Identifier
Mp00096: Binary words Foata bijectionBinary words
Mp00104: Binary words reverseBinary words
Mp00105: Binary words complementBinary words
Images
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0101100111=>0110010111=>1110100110=>0001011001 0101001111=>0110001111=>1111000110=>0000111001 101000110001=>001001100101=>101001100100=>010110011011 100101100001=>000110100101=>101001011000=>010110100111 101011000001=>000011100101=>101001110000=>010110001111 101110111011=>111011011011=>110110110111=>001001001000 101001111011=>101100111011=>110111001101=>001000110010 101110110011=>011101101011=>110101101110=>001010010001 101101110111=>111010110111=>111011010111=>000100101000 101100101111=>101101001111=>111100101101=>000011010010 101001101111=>101100101111=>111101001101=>000010110010 101011101111=>111001101111=>111101100111=>000010011000 101011001111=>011100101111=>111101001110=>000010110001 0001010000=>0001100000=>0000011000=>1111100111 0001000100=>0100100000=>0000010010=>1111101101 0111011100=>0110110110=>0110110110=>1001001001 0110110101=>1110101001=>1001010111=>0110101000 0110001001=>0100101001=>1001010010=>0110101101 0111010011=>0110110011=>1100110110=>0011001001 010100000000=>000000011000=>000110000000=>111001111111 000101000000=>000001100000=>000001100000=>111110011111 010001000000=>000001001000=>000100100000=>111011011111 000001010000=>000110000000=>000000011000=>111111100111 010101010000=>000111100000=>000001111000=>111110000111 000100010000=>000100100000=>000001001000=>111110110111 011010010000=>000101101000=>000101101000=>111010010111 000000010100=>011000000000=>000000000110=>111111111001 010100010100=>011001100000=>000001100110=>111110011001 000101010100=>011110000000=>000000011110=>111111100001 000100000100=>010000100000=>000001000010=>111110111101 011010000100=>010001101000=>000101100010=>111010011101 000110100100=>010110001000=>000100011010=>111011100101 011000100100=>010100101000=>000101001010=>111010110101 010001000101=>100100100001=>100001001001=>011110110110 000001101001=>011000001001=>100100000110=>011011111001 011000001001=>010000101001=>100101000010=>011010111101 011001010011=>011010100011=>110001010110=>001110101001 0011011110=>1100101110=>0111010011=>1000101100 0011101110=>1100110110=>0110110011=>1001001100 0011110110=>1100111010=>0101110011=>1010001100 00101111110=>11000111110=>01111100011=>10000011100 11000000000=>00000000110=>01100000000=>10011111111 111100000111=>000011110111=>111011110000=>000100001111 011101100011=>001101101011=>110101101100=>001010010011 011101000111=>001101100111=>111001101100=>000110010011 111000001111=>000011101111=>111101110000=>000010001111 001111010011=>011001110011=>110011100110=>001100011001 000111110101=>110001111001=>100111100011=>011000011100 0111001101=>1010110101=>1010110101=>0101001010 011100011101=>100101101101=>101101101001=>010010010110 011101010101=>111101100001=>100001101111=>011110010000 011101010011=>011101100011=>110001101110=>001110010001 011100110101=>110101101001=>100101101011=>011010010100 001111010101=>111001110001=>100011100111=>011100011000 110110000111=>000111010111=>111010111000=>000101000111 111000011101=>100011101101=>101101110001=>010010001110 011101101001=>011101101001=>100101101110=>011010010001 011101011001=>011101100101=>101001101110=>010110010001 011110010101=>110101110001=>100011101011=>011100010100 011110101001=>011101110001=>100011101110=>011100010001 011110100011=>001101110011=>110011101100=>001100010011 011111000101=>100101111001=>100111101001=>011000010110 00001101111=>10000101111=>11110100001=>00001011110
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 $\phi(4154223) = 1254423$.
Map
reverse
Description
Return the reversal of a binary word.
Map
complement
Description
Send a binary word to the word obtained by interchanging the two letters.