Identifier
Mp00096: Binary words Foata bijectionBinary words
Mp00136: Binary words rotate back-to-frontBinary words
Images
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11101000000101=>10000011110001=>11000001111000 11101000011101=>10001111001101=>11000111100110 0011000011=>0001001011=>1000100101 0011011011=>1100101011=>1110010101 0001111011=>1000111011=>1100011101 111100111111=>011110111111=>101111011111 111100001111=>000111101111=>100011110111 110011110011=>010110111011=>101011011101 110011001111=>010110101111=>101011010111 110000111111=>000110111111=>100011011111 110000000011=>000000011011=>100000001101 110001101111=>100110101111=>110011010111 110001111011=>100110111011=>110011011101 110111001011=>101110110011=>110111011001 110100111011=>101110011011=>110111001101 001111111111=>001111111111=>100111111111 001111001111=>010011101111=>101001110111 001100111111=>010010111111=>101001011111 001100000011=>000001001011=>100000100101 000011111111=>000011111111=>100001111111 000011000011=>000100001011=>100010000101 000000110011=>010000001011=>101000000101 000000001111=>000000001111=>100000000111 000000000011=>000000000011=>100000000001 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11011111000=>00111011110=>00011101111 01110111110=>11011011110=>01101101111 01111101110=>11011110110=>01101111011 0000100001=>0001000001=>1000100000 000000001001=>010000000001=>101000000000 000000100001=>000100000001=>100010000000 000010011001=>010100000101=>101010000010 000010000001=>000001000001=>100000100000 000010100101=>101100000001=>110110000000 001001001001=>010101000001=>101010100000 001001100001=>000101000101=>100010100010 001000011001=>010001000101=>101000100010 001000000001=>000000010001=>100000001000 001000100101=>101001000001=>110100100000 001010000101=>100011000001=>110001100000 011010011111=>011010011111=>101101001111 011010001101=>100110100101=>110011010010 011000111001=>010010101101=>101001010110 011010111101=>111010011101=>111101001110 110110111001=>011110101101=>101111010110 011000101101=>110010100101=>111001010010 110110101101=>111110100101=>111111010010 011110101101=>111011100101=>111101110010
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
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.