Identifier
Mp00278: Binary words rowmotionBinary words
Mp00104: Binary words reverseBinary words
Mp00096: Binary words Foata bijectionBinary words
Images
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110100101010=>111001010100=>001010100111=>011100000111 110100110010=>111001001100=>001100100111=>010100100111 110100110100=>111001011000=>000110100111=>011000100111 110101001010=>111010010100=>001010010111=>101100000111 110101001100=>111010010001=>100010010111=>101001000111 110101010010=>111010100100=>001001010111=>110100000111 110101010100=>111010101000=>000101010111=>111000000111 110101100010=>111010001100=>001100010111=>100100100111 110101100100=>111010011000=>000110010111=>101000100111 110101101000=>111010110000=>000011010111=>110000100111 110110001010=>111000110100=>001011000111=>001100010111 110110001100=>111000110001=>100011000111=>001001010111 110110010010=>111001100100=>001001100111=>010100010111 110110010100=>111001101000=>000101100111=>011000010111 110110011000=>111001100001=>100001100111=>010001010111 110110100010=>111011000100=>001000110111=>100100010111 110110100100=>111011001000=>000100110111=>101000010111 110110101000=>111011010000=>000010110111=>110000010111 110110110000=>111011000001=>100000110111=>100001010111 110111000100=>111000111000=>000111000111=>001000110111 110111001000=>111001110000=>000011100111=>010000110111 110111010000=>111011100000=>000001110111=>100000110111 110111100000=>111000000111=>111000000111=>000001110111 111010001010=>111100010100=>001010001111=>001100001111 111010010010=>111100100100=>001001001111=>010100001111 111010010100=>111100101000=>000101001111=>011000001111 111010100010=>111101000100=>001000101111=>100100001111 111010100100=>111101001000=>000100101111=>101000001111 111010101000=>111101010000=>000010101111=>110000001111 111011000010=>111100001100=>001100001111=>000100101111 111011000100=>111100011000=>000110001111=>001000101111 111011001000=>111100110000=>000011001111=>010000101111 111011010000=>111101100000=>000001101111=>100000101111 111101000010=>111110000100=>001000011111=>000100011111 111101000100=>111110001000=>000100011111=>001000011111 111101001000=>111110010000=>000010011111=>010000011111 111101010000=>111110100000=>000001011111=>100000011111 111101100000=>111110000001=>100000011111=>000001011111 111110100000=>111111000000=>000000111111=>000000111111 111111000000=>000000111111=>111111000000=>000001111110 1111111111=>1111111111=>1111111111=>1111111111 1111111110=>0111111111=>1111111110=>1111111110 1111111101=>1111111110=>0111111111=>0111111111 1111111001=>0111111110=>0111111110=>1011111110 1111101101=>1111110110=>0110111111=>1010111111 1111100111=>0111111011=>1101111110=>1110111110 1110111101=>1111011110=>0111101111=>1011101111 1110100101=>1111001010=>0101001111=>0110001111 1110100001=>1111000010=>0100001111=>0001001111 1101111110=>1110011111=>1111100111=>0111110111 1101111011=>1110111101=>1011110111=>1101110111 1101010011=>1110100101=>1010010111=>1011000111 1101010001=>1110100010=>0100010111=>1001000111 1101001001=>1110010010=>0100100111=>0101000111 1101000101=>1110001010=>0101000111=>0011000111 1100001011=>0001110101=>1010111000=>0011100110 1011111101=>1101111110=>0111111011=>1011111011 1011110001=>1100011110=>0111100011=>0010111011 1011100101=>1100111010=>0101110011=>0110011011 1011011001=>1101100110=>0110011011=>1010101011 1011010001=>1101100010=>0100011011=>1001001011 1011001101=>1100110110=>0110110011=>0110101011 1011001001=>1100110010=>0100110011=>0101001011 1011000111=>1100011011=>1101100011=>0011101011 1011000101=>1100011010=>0101100011=>0011001011 1011000001=>1100000110=>0110000011=>0000101011 1010101001=>1101010010=>0100101011=>1101000011 1010100101=>1101001010=>0101001011=>1011000011 1010011101=>1100101110=>0111010011=>0110110011 1010011010=>1100101100=>0011010011=>0110010011 1010011001=>1100100110=>0110010011=>0101010011 1010010111=>1100101011=>1101010011=>0111100011 1010010101=>1100101010=>0101010011=>0111000011 1010010001=>1100100010=>0100010011=>0100100011 1010000101=>1100001010=>0101000011=>0001100011 1000111110=>0011001111=>1111001100=>0101111010 1000111100=>0011000111=>1110001100=>0100111010 1000110001=>0011000110=>0110001100=>0100101010 1000101110=>0011010011=>1100101100=>0110110010 1000101011=>0011010101=>1010101100=>0111100010 1000011111=>0001101111=>1111011000=>0011111010 1000011110=>0001100111=>1110011000=>0010111010 1000010110=>0001101001=>1001011000=>0011010010 1000001101=>0000110110=>0110110000=>0001101010 1000000000=>0000000001=>1000000000=>0000000010 10000000000=>00000000001=>10000000000=>00000000010 11111111110=>01111111111=>11111111110=>11111111110 100001111100=>000110001111=>111100011000=>001001111010 0111111101=>1011111110=>0111111101=>1011111101 0111111001=>1001111110=>0111111001=>0101111101 0111110001=>1000111110=>0111110001=>0010111101 0111100001=>1000011110=>0111100001=>0001011101 0111000001=>1000001110=>0111000001=>0000101101 0110000001=>1000000110=>0110000001=>0000010101 0100000001=>1000000010=>0100000001=>0000001001 01111111101=>10111111110=>01111111101=>10111111101 0111111011=>1011111101=>1011111101=>1101111101 01111111001=>10011111110=>01111111001=>01011111101 01111110001=>10001111110=>01111110001=>00101111101 01111100001=>10000111110=>01111100001=>00010111101 0111010001=>1011100010=>0100011101=>1001001101 01111000001=>10000011110=>01111000001=>00001011101 01110000001=>10000001110=>01110000001=>00000101101 01100000001=>10000000110=>01100000001=>00000010101 01000000001=>10000000010=>01000000001=>00000001001 0101010101=>1010101010=>0101010101=>1111000001 0100011101=>1000101110=>0111010001=>0011011001 0011001011=>0100110101=>1010110010=>1011100100 0011000111=>0100011011=>1101100010=>1001110100 0010110011=>0101001101=>1011001010=>1101101000 0010101011=>0101010101=>1010101010=>1111100000 0010100111=>0101001011=>1101001010=>1101110000 0010011011=>0100101101=>1011010010=>1011101000 0010010111=>0100101011=>1101010010=>1011110000 0010001111=>0100010111=>1110100010=>1001111000 0001110011=>0010011101=>1011100100=>0101101100 0001101011=>0010110101=>1010110100=>0111100100 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011000011110=>100001100111=>111001100001=>000101110101 011001100000=>100110000001=>100000011001=>010000010101 011001100110=>100110011001=>100110011001=>010101010101 011001111000=>100110000111=>111000011001=>010001110101 011001111110=>100110011111=>111110011001=>010111110101 011110000000=>100000000111=>111000000001=>000000011101 011110000110=>100001111001=>100111100001=>000101011101 011110011000=>100111100001=>100001111001=>010001011101 011110011110=>100111100111=>111001111001=>010111011101 011111100000=>100000011111=>111110000001=>000001111101 011111100110=>100111111001=>100111111001=>010101111101 011111111000=>100001111111=>111111100001=>000111111101 011111111110=>100111111111=>111111111001=>011111111101 011101101000=>101110110000=>000011011101=>110000101101 011101100010=>101110001100=>001100011101=>100100101101 011101001010=>101110010100=>001010011101=>101100001101 010000100110=>100001001001=>100100100001=>000101010001 010110110010=>101011001100=>001100110101=>110100101001 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001100110010=>010011001100=>001100110010=>101010010100 0100000000=>1000000000=>0000000001=>0000000001 0100011111=>1000101111=>1111010001=>0011111001 01000000000=>10000000000=>00000000001=>00000000001 0100000101=>1000001010=>0101000001=>0000110001 0100101111=>1001010111=>1110101001=>0111110001 0110111111=>1011011111=>1111101101=>1111110101 0101011001=>1010100110=>0110010101=>1101010001 0101011010=>1010101100=>0011010101=>1110010001 0101101001=>1010110010=>0100110101=>1101001001 0101101010=>1010110100=>0010110101=>1110001001 0101101100=>1010110001=>1000110101=>1100101001 0101110100=>1010111000=>0001110101=>1100011001 0110011001=>1001100110=>0110011001=>0101010101 0110100101=>1011001010=>0101001101=>1011000101 0110101001=>1011010010=>0100101101=>1101000101 0110101010=>1011010100=>0010101101=>1110000101 0110110010=>1011001100=>0011001101=>1010010101 0110110100=>1011011000=>0001101101=>1100010101 0111000101=>1000111010=>0101110001=>0011001101 0111001001=>1001110010=>0100111001=>0101001101 0111010010=>1011100100=>0010011101=>1010001101 0111101000=>1011110000=>0000111101=>1000011101 010101011010=>101010101100=>001101010101=>111100100001 010101101001=>101010110010=>010011010101=>111010010001 010101110100=>101010111000=>000111010101=>111000110001 010110100101=>101011001010=>010100110101=>110110001001 010110101001=>101011010010=>010010110101=>111010001001 010110101010=>101011010100=>001010110101=>111100001001 010110101100=>101011010001=>100010110101=>111001001001 010110110100=>101011011000=>000110110101=>111000101001 010111001100=>101001110001=>100011100101=>101001011001 010111010001=>101011100010=>010001110101=>110010011001 010111010010=>101011100100=>001001110101=>110100011001 010111010100=>101011101000=>000101110101=>111000011001 011001101001=>100110110010=>010011011001=>011010010101 011001101100=>100110110001=>100011011001=>011001010101 011001110001=>100110001110=>011100011001=>010010110101 011001110100=>100110111000=>000111011001=>011000110101 011010010101=>101100101010=>010101001101=>101110000101 011010011010=>101100101100=>001101001101=>101100100101 011010100101=>101101001010=>010100101101=>110110000101 011010101001=>101101010010=>010010101101=>111010000101 011010101010=>101101010100=>001010101101=>111100000101 011010110010=>101101001100=>001100101101=>110100100101 011010110100=>101101011000=>000110101101=>111000100101 011011000110=>101100011001=>100110001101=>100101010101 011011001001=>101100110010=>010011001101=>101010010101 011011001010=>101100110100=>001011001101=>101100010101 011011001100=>101100110001=>100011001101=>101001010101 011011010001=>101101100010=>010001101101=>110010010101 011011010010=>101101100100=>001001101101=>110100010101 011011010100=>101101101000=>000101101101=>111000010101 011011011000=>101101100001=>100001101101=>110001010101 011011100100=>101100111000=>000111001101=>101000110101 011011101000=>101101110000=>000011101101=>110000110101 011011110000=>101100000111=>111000001101=>100001110101 011101000101=>101110001010=>010100011101=>100110001101 011101001001=>101110010010=>010010011101=>101010001101 011101001100=>101110010001=>100010011101=>101001001101 011101010001=>101110100010=>010001011101=>110010001101 011101010010=>101110100100=>001001011101=>110100001101 011101010100=>101110101000=>000101011101=>111000001101 011101100100=>101110011000=>000110011101=>101000101101 011110001100=>100011110001=>100011110001=>001001011101 011110100001=>101111000010=>010000111101=>100010011101 011110100010=>101111000100=>001000111101=>100100011101 011110100100=>101111001000=>000100111101=>101000011101 011110101000=>101111010000=>000010111101=>110000011101 011110110000=>101111000001=>100000111101=>100001011101 011111010000=>101111100000=>000001111101=>100000111101 0100000100=>1000001000=>0001000001=>0000100001 0001101110=>0010110011=>1100110100=>0110110100 0011111000=>0100001111=>1111000010=>1000111100 0001111100=>0010001111=>1111000100=>0100111100 101001110100=>110010111000=>000111010011=>011000110011 001111110000=>010000011111=>111110000010=>100001111100 100011111000=>001100001111=>111100001100=>010001111010 001011111000=>010100001111=>111100001010=>110001111000 001001111100=>010010001111=>111100010010=>101001111000 000101111100=>001010001111=>111100010100=>011001111000 000011111100=>000100011111=>111110001000=>001001111100 0000000101=>0000001010=>0101000000=>0000011000 0101111101=>1010111110=>0111110101=>1101111001 11111111101=>11111111110=>01111111111=>01111111111 101001010101=>110010101010=>010101010011=>011110000011 101001101001=>110010110010=>010011010011=>011010010011 101010100101=>110101001010=>010100101011=>110110000011 101010101001=>110101010010=>010010101011=>111010000011 101011001001=>110100110010=>010011001011=>101010010011 101100101001=>110011010010=>010010110011=>011010001011 101101000101=>110110001010=>010100011011=>100110001011 101101001001=>110110010010=>010010011011=>101010001011 101101010001=>110110100010=>010001011011=>110010001011 101110100001=>110111000010=>010000111011=>100010011011 110001110001=>001110001110=>011100011100=>010010110110 110100010101=>111000101010=>010101000111=>001110000111 110100011001=>111000100110=>011001000111=>001010100111 110100100101=>111001001010=>010100100111=>010110000111 110100101001=>111001010010=>010010100111=>011010000111 110101000101=>111010001010=>010100010111=>100110000111 110101001001=>111010010010=>010010010111=>101010000111 110101010001=>111010100010=>010001010111=>110010000111 110101100001=>111010000110=>011000010111=>100010100111 110110001001=>111000110010=>010011000111=>001010010111 110110010001=>111001100010=>010001100111=>010010010111 110110100001=>111011000010=>010000110111=>100010010111 111010001001=>111100010010=>010010001111=>001010001111 111010100001=>111101000010=>010000101111=>100010001111 111101000001=>111110000010=>010000011111=>000010011111
Map
rowmotion
Description
Return the rowmotion of the binary word, regarded as an order ideal in the product of two chains.
In particular, this operation preserves the number of ones, and its order is the length of the word, see section 3.3 of [1].
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 $\phi(4154223) = 1254423$.