- FindStat

Identifier

Mp00096: Binary words —Foata bijection⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words

Images

0 => 0 => 0 => 0 => 0

1 => 1 => 1 => 1 => 1

00 => 00 => 00 => 00 => 00

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

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

11 => 11 => 11 => 11 => 11

000 => 000 => 000 => 000 => 000

001 => 001 => 100 => 010 => 100

010 => 100 => 010 => 001 => 001

011 => 011 => 101 => 110 => 110

100 => 010 => 001 => 100 => 010

101 => 101 => 110 => 011 => 011

110 => 110 => 011 => 101 => 101

111 => 111 => 111 => 111 => 111

0000 => 0000 => 0000 => 0000 => 0000

0001 => 0001 => 1000 => 0100 => 0100

0010 => 1000 => 0100 => 0010 => 1000

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

0100 => 0100 => 0010 => 0001 => 0001

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

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

0111 => 0111 => 1011 => 1101 => 1101

1000 => 0010 => 0001 => 1000 => 0010

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

1010 => 1100 => 0110 => 0011 => 0011

1011 => 1011 => 1101 => 1110 => 1110

1100 => 0110 => 0011 => 1001 => 0101

1101 => 1101 => 1110 => 0111 => 0111

1110 => 1110 => 0111 => 1011 => 1011

1111 => 1111 => 1111 => 1111 => 1111

00000 => 00000 => 00000 => 00000 => 00000

00001 => 00001 => 10000 => 01000 => 00100

00010 => 10000 => 01000 => 00100 => 01000

00011 => 00011 => 10001 => 11000 => 01010

00100 => 01000 => 00100 => 00010 => 10000

00101 => 10001 => 11000 => 01100 => 10100

00110 => 10010 => 01001 => 10100 => 10010

00111 => 00111 => 10011 => 11001 => 10101

01000 => 00100 => 00010 => 00001 => 00001

01001 => 01001 => 10100 => 01010 => 01100

01010 => 11000 => 01100 => 00110 => 11000

01011 => 10011 => 11001 => 11100 => 11010

01100 => 01010 => 00101 => 10010 => 00110

01101 => 10101 => 11010 => 01101 => 11001

01110 => 10110 => 01011 => 10101 => 01101

01111 => 01111 => 10111 => 11011 => 11011

10000 => 00010 => 00001 => 10000 => 00010

10001 => 00101 => 10010 => 01001 => 01001

10010 => 10100 => 01010 => 00101 => 10001

10011 => 01011 => 10101 => 11010 => 10110

10100 => 01100 => 00110 => 00011 => 00011

10101 => 11001 => 11100 => 01110 => 11100

10110 => 11010 => 01101 => 10110 => 01110

10111 => 10111 => 11011 => 11101 => 11101

11000 => 00110 => 00011 => 10001 => 00101

11001 => 01101 => 10110 => 01011 => 10011

11010 => 11100 => 01110 => 00111 => 00111

11011 => 11011 => 11101 => 11110 => 11110

11100 => 01110 => 00111 => 10011 => 01011

11101 => 11101 => 11110 => 01111 => 01111

11110 => 11110 => 01111 => 10111 => 10111

11111 => 11111 => 11111 => 11111 => 11111

000000 => 000000 => 000000 => 000000 => 000000

000001 => 000001 => 100000 => 010000 => 000100

000010 => 100000 => 010000 => 001000 => 001000

000011 => 000011 => 100001 => 110000 => 001010

000100 => 010000 => 001000 => 000100 => 010000

000101 => 100001 => 110000 => 011000 => 010100

000110 => 100010 => 010001 => 101000 => 010010

000111 => 000111 => 100011 => 110001 => 010101

001000 => 001000 => 000100 => 000010 => 100000

001001 => 010001 => 101000 => 010100 => 100100

001010 => 110000 => 011000 => 001100 => 101000

001011 => 100011 => 110001 => 111000 => 101010

001100 => 010010 => 001001 => 100100 => 100010

001101 => 100101 => 110010 => 011001 => 101001

001110 => 100110 => 010011 => 101001 => 100101

001111 => 001111 => 100111 => 110011 => 101011

010000 => 000100 => 000010 => 000001 => 000001

010001 => 001001 => 100100 => 010010 => 001100

010010 => 101000 => 010100 => 001010 => 011000

010011 => 010011 => 101001 => 110100 => 011010

010100 => 011000 => 001100 => 000110 => 110000

010101 => 110001 => 111000 => 011100 => 110100

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

010111 => 100111 => 110011 => 111001 => 110101

011000 => 001010 => 000101 => 100010 => 000110

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

011010 => 110100 => 011010 => 001101 => 110001

011011 => 101011 => 110101 => 111010 => 110110

011100 => 010110 => 001011 => 100101 => 001101

011101 => 101101 => 110110 => 011011 => 110011

011110 => 101110 => 010111 => 101011 => 011011

011111 => 011111 => 101111 => 110111 => 110111

100000 => 000010 => 000001 => 100000 => 000010

100001 => 000101 => 100010 => 010001 => 001001

100010 => 100100 => 010010 => 001001 => 010001

100011 => 001011 => 100101 => 110010 => 010110

100100 => 010100 => 001010 => 000101 => 100001

100101 => 101001 => 110100 => 011010 => 101100

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

>>> Load all 1043 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

rotate back-to-front

Description

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

Map

inverse Foata bijection

Description

The inverse of Foata's bijection.
See Mp00096Foata bijection.