Identifier
Values
[1,0] => 10 => 11 => 11 => 0
[1,0,1,0] => 1010 => 1101 => 1101 => 1
[1,1,0,0] => 1100 => 1101 => 1101 => 1
[1,0,1,0,1,0] => 101010 => 110101 => 101101 => 1
[1,0,1,1,0,0] => 101100 => 110101 => 101101 => 1
[1,1,0,0,1,0] => 110010 => 110101 => 101101 => 1
[1,1,0,1,0,0] => 110100 => 111001 => 110101 => 2
[1,1,1,0,0,0] => 111000 => 111001 => 110101 => 2
[1,0,1,0,1,0,1,0] => 10101010 => 11010101 => 01101101 => 1
[1,0,1,0,1,1,0,0] => 10101100 => 11010101 => 01101101 => 1
[1,0,1,1,0,0,1,0] => 10110010 => 11010101 => 01101101 => 1
[1,0,1,1,0,1,0,0] => 10110100 => 11011001 => 01110101 => 2
[1,0,1,1,1,0,0,0] => 10111000 => 11011001 => 01110101 => 2
[1,1,0,0,1,0,1,0] => 11001010 => 11010101 => 01101101 => 1
[1,1,0,0,1,1,0,0] => 11001100 => 11010101 => 01101101 => 1
[1,1,0,1,0,0,1,0] => 11010010 => 11100101 => 10101101 => 2
[1,1,0,1,0,1,0,0] => 11010100 => 11101001 => 10110101 => 2
[1,1,0,1,1,0,0,0] => 11011000 => 11101001 => 10110101 => 2
[1,1,1,0,0,0,1,0] => 11100010 => 11100101 => 10101101 => 2
[1,1,1,0,0,1,0,0] => 11100100 => 11101001 => 10110101 => 2
[1,1,1,0,1,0,0,0] => 11101000 => 11110001 => 11010101 => 3
[1,1,1,1,0,0,0,0] => 11110000 => 11110001 => 11010101 => 3
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       
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the semilength of the longest Dyck word in this factorisation.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
valleys-to-peaks
Description
Return the binary word with every valley replaced by a peak.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. This map replaces every valley with a peak.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00096Foata bijection.