Identifier
Mp00104: Binary words reverseBinary words
Mp00096: Binary words Foata bijectionBinary words
Mp00135: Binary words rotate front-to-backBinary words
Images
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111110000100=>001000011111=>000100011111=>001000111110 111110001000=>000100011111=>001000011111=>010000111110 111110010000=>000010011111=>010000011111=>100000111110 111110100000=>000001011111=>100000011111=>000000111111 111111000000=>000000111111=>000000111111=>000001111110 1111111111=>1111111111=>1111111111=>1111111111 1111111110=>0111111111=>0111111111=>1111111110 1111111011=>1101111111=>1101111111=>1011111111 1111110111=>1110111111=>1110111111=>1101111111 1111110000=>0000111111=>0000111111=>0001111110 1111101101=>1011011111=>1101011111=>1010111111 1111101011=>1101011111=>1110011111=>1100111111 1111000011=>1100001111=>0001101111=>0011011110 1110111000=>0001110111=>1000110111=>0001101111 1110110010=>0100110111=>1010010111=>0100101111 1110100101=>1010010111=>1011000111=>0110001111 1110001101=>1011000111=>0011010111=>0110101110 1110000111=>1110000111=>0001110111=>0011101110 1101010110=>0110101011=>1110100011=>1101000111 1100111010=>0101110011=>0110011011=>1100110110 1100110110=>0110110011=>0110101011=>1101010110 1100101110=>0111010011=>0110110011=>1101100110 1100101011=>1101010011=>0111100011=>1111000110 1100100111=>1110010011=>0101110011=>1011100110 1100100110=>0110010011=>0101010011=>1010100110 1100100010=>0100010011=>0100100011=>1001000110 1100011010=>0101100011=>0011001011=>0110010110 1100010111=>1110100011=>0011110011=>0111100110 1100010110=>0110100011=>0011010011=>0110100110 1100001010=>0101000011=>0001100011=>0011000110 1100000000=>0000000011=>0000000011=>0000000110 1011111110=>0111111101=>1011111101=>0111111011 1011100101=>1010011101=>1011001101=>0110011011 1011011001=>1001101101=>1101010101=>1010101011 1011001101=>1011001101=>1011010101=>0110101011 1011001011=>1101001101=>1011100101=>0111001011 1011000001=>1000001101=>1000010101=>0000101011 1011000000=>0000001101=>1000000101=>0000001011 1010110101=>1010110101=>1111001001=>1110010011 1010110011=>1100110101=>1101101001=>1011010011 1010101101=>1011010101=>1111010001=>1110100011 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111001101101=>101101100111=>011101010111=>111010101110 111000011001=>100110000111=>000101010111=>001010101110 111000000111=>111000000111=>000001110111=>000011101110 111000001101=>101100000111=>000011010111=>000110101110 111000110001=>100011000111=>001001010111=>010010101110 111000100101=>101001000111=>001011000111=>010110001110 111011110001=>100011110111=>100101110111=>001011101111 111011100101=>101001110111=>101100110111=>011001101111 111010010111=>111010010111=>101111000111=>011110001111 111010010001=>100010010111=>101001000111=>010010001111 111010011101=>101110010111=>101101100111=>011011001111 111010000101=>101000010111=>100011000111=>000110001111 100111100001=>100001111001=>010001011101=>100010111010 100110011001=>100110011001=>010101010101=>101010101010 100110000111=>111000011001=>010001110101=>100011101010 100110000001=>100000011001=>010000010101=>100000101010 100110001101=>101100011001=>010011010101=>100110101010 100110111101=>101111011001=>011101110101=>111011101010 100110100101=>101001011001=>011011000101=>110110001010 100001111001=>100111100001=>000101011101=>001010111010 100001100111=>111001100001=>000101110101=>001011101010 100001100001=>100001100001=>000100010101=>001000101010 100001101101=>101101100001=>000111010101=>001110101010 100000011111=>111110000001=>000001111101=>000011111010 100000011001=>100110000001=>000001010101=>000010101010 100000000001=>100000000001=>000000000101=>000000001010 100000110111=>111011000001=>000011110101=>000111101010 100000111101=>101111000001=>000011011101=>000110111010 100011011001=>100110110001=>001101010101=>011010101010 100011000111=>111000110001=>001001110101=>010011101010 100011001101=>101100110001=>001011010101=>010110101010 100011110001=>100011110001=>001001011101=>010010111010 100011100101=>101001110001=>001011001101=>010110011010 101100100101=>101001001101=>101011000101=>010110001011 101111100101=>101001111101=>101100111101=>011001111011 101110010111=>111010011101=>101111001101=>011110011011 101110010001=>100010011101=>101001001101=>010010011011 101110011101=>101110011101=>101101101101=>011011011011 101110000101=>101000011101=>100011001101=>000110011011 101001011111=>111110100101=>101111110001=>011111100011 101001000111=>111000100101=>101001110001=>010011100011 101001000001=>100000100101=>101000010001=>010000100011 101001001101=>101100100101=>101011010001=>010110100011 101001110111=>111011100101=>101111011001=>011110110011 101001110001=>100011100101=>101001011001=>010010110011 101001111101=>101111100101=>101101111001=>011011110011 101000010111=>111010000101=>100011110001=>000111100011 101000010001=>100010000101=>100010010001=>000100100011 101000011101=>101110000101=>100011011001=>000110110011 101000000101=>101000000101=>100000110001=>000001100011 101010010101=>101010010101=>110111000001=>101110000011 0000000000=>0000000000=>0000000000=>0000000000 0111100000=>0000011110=>1000001110=>0000011101 0111111000=>0001111110=>1000111110=>0001111101 0111111110=>0111111110=>1011111110=>0111111101 000000000000=>000000000000=>000000000000=>000000000000 0000001010=>0101000000=>0000011000=>0000110000 01111111110=>01111111110=>10111111110=>01111111101 0110001100=>0011000110=>1001001010=>0010010101 001010101010=>010101010100=>011111000000=>111110000000 001010101100=>001101010100=>011110010000=>111100100000 001010110010=>010011010100=>011101001000=>111010010000 001010110100=>001011010100=>011110001000=>111100010000 001011001010=>010100110100=>011011000100=>110110001000 001011001100=>001100110100=>011010010100=>110100101000 001011010010=>010010110100=>011101000100=>111010001000 001011010100=>001010110100=>011110000100=>111100001000 001100101010=>010101001100=>010111000010=>101110000100 001100101100=>001101001100=>010110010010=>101100100100 001100110010=>010011001100=>010101001010=>101010010100 001100110100=>001011001100=>010110001010=>101100010100 001101001010=>010100101100=>011011000010=>110110000100 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100010101111=>111101010001=>001111110001=>011111100010 100001101111=>111101100001=>000111110101=>001111101010 100001011111=>111110100001=>000111111001=>001111110010 100000111111=>111111000001=>000011111101=>000111111010 0000111111=>1111110000=>0001111110=>0011111100 0011111100=>0011111100=>0100111110=>1001111100 0010011001=>1001100100=>0101010100=>1010101000 0010100101=>1010010100=>0110110000=>1101100000 0001101111=>1111011000=>0011111010=>0111110100 0100101001=>1001010010=>1011010000=>0110100001 110010001110=>011100010011=>010010110011=>100101100110 110100010110=>011010001011=>100110100011=>001101000111 111101110010=>010011101111=>101001101111=>010011011111 111101001110=>011100101111=>101011001111=>010110011111 0100010101=>1010100010=>1001110000=>0011100001 0101010111=>1110101010=>1111110000=>1111100001 0001010101=>1010101000=>0011110000=>0111100000 0001110111=>1110111000=>0011110110=>0111101100 111000101011=>110101000111=>001111000111=>011110001110 111010001101=>101100010111=>100110100111=>001101001111 101011100011=>110001110101=>110011011001=>100110110011 100110111000=>000111011001=>011000110101=>110001101010 100110110010=>010011011001=>011010010101=>110100101010 111000100110=>011001000111=>001010100111=>010101001110 100001001001=>100100100001=>000101010001=>001010100010 0100011001=>1001100010=>1001010100=>0010101001 0100110001=>1000110010=>1010010100=>0100101001 101000101000=>000101000101=>100110000001=>001100000011 100001011000=>000110100001=>000110001001=>001100010010 100101001000=>000100101001=>011010000001=>110100000010 100100011000=>000110001001=>010010001001=>100100010010 110010100110=>011001010011=>011010100011=>110101000110 11000000000=>00000000011=>00000000011=>00000000110 101001111100=>001111100101=>101001111001=>010011110011 111111100000=>000001111111=>000001111111=>000011111110
Map
reverse
Description
Return the reversal of a binary word.
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 front-to-back
Description
The rotation of a binary word, first letter last.
This is the word obtained by moving the first letter to the end.