- FindStat

Identifier

Mp00278: Binary words —rowmotion⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words

Images

0 => 0 => 0

1 => 1 => 1

00 => 00 => 00

01 => 10 => 10

10 => 01 => 01

11 => 11 => 11

000 => 000 => 000

001 => 010 => 100

010 => 100 => 010

011 => 101 => 101

100 => 001 => 001

101 => 110 => 110

110 => 011 => 011

111 => 111 => 111

0000 => 0000 => 0000

0001 => 0010 => 1000

0010 => 0100 => 0100

0011 => 0101 => 1001

0100 => 1000 => 0010

0101 => 1010 => 1100

0110 => 1001 => 0101

0111 => 1011 => 1011

1000 => 0001 => 0001

1001 => 0110 => 1010

1010 => 1100 => 0110

1011 => 1101 => 1101

1100 => 0011 => 0011

1101 => 1110 => 1110

1110 => 0111 => 0111

1111 => 1111 => 1111

00000 => 00000 => 00000

00001 => 00010 => 10000

00010 => 00100 => 01000

00011 => 00101 => 10001

00100 => 01000 => 00100

00101 => 01010 => 11000

00110 => 01001 => 01001

00111 => 01011 => 10011

01000 => 10000 => 00010

01001 => 10010 => 10100

01010 => 10100 => 01100

01011 => 10101 => 11001

01100 => 10001 => 00101

01101 => 10110 => 11010

01110 => 10011 => 01011

01111 => 10111 => 10111

10000 => 00001 => 00001

10001 => 00110 => 10010

10010 => 01100 => 01010

10011 => 01101 => 10101

10100 => 11000 => 00110

10101 => 11010 => 11100

10110 => 11001 => 01101

10111 => 11011 => 11011

11000 => 00011 => 00011

11001 => 01110 => 10110

11010 => 11100 => 01110

11011 => 11101 => 11101

11100 => 00111 => 00111

11101 => 11110 => 11110

11110 => 01111 => 01111

11111 => 11111 => 11111

000000 => 000000 => 000000

000001 => 000010 => 100000

000010 => 000100 => 010000

000011 => 000101 => 100001

000100 => 001000 => 001000

000101 => 001010 => 110000

000110 => 001001 => 010001

000111 => 001011 => 100011

001000 => 010000 => 000100

001001 => 010010 => 101000

001010 => 010100 => 011000

001011 => 010101 => 110001

001100 => 010001 => 001001

001101 => 010110 => 110010

001110 => 010011 => 010011

001111 => 010111 => 100111

010000 => 100000 => 000010

010001 => 100010 => 100100

010010 => 100100 => 010100

010011 => 100101 => 101001

010100 => 101000 => 001100

010101 => 101010 => 111000

010110 => 101001 => 011001

010111 => 101011 => 110011

011000 => 100001 => 000101

011001 => 100110 => 101010

011010 => 101100 => 011010

011011 => 101101 => 110101

011100 => 100011 => 001011

011101 => 101110 => 110110

011110 => 100111 => 010111

011111 => 101111 => 101111

100000 => 000001 => 000001

100001 => 000110 => 100010

100010 => 001100 => 010010

100011 => 001101 => 100101

100100 => 011000 => 001010

100101 => 011010 => 110100

100110 => 011001 => 010101

>>> Load all 2349 entries. <<<

Map

rowmotion

Description

Return the rowmotion of the binary word, regarded as an order ideal in the product of two chains.
In particular, this operation preserves the number of ones, and its order is the length of the word, see section 3.3 of [1].

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 $\phi(4154223) = 1254423$.