Identifier
Mp00096: Binary words Foata bijectionBinary words
Mp00105: Binary words complementBinary words
Mp00136: Binary words rotate back-to-frontBinary words
Images
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100000011111=>000001011111=>111110100000=>011111010000 100000110111=>100001010111=>011110101000=>001111010100 100011000111=>001001010111=>110110101000=>011011010100 100011110001=>001001011101=>110110100010=>011011010001 101101100001=>000111010101=>111000101010=>011100010101 101100011001=>010011010101=>101100101010=>010110010101 101100000111=>000011010111=>111100101000=>011110010100 101100110001=>001011010101=>110100101010=>011010010101 101111000001=>000011011101=>111100100010=>011110010001 101001000111=>001011000111=>110100111000=>011010011100 101001110001=>001011001101=>110100110010=>011010011001 0000000000=>0000000000=>1111111111=>1111111111 0000011110=>1000001110=>0111110001=>1011111000 0001111110=>1000111110=>0111000001=>1011100000 0111111110=>1011111110=>0100000001=>1010000000 0100111110=>1010011110=>0101100001=>1010110000 000000000000=>000000000000=>111111111111=>111111111111 0101010100=>0111100000=>1000011111=>1100001111 0000010100=>0110000000=>1001111111=>1100111111 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011001000111=>001010100111=>110101011000=>011010101100 011101000011=>000110110011=>111001001100=>011100100110 0111011001=>0110110101=>1001001010=>0100100101 0101111001=>0110011101=>1001100010=>0100110001 0101110011=>0110011011=>1001100100=>0100110010 0110100011=>0011010011=>1100101100=>0110010110 0101100111=>0110010111=>1001101000=>0100110100 0101001111=>0110001111=>1001110000=>0100111000 101100101111=>101101001111=>010010110000=>001001011000 0001010000=>0001100000=>1110011111=>1111001111 0001000100=>0100100000=>1011011111=>1101101111 0111010011=>0110110011=>1001001100=>0100100110 010100000000=>000000011000=>111111100111=>111111110011 000101000000=>000001100000=>111110011111=>111111001111 010001000000=>000001001000=>111110110111=>111111011011 000001010000=>000110000000=>111001111111=>111100111111 010101010000=>000111100000=>111000011111=>111100001111 000100010000=>000100100000=>111011011111=>111101101111 011010010000=>000101101000=>111010010111=>111101001011 000000010100=>011000000000=>100111111111=>110011111111 010100010100=>011001100000=>100110011111=>110011001111 000101010100=>011110000000=>100001111111=>110000111111 000100000100=>010000100000=>101111011111=>110111101111 011010000100=>010001101000=>101110010111=>110111001011 000110100100=>010110001000=>101001110111=>110100111011 011000100100=>010100101000=>101011010111=>110101101011 011100011101=>100101101101=>011010010010=>001101001001 111000011101=>100011101101=>011100010010=>001110001001
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
complement
Description
Send a binary word to the word obtained by interchanging the two letters.
Map
rotate back-to-front
Description
The rotation of a binary word, last letter first.
This is the word obtained by moving the last letter to the beginnig.