Identifier
Mp00105: Binary words complementBinary words
Mp00096: Binary words Foata bijectionBinary words
Images
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0=>1=>1 1=>0=>0 00=>11=>11 01=>10=>10 10=>01=>01 11=>00=>00 000=>111=>111 001=>110=>110 010=>101=>101 011=>100=>010 100=>011=>011 101=>010=>100 110=>001=>001 111=>000=>000 0000=>1111=>1111 0001=>1110=>1110 0010=>1101=>1101 0011=>1100=>0110 0100=>1011=>1011 0101=>1010=>1100 0110=>1001=>0101 0111=>1000=>0010 1000=>0111=>0111 1001=>0110=>1010 1010=>0101=>1001 1011=>0100=>0100 1100=>0011=>0011 1101=>0010=>1000 1110=>0001=>0001 1111=>0000=>0000 00000=>11111=>11111 00001=>11110=>11110 00010=>11101=>11101 00011=>11100=>01110 00100=>11011=>11011 00101=>11010=>11100 00110=>11001=>01101 00111=>11000=>00110 01000=>10111=>10111 01001=>10110=>11010 01010=>10101=>11001 01011=>10100=>01100 01100=>10011=>01011 01101=>10010=>10100 01110=>10001=>00101 01111=>10000=>00010 10000=>01111=>01111 10001=>01110=>10110 10010=>01101=>10101 10011=>01100=>01010 10100=>01011=>10011 10101=>01010=>11000 10110=>01001=>01001 10111=>01000=>00100 11000=>00111=>00111 11001=>00110=>10010 11010=>00101=>10001 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110100110011=>001011001100=>010110001010 101100110011=>010011001100=>010101001010 110010110011=>001101001100=>010110010010 101010110011=>010101001100=>010111000010 100110110011=>011001001100=>010101010010 110001110011=>001110001100=>010010011010 101001110011=>010110001100=>010011001010 100101110011=>011010001100=>010011010010 100011110011=>011100001100=>010001011010 110101001011=>001010110100=>011110000100 101101001011=>010010110100=>011101000100 110011001011=>001100110100=>011010010100 101011001011=>010100110100=>011011000100 100111001011=>011000110100=>011001010100 110100101011=>001011010100=>011110001000 101100101011=>010011010100=>011101001000 110010101011=>001101010100=>011110010000 101010101011=>010101010100=>011111000000 100110101011=>011001010100=>011101010000 110001101011=>001110010100=>011010011000 101001101011=>010110010100=>011011001000 100101101011=>011010010100=>011011010000 100011101011=>011100010100=>011001011000 110100011011=>001011100100=>010110001100 101100011011=>010011100100=>010101001100 110010011011=>001101100100=>010110010100 101010011011=>010101100100=>010111000100 100110011011=>011001100100=>010101010100 110001011011=>001110100100=>010110011000 101001011011=>010110100100=>010111001000 100101011011=>011010100100=>010111010000 100011011011=>011100100100=>010101011000 110000111011=>001111000100=>010010011100 101000111011=>010111000100=>010011001100 100100111011=>011011000100=>010011010100 100010111011=>011101000100=>010011011000 100001111011=>011110000100=>010001011100 110101000111=>001010111000=>001110000110 101101000111=>010010111000=>001101000110 110011000111=>001100111000=>001010010110 101011000111=>010100111000=>001011000110 100111000111=>011000111000=>001001010110 110100100111=>001011011000=>001110001010 101100100111=>010011011000=>001101001010 110010100111=>001101011000=>001110010010 101010100111=>010101011000=>001111000010 100110100111=>011001011000=>001101010010 110001100111=>001110011000=>001010011010 101001100111=>010110011000=>001011001010 100101100111=>011010011000=>001011010010 100011100111=>011100011000=>001001011010 110100010111=>001011101000=>001110001100 101100010111=>010011101000=>001101001100 110010010111=>001101101000=>001110010100 101010010111=>010101101000=>001111000100 100110010111=>011001101000=>001101010100 110001010111=>001110101000=>001110011000 101001010111=>010110101000=>001111001000 100101010111=>011010101000=>001111010000 100011010111=>011100101000=>001101011000 110000110111=>001111001000=>001010011100 101000110111=>010111001000=>001011001100 100100110111=>011011001000=>001011010100 100010110111=>011101001000=>001011011000 100001110111=>011110001000=>001001011100 110100001111=>001011110000=>000110001110 101100001111=>010011110000=>000101001110 110010001111=>001101110000=>000110010110 101010001111=>010101110000=>000111000110 100110001111=>011001110000=>000101010110 110001001111=>001110110000=>000110011010 101001001111=>010110110000=>000111001010 100101001111=>011010110000=>000111010010 100011001111=>011100110000=>000101011010 110000101111=>001111010000=>000110011100 101000101111=>010111010000=>000111001100 100100101111=>011011010000=>000111010100 100010101111=>011101010000=>000111011000 100001101111=>011110010000=>000101011100 110000011111=>001111100000=>000010011110 101000011111=>010111100000=>000011001110 100100011111=>011011100000=>000011010110 100010011111=>011101100000=>000011011010 100001011111=>011110100000=>000011011100 100000111111=>011111000000=>000001011110 0011111111=>1100000000=>0000000110 0000010111=>1111101000=>0011111100 00111111111=>11000000000=>00000000110 010100111100=>101011000011=>000111001011 010010111100=>101101000011=>000111010011 0011111001=>1100000110=>1000011010 0000110011=>1111001100=>0101111010 0000100111=>1111011000=>0011111010 0110100111=>1001011000=>0011010010 0110101101=>1001010010=>1011010000 0110110011=>1001001100=>0101010010 0100111011=>1011000100=>0100110100 0101010111=>1010101000=>0011110000 0001010101=>1110101010=>1111110000 0000110101=>1111001010=>1101111000 0111010101=>1000101010=>1110010000 0101110101=>1010001010=>1100110000 0011110101=>1100001010=>1100011000 0000011011=>1111100100=>0101111100 0001001011=>1110110100=>0111110100 0110001111=>1001110000=>0001010110 0110011011=>1001100100=>0101010100 111110110100=>000001001011=>101000000011 111011110100=>000100001011=>100010000011 100110111000=>011001000111=>001010100111 100101011100=>011010100011=>001110100011 101100100110=>010011011001=>011010010101 111000100110=>000111011001=>011000110101 0110101011=>1001010100=>0111010000 11110010000=>00001101111=>10000101111 100101101111=>011010010000=>000101101000 101101110011=>010010001100=>010010100010 100101111011=>011010000100=>010001101000 0101110011=>1010001100=>0100110010 0110010111=>1001101000=>0011010100 0100110111=>1011001000=>0010110100 0101100111=>1010011000=>0010110010 0101001111=>1010110000=>0001110010 101101001111=>010010110000=>000110100010 0110110101=>1001001010=>1101010000 0101011011=>1010100100=>0101110000 0111010011=>1000101100=>0110010010 0111000111=>1000111000=>0010010110 011010010000=>100101101111=>110100101111 010010110000=>101101001111=>011101001111 011010000100=>100101111011=>110100111011 010010001100=>101101110011=>011101011011 101010001000=>010101110111=>111000110111 100010101000=>011101010111=>111011000111 100001011000=>011110100111=>011011100111 100101001000=>011010110111=>111010010111 111011101010=>000100010101=>110010000001 111011010110=>000100101001=>011010000001 111010101110=>000101010001=>001110000001 111001011110=>000110100001=>000110001001 011011011110=>100100100001=>000101010001 010001011000=>101110100111=>011101100111 010100011000=>101011100111=>011100110111 010001010000=>101110101111=>111011001111 010010010000=>101101101111=>111010101111 010100010000=>101011101111=>111001101111 111110010110=>000001101001=>011000001001 111001110110=>000110001001=>010010001001 110000101100=>001111010011=>011001110011 110000100110=>001111011001=>011001110101 110000110010=>001111001101=>101001110101 110000010110=>001111101001=>011001111001 110000011010=>001111100101=>101001111001 111000001010=>000111110101=>110001111001 00100000000=>11011111111=>11011111111 11000000000=>00111111111=>00111111111 11111111100=>00000000011=>00000000011 111000001100=>000111110011=>010001111011 0011010001=>1100101110=>1101100110 0001000111=>1110111000=>0011110110 0001001101=>1110110010=>1011110100 0001010011=>1110101100=>0111110010 0000101011=>1111010100=>0111111000 0110011101=>1001100010=>1001010100 0111001101=>1000110010=>1010010100 100011001100=>011100110011=>010101101011 100010111000=>011101000111=>001101100111 0100111101=>1011000010=>1000110100 00111100001=>11000011110=>10001101110 100010101100=>011101010011=>011101100011 100010110010=>011101001101=>101101100101 100011001010=>011100110101=>110101101001 100011100010=>011100011101=>100101101101 100011110000=>011100001111=>000101101111 11111001000=>00000110111=>10000010111
Map
complement
Description
Send a binary word to the word obtained by interchanging the two letters.
Map
Foata bijection
Description
The Foata bijection $\phi$ is a bijection on the set of words of given content (by a slight generalization of Section 2 in [1]).
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$. At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
  • If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
  • If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
For instance, to compute $\phi(4154223)$, the sequence of words is
  • 4,
  • |4|1 -- > 41,
  • |4|1|5 -- > 415,
  • |415|4 -- > 5414,
  • |5|4|14|2 -- > 54412,
  • |5441|2|2 -- > 154422,
  • |1|5442|2|3 -- > 1254423.
So $\phi(4154223) = 1254423$.