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Identifier
Mp00096: Binary words —Foata bijection⟶ Binary words
Images
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=> 0=>0 1=>1 00=>00 01=>01 10=>10 11=>11 000=>000 001=>001 010=>100 011=>011 100=>010 101=>101 110=>110 111=>111 0000=>0000 0001=>0001 0010=>1000 0011=>0011 0100=>0100 0101=>1001 0110=>1010 0111=>0111 1000=>0010 1001=>0101 1010=>1100 1011=>1011 1100=>0110 1101=>1101 1110=>1110 1111=>1111 00000=>00000 00001=>00001 00010=>10000 00011=>00011 00100=>01000 00101=>10001 00110=>10010 00111=>00111 01000=>00100 01001=>01001 01010=>11000 01011=>10011 01100=>01010 01101=>10101 01110=>10110 01111=>01111 10000=>00010 10001=>00101 10010=>10100 10011=>01011 10100=>01100 10101=>11001 10110=>11010 10111=>10111 11000=>00110 11001=>01101 11010=>11100 11011=>11011 11100=>01110 11101=>11101 11110=>11110 11111=>11111 000000=>000000 000001=>000001 000010=>100000 000011=>000011 000100=>010000 000101=>100001 000110=>100010 000111=>000111 001000=>001000 001001=>010001 001010=>110000 001011=>100011 001100=>010010 001101=>100101 001110=>100110 001111=>001111 010000=>000100 010001=>001001 010010=>101000 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11010101010100101010=>11101111111000000000 11010101010101001010=>11011111111000000000 11010101010101010010=>10111111111000000000 01010101010101010101=>11111111100000000001 00110101010101010101=>11111111001000000001 00101101010101010101=>11111111000100000001 00101011010101010101=>11111111000010000001 00101010110101010101=>11111111000001000001 00101010101101010101=>11111111000000100001 00101010101011010101=>11111111000000010001 00101010101010110101=>11111111000000001001 00101010101010101101=>11111111000000000101 00101010101010101011=>11111111000000000011 01010101010101010011=>01111111100000000011 01010101010101001011=>10111111100000000011 01010101010100101011=>11011111100000000011 01010101010010101011=>11101111100000000011 01010101001010101011=>11110111100000000011 01010100101010101011=>11111011100000000011 01010010101010101011=>11111101100000000011 01001010101010101011=>11111110100000000011 11100101010101010100=>01111111011100000000 11100010101010101010=>11111110011100000000 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11011101010101010000=>00011111111011000000 11110101010101010000=>00011111111110000000 11111111100000001000=>00100000011111111100 11111111100000010000=>00010000011111111100 11111111100000100000=>00001000011111111100 11111111100001000000=>00000100011111111100 11111111100010000000=>00000010011111111100 11001010101010110100=>01111111011000000100 11001010101010101100=>01111111011000000010 11111111011000000000=>00000000111111111010 11111110101100000000=>00000001111111110010 11111111001010000000=>00000011011111111000 10111111110000000010=>10000000110111111100 10101010101100101010=>11101111110000010000 10101010101101001010=>11011111110000010000 10101010101101010010=>10111111110000010000 11101010100101010100=>01111011111100000000 11011010100101010100=>01111011111010000000 11010110100101010100=>01111011111001000000 11010101100101010100=>01111011111000100000 11001111111100000000=>00000001011011111110 10101010110101010111=>11111111000010000111 11101010101101010101=>11111111110000100001 10101010000101010111=>11100011110000000111 11101010100001010101=>11100011111100000001 11111111000000001100=>01000000011111111010 11111111010000000100=>01000000111111111000 11100011111110000000=>00000010011101111110 11110000111111000000=>00000100011110111110 11111000001111100000=>00001000011111011110 11111100000011110000=>00010000011111101110 11111110000000111000=>00100000011111110110 11111111111111111111=>11111111111111111111 11111111111111111001=>01111111111111111101 11111111111111100001=>00011111111111111101 11111111111110000001=>00000111111111111101 11111111111000000001=>00000001111111111101 11111111100000000001=>00000000011111111101 11111110000000000001=>00000000000111111101 11111000000000000001=>00000000000001111101 11100000000000000001=>00000000000000011101 10000000000000000001=>00000000000000000101 11111010101011000000=>00000111111111000010 11111100101010100000=>00001111011111100000 11110101010101100000=>00001111111110000010 11111001010101010000=>00011111011111000000 11111110111000000000=>00000000111111110110 11111101111000000000=>00000000111111101110 11111011111000000000=>00000000111111011110 11110111111000000000=>00000000111110111110 11101111111000000000=>00000000111101111110 11101010101010100100=>01011111111100000000 1111111111000000000010=>1000000000111111111100 1011111111110000000000=>0000000001101111111110 0000000000111111111101=>1000000000011111111101 0100000000001111111111=>0000000001001111111111 1101010101010101010100=>0111111111110000000000 1010101010101010101100=>0111111111100000000010 1111111111001000000000=>0000000010111111111100 1111111111010000000000=>0000000001111111111100 1010101010101010110100=>0111111111100000000100 1111111110110000000000=>0000000001111111111010 1010101010010101010111=>1111011111000000000111 1110101010100101010101=>1111011111110000000001 1100101010101010101010=>1111111110110000000000 1111111111000000000100=>0100000000111111111100 1111101010101011000000=>0000011111111110000010 1111110010101010100000=>0000111110111111000000
click to show experimental identities (only identities of compositions of up to three maps are shown)
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 $\phi(4154223) = 1254423$.
References
[1] Foata, D., Schützenberger, M.-P. Major index and inversion number of permutations MathSciNet:0506852
[2] Foata bijection Mp00067
Properties
bijective, graded
Sage code
def mapping(w):
    return w.foata_bijection()
Weight
26
Created
Jan 29, 2020 at 13:26 by FindStatCrew
Updated
Jan 29, 2020 at 13:26 by Martin Rubey