Identifier
Mp00104: Binary words reverseBinary words
Mp00096: Binary words Foata bijectionBinary words
Mp00261: Binary words Burrows-WheelerBinary words
Images
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111011100000=>000001110111=>100000110111=>100001111010 111100001010=>010100001111=>000110001111=>110000111010 111100001100=>001100001111=>000100101111=>101001100110 111100010010=>010010001111=>001010001111=>110100010110 111100010100=>001010001111=>001100001111=>101000110110 111100011000=>000110001111=>001000101111=>110001010110 111100100010=>010001001111=>010010001111=>110010001110 111100100100=>001001001111=>010100001111=>100110001110 111100110000=>000011001111=>010000101111=>100101010110 111110000010=>010000011111=>000010011111=>100100101110 111110000100=>001000011111=>000100011111=>110000101110 111110001000=>000100011111=>001000011111=>101000011110 111110010000=>000010011111=>010000011111=>100010011110 111110100000=>000001011111=>100000011111=>100000111110 111111000000=>000000111111=>000000111111=>100000111110 1111111111=>1111111111=>1111111111=>1111111111 1111111110=>0111111111=>0111111111=>1111111110 1111101101=>1011011111=>1101011111=>1110111110 1111101011=>1101011111=>1110011111=>1011111110 1110111000=>0001110111=>1000110111=>1001111010 1110110010=>0100110111=>1010010111=>1101010110 1110100101=>1010010111=>1011000111=>1010110110 1101100110=>0110011011=>1010101011=>1111100010 1100100110=>0110010011=>0101010011=>1111000010 1100011010=>0101100011=>0011001011=>1100111000 1100010110=>0110100011=>0011010011=>1110001100 1011111110=>0111111101=>1011111101=>1111101110 1011100101=>1010011101=>1011001101=>1110111000 1011011001=>1001101101=>1101010101=>1111100010 1011001011=>1101001101=>1011100101=>1011101100 1010101011=>1101010101=>1111100001=>1000111110 1010101000=>0001010101=>1110000001=>1000001110 1010100110=>0110010101=>1101010001=>1011000110 1010011101=>1011100101=>1011011001=>1110111000 1010010111=>1110100101=>1011110001=>1001111010 1010010001=>1000100101=>1010010001=>1100100010 1010001111=>1111000101=>1001111001=>1100111010 1010001010=>0101000101=>1001100001=>1010001100 1010000101=>1010000101=>1000110001=>1100001100 1001111111=>1111111001=>0111111101=>1110111110 1001110010=>0100111001=>0101001101=>1111000010 1001100110=>0110011001=>0101010101=>1111100000 1001011011=>1101101001=>0111101001=>1101010110 1001010111=>1110101001=>0111110001=>1001101110 1001010010=>0100101001=>0110100001=>1001010100 1000111010=>0101110001=>0011001101=>1110001100 1000110110=>0110110001=>0011010101=>1111000010 1000100000=>0000010001=>0010000001=>1000100000 1000011110=>0111100001=>0001011101=>1010101010 1000000000=>0000000001=>0000000001=>1000000000 10000000000=>00000000001=>00000000001=>10000000000 11111111110=>01111111111=>01111111111=>11111111110 100001101110=>011101100001=>000110110101=>101110001100 110000101110=>011101000011=>000110110011=>110010111000 100110001110=>011100011001=>010010110101=>111101000100 0111111111=>1111111110=>1111111110=>1111111110 0111111101=>1011111110=>1101111110=>1111101110 0111111001=>1001111110=>1010111110=>1111001110 0111110001=>1000111110=>1001011110=>1101010110 0111100001=>1000011110=>1000101110=>1010101010 0111000001=>1000001110=>1000010110=>1001010100 0110000001=>1000000110=>1000001010=>1000110000 0100000001=>1000000010=>1000000100=>1000100000 01111111111=>11111111110=>11111111110=>11111111110 0101010101=>1010101010=>1111100000=>1000011110 0101010011=>1100101010=>1110110000=>1001011010 0101001101=>1011001010=>1101101000=>1011001100 0101001011=>1101001010=>1101110000=>1000111100 0101000111=>1110001010=>1100111000=>1010011100 0100110101=>1010110010=>1011100100=>1100110010 0100110011=>1100110010=>1010110100=>1101101000 0100101101=>1011010010=>1011101000=>1010101010 0100101011=>1101010010=>1011110000=>1000110110 0100100111=>1110010010=>1010111000=>1001110010 010101010101=>101010101010=>111111000000=>100000111110 010101001101=>101100101010=>111011010000=>100110011010 010101001011=>110100101010=>111011100000=>100010111100 010100110101=>101011001010=>110111001000=>110001011100 010100101101=>101101001010=>110111010000=>100101011100 010100101011=>110101001010=>110111100000=>100001111010 010100100111=>111001001010=>110101110000=>100101110100 010100011011=>110110001010=>110011101000=>101100011010 010100010111=>111010001010=>110011110000=>100100111010 010011010101=>101010110010=>101111000100=>101001100110 010011010011=>110010110010=>101101100100=>110011101000 010011001101=>101100110010=>101011010100=>111011001000 010011001011=>110100110010=>101011100100=>110011100010 010010110101=>101011010010=>101111001000=>110001010110 010010101101=>101101010010=>101111010000=>100101010110 010010101011=>110101010010=>101111100000=>100001101110 010010011101=>101110010010=>101011011000=>100111101000 010010011011=>110110010010=>101011101000=>101011010010 010010010111=>111010010010=>101011110000=>100011100110 111111111111=>111111111111=>111111111111=>111111111111 111110000001=>100000011111=>000001011111=>100001101110 111110100101=>101001011111=>101100011111=>101011011110 111001100001=>100001100111=>010001010111=>101011010010 111000011001=>100110000111=>000101010111=>100111100010 111000000111=>111000000111=>000001110111=>100010111100 111000001101=>101100000111=>000011010111=>100101110100 111000110001=>100011000111=>001001010111=>110011100010 111000100101=>101001000111=>001011000111=>110010110010 111011100101=>101001110111=>101100110111=>110111101010 111010010111=>111010010111=>101111000111=>101011111100 111010010001=>100010010111=>101001000111=>110010001110 111010011101=>101110010111=>101101100111=>111011100110 111010000101=>101000010111=>100011000111=>110000111010 100111100001=>100001111001=>010001011101=>101101001010 100110011001=>100110011001=>010101010101=>111111000000 100110000111=>111000011001=>010001110101=>101110000110 100110001101=>101100011001=>010011010101=>111110000010 100001111001=>100111100001=>000101011101=>101011010010 100001100111=>111001100001=>000101110101=>101101001010 100001101101=>101101100001=>000111010101=>101110000110 100000011111=>111110000001=>000001111101=>100010011110 100000110111=>111011000001=>000011110101=>100110001110 100000111101=>101111000001=>000011011101=>100101011100 100011011001=>100110110001=>001101010101=>111110000010 100011000111=>111000110001=>001001110101=>111010000110 100011001101=>101100110001=>001011010101=>111101000100 100011110001=>100011110001=>001001011101=>111001001010 100011100101=>101001110001=>001011001101=>111010011000 101100100101=>101001001101=>101011000101=>101011110000 101111100101=>101001111101=>101100111101=>111011100110 101110010111=>111010011101=>101111001101=>101111110010 101110010001=>100010011101=>101001001101=>111010001100 101110011101=>101110011101=>101101101101=>111111110000 101110000101=>101000011101=>100011001101=>101100111000 101001011111=>111110100101=>101111110001=>100111101110 101001000111=>111000100101=>101001110001=>101100101100 101001000001=>100000100101=>101000010001=>101001000010 101001001101=>101100100101=>101011010001=>101110011000 101001110111=>111011100101=>101111011001=>110111101010 101001110001=>100011100101=>101001011001=>111010011000 101001111101=>101111100101=>101101111001=>101111110010 101000010111=>111010000101=>100011110001=>110000111010 101000010001=>100010000101=>100010010001=>111000001000 101000011101=>101110000101=>100011011001=>110010111000 101000000101=>101000000101=>100000110001=>101000001100 0111100000=>0000011110=>1000001110=>1000100110 0111111000=>0001111110=>1000111110=>1010011110 0111111110=>0111111110=>1011111110=>1110111110 01111111110=>01111111110=>10111111110=>11101111110 0100000000=>0000000010=>1000000000=>1000000000 01000000000=>00000000010=>10000000000=>10000000000 0111110000=>0000111110=>1000011110=>1001001110 110010011001=>100110010011=>010101010011=>111110000010 111000010101=>101010000111=>000111000111=>110000111100 111000101001=>100101000111=>001101000111=>110100011100 110000110101=>101011000011=>000111001011=>110010110010 110010001101=>101100010011=>010011010011=>111100001100 110000101101=>101101000011=>000111010011=>110100101010 110010001110=>011100010011=>010010110011=>111010011000 111000100110=>011001000111=>001010100111=>111100100010 101000101000=>000101000101=>100110000001=>100010001100
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
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.