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Identifier
Mp00096: Binary words Foata bijectionBinary 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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110110110110110100000010=>100000111111101010101000 101110110110110100000100=>010000111111011010101000 101101110110110100001000=>001000111111010110101000 101101101110110100010000=>000100111111010101101000 101101101101110100100000=>000010111111010101011000 101101101101101101000000=>000001111111010101010100 100001001001001001000010=>100010101010100010000000 110111111111100000000000=>000000000011101111111110 111111111110000000000100=>010000000001111111111100 1000000100001001000010010000100000010=>1000001000101000101000100000100000000
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()

Created
Jan 29, 2020 at 13:26 by FindStatCrew
Updated
Jan 29, 2020 at 13:26 by Martin Rubey