000 => 0001 => 0010 => [3,2] => 2
1001 => 1001 => 0110 => [2,1,2] => 2
1010 => 0101 => 1010 => [1,2,2] => 2
1011 => 1101 => 1110 => [1,1,1,2] => 2
1100 => 0011 => 0100 => [2,3] => 3
1101 => 1011 => 1100 => [1,1,3] => 3
1110 => 0111 => 1000 => [1,4] => 4
1111 => 1111 => 0000 => [5] => 5
00000 => 00000 => 00001 => [5,1] => 1
00001 => 10000 => 00011 => [4,1,1] => 1
00010 => 01000 => 10001 => [1,4,1] => 1
00011 => 11000 => 00111 => [3,1,1,1] => 1
00100 => 00100 => 01001 => [2,3,1] => 1
00101 => 10100 => 11001 => [1,1,3,1] => 1
00110 => 01100 => 10011 => [1,3,1,1] => 1
00111 => 11100 => 01111 => [2,1,1,1,1] => 1
01000 => 00010 => 00101 => [3,2,1] => 1
01001 => 10010 => 01101 => [2,1,2,1] => 1
01010 => 01010 => 10101 => [1,2,2,1] => 1
01011 => 11010 => 11101 => [1,1,1,2,1] => 1
01100 => 00110 => 01011 => [2,2,1,1] => 1
01101 => 10110 => 11011 => [1,1,2,1,1] => 1
01110 => 01110 => 10111 => [1,2,1,1,1] => 1
01111 => 11110 => 11111 => [1,1,1,1,1,1] => 1
10000 => 00001 => 00010 => [4,2] => 2
10001 => 10001 => 00110 => [3,1,2] => 2
10010 => 01001 => 10010 => [1,3,2] => 2
10011 => 11001 => 01110 => [2,1,1,2] => 2
10100 => 00101 => 01010 => [2,2,2] => 2
10101 => 10101 => 11010 => [1,1,2,2] => 2
10110 => 01101 => 10110 => [1,2,1,2] => 2
10111 => 11101 => 11110 => [1,1,1,1,2] => 2
11000 => 00011 => 00100 => [3,3] => 3
11001 => 10011 => 01100 => [2,1,3] => 3
11010 => 01011 => 10100 => [1,2,3] => 3
11011 => 11011 => 11100 => [1,1,1,3] => 3
11100 => 00111 => 01000 => [2,4] => 4
11101 => 10111 => 11000 => [1,1,4] => 4
11110 => 01111 => 10000 => [1,5] => 5
11111 => 11111 => 00000 => [6] => 6
000000 => 000000 => 000001 => [6,1] => 1
000001 => 100000 => 000011 => [5,1,1] => 1
000010 => 010000 => 100001 => [1,5,1] => 1
000011 => 110000 => 000111 => [4,1,1,1] => 1
000100 => 001000 => 010001 => [2,4,1] => 1
000101 => 101000 => 110001 => [1,1,4,1] => 1
000110 => 011000 => 100011 => [1,4,1,1] => 1
000111 => 111000 => 001111 => [3,1,1,1,1] => 1
001000 => 000100 => 001001 => [3,3,1] => 1
001001 => 100100 => 011001 => [2,1,3,1] => 1
001010 => 010100 => 101001 => [1,2,3,1] => 1
001011 => 110100 => 111001 => [1,1,1,3,1] => 1
001100 => 001100 => 010011 => [2,3,1,1] => 1
001101 => 101100 => 110011 => [1,1,3,1,1] => 1
001110 => 011100 => 100111 => [1,3,1,1,1] => 1
001111 => 111100 => 011111 => [2,1,1,1,1,1] => 1
010000 => 000010 => 000101 => [4,2,1] => 1
010001 => 100010 => 001101 => [3,1,2,1] => 1
010010 => 010010 => 100101 => [1,3,2,1] => 1
010011 => 110010 => 011101 => [2,1,1,2,1] => 1
010100 => 001010 => 010101 => [2,2,2,1] => 1
010101 => 101010 => 110101 => [1,1,2,2,1] => 1
010110 => 011010 => 101101 => [1,2,1,2,1] => 1
010111 => 111010 => 111101 => [1,1,1,1,2,1] => 1
011000 => 000110 => 001011 => [3,2,1,1] => 1
011001 => 100110 => 011011 => [2,1,2,1,1] => 1
011010 => 010110 => 101011 => [1,2,2,1,1] => 1
011011 => 110110 => 111011 => [1,1,1,2,1,1] => 1
011100 => 001110 => 010111 => [2,2,1,1,1] => 1
011101 => 101110 => 110111 => [1,1,2,1,1,1] => 1
011110 => 011110 => 101111 => [1,2,1,1,1,1] => 1
011111 => 111110 => 111111 => [1,1,1,1,1,1,1] => 1
100000 => 000001 => 000010 => [5,2] => 2
100001 => 100001 => 000110 => [4,1,2] => 2
100010 => 010001 => 100010 => [1,4,2] => 2
100011 => 110001 => 001110 => [3,1,1,2] => 2
100100 => 001001 => 010010 => [2,3,2] => 2
100101 => 101001 => 110010 => [1,1,3,2] => 2
100110 => 011001 => 100110 => [1,3,1,2] => 2
>>> Load all 313 entries. <<<
100111 => 111001 => 011110 => [2,1,1,1,2] => 2
101000 => 000101 => 001010 => [3,2,2] => 2
101001 => 100101 => 011010 => [2,1,2,2] => 2
101010 => 010101 => 101010 => [1,2,2,2] => 2
101011 => 110101 => 111010 => [1,1,1,2,2] => 2
101100 => 001101 => 010110 => [2,2,1,2] => 2
101101 => 101101 => 110110 => [1,1,2,1,2] => 2
101110 => 011101 => 101110 => [1,2,1,1,2] => 2
101111 => 111101 => 111110 => [1,1,1,1,1,2] => 2
110000 => 000011 => 000100 => [4,3] => 3
110001 => 100011 => 001100 => [3,1,3] => 3
110010 => 010011 => 100100 => [1,3,3] => 3
110011 => 110011 => 011100 => [2,1,1,3] => 3
110100 => 001011 => 010100 => [2,2,3] => 3
110101 => 101011 => 110100 => [1,1,2,3] => 3
110110 => 011011 => 101100 => [1,2,1,3] => 3
110111 => 111011 => 111100 => [1,1,1,1,3] => 3
111000 => 000111 => 001000 => [3,4] => 4
111001 => 100111 => 011000 => [2,1,4] => 4
111010 => 010111 => 101000 => [1,2,4] => 4
111011 => 110111 => 111000 => [1,1,1,4] => 4
111100 => 001111 => 010000 => [2,5] => 5
111101 => 101111 => 110000 => [1,1,5] => 5
111110 => 011111 => 100000 => [1,6] => 6
111111 => 111111 => 000000 => [7] => 7
0000000 => 0000000 => 0000001 => [7,1] => 1
0000001 => 1000000 => 0000011 => [6,1,1] => 1
0000010 => 0100000 => 1000001 => [1,6,1] => 1
0000011 => 1100000 => 0000111 => [5,1,1,1] => 1
0000101 => 1010000 => 1100001 => [1,1,5,1] => 1
0000110 => 0110000 => 1000011 => [1,5,1,1] => 1
0001000 => 0001000 => 0010001 => [3,4,1] => 1
0001001 => 1001000 => 0110001 => [2,1,4,1] => 1
0001010 => 0101000 => 1010001 => [1,2,4,1] => 1
0010000 => 0000100 => 0001001 => [4,3,1] => 1
0010001 => 1000100 => 0011001 => [3,1,3,1] => 1
0010010 => 0100100 => 1001001 => [1,3,3,1] => 1
0010011 => 1100100 => 0111001 => [2,1,1,3,1] => 1
0010100 => 0010100 => 0101001 => [2,2,3,1] => 1
0010101 => 1010100 => 1101001 => [1,1,2,3,1] => 1
0010110 => 0110100 => 1011001 => [1,2,1,3,1] => 1
0010111 => 1110100 => 1111001 => [1,1,1,1,3,1] => 1
0011000 => 0001100 => 0010011 => [3,3,1,1] => 1
0011001 => 1001100 => 0110011 => [2,1,3,1,1] => 1
0011010 => 0101100 => 1010011 => [1,2,3,1,1] => 1
0011011 => 1101100 => 1110011 => [1,1,1,3,1,1] => 1
0011100 => 0011100 => 0100111 => [2,3,1,1,1] => 1
0011101 => 1011100 => 1100111 => [1,1,3,1,1,1] => 1
0011110 => 0111100 => 1001111 => [1,3,1,1,1,1] => 1
0100000 => 0000010 => 0000101 => [5,2,1] => 1
0100001 => 1000010 => 0001101 => [4,1,2,1] => 1
0100010 => 0100010 => 1000101 => [1,4,2,1] => 1
0100100 => 0010010 => 0100101 => [2,3,2,1] => 1
0100101 => 1010010 => 1100101 => [1,1,3,2,1] => 1
0100110 => 0110010 => 1001101 => [1,3,1,2,1] => 1
0100111 => 1110010 => 0111101 => [2,1,1,1,2,1] => 1
0101000 => 0001010 => 0010101 => [3,2,2,1] => 1
0101001 => 1001010 => 0110101 => [2,1,2,2,1] => 1
0101010 => 0101010 => 1010101 => [1,2,2,2,1] => 1
0101011 => 1101010 => 1110101 => [1,1,1,2,2,1] => 1
0101100 => 0011010 => 0101101 => [2,2,1,2,1] => 1
0101101 => 1011010 => 1101101 => [1,1,2,1,2,1] => 1
0101110 => 0111010 => 1011101 => [1,2,1,1,2,1] => 1
0101111 => 1111010 => 1111101 => [1,1,1,1,1,2,1] => 1
0110010 => 0100110 => 1001011 => [1,3,2,1,1] => 1
0110011 => 1100110 => 0111011 => [2,1,1,2,1,1] => 1
0110100 => 0010110 => 0101011 => [2,2,2,1,1] => 1
0110101 => 1010110 => 1101011 => [1,1,2,2,1,1] => 1
0110110 => 0110110 => 1011011 => [1,2,1,2,1,1] => 1
0110111 => 1110110 => 1111011 => [1,1,1,1,2,1,1] => 1
0111001 => 1001110 => 0110111 => [2,1,2,1,1,1] => 1
0111010 => 0101110 => 1010111 => [1,2,2,1,1,1] => 1
0111011 => 1101110 => 1110111 => [1,1,1,2,1,1,1] => 1
0111100 => 0011110 => 0101111 => [2,2,1,1,1,1] => 1
0111101 => 1011110 => 1101111 => [1,1,2,1,1,1,1] => 1
0111110 => 0111110 => 1011111 => [1,2,1,1,1,1,1] => 1
0111111 => 1111110 => 1111111 => [1,1,1,1,1,1,1,1] => 1
1000010 => 0100001 => 1000010 => [1,5,2] => 2
1000100 => 0010001 => 0100010 => [2,4,2] => 2
1000110 => 0110001 => 1000110 => [1,4,1,2] => 2
1001000 => 0001001 => 0010010 => [3,3,2] => 2
1001001 => 1001001 => 0110010 => [2,1,3,2] => 2
1001010 => 0101001 => 1010010 => [1,2,3,2] => 2
1001011 => 1101001 => 1110010 => [1,1,1,3,2] => 2
1001100 => 0011001 => 0100110 => [2,3,1,2] => 2
1001101 => 1011001 => 1100110 => [1,1,3,1,2] => 2
1001110 => 0111001 => 1001110 => [1,3,1,1,2] => 2
1001111 => 1111001 => 0111110 => [2,1,1,1,1,2] => 2
1010001 => 1000101 => 0011010 => [3,1,2,2] => 2
1010010 => 0100101 => 1001010 => [1,3,2,2] => 2
1010011 => 1100101 => 0111010 => [2,1,1,2,2] => 2
1010100 => 0010101 => 0101010 => [2,2,2,2] => 2
1010101 => 1010101 => 1101010 => [1,1,2,2,2] => 2
1010110 => 0110101 => 1011010 => [1,2,1,2,2] => 2
1010111 => 1110101 => 1111010 => [1,1,1,1,2,2] => 2
1011000 => 0001101 => 0010110 => [3,2,1,2] => 2
1011001 => 1001101 => 0110110 => [2,1,2,1,2] => 2
1011010 => 0101101 => 1010110 => [1,2,2,1,2] => 2
1011011 => 1101101 => 1110110 => [1,1,1,2,1,2] => 2
1011100 => 0011101 => 0101110 => [2,2,1,1,2] => 2
1011101 => 1011101 => 1101110 => [1,1,2,1,1,2] => 2
1011110 => 0111101 => 1011110 => [1,2,1,1,1,2] => 2
1100010 => 0100011 => 1000100 => [1,4,3] => 3
1100011 => 1100011 => 0011100 => [3,1,1,3] => 3
1100100 => 0010011 => 0100100 => [2,3,3] => 3
1100101 => 1010011 => 1100100 => [1,1,3,3] => 3
1100110 => 0110011 => 1001100 => [1,3,1,3] => 3
1101000 => 0001011 => 0010100 => [3,2,3] => 3
1101001 => 1001011 => 0110100 => [2,1,2,3] => 3
1101010 => 0101011 => 1010100 => [1,2,2,3] => 3
1101100 => 0011011 => 0101100 => [2,2,1,3] => 3
1110000 => 0000111 => 0001000 => [4,4] => 4
1110110 => 0110111 => 1011000 => [1,2,1,4] => 4
1111110 => 0111111 => 1000000 => [1,7] => 7
00000000 => 00000000 => 00000001 => [8,1] => 1
00000001 => 10000000 => 00000011 => [7,1,1] => 1
00000010 => 01000000 => 10000001 => [1,7,1] => 1
00000101 => 10100000 => 11000001 => [1,1,6,1] => 1
00000110 => 01100000 => 10000011 => [1,6,1,1] => 1
00001001 => 10010000 => 01100001 => [2,1,5,1] => 1
00001010 => 01010000 => 10100001 => [1,2,5,1] => 1
00010001 => 10001000 => 00110001 => [3,1,4,1] => 1
00010010 => 01001000 => 10010001 => [1,3,4,1] => 1
00010100 => 00101000 => 01010001 => [2,2,4,1] => 1
00010111 => 11101000 => 11110001 => [1,1,1,1,4,1] => 1
00011110 => 01111000 => 10001111 => [1,4,1,1,1,1] => 1
00100010 => 01000100 => 10001001 => [1,4,3,1] => 1
00101011 => 11010100 => 11101001 => [1,1,1,2,3,1] => 1
00101101 => 10110100 => 11011001 => [1,1,2,1,3,1] => 1
00101110 => 01110100 => 10111001 => [1,2,1,1,3,1] => 1
01000000 => 00000010 => 00000101 => [6,2,1] => 1
01000010 => 01000010 => 10000101 => [1,5,2,1] => 1
01001110 => 01110010 => 10011101 => [1,3,1,1,2,1] => 1
01010110 => 01101010 => 10110101 => [1,2,1,2,2,1] => 1
01011010 => 01011010 => 10101101 => [1,2,2,1,2,1] => 1
01100110 => 01100110 => 10011011 => [1,3,1,2,1,1] => 1
01110010 => 01001110 => 10010111 => [1,3,2,1,1,1] => 1
10000010 => 01000001 => 10000010 => [1,6,2] => 2
10000110 => 01100001 => 10000110 => [1,5,1,2] => 2
10100010 => 01000101 => 10001010 => [1,4,2,2] => 2
11000110 => 01100011 => 10001100 => [1,4,1,3] => 3
11111110 => 01111111 => 10000000 => [1,8] => 8
000000000 => 000000000 => 000000001 => [9,1] => 1
000000001 => 100000000 => 000000011 => [8,1,1] => 1
000000010 => 010000000 => 100000001 => [1,8,1] => 1
000000101 => 101000000 => 110000001 => [1,1,7,1] => 1
000000110 => 011000000 => 100000011 => [1,7,1,1] => 1
000001001 => 100100000 => 011000001 => [2,1,6,1] => 1
000001010 => 010100000 => 101000001 => [1,2,6,1] => 1
000010010 => 010010000 => 100100001 => [1,3,5,1] => 1
000010111 => 111010000 => 111100001 => [1,1,1,1,5,1] => 1
000011110 => 011110000 => 100001111 => [1,5,1,1,1,1] => 1
000100010 => 010001000 => 100010001 => [1,4,4,1] => 1
000101110 => 011101000 => 101110001 => [1,2,1,1,4,1] => 1
000110000 => 000011000 => 000100011 => [4,4,1,1] => 1
001000010 => 010000100 => 100001001 => [1,5,3,1] => 1
001001110 => 011100100 => 100111001 => [1,3,1,1,3,1] => 1
001100011 => 110001100 => 001110011 => [3,1,1,3,1,1] => 1
001100101 => 101001100 => 110010011 => [1,1,3,3,1,1] => 1
001101001 => 100101100 => 011010011 => [2,1,2,3,1,1] => 1
001111000 => 000111100 => 001001111 => [3,3,1,1,1,1] => 1
010000010 => 010000010 => 100000101 => [1,6,2,1] => 1
010001110 => 011100010 => 100011101 => [1,4,1,1,2,1] => 1
011001111 => 111100110 => 011111011 => [2,1,1,1,1,2,1,1] => 1
011010100 => 001010110 => 010101011 => [2,2,2,2,1,1] => 1
011010111 => 111010110 => 111101011 => [1,1,1,1,2,2,1,1] => 1
011011000 => 000110110 => 001011011 => [3,2,1,2,1,1] => 1
011011101 => 101110110 => 110111011 => [1,1,2,1,1,2,1,1] => 1
011110011 => 110011110 => 011101111 => [2,1,1,2,1,1,1,1] => 1
011110101 => 101011110 => 110101111 => [1,1,2,2,1,1,1,1] => 1
011111100 => 001111110 => 010111111 => [2,2,1,1,1,1,1,1] => 1
011111111 => 111111110 => 111111111 => [1,1,1,1,1,1,1,1,1,1] => 1
100000110 => 011000001 => 100000110 => [1,6,1,2] => 2
100110000 => 000011001 => 000100110 => [4,3,1,2] => 2
100111111 => 111111001 => 011111110 => [2,1,1,1,1,1,1,2] => 2
101001000 => 000100101 => 001001010 => [3,3,2,2] => 2
101010011 => 110010101 => 011101010 => [2,1,1,2,2,2] => 2
101010101 => 101010101 => 110101010 => [1,1,2,2,2,2] => 2
101011100 => 001110101 => 010111010 => [2,2,1,1,2,2] => 2
101011111 => 111110101 => 111111010 => [1,1,1,1,1,1,2,2] => 2
101100011 => 110001101 => 001110110 => [3,1,1,2,1,2] => 2
101100101 => 101001101 => 110010110 => [1,1,3,2,1,2] => 2
101101001 => 100101101 => 011010110 => [2,1,2,2,1,2] => 2
101110100 => 001011101 => 010101110 => [2,2,2,1,1,2] => 2
101110111 => 111011101 => 111101110 => [1,1,1,1,2,1,1,2] => 2
101111000 => 000111101 => 001011110 => [3,2,1,1,1,2] => 2
101111101 => 101111101 => 110111110 => [1,1,2,1,1,1,1,2] => 2
110001111 => 111100011 => 001111100 => [3,1,1,1,1,3] => 3
110010100 => 001010011 => 010100100 => [2,2,3,3] => 3
110010111 => 111010011 => 111100100 => [1,1,1,1,3,3] => 3
110011101 => 101110011 => 110011100 => [1,1,3,1,1,3] => 3
110100111 => 111001011 => 011110100 => [2,1,1,1,2,3] => 3
110101000 => 000101011 => 001010100 => [3,2,2,3] => 3
110101101 => 101101011 => 110110100 => [1,1,2,1,2,3] => 3
110110000 => 000011011 => 000101100 => [4,2,1,3] => 3
110111001 => 100111011 => 011011100 => [2,1,2,1,1,3] => 3
111000011 => 110000111 => 000111000 => [4,1,1,4] => 4
111000101 => 101000111 => 110001000 => [1,1,4,4] => 4
111001001 => 100100111 => 011001000 => [2,1,3,4] => 4
111010001 => 100010111 => 001101000 => [3,1,2,4] => 4
111100000 => 000001111 => 000010000 => [5,5] => 5
111111110 => 011111111 => 100000000 => [1,9] => 9
1111111110 => 0111111111 => 1000000000 => [1,10] => 10
=> => => [1] => 1
01111111111 => 11111111110 => 11111111111 => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
11010110100 => 00101101011 => 01010110100 => [2,2,2,1,2,3] => 3
10101010100 => 00101010101 => 01010101010 => [2,2,2,2,2,2] => 2
0000000000 => 0000000000 => 0000000001 => [10,1] => 1
0000000110 => 0110000000 => 1000000011 => [1,8,1,1] => 1
0100000010 => 0100000010 => 1000000101 => [1,7,2,1] => 1
10100101001 => 10010100101 => 01101001010 => [2,1,2,3,2,2] => 2
0010000010 => 0100000100 => 1000001001 => [1,6,3,1] => 1
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Description
The last part of an integer composition.
Map
reverse
Description
Return the reversal of a binary word.
Map
path rowmotion
Description
Return the rowmotion of the binary word, regarded as a lattice path.
Consider the binary word of length $n$ as a lattice path with $n$ steps, where a 1 corresponds to an up step and a 0 corresponds to a down step.
This map returns the path whose peaks are the valleys of the original path with an up step appended.
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.