- FindStat

Identifier

Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000391: Binary words ⟶ ℤ

Values

[1,0] => 10 => 00 => 00 => 0
[1,0,1,0] => 1010 => 0000 => 0000 => 0
[1,1,0,0] => 1100 => 0101 => 1010 => 4
[1,0,1,0,1,0] => 101010 => 000000 => 000000 => 0
[1,0,1,1,0,0] => 101100 => 010100 => 001010 => 8
[1,1,0,0,1,0] => 110010 => 000101 => 101000 => 4
[1,1,0,1,0,0] => 110100 => 010001 => 100010 => 6
[1,1,1,0,0,0] => 111000 => 011011 => 110110 => 12
[1,0,1,0,1,0,1,0] => 10101010 => 00000000 => 00000000 => 0
[1,0,1,0,1,1,0,0] => 10101100 => 01010000 => 00001010 => 12
[1,0,1,1,0,0,1,0] => 10110010 => 00010100 => 00101000 => 8
[1,0,1,1,0,1,0,0] => 10110100 => 01000100 => 00100010 => 10
[1,0,1,1,1,0,0,0] => 10111000 => 01101100 => 00110110 => 20
[1,1,0,0,1,0,1,0] => 11001010 => 00000101 => 10100000 => 4
[1,1,0,0,1,1,0,0] => 11001100 => 01010101 => 10101010 => 16
[1,1,0,1,0,0,1,0] => 11010010 => 00010001 => 10001000 => 6
[1,1,0,1,0,1,0,0] => 11010100 => 01000001 => 10000010 => 8
[1,1,0,1,1,0,0,0] => 11011000 => 01101001 => 10010110 => 18
[1,1,1,0,0,0,1,0] => 11100010 => 00011011 => 11011000 => 12
[1,1,1,0,0,1,0,0] => 11100100 => 01001011 => 11010010 => 14
[1,1,1,0,1,0,0,0] => 11101000 => 01100011 => 11000110 => 16
[1,1,1,1,0,0,0,0] => 11110000 => 01110111 => 11101110 => 24
[1,0,1,0,1,0,1,0,1,0] => 1010101010 => 0000000000 => 0000000000 => 0
[1,0,1,0,1,0,1,1,0,0] => 1010101100 => 0101000000 => 0000001010 => 16
[1,0,1,0,1,1,0,0,1,0] => 1010110010 => 0001010000 => 0000101000 => 12
[1,0,1,0,1,1,0,1,0,0] => 1010110100 => 0100010000 => 0000100010 => 14
[1,0,1,0,1,1,1,0,0,0] => 1010111000 => 0110110000 => 0000110110 => 28
[1,0,1,1,0,0,1,0,1,0] => 1011001010 => 0000010100 => 0010100000 => 8
[1,0,1,1,0,0,1,1,0,0] => 1011001100 => 0101010100 => 0010101010 => 24
[1,0,1,1,0,1,0,0,1,0] => 1011010010 => 0001000100 => 0010001000 => 10
[1,0,1,1,0,1,0,1,0,0] => 1011010100 => 0100000100 => 0010000010 => 12
[1,0,1,1,0,1,1,0,0,0] => 1011011000 => 0110100100 => 0010010110 => 26

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$F_{1} = 1$

$F_{2} = 1 + q^{4}$

$F_{3} = 1 + q^{4} + q^{6} + q^{8} + q^{12}$

$F_{4} = 1 + q^{4} + q^{6} + 2\ q^{8} + q^{10} + 2\ q^{12} + q^{14} + 2\ q^{16} + q^{18} + q^{20} + q^{24}$

Description

The sum of the positions of the ones in a binary word.

Map

to binary word

Description

Return the Dyck word as binary word.

Map

flag zeros to zeros

Description

Return a binary word of the same length, such that the number of zeros equals the number of occurrences of

$10$ in the word obtained from the original word by prepending the reverse of the complement.
For example, the image of the word

$w=1\dots 1$ is

$1\dots 1$ , because

$0\dots 01\dots 1$ has no occurrences of

$10$ . The words

$10\dots 10$ and

$010\dots 10$ have image

$0\dots 0$ .

Map

reverse

Description

Return the reversal of a binary word.