Identifier
Mp00096: Binary words Foata bijectionBinary words
Mp00261: Binary words Burrows-WheelerBinary words
Images
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100001000=>001000100=>110000000 100001001=>010001001=>101100000 100001010=>110001000=>110000100 100001011=>100010011=>101001010 100001100=>010001010=>110010000 100001101=>100010101=>100111000 100001110=>100010110=>101010100 100001111=>000101111=>100110110 100010000=>000100100=>100100000 100010001=>001001001=>111000000 100010010=>101001000=>110010000 100010011=>010010011=>111000010 100010100=>011001000=>101000010 100010101=>110010001=>110000110 100010110=>110010010=>110011000 100010111=>100100111=>110010110 100011000=>001001010=>101100000 100011001=>010010101=>111100000 100011010=>110010100=>111001000 100011011=>100101011=>101110010 100011100=>010010110=>110011000 100011101=>100101101=>101111000 100011110=>100101110=>110101010 100011111=>001011111=>101101110 100100000=>000010100=>100001000 100100001=>000101001=>110010000 100100010=>100101000=>101100000 100100011=>001010011=>111001000 100100100=>010101000=>100110000 100100101=>101010001=>101100010 100100110=>101010010=>111100000 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101000110=>100110010=>111001000 101000111=>001100111=>110011100 101001000=>001011000=>100001100 101001001=>010110001=>100111000 101001010=>110110000=>100101100 101001011=>101100011=>101011010 101001100=>010110010=>110011000 101001101=>101100101=>101111000 101001110=>101100110=>111011000 101001111=>011001111=>110110110 101010000=>000111000=>100000110 101010001=>001110001=>101001010 101010010=>101110000=>100011010 101010011=>011100011=>100111100 101010100=>011110000=>100001110 101010101=>111100001=>100011110 101010110=>111100010=>100110110 101010111=>111000111=>100111110 101011000=>001110010=>110000110 101011001=>011100101=>101110010 101011010=>111100100=>110010110 101011011=>111001011=>101101110 101011100=>011100110=>110011100 101011101=>111001101=>101111010 101011110=>111001110=>110111100 101011111=>110011111=>101111110 101100000=>000011010=>100010010 101100001=>000110101=>101100010 101100010=>100110100=>111000010 101100011=>001101011=>111011000 101100100=>010110100=>101010100 101100101=>101101001=>111001100 101100110=>101101010=>111110000 101100111=>011010111=>111110010 101101000=>001110100=>100100110 101101001=>011101001=>110101010 101101010=>111101000=>101001110 101101011=>111010011=>110011110 101101100=>011101010=>111000110 101101101=>111010101=>111100110 101101110=>111010110=>111101100 101101111=>110101111=>111011110 101110000=>000110110=>100101100 101110001=>001101101=>111001100 101110010=>101101100=>101111000 101110011=>011011011=>111111000 101110100=>011101100=>101011010 101110101=>111011001=>110110110 101110110=>111011010=>111110010 101110111=>110110111=>111110110 101111000=>001101110=>100111100 101111001=>011011101=>111101100 101111010=>111011100=>110111100 101111011=>110111011=>111111100 101111100=>011011110=>101111010 101111101=>110111101=>111111100 101111110=>110111110=>111110110 101111111=>101111111=>111111110 110000000=>000000110=>100000010 110000001=>000001101=>100010010 110000010=>100001100=>101000010 110000011=>000011011=>100101100 110000100=>010001100=>110000100 110000101=>100011001=>110001100 110000110=>100011010=>101100010 110000111=>000110111=>100111100 110001000=>001001100=>100100100 110001001=>010011001=>111001000 110001010=>110011000=>110001100 110001011=>100110011=>110011100 110001100=>010011010=>111000010 110001101=>100110101=>111011000 110001110=>100110110=>111001100 110001111=>001101111=>101111010 110010000=>000101100=>100001100 110010001=>001011001=>110011000 110010010=>101011000=>100111000 110010011=>010110011=>111011000 110010100=>011011000=>100101100 110010101=>110110001=>101011010 110010110=>110110010=>101111000 110010111=>101100111=>110110110 110011000=>001011010=>101010100 110011001=>010110101=>111110000 110011010=>110110100=>111001100 110011011=>101101011=>111110010 110011100=>010110110=>101111000 110011101=>101101101=>111111000 110011110=>101101110=>111101100 110011111=>011011111=>111110110 110100000=>000011100=>100000110 110100001=>000111001=>110000110 110100010=>100111000=>101001010 110100011=>001110011=>110011100 110100100=>010111000=>100011010 110100101=>101110001=>100111100 110100110=>101110010=>101110010 110100111=>011100111=>110111100 110101000=>001111000=>100001110 110101001=>011110001=>100110110 110101010=>111110000=>100011110 110101011=>111100011=>100111110 110101100=>011110010=>110010110 110101101=>111100101=>101101110 110101110=>111100110=>101111010 110101111=>111001111=>101111110 110110000=>000111010=>100100110 110110001=>001110101=>111000110 110110010=>101110100=>110101010 110110011=>011101011=>111101100 110110100=>011110100=>101001110 110110101=>111101001=>110011110 110110110=>111101010=>111100110 110110111=>111010111=>111011110 110111000=>001110110=>101011010 110111001=>011101101=>111110010 110111010=>111101100=>110110110 110111011=>111011011=>111110110 110111100=>011101110=>110111100 110111101=>111011101=>111111100 110111110=>111011110=>111111100 110111111=>110111111=>111111110 111000000=>000001110=>100000110 111000001=>000011101=>100100110 111000010=>100011100=>110000110 111000011=>000111011=>101011010 111000100=>010011100=>101001010 111000101=>100111001=>110011100 111000110=>100111010=>111000110 111000111=>001110111=>110111100 111001000=>001011100=>100011010 111001001=>010111001=>101110010 111001010=>110111000=>100111100 111001011=>101110011=>110111100 111001100=>010111010=>110101010 111001101=>101110101=>111101100 111001110=>101110110=>111110010 111001111=>011101111=>111111100 111010000=>000111100=>100001110 111010001=>001111001=>110010110 111010010=>101111000=>100110110 111010011=>011110011=>101111010 111010100=>011111000=>100011110 111010101=>111110001=>100111110 111010110=>111110010=>101101110 111010111=>111100111=>101111110 111011000=>001111010=>101001110 111011001=>011110101=>111100110 111011010=>111110100=>110011110 111011011=>111101011=>111011110 111011100=>011110110=>110110110 111011101=>111101101=>111110110 111011110=>111101110=>111111100 111011111=>111011111=>111111110 111100000=>000011110=>100001110 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1010101010=>1111100000=>1000011110 1010110010=>1011100100=>1100110010 1011001010=>1101101000=>1011001100 1011010010=>1011101000=>1010101010 1100101010=>1110110000=>1001011010 1100110010=>1010110100=>1101101000 1101001010=>1101110000=>1000111100 1101010010=>1011110000=>1000110110 1110001010=>1100111000=>1010011100 1110010010=>1010111000=>1001110010 101010101010=>111111000000=>100000111110 101010110010=>101111000100=>101001100110 101011001010=>110111001000=>110001011100 101011010010=>101111001000=>110001010110 101100101010=>111011010000=>100110011010 101100110010=>101011010100=>111011001000 101101001010=>110111010000=>100101011100 101101010010=>101111010000=>100101010110 101110001010=>110011011000=>101100111000 101110010010=>101011011000=>100111101000 110010101010=>111101100000=>100010110110 110010110010=>101101100100=>110011101000 110011001010=>110101101000=>101110011000 110011010010=>101101101000=>101011011000 110100101010=>111011100000=>100010111100 110100110010=>101011100100=>110011100010 110101001010=>110111100000=>100001111010 110101010010=>101111100000=>100001101110 110110001010=>110011101000=>101100011010 110110010010=>101011101000=>101011010010 111000101010=>111001110000=>101000111100 111001001010=>110101110000=>100101110100 111001010010=>101101110000=>100011101100 111010001010=>110011110000=>100100111010 111010010010=>101011110000=>100011100110 1111111111=>1111111111=>1111111111 1111111110=>1111111110=>1111111110 1111111001=>0111111101=>1110111110 1111000101=>1001111001=>1100111010 1110101001=>0111110001=>1001101110 1110100101=>1011110001=>1001111010 1101101001=>0111101001=>1101010110 1101011111=>1110011111=>1011111110 1101010101=>1111100001=>1000111110 1101001101=>1011100101=>1011101100 1100011101=>1001101101=>1110111000 1011111110=>1101111110=>1111101110 1011100101=>1011011001=>1110111000 1011011111=>1101011111=>1110111110 1010011101=>1011001101=>1110111000 1010010111=>1011000111=>1010110110 1010000101=>1000110001=>1100001100 1001111110=>1010111110=>1111001110 1001101101=>1101010101=>1111100010 1000111110=>1001011110=>1101010110 1000100101=>1010010001=>1100100010 1000011110=>1000101110=>1010101010 1000001110=>1000010110=>1001010100 1000000110=>1000001010=>1000110000 1000000010=>1000000100=>1000100000 11111111110=>11111111110=>11111111110 0111111111=>0111111111=>1111111110 0111111101=>1011111101=>1111101110 0111100001=>0001011101=>1010101010 0000000001=>0000000001=>1000000000 01111111111=>01111111111=>11111111110 00000000001=>00000000001=>10000000000 0101000111=>0011000111=>1100011010 0100110011=>0101001011=>1110100100 0100101011=>1101000011=>1001001110 0100100111=>0101000111=>1011000110 0100001111=>0001001111=>1010010110 0011000111=>0010010111=>1100110010 0010001111=>0010001111=>1100001110 0001110011=>0100011011=>1011001100 0001100111=>0100010111=>1010101010 0001011011=>1100001011=>1000110110 0001001111=>0100001111=>1001001110 0000011111=>0000011111=>1000011110 010101010101=>111110000001=>100000111110 010101001101=>101110000101=>100011101100 010100011101=>100110001101=>110010111000 010100010111=>100110000111=>101000110110 010100001111=>000110001111=>110000111010 010011010101=>111010010001=>110010001110 010011001101=>101010010101=>111011001000 010011000111=>001010010111=>111001100010 010010101101=>111010000101=>100101010110 010010001111=>001010001111=>110100010110 010001110011=>010010011011=>111010001100 010001100111=>010010010111=>111001001010 010001001111=>010010001111=>110010001110 010000111101=>100010011101=>101010101010 010000110111=>100010010111=>101001100110 010000011111=>000010011111=>100100101110 001101001101=>101100100101=>110011101000 001101000111=>001100100111=>111000110010 001100110011=>010100101011=>111011001000 001100100111=>010100100111=>111010000110 001100001111=>000100101111=>101001100110 001011000111=>001100010111=>101001110010 001010011101=>101100001101=>100110111000 001010010111=>101100000111=>100010110110 001010001111=>001100001111=>101000110110 001001110011=>010100011011=>101110001100 001001100111=>010100010111=>101101001010 001001001111=>010100001111=>100110001110 001000011111=>000100011111=>110000101110 000111010101=>111000110001=>110000111010 000111001101=>101000110101=>101110011000 000111000111=>001000110111=>110001011100 000110101101=>111000100101=>101001100110 000110001111=>001000101111=>110001010110 000101110101=>111000011001=>100100111010 000101011101=>111000001101=>100001111010 000101010111=>111000000111=>100000111110 000100111101=>101000011101=>100110011010 000100110111=>101000010111=>100101010110 000100011111=>001000011111=>101000011110 000011110011=>010000111011=>100110011010 000011100111=>010000110111=>100101011100 000011001111=>010000101111=>100101010110 000010011111=>010000011111=>100010011110 000001111101=>100000111101=>100010110110 000001110111=>100000110111=>100001111010 000001011111=>100000011111=>100000111110 000000111111=>000000111111=>100000111110 111111111111=>111111111111=>111111111111 111110000001=>000001111101=>100010011110 111110100101=>101111110001=>100111101110 111001100001=>000101110101=>101101001010 111000011001=>010001110101=>101110000110 111000000111=>000001110111=>100010111100 111000110001=>001001110101=>111010000110 111000100101=>101001110001=>101100101100 111011000001=>000011110101=>100110001110 111011100101=>101111011001=>110111101010 111010010111=>101111000111=>101011111100 111010011101=>101111001101=>101111110010 111010000101=>100011110001=>110000111010 100111100001=>000101011101=>101011010010 100110011001=>010101010101=>111111000000 100110000111=>000101010111=>100111100010 100110110001=>001101010101=>111110000010 100001111001=>010001011101=>101101001010 100001100111=>010001010111=>101011010010 100000011111=>000001011111=>100001101110 100000100101=>101000010001=>101001000010 100011000111=>001001010111=>110011100010 100011110001=>001001011101=>111001001010 100011100101=>101001011001=>111010011000 100010010111=>101001000111=>110010001110 100010011101=>101001001101=>111010001100 100010000101=>100010010001=>111000001000 101101100001=>000111010101=>101110000110 101100011001=>010011010101=>111110000010 101100000111=>000011010111=>100101110100 101100110001=>001011010101=>111101000100 101100100101=>101011010001=>101110011000 101111000001=>000011011101=>100101011100 101111100101=>101101111001=>101111110010 101110010111=>101101100111=>111011100110 101110011101=>101101101101=>111111110000 101110000101=>100011011001=>110010111000 101001011111=>101100011111=>101011011110 101001000111=>001011000111=>110010110010 101001001101=>101011000101=>101011110000 101001110111=>101100110111=>110111101010 101001110001=>001011001101=>111010011000 101001111101=>101100111101=>111011100110 101000010111=>100011000111=>110000111010 101000011101=>100011001101=>101100111000 101000000101=>100000110001=>101000001100 0000011110=>1000001110=>1000100110 0001111110=>1000111110=>1010011110 0111111110=>1011111110=>1110111110 0000000010=>1000000000=>1000000000 0000001110=>1000000110=>1000010010 0000111110=>1000011110=>1001001110 0011111110=>1001111110=>1100111110 00000000010=>10000000000=>10000000000 00000001110=>10000000110=>10000010010 00000111110=>10000011110=>10001001110 00011111110=>10001111110=>10100111110 01111111110=>10111111110=>11101111110 0101110001=>0011001101=>1110001100 0110010101=>1101010001=>1011000110 0110011001=>0101010101=>1111100000 0110110001=>0011010101=>1111000010 011100011001=>010010110101=>111101000100 011101100001=>000110110101=>101110001100 0000000101=>1000000001=>1000000010 0001111101=>1000111101=>1010110110 100101110001=>001101001101=>111100001100 101011100001=>000111001101=>110100011100 0000010001=>0010000001=>1000100000 101000111001=>010011001101=>111100010100 100100111001=>010101001101=>111110000010 100110100011=>001101010011=>111100010100 101001100011=>001011001011=>110011110000 100101100011=>001101001011=>111010011000 100110010011=>010101010011=>111110000010 101000110011=>010011001011=>111001101000 100100110011=>010101001011=>111101000100 101010000111=>000111000111=>110000111100 100101000111=>001101000111=>110100011100 101000100111=>010011000111=>110100101010 100100100111=>010101000111=>101110000110 0110010011=>0101010011=>1111000010 011100010011=>010010110011=>111010011000 101100100011=>001011010011=>111001101000 101100010011=>010011010011=>111100001100 011100000111=>000010110111=>100011101100 0100101001=>0110100001=>1001010100 011010110111=>111010010111=>110101011110 101011000011=>000111001011=>110010110010 101101000011=>000111010011=>110100101010 0001010101=>1110000001=>1000001110 0111010101=>1110110001=>1010110110 0001110111=>1000110111=>1001111010 000000000101=>100000000001=>100000000010 010101110111=>111000110111=>100111101110 000001010101=>111000000001=>100000001110 011101010111=>111011000111=>101011011110 011101110101=>111011011001=>111011100110 011111011111=>101111011111=>111111111010 0110011011=>1010101011=>1111100010 0110110111=>1101010111=>1111001110 0100111001=>0101001101=>1111000010 0101100011=>0011001011=>1100111000 011001000111=>001010100111=>111100100010 011101000011=>000110110011=>110010111000 0110111011=>1101011011=>1111011010 0110100011=>0011010011=>1110001100 0100110111=>1010010111=>1101010110 0101000101=>1001100001=>1010001100 010100000101=>100001100001=>110000001100 000101000101=>100110000001=>100010001100 0111001101=>1010110101=>1111110000
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
Burrows-Wheeler
Description
The Burrows-Wheeler transform of a binary word.
The Burrows-Wheeler transform of a finite word $w$ is obtained from $w$ by first listing the conjugates of $w$ in lexicographic order and then concatenating the final letters of the conjugates in this order.