- FindStat

Identifier

Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000291: Binary words ⟶ ℤ

Values

[1,0] => 10 => 10 => 1

[1,0,1,0] => 1010 => 1100 => 1

[1,1,0,0] => 1100 => 0110 => 1

[1,0,1,0,1,0] => 101010 => 111000 => 1

[1,0,1,1,0,0] => 101100 => 011010 => 2

[1,1,0,0,1,0] => 110010 => 101100 => 2

[1,1,0,1,0,0] => 110100 => 011100 => 1

[1,1,1,0,0,0] => 111000 => 001110 => 1

[1,0,1,0,1,0,1,0] => 10101010 => 11110000 => 1

[1,0,1,0,1,1,0,0] => 10101100 => 01110010 => 2

[1,0,1,1,0,0,1,0] => 10110010 => 10110100 => 3

[1,0,1,1,0,1,0,0] => 10110100 => 01110100 => 2

[1,0,1,1,1,0,0,0] => 10111000 => 00110110 => 2

[1,1,0,0,1,0,1,0] => 11001010 => 11011000 => 2

[1,1,0,0,1,1,0,0] => 11001100 => 01011010 => 3

[1,1,0,1,0,0,1,0] => 11010010 => 10111000 => 2

[1,1,0,1,0,1,0,0] => 11010100 => 01111000 => 1

[1,1,0,1,1,0,0,0] => 11011000 => 00111010 => 2

[1,1,1,0,0,0,1,0] => 11100010 => 10011100 => 2

[1,1,1,0,0,1,0,0] => 11100100 => 01011100 => 2

[1,1,1,0,1,0,0,0] => 11101000 => 00111100 => 1

[1,1,1,1,0,0,0,0] => 11110000 => 00011110 => 1

[1,0,1,0,1,0,1,0,1,0] => 1010101010 => 1111100000 => 1

[1,0,1,0,1,0,1,1,0,0] => 1010101100 => 0111100010 => 2

[1,0,1,0,1,1,0,0,1,0] => 1010110010 => 1011100100 => 3

[1,0,1,0,1,1,0,1,0,0] => 1010110100 => 0111100100 => 2

[1,0,1,0,1,1,1,0,0,0] => 1010111000 => 0011100110 => 2

[1,0,1,1,0,0,1,0,1,0] => 1011001010 => 1101101000 => 3

[1,0,1,1,0,0,1,1,0,0] => 1011001100 => 0101101010 => 4

[1,0,1,1,0,1,0,0,1,0] => 1011010010 => 1011101000 => 3

[1,0,1,1,0,1,0,1,0,0] => 1011010100 => 0111101000 => 2

[1,0,1,1,0,1,1,0,0,0] => 1011011000 => 0011101010 => 3

[1,0,1,1,1,0,0,0,1,0] => 1011100010 => 1001101100 => 3

[1,0,1,1,1,0,0,1,0,0] => 1011100100 => 0101101100 => 3

[1,0,1,1,1,0,1,0,0,0] => 1011101000 => 0011101100 => 2

[1,0,1,1,1,1,0,0,0,0] => 1011110000 => 0001101110 => 2

[1,1,0,0,1,0,1,0,1,0] => 1100101010 => 1110110000 => 2

[1,1,0,0,1,0,1,1,0,0] => 1100101100 => 0110110010 => 3

[1,1,0,0,1,1,0,0,1,0] => 1100110010 => 1010110100 => 4

[1,1,0,0,1,1,0,1,0,0] => 1100110100 => 0110110100 => 3

[1,1,0,0,1,1,1,0,0,0] => 1100111000 => 0010110110 => 3

[1,1,0,1,0,0,1,0,1,0] => 1101001010 => 1101110000 => 2

[1,1,0,1,0,0,1,1,0,0] => 1101001100 => 0101110010 => 3

[1,1,0,1,0,1,0,0,1,0] => 1101010010 => 1011110000 => 2

[1,1,0,1,0,1,0,1,0,0] => 1101010100 => 0111110000 => 1

[1,1,0,1,0,1,1,0,0,0] => 1101011000 => 0011110010 => 2

[1,1,0,1,1,0,0,0,1,0] => 1101100010 => 1001110100 => 3

[1,1,0,1,1,0,0,1,0,0] => 1101100100 => 0101110100 => 3

[1,1,0,1,1,0,1,0,0,0] => 1101101000 => 0011110100 => 2

[1,1,0,1,1,1,0,0,0,0] => 1101110000 => 0001110110 => 2

[1,1,1,0,0,0,1,0,1,0] => 1110001010 => 1100111000 => 2

[1,1,1,0,0,0,1,1,0,0] => 1110001100 => 0100111010 => 3

[1,1,1,0,0,1,0,0,1,0] => 1110010010 => 1010111000 => 3

[1,1,1,0,0,1,0,1,0,0] => 1110010100 => 0110111000 => 2

[1,1,1,0,0,1,1,0,0,0] => 1110011000 => 0010111010 => 3

[1,1,1,0,1,0,0,0,1,0] => 1110100010 => 1001111000 => 2

[1,1,1,0,1,0,0,1,0,0] => 1110100100 => 0101111000 => 2

[1,1,1,0,1,0,1,0,0,0] => 1110101000 => 0011111000 => 1

[1,1,1,0,1,1,0,0,0,0] => 1110110000 => 0001111010 => 2

[1,1,1,1,0,0,0,0,1,0] => 1111000010 => 1000111100 => 2

[1,1,1,1,0,0,0,1,0,0] => 1111000100 => 0100111100 => 2

[1,1,1,1,0,0,1,0,0,0] => 1111001000 => 0010111100 => 2

[1,1,1,1,0,1,0,0,0,0] => 1111010000 => 0001111100 => 1

[1,1,1,1,1,0,0,0,0,0] => 1111100000 => 0000111110 => 1

[1,0,1,0,1,0,1,0,1,0,1,0] => 101010101010 => 111111000000 => 1

[1,0,1,0,1,0,1,0,1,1,0,0] => 101010101100 => 011111000010 => 2

[1,0,1,0,1,0,1,1,0,0,1,0] => 101010110010 => 101111000100 => 3

[1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 011111000100 => 2

[1,0,1,0,1,0,1,1,1,0,0,0] => 101010111000 => 001111000110 => 2

[1,0,1,0,1,1,0,0,1,0,1,0] => 101011001010 => 110111001000 => 3

[1,0,1,0,1,1,0,0,1,1,0,0] => 101011001100 => 010111001010 => 4

[1,0,1,0,1,1,0,1,0,0,1,0] => 101011010010 => 101111001000 => 3

[1,0,1,0,1,1,0,1,0,1,0,0] => 101011010100 => 011111001000 => 2

[1,0,1,0,1,1,0,1,1,0,0,0] => 101011011000 => 001111001010 => 3

[1,0,1,0,1,1,1,0,0,0,1,0] => 101011100010 => 100111001100 => 3

[1,0,1,0,1,1,1,0,0,1,0,0] => 101011100100 => 010111001100 => 3

[1,0,1,0,1,1,1,0,1,0,0,0] => 101011101000 => 001111001100 => 2

[1,0,1,0,1,1,1,1,0,0,0,0] => 101011110000 => 000111001110 => 2

[1,0,1,1,0,0,1,0,1,0,1,0] => 101100101010 => 111011010000 => 3

[1,0,1,1,0,0,1,0,1,1,0,0] => 101100101100 => 011011010010 => 4

[1,0,1,1,0,0,1,1,0,0,1,0] => 101100110010 => 101011010100 => 5

[1,0,1,1,0,0,1,1,0,1,0,0] => 101100110100 => 011011010100 => 4

[1,0,1,1,0,0,1,1,1,0,0,0] => 101100111000 => 001011010110 => 4

[1,0,1,1,0,1,0,0,1,0,1,0] => 101101001010 => 110111010000 => 3

[1,0,1,1,0,1,0,0,1,1,0,0] => 101101001100 => 010111010010 => 4

[1,0,1,1,0,1,0,1,0,0,1,0] => 101101010010 => 101111010000 => 3

[1,0,1,1,0,1,0,1,0,1,0,0] => 101101010100 => 011111010000 => 2

[1,0,1,1,0,1,0,1,1,0,0,0] => 101101011000 => 001111010010 => 3

[1,0,1,1,0,1,1,0,0,0,1,0] => 101101100010 => 100111010100 => 4

[1,0,1,1,0,1,1,0,0,1,0,0] => 101101100100 => 010111010100 => 4

[1,0,1,1,0,1,1,0,1,0,0,0] => 101101101000 => 001111010100 => 3

[1,0,1,1,0,1,1,1,0,0,0,0] => 101101110000 => 000111010110 => 3

[1,0,1,1,1,0,0,0,1,0,1,0] => 101110001010 => 110011011000 => 3

[1,0,1,1,1,0,0,0,1,1,0,0] => 101110001100 => 010011011010 => 4

[1,0,1,1,1,0,0,1,0,0,1,0] => 101110010010 => 101011011000 => 4

[1,0,1,1,1,0,0,1,0,1,0,0] => 101110010100 => 011011011000 => 3

[1,0,1,1,1,0,0,1,1,0,0,0] => 101110011000 => 001011011010 => 4

[1,0,1,1,1,0,1,0,0,0,1,0] => 101110100010 => 100111011000 => 3

[1,0,1,1,1,0,1,0,0,1,0,0] => 101110100100 => 010111011000 => 3

[1,0,1,1,1,0,1,0,1,0,0,0] => 101110101000 => 001111011000 => 2

[1,0,1,1,1,0,1,1,0,0,0,0] => 101110110000 => 000111011010 => 3

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$F_{1} = q$

$F_{2} = 2\ q$

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

$F_{4} = 4\ q + 8\ q^{2} + 2\ q^{3}$

$F_{5} = 5\ q + 20\ q^{2} + 15\ q^{3} + 2\ q^{4}$

$F_{6} = 6\ q + 40\ q^{2} + 60\ q^{3} + 24\ q^{4} + 2\ q^{5}$

Description

The number of descents 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.

$\phi(4154223) = 1254423$ .

Map

to binary word

Description

Return the Dyck word as binary word.