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=>1100
0111=>0111
1000=>0010
1001=>0101
1010=>0110
1011=>1011
1100=>1010
1101=>1101
1110=>1110
1111=>1111
00000=>00000
00001=>00001
00010=>10000
00011=>00011
00100=>01000
00101=>10001
00110=>11000
00111=>00111
01000=>00100
01001=>01001
01010=>01100
01011=>10011
01100=>10100
01101=>11001
01110=>11100
01111=>01111
10000=>00010
10001=>00101
10010=>00110
10011=>01011
10100=>10010
10101=>01101
10110=>01110
10111=>10111
11000=>01010
11001=>10101
11010=>10110
11011=>11011
11100=>11010
11101=>11101
11110=>11110
11111=>11111
000000=>000000
000001=>000001
000010=>100000
000011=>000011
000100=>010000
000101=>100001
000110=>110000
000111=>000111
001000=>001000
001001=>010001
001010=>011000
001011=>100011
001100=>101000
001101=>110001
001110=>111000
001111=>001111
010000=>000100
010001=>001001
010010=>001100
010011=>010011
010100=>100100
010101=>011001
010110=>011100
010111=>100111
011000=>010100
011001=>101001
011010=>101100
011011=>110011
011100=>110100
011101=>111001
011110=>111100
011111=>011111
100000=>000010
100001=>000101
100010=>000110
100011=>001011
100100=>100010
100101=>001101
100110=>001110
100111=>010111
101000=>010010
101001=>100101
101010=>100110
101011=>011011
101100=>110010
101101=>011101
101110=>011110
101111=>101111
110000=>001010
110001=>010101
110010=>010110
110011=>101011
110100=>011010
110101=>101101
110110=>101110
110111=>110111
111000=>101010
111001=>110101
111010=>110110
111011=>111011
111100=>111010
111101=>111101
111110=>111110
111111=>111111
0000000=>0000000
0000001=>0000001
0000010=>1000000
0000011=>0000011
0000100=>0100000
0000101=>1000001
0000110=>1100000
0000111=>0000111
0001000=>0010000
0001001=>0100001
0001010=>0110000
0001011=>1000011
0001100=>1010000
0001101=>1100001
0001110=>1110000
0001111=>0001111
0010000=>0001000
0010001=>0010001
0010010=>0011000
0010011=>0100011
0010100=>1001000
0010101=>0110001
0010110=>0111000
0010111=>1000111
0011000=>0101000
0011001=>1010001
0011010=>1011000
0011011=>1100011
0011100=>1101000
0011101=>1110001
0011110=>1111000
0011111=>0011111
0100000=>0000100
0100001=>0001001
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0100011=>0010011
0100100=>1000100
0100101=>0011001
0100110=>0011100
0100111=>0100111
0101000=>0100100
0101001=>1001001
0101010=>1001100
0101011=>0110011
0101100=>1100100
0101101=>0111001
0101110=>0111100
0101111=>1001111
0110000=>0010100
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0110011=>1010011
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0111001=>1101001
0111010=>1101100
0111011=>1110011
0111100=>1110100
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0111110=>1111100
0111111=>0111111
1000000=>0000010
1000001=>0000101
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1000011=>0001011
1000100=>1000010
1000101=>0001101
1000110=>0001110
1000111=>0010111
1001000=>0100010
1001001=>1000101
1001010=>1000110
1001011=>0011011
1001100=>1100010
1001101=>0011101
1001110=>0011110
1001111=>0101111
1010000=>0010010
1010001=>0100101
1010010=>0100110
1010011=>1001011
1010100=>0110010
1010101=>1001101
1010110=>1001110
1010111=>0110111
1011000=>1010010
1011001=>1100101
1011010=>1100110
1011011=>0111011
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1011110=>0111110
1011111=>1011111
1100000=>0001010
1100001=>0010101
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1100011=>0101011
1100100=>0011010
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1100110=>0101110
1100111=>1010111
1101000=>1001010
1101001=>0110101
1101010=>0110110
1101011=>1011011
1101100=>0111010
1101101=>1011101
1101110=>1011110
1101111=>1101111
1110000=>0101010
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1110010=>1010110
1110011=>1101011
1110100=>1011010
1110101=>1101101
1110110=>1101110
1110111=>1110111
1111000=>1101010
1111001=>1110101
1111010=>1110110
1111011=>1111011
1111100=>1111010
1111101=>1111101
1111110=>1111110
1111111=>1111111
00000000=>00000000
00000001=>00000001
00000010=>10000000
00000011=>00000011
00000100=>01000000
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00000110=>11000000
00000111=>00000111
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00001001=>01000001
00001010=>01100000
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00010000=>00010000
00010001=>00100001
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00100001=>00010001
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00101000=>01001000
00101001=>10010001
00101010=>10011000
00101011=>01100011
00101100=>11001000
00101101=>01110001
00101110=>01111000
00101111=>10001111
00110000=>00101000
00110001=>01010001
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00110011=>10100011
00110100=>01101000
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00111001=>11010001
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00111011=>11100011
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00111111=>00111111
01000000=>00000100
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01000010=>00001100
01000011=>00010011
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01000101=>00011001
01000110=>00011100
01000111=>00100111
01001000=>01000100
01001001=>10001001
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01001011=>00110011
01001100=>11000100
01001101=>00111001
01001110=>00111100
01001111=>01001111
01010000=>00100100
01010001=>01001001
01010010=>01001100
01010011=>10010011
01010100=>01100100
01010101=>10011001
01010110=>10011100
01010111=>01100111
01011000=>10100100
01011001=>11001001
01011010=>11001100
01011011=>01110011
01011100=>11100100
01011101=>01111001
01011110=>01111100
01011111=>10011111
01100000=>00010100
01100001=>00101001
01100010=>00101100
01100011=>01010011
01100100=>00110100
01100101=>01011001
01100110=>01011100
01100111=>10100111
01101000=>10010100
01101001=>01101001
01101010=>01101100
01101011=>10110011
01101100=>01110100
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01101110=>10111100
01101111=>11001111
01110000=>01010100
01110001=>10101001
01110010=>10101100
01110011=>11010011
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01111000=>11010100
01111001=>11101001
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01111011=>11110011
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01111101=>11111001
01111110=>11111100
01111111=>01111111
10000000=>00000010
10000001=>00000101
10000010=>00000110
10000011=>00001011
10000100=>10000010
10000101=>00001101
10000110=>00001110
10000111=>00010111
10001000=>01000010
10001001=>10000101
10001010=>10000110
10001011=>00011011
10001100=>11000010
10001101=>00011101
10001110=>00011110
10001111=>00101111
10010000=>00100010
10010001=>01000101
10010010=>01000110
10010011=>10001011
10010100=>01100010
10010101=>10001101
10010110=>10001110
10010111=>00110111
10011000=>10100010
10011001=>11000101
10011010=>11000110
10011011=>00111011
10011100=>11100010
10011101=>00111101
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10011111=>01011111
10100000=>00010010
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10100010=>00100110
10100011=>01001011
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10100101=>01001101
10100110=>01001110
10100111=>10010111
10101000=>10010010
10101001=>01100101
10101010=>01100110
10101011=>10011011
10101100=>01110010
10101101=>10011101
10101110=>10011110
10101111=>01101111
10110000=>01010010
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10110010=>10100110
10110011=>11001011
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10110110=>11001110
10110111=>01110111
10111000=>11010010
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10111010=>11100110
10111011=>01111011
10111100=>11110010
10111101=>01111101
10111110=>01111110
10111111=>10111111
11000000=>00001010
11000001=>00010101
11000010=>00010110
11000011=>00101011
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11000101=>00101101
11000110=>00101110
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11001000=>10001010
11001001=>00110101
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11001011=>01011011
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11001101=>01011101
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11001111=>10101111
11010000=>01001010
11010001=>10010101
11010010=>10010110
11010011=>01101011
11010100=>10011010
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11010110=>01101110
11010111=>10110111
11011000=>11001010
11011001=>01110101
11011010=>01110110
11011011=>10111011
11011100=>01111010
11011101=>10111101
11011110=>10111110
11011111=>11011111
11100000=>00101010
11100001=>01010101
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11100011=>10101011
11100100=>01011010
11100101=>10101101
11100110=>10101110
11100111=>11010111
11101000=>01101010
11101001=>10110101
11101010=>10110110
11101011=>11011011
11101100=>10111010
11101101=>11011101
11101110=>11011110
11101111=>11101111
11110000=>10101010
11110001=>11010101
11110010=>11010110
11110011=>11101011
11110100=>11011010
11110101=>11101101
11110110=>11101110
11110111=>11110111
11111000=>11101010
11111001=>11110101
11111010=>11110110
11111011=>11111011
11111100=>11111010
11111101=>11111101
11111110=>11111110
11111111=>11111111
000000000=>000000000
000000001=>000000001
000000010=>100000000
000000011=>000000011
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000001000=>001000000
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000001010=>011000000
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111010101=>101101101
111010110=>101101110
111010111=>110110111
111011000=>011101010
111011001=>101110101
111011010=>101110110
111011011=>110111011
111011100=>101111010
111011101=>110111101
111011110=>110111110
111011111=>111011111
111100000=>010101010
111100001=>101010101
111100010=>101010110
111100011=>110101011
111100100=>101011010
111100101=>110101101
111100110=>110101110
111100111=>111010111
111101000=>101101010
111101001=>110110101
111101010=>110110110
111101011=>111011011
111101100=>110111010
111101101=>111011101
111101110=>111011110
111101111=>111101111
111110000=>110101010
111110001=>111010101
111110010=>111010110
111110011=>111101011
111110100=>111011010
111110101=>111101101
111110110=>111101110
111110111=>111110111
111111000=>111101010
111111001=>111110101
111111010=>111110110
111111011=>111111011
111111100=>111111010
111111101=>111111101
111111110=>111111110
111111111=>111111111
1111111110=>1111111110
1101111110=>1011111110
1011111110=>0111111110
1010111110=>1001111110
1001111110=>0011111110
1001011110=>1000111110
1000111110=>0001111110
1000101110=>1000011110
1000011110=>0000111110
1000010110=>1000001110
1000001110=>0000011110
1000001010=>1000000110
1000000110=>0000001110
1000000100=>1000000010
1000000010=>0000000110
1000000000=>0000000010
0111111111=>0111111111
0111111101=>1111111001
0111111001=>1111101001
0111110001=>1110101001
0111100001=>1010101001
0111000001=>0010101001
0110000001=>0000101001
0100000001=>0000001001
0000000001=>0000000001
0111111011=>1111110011
0111110101=>1111011001
0111101001=>1101101001
0111010001=>0110101001
0110100001=>0100101001
0101000001=>0001001001
0010000001=>0000010001
10000000000=>00000000010
10000000010=>00000000110
10000000110=>00000001110
10000001110=>00000011110
10000011110=>00000111110
10000111110=>00001111110
10001111110=>00011111110
10011111110=>00111111110
10111111110=>01111111110
11111111110=>11111111110
01111111111=>01111111111
01111111101=>11111111001
01111111001=>11111101001
01111110001=>11110101001
01111100001=>11010101001
01111000001=>01010101001
01110000001=>00010101001
01100000001=>00000101001
01000000001=>00000001001
00000000001=>00000000001
click to show experimental identities
(only identities of compositions of up to three maps are shown)
Description
The inverse of Foata's bijection.
See Mp00096Foata bijection.
See Mp00096Foata bijection.
References
[1] Foata, D., Schützenberger, M.-P. Major index and inversion number of permutations MathSciNet:0506852
Properties
bijective, graded
Sage code
def mapping(self): """ sage: all(mapping(pi.foata_bijection()) == pi for n in range(9) for pi in Words([0,1], n)) True """ self = Words([0,1])(self) s = self.standard_permutation() ordered_alphabet = sorted(self.letters(), key=self.parent().sortkey_letters) eval_dict = self.evaluation_dict() weight = [eval_dict[a] for a in ordered_alphabet] return Words([0,1])((s.foata_bijection_inverse()).destandardize(weight, ordered_alphabet=ordered_alphabet))
Weight
21
Created
Sep 12, 2023 at 11:34 by Martin Rubey
Updated
Sep 12, 2023 at 11:34 by Martin Rubey
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