Identifier
Images
=>
=>
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
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000100=>010000
000101=>100001
000110=>100010
000111=>000111
001000=>001000
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001011=>100011
001100=>010010
001101=>100101
001110=>100110
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010000=>000100
010001=>001001
010010=>101000
010011=>010011
010100=>011000
010101=>110001
010110=>110010
010111=>100111
011000=>001010
011001=>010101
011010=>110100
011011=>101011
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011101=>101101
011110=>101110
011111=>011111
100000=>000010
100001=>000101
100010=>100100
100011=>001011
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100110=>101010
100111=>010111
101000=>001100
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101100=>011010
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101110=>110110
101111=>101111
110000=>000110
110001=>001101
110010=>101100
110011=>011011
110100=>011100
110101=>111001
110110=>111010
110111=>110111
111000=>001110
111001=>011101
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111100=>011110
111101=>111101
111110=>111110
111111=>111111
0000000=>0000000
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1111000=>0011110
1111001=>0111101
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1111101=>1111101
1111110=>1111110
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00000000=>00000000
00000001=>00000001
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01011010=>11100100
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01011111=>10011111
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10000000=>00000010
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10010010010010010010=>10101010101010000000
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111010011101101000100010=>100100111011110011010000
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110100110110110110100000=>000011111011100101010100
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110110100011011011010000=>000111100111101001010100
111101110100100010010010=>101010010111111011000000
111011101001000100110100=>011010010111110110000100
111011101001000110100010=>100110010111110110001000
110111010010001110100100=>010110010111101100011000
110111010010001101101000=>001110010111101100010100
111011110100100100010010=>101001010111110111000000
110111101001001000110100=>011001010111101110000100
110111110100100100100010=>100101010111101111000000
101111110100100100100100=>010101010111011111000000
101111101001001001101000=>001101010111011110000100
110111101001001101000010=>100011010111101110001000
101111101001001101000100=>010011010111011110001000
101111010010011101001000=>001011010111011100011000
101111010010011011010000=>000111010111011100010100
111011101001101000010010=>101000110111110110010000
110111010011010000110100=>011000110111101100100100
110111101001101000100010=>100100110111101110010000
101111101001101000100100=>010100110111011110010000
101111010011010001101000=>001100110111011100100100
110111010011101001000010=>100010110111101100110000
101111010011101001000100=>010010110111011100110000
101110100111101001001000=>001010110111011001110000
101110100111010011010000=>000110110111011001100100
110111010011011010000010=>100001110111101100101000
101111010011011010000100=>010001110111011100101000
101110100111011010001000=>001001110111011001101000
101110100110111010010000=>000101110111011001011000
101110100110110110100000=>000011110111011001010100
111101101101000010010010=>101010001111111010100000
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111011011010000110100010=>100110001111110101001000
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101111011010001101000100=>010011001111011101001000
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101110110100011011010000=>000111001111011010010100
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110110111010010000110100=>011000101111101011000100
110111011101001000100010=>100100101111101101100000
101111011101001000100100=>010100101111011101100000
101110111010010001101000=>001100101111011011000100
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101101111010011010001000=>001001101111010111001000
101101110100111010010000=>000101101111010110011000
101101110100110110100000=>000011101111010110010100
111011011011010000010010=>101000011111110101010000
110110110110100000110100=>011000011111101010100100
110111011011010000100010=>100100011111101101010000
101111011011010000100100=>010100011111011101010000
101110110110100001101000=>001100011111011010100100
110110111011010001000010=>100010011111101011010000
101110111011010001000100=>010010011111011011010000
101101111011010001001000=>001010011111010111010000
101101110110100011010000=>000110011111010110100100
110110110111010010000010=>100001011111101010110000
101110110111010010000100=>010001011111011010110000
101101110111010010001000=>001001011111010110110000
101101101111010010010000=>000101011111010101110000
101101101110100110100000=>000011011111010101100100
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.
For instance, to compute $\phi(4154223)$, the sequence of words is
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 $w_{i+1} < v_k$.
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.
References
[1] Foata, D., Schützenberger, M.-P. Major index and inversion number of permutations MathSciNet:0506852
[2] Foata bijection Mp00067
[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
searching the database
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