- FindStat

Identifier

Mp00096: Binary words —Foata bijection⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words

Images

0 => 0 => 0 => 1 => 1

1 => 1 => 1 => 0 => 0

00 => 00 => 00 => 11 => 11

01 => 01 => 10 => 01 => 10

10 => 10 => 01 => 10 => 01

11 => 11 => 11 => 00 => 00

000 => 000 => 000 => 111 => 111

001 => 001 => 100 => 011 => 110

010 => 100 => 001 => 110 => 101

011 => 011 => 110 => 001 => 010

100 => 010 => 010 => 101 => 011

101 => 101 => 101 => 010 => 100

110 => 110 => 011 => 100 => 001

111 => 111 => 111 => 000 => 000

0000 => 0000 => 0000 => 1111 => 1111

0001 => 0001 => 1000 => 0111 => 1110

0010 => 1000 => 0001 => 1110 => 1101

0011 => 0011 => 1100 => 0011 => 0110

0100 => 0100 => 0010 => 1101 => 1011

0101 => 1001 => 1001 => 0110 => 1100

0110 => 1010 => 0101 => 1010 => 0101

0111 => 0111 => 1110 => 0001 => 0010

1000 => 0010 => 0100 => 1011 => 0111

1001 => 0101 => 1010 => 0101 => 1010

1010 => 1100 => 0011 => 1100 => 1001

1011 => 1011 => 1101 => 0010 => 0100

1100 => 0110 => 0110 => 1001 => 0011

1101 => 1101 => 1011 => 0100 => 1000

1110 => 1110 => 0111 => 1000 => 0001

1111 => 1111 => 1111 => 0000 => 0000

00000 => 00000 => 00000 => 11111 => 11111

00001 => 00001 => 10000 => 01111 => 11110

00010 => 10000 => 00001 => 11110 => 11101

00011 => 00011 => 11000 => 00111 => 01110

00100 => 01000 => 00010 => 11101 => 11011

00101 => 10001 => 10001 => 01110 => 11100

00110 => 10010 => 01001 => 10110 => 01101

00111 => 00111 => 11100 => 00011 => 00110

01000 => 00100 => 00100 => 11011 => 10111

01001 => 01001 => 10010 => 01101 => 11010

01010 => 11000 => 00011 => 11100 => 11001

01011 => 10011 => 11001 => 00110 => 01100

01100 => 01010 => 01010 => 10101 => 01011

01101 => 10101 => 10101 => 01010 => 10100

01110 => 10110 => 01101 => 10010 => 00101

01111 => 01111 => 11110 => 00001 => 00010

10000 => 00010 => 01000 => 10111 => 01111

10001 => 00101 => 10100 => 01011 => 10110

10010 => 10100 => 00101 => 11010 => 10101

10011 => 01011 => 11010 => 00101 => 01010

10100 => 01100 => 00110 => 11001 => 10011

10101 => 11001 => 10011 => 01100 => 11000

10110 => 11010 => 01011 => 10100 => 01001

10111 => 10111 => 11101 => 00010 => 00100

11000 => 00110 => 01100 => 10011 => 00111

11001 => 01101 => 10110 => 01001 => 10010

11010 => 11100 => 00111 => 11000 => 10001

11011 => 11011 => 11011 => 00100 => 01000

11100 => 01110 => 01110 => 10001 => 00011

11101 => 11101 => 10111 => 01000 => 10000

11110 => 11110 => 01111 => 10000 => 00001

11111 => 11111 => 11111 => 00000 => 00000

000000 => 000000 => 000000 => 111111 => 111111

000001 => 000001 => 100000 => 011111 => 111110

000010 => 100000 => 000001 => 111110 => 111101

000011 => 000011 => 110000 => 001111 => 011110

000100 => 010000 => 000010 => 111101 => 111011

000101 => 100001 => 100001 => 011110 => 111100

000110 => 100010 => 010001 => 101110 => 011101

000111 => 000111 => 111000 => 000111 => 001110

001000 => 001000 => 000100 => 111011 => 110111

001001 => 010001 => 100010 => 011101 => 111010

001010 => 110000 => 000011 => 111100 => 111001

001011 => 100011 => 110001 => 001110 => 011100

001100 => 010010 => 010010 => 101101 => 011011

001101 => 100101 => 101001 => 010110 => 101100

001110 => 100110 => 011001 => 100110 => 001101

001111 => 001111 => 111100 => 000011 => 000110

010000 => 000100 => 001000 => 110111 => 101111

010001 => 001001 => 100100 => 011011 => 110110

010010 => 101000 => 000101 => 111010 => 110101

010011 => 010011 => 110010 => 001101 => 011010

010100 => 011000 => 000110 => 111001 => 110011

010101 => 110001 => 100011 => 011100 => 111000

010110 => 110010 => 010011 => 101100 => 011001

010111 => 100111 => 111001 => 000110 => 001100

011000 => 001010 => 010100 => 101011 => 010111

011001 => 010101 => 101010 => 010101 => 101010

011010 => 110100 => 001011 => 110100 => 101001

011011 => 101011 => 110101 => 001010 => 010100

011100 => 010110 => 011010 => 100101 => 001011

011101 => 101101 => 101101 => 010010 => 100100

011110 => 101110 => 011101 => 100010 => 000101

011111 => 011111 => 111110 => 000001 => 000010

100000 => 000010 => 010000 => 101111 => 011111

100001 => 000101 => 101000 => 010111 => 101110

100010 => 100100 => 001001 => 110110 => 101101

100011 => 001011 => 110100 => 001011 => 010110

100100 => 010100 => 001010 => 110101 => 101011

100101 => 101001 => 100101 => 011010 => 110100

100110 => 101010 => 010101 => 101010 => 010101

>>> Load all 2990 entries. <<<

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 $\phi(4154223) = 1254423$.

Map

reverse

Description

Return the reversal of a binary word.

Map

complement

Description

Send a binary word to the word obtained by interchanging the two letters.

Map

rotate front-to-back

Description

The rotation of a binary word, first letter last.
This is the word obtained by moving the first letter to the end.