Identifier
Mp00096: Binary words Foata bijectionBinary words
Mp00136: Binary words rotate back-to-frontBinary words
Images
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010101011000=>001111000010=>000111100001 010101100100=>010111000100=>001011100010 010101101000=>001111000100=>000111100010 010110001100=>010011001010=>001001100101 010110010100=>011011001000=>001101100100 010110011000=>001011001010=>000101100101 010110100100=>010111001000=>001011100100 010110101000=>001111001000=>000111100100 010111000100=>010011001100=>001001100110 010111001000=>001011001100=>000101100110 011000101100=>011001010010=>001100101001 011000110100=>011001010100=>001100101010 011001001100=>010101010010=>001010101001 011001010100=>011101010000=>001110101000 011001011000=>001101010010=>000110101001 011001100100=>010101010100=>001010101010 011001101000=>001101010100=>000110101010 011010001100=>010011010010=>001001101001 011010010100=>011011010000=>001101101000 011010011000=>001011010010=>000101101001 011010100100=>010111010000=>001011101000 011010101000=>001111010000=>000111101000 011011000100=>010011010100=>001001101010 011011001000=>001011010100=>000101101010 011100010100=>011001011000=>001100101100 011100100100=>010101011000=>001010101100 011100101000=>001101011000=>000110101100 011101000100=>010011011000=>001001101100 011101001000=>001011011000=>000101101100 0101110001=>0011001101=>1001100110 0110010101=>1101010001=>1110101000 0110011001=>0101010101=>1010101010 0110110001=>0011010101=>1001101010 011010010101=>110110100001=>111011010000 011010101001=>011110100001=>101111010000 011011010100=>011110101000=>001111010100 011100011001=>010010110101=>101001011010 011101100001=>000110110101=>100011011010 0000000101=>1000000001=>1100000000 0000001101=>1000000101=>1100000010 0001111101=>1000111101=>1100011110 00000001101=>10000000101=>11000000010 100101110001=>001101001101=>100110100110 101011100001=>000111001101=>100011100110 101110100001=>000111011001=>100011101100 0011000110=>1001001010=>0100100101 0001101001=>0110001001=>1011000100 0000010001=>0010000001=>1001000000 11111111111=>11111111111=>11111111111 101000111001=>010011001101=>101001100110 100100111001=>010101001101=>101010100110 100110100011=>001101010011=>100110101001 101001100011=>001011001011=>100101100101 100101100011=>001101001011=>100110100101 100110010011=>010101010011=>101010101001 101000110011=>010011001011=>101001100101 100100110011=>010101001011=>101010100101 101010000111=>000111000111=>100011100011 100101000111=>001101000111=>100110100011 101000100111=>010011000111=>101001100011 100100100111=>010101000111=>101010100011 100100010111=>100101000111=>110010100011 110101010101=>111111000001=>111111100000 110101011001=>011111000101=>101111100010 110101101001=>011111001001=>101111100100 110101110001=>001111001101=>100111100110 110110010101=>110111010001=>111011101000 110110101001=>011111010001=>101111101000 110110110001=>001111010101=>100111101010 110111010001=>001111011001=>100111101100 111001010101=>111011100001=>111101110000 111010010101=>110111100001=>111011110000 111010101001=>011111100001=>101111110000 111010110001=>001111100101=>100111110010 111011010001=>001111101001=>100111110100 111100010101=>110011110001=>111001111000 111100100101=>101011110001=>110101111000 111101010001=>001111110001=>100111111000 111101100001=>000111110101=>100011111010 111110100001=>000111111001=>100011111100 111111000001=>000011111101=>100001111110 0110010011=>0101010011=>1010101001 011100010011=>010010110011=>101001011001 011010001011=>100110100011=>110011010001 101100100011=>001011010011=>100101101001 101100010011=>010011010011=>101001101001 011100000111=>000010110111=>100001011011 0101000000=>0000011000=>0000001100 110011010101=>111011010001=>111101101000 110100110101=>110111001001=>111011100100 110001110101=>110011011001=>111001101100 110101010011=>011111000011=>101111100001 101000111011=>100110011011=>110011001101 110101000111=>001111000111=>100111100011 101100010111=>100110100111=>110011010011 101000110111=>100110010111=>110011001011 100100101111=>101010001111=>110101000111 0000000011=>0000000011=>1000000001 0000111111=>0000111111=>1000011111 0011111111=>0011111111=>1001111111 00000000011=>00000000011=>10000000001 00111111111=>00111111111=>10011111111 11011111111=>11011111111=>11101111111 0011111100=>0100111110=>0010011111 0011110011=>0100111011=>1010011101 0110100111=>0110100111=>1011010011 0110110011=>0110101011=>1011010101 0100101001=>0110100001=>1011010000 011010110111=>111010010111=>111101001011 011000100001=>000100101001=>100010010100 000110100001=>000110001001=>100011000100 010010100001=>000110100001=>100011010000 011100101111=>101011001111=>110101100111 011010000001=>000001101001=>100000110100 000100101001=>011010000001=>101101000000 010000101001=>011000100001=>101100010000 000110001001=>010010001001=>101001000100 010010001001=>010010100001=>101001010000 101011000011=>000111001011=>100011100101 101101000011=>000111010011=>100011101001 0001010101=>1110000001=>1111000000 0111010101=>1110110001=>1111011000 0001110111=>1000110111=>1100011011 0011110101=>1100111001=>1110011100 000000000101=>100000000001=>110000000000 010101110111=>111000110111=>111100011011 000001010101=>111000000001=>111100000000 011101010111=>111011000111=>111101100011 011101110101=>111011011001=>111101101100 011111011111=>101111011111=>110111101111 010011101111=>101001101111=>110100110111 110110100011=>001111010011=>100111101001 111000110101=>110011101001=>111001110100 111010100011=>001111100011=>100111110001 111101000011=>000111110011=>100011111001 000001111111=>000001111111=>100000111111 001111100101=>101001111001=>110100111100 011110100101=>101101110001=>110110111000 0110101011=>1110100011=>1111010001 0100111001=>0101001101=>1010100110 0101100011=>0011001011=>1001100101 000111011001=>011000110101=>101100011010 010011011001=>011010010101=>101101001010 011001000111=>001010100111=>100101010011 011101000011=>000110110011=>100011011001 0111011001=>0110110101=>1011011010 100100100001=>000101010001=>100010101000 0100010011=>0100100011=>1010010001 0101110011=>0110011011=>1011001101 0110100011=>0011010011=>1001101001 0101000011=>0001100011=>1000110001 0100110111=>1010010111=>1101001011 0101100111=>0110010111=>1011001011 0110101111=>1101001111=>1110100111 101001111011=>101100111011=>110110011101 101110110011=>011101101011=>101110110101 101100101111=>101101001111=>110110100111 101001101111=>101100101111=>110110010111 101011001111=>011100101111=>101110010111 0101000101=>1001100001=>1100110000 0110110101=>1110101001=>1111010100 0110001001=>0100101001=>1010010100 0111010011=>0110110011=>1011011001 010100000101=>100001100001=>110000110000 000101000101=>100110000001=>110011000000 010001000101=>100100100001=>110010010000 000001101001=>011000001001=>101100000100 011000001001=>010000101001=>101000010100 011001010011=>011010100011=>101101010001 0011011110=>1100101110=>0110010111 000111110101=>110001111001=>111000111100 0111001101=>1010110101=>1101011010 011101010101=>111101100001=>111110110000 011101010011=>011101100011=>101110110001 011100110101=>110101101001=>111010110100 001111010101=>111001110001=>111100111000 011101101001=>011101101001=>101110110100 011101011001=>011101100101=>101110110010 011110010101=>110101110001=>111010111000 011110101001=>011101110001=>101110111000 011111000101=>100101111001=>110010111100
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.