Identifier
-
Mp00093:
Dyck paths
—to binary word⟶
Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000628: Binary words ⟶ ℤ
Values
[1,0] => 10 => 11 => 0
[1,0,1,0] => 1010 => 1101 => 1
[1,1,0,0] => 1100 => 1101 => 1
[1,0,1,0,1,0] => 101010 => 110101 => 1
[1,0,1,1,0,0] => 101100 => 110101 => 1
[1,1,0,0,1,0] => 110010 => 110101 => 1
[1,1,0,1,0,0] => 110100 => 111001 => 2
[1,1,1,0,0,0] => 111000 => 111001 => 2
[1,0,1,0,1,0,1,0] => 10101010 => 11010101 => 1
[1,0,1,0,1,1,0,0] => 10101100 => 11010101 => 1
[1,0,1,1,0,0,1,0] => 10110010 => 11010101 => 1
[1,0,1,1,0,1,0,0] => 10110100 => 11011001 => 2
[1,0,1,1,1,0,0,0] => 10111000 => 11011001 => 2
[1,1,0,0,1,0,1,0] => 11001010 => 11010101 => 1
[1,1,0,0,1,1,0,0] => 11001100 => 11010101 => 1
[1,1,0,1,0,0,1,0] => 11010010 => 11100101 => 2
[1,1,0,1,0,1,0,0] => 11010100 => 11101001 => 2
[1,1,0,1,1,0,0,0] => 11011000 => 11101001 => 2
[1,1,1,0,0,0,1,0] => 11100010 => 11100101 => 2
[1,1,1,0,0,1,0,0] => 11100100 => 11101001 => 2
[1,1,1,0,1,0,0,0] => 11101000 => 11110001 => 3
[1,1,1,1,0,0,0,0] => 11110000 => 11110001 => 3
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Description
The balance of a binary word.
The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1].
A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.
The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1].
A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.
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.
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
to binary word
Description
Return the Dyck word as binary word.
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