- FindStat

Identifier

Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000877: Binary words ⟶ ℤ

Values

[1,0] => 10 => 01 => 01 => 1
[1,0,1,0] => 1010 => 0101 => 1001 => 1
[1,1,0,0] => 1100 => 0011 => 0011 => 2
[1,0,1,0,1,0] => 101010 => 010101 => 011001 => 1
[1,0,1,1,0,0] => 101100 => 010011 => 010011 => 2
[1,1,0,0,1,0] => 110010 => 001101 => 110001 => 1
[1,1,0,1,0,0] => 110100 => 001011 => 100011 => 2
[1,1,1,0,0,0] => 111000 => 000111 => 000111 => 3
[1,0,1,0,1,0,1,0] => 10101010 => 01010101 => 10011001 => 1
[1,0,1,0,1,1,0,0] => 10101100 => 01010011 => 10010011 => 2
[1,0,1,1,0,0,1,0] => 10110010 => 01001101 => 00111001 => 2
[1,0,1,1,0,1,0,0] => 10110100 => 01001011 => 00110011 => 2
[1,0,1,1,1,0,0,0] => 10111000 => 01000111 => 00100111 => 3
[1,1,0,0,1,0,1,0] => 11001010 => 00110101 => 10110001 => 1
[1,1,0,0,1,1,0,0] => 11001100 => 00110011 => 10100011 => 2
[1,1,0,1,0,0,1,0] => 11010010 => 00101101 => 01110001 => 1
[1,1,0,1,0,1,0,0] => 11010100 => 00101011 => 01100011 => 2
[1,1,0,1,1,0,0,0] => 11011000 => 00100111 => 01000111 => 3
[1,1,1,0,0,0,1,0] => 11100010 => 00011101 => 11100001 => 1
[1,1,1,0,0,1,0,0] => 11100100 => 00011011 => 11000011 => 2
[1,1,1,0,1,0,0,0] => 11101000 => 00010111 => 10000111 => 3
[1,1,1,1,0,0,0,0] => 11110000 => 00001111 => 00001111 => 4

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q$

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

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

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

Description

The depth of the binary word interpreted as a path.
This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2].
The number of binary words of length

$n$ with depth

$k$ is

$\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$ , see [2].

Map

inverse Foata bijection

Description

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

Map

to binary word

Description

Return the Dyck word as binary word.

Map

complement

Description

Send a binary word to the word obtained by interchanging the two letters.