- FindStat

Identifier

Mp00096: Binary words —Foata bijection⟶ Binary words
St000293: Binary words ⟶ ℤ

Values

0 => 0 => 0

1 => 1 => 0

00 => 00 => 0

01 => 01 => 0

10 => 10 => 1

11 => 11 => 0

000 => 000 => 0

001 => 001 => 0

010 => 100 => 2

011 => 011 => 0

100 => 010 => 1

101 => 101 => 1

110 => 110 => 2

111 => 111 => 0

0000 => 0000 => 0

0001 => 0001 => 0

0010 => 1000 => 3

0011 => 0011 => 0

0100 => 0100 => 2

0101 => 1001 => 2

0110 => 1010 => 3

0111 => 0111 => 0

1000 => 0010 => 1

1001 => 0101 => 1

1010 => 1100 => 4

1011 => 1011 => 1

1100 => 0110 => 2

1101 => 1101 => 2

1110 => 1110 => 3

1111 => 1111 => 0

00000 => 00000 => 0

00001 => 00001 => 0

00010 => 10000 => 4

00011 => 00011 => 0

00100 => 01000 => 3

00101 => 10001 => 3

00110 => 10010 => 4

00111 => 00111 => 0

01000 => 00100 => 2

01001 => 01001 => 2

01010 => 11000 => 6

01011 => 10011 => 2

01100 => 01010 => 3

01101 => 10101 => 3

01110 => 10110 => 4

01111 => 01111 => 0

10000 => 00010 => 1

10001 => 00101 => 1

10010 => 10100 => 5

10011 => 01011 => 1

10100 => 01100 => 4

10101 => 11001 => 4

10110 => 11010 => 5

10111 => 10111 => 1

11000 => 00110 => 2

11001 => 01101 => 2

11010 => 11100 => 6

11011 => 11011 => 2

11100 => 01110 => 3

11101 => 11101 => 3

11110 => 11110 => 4

11111 => 11111 => 0

000000 => 000000 => 0

000001 => 000001 => 0

000010 => 100000 => 5

000011 => 000011 => 0

000100 => 010000 => 4

000101 => 100001 => 4

000110 => 100010 => 5

000111 => 000111 => 0

001000 => 001000 => 3

001001 => 010001 => 3

001010 => 110000 => 8

001011 => 100011 => 3

001100 => 010010 => 4

001101 => 100101 => 4

001110 => 100110 => 5

001111 => 001111 => 0

010000 => 000100 => 2

010001 => 001001 => 2

010010 => 101000 => 7

010011 => 010011 => 2

010100 => 011000 => 6

010101 => 110001 => 6

010110 => 110010 => 7

010111 => 100111 => 2

011000 => 001010 => 3

011001 => 010101 => 3

011010 => 110100 => 8

011011 => 101011 => 3

011100 => 010110 => 4

011101 => 101101 => 4

011110 => 101110 => 5

011111 => 011111 => 0

100000 => 000010 => 1

100001 => 000101 => 1

100010 => 100100 => 6

100011 => 001011 => 1

100100 => 010100 => 5

100101 => 101001 => 5

100110 => 101010 => 6

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$F_{1} = 2$

$F_{2} = 3 + q$

$F_{3} = 4 + 2\ q + 2\ q^{2}$

$F_{4} = 5 + 3\ q + 4\ q^{2} + 3\ q^{3} + q^{4}$

$F_{5} = 6 + 4\ q + 6\ q^{2} + 6\ q^{3} + 6\ q^{4} + 2\ q^{5} + 2\ q^{6}$

$F_{6} = 7 + 5\ q + 8\ q^{2} + 9\ q^{3} + 11\ q^{4} + 9\ q^{5} + 7\ q^{6} + 4\ q^{7} + 3\ q^{8} + q^{9}$

$F_{7} = 8 + 6\ q + 10\ q^{2} + 12\ q^{3} + 16\ q^{4} + 16\ q^{5} + 18\ q^{6} + 12\ q^{7} + 12\ q^{8} + 8\ q^{9} + 6\ q^{10} + 2\ q^{11} + 2\ q^{12}$

$F_{8} = 9 + 7\ q + 12\ q^{2} + 15\ q^{3} + 21\ q^{4} + 23\ q^{5} + 29\ q^{6} + 27\ q^{7} + 26\ q^{8} + 23\ q^{9} + 21\ q^{10} + 15\ q^{11} + 13\ q^{12} + 7\ q^{13} + 4\ q^{14} + 3\ q^{15} + q^{16}$

$F_{9} = 10 + 8\ q + 14\ q^{2} + 18\ q^{3} + 26\ q^{4} + 30\ q^{5} + 40\ q^{6} + 42\ q^{7} + 48\ q^{8} + 44\ q^{9} + 46\ q^{10} + 40\ q^{11} + 40\ q^{12} + 30\ q^{13} + 26\ q^{14} + 18\ q^{15} + 14\ q^{16} + 8\ q^{17} + 6\ q^{18} + 2\ q^{19} + 2\ q^{20}$

Description

The number of inversions of a binary word.

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.

$\phi(4154223) = 1254423$ .