Identifier
Values
[1,0] => 10 => 11 => 00 => 0
[1,0,1,0] => 1010 => 1101 => 0010 => 1
[1,1,0,0] => 1100 => 1101 => 0010 => 1
[1,0,1,0,1,0] => 101010 => 110101 => 001010 => 1
[1,0,1,1,0,0] => 101100 => 110101 => 001010 => 1
[1,1,0,0,1,0] => 110010 => 110101 => 001010 => 1
[1,1,0,1,0,0] => 110100 => 111001 => 000110 => 2
[1,1,1,0,0,0] => 111000 => 111001 => 000110 => 2
[1,0,1,0,1,0,1,0] => 10101010 => 11010101 => 00101010 => 1
[1,0,1,0,1,1,0,0] => 10101100 => 11010101 => 00101010 => 1
[1,0,1,1,0,0,1,0] => 10110010 => 11010101 => 00101010 => 1
[1,0,1,1,0,1,0,0] => 10110100 => 11011001 => 00100110 => 2
[1,0,1,1,1,0,0,0] => 10111000 => 11011001 => 00100110 => 2
[1,1,0,0,1,0,1,0] => 11001010 => 11010101 => 00101010 => 1
[1,1,0,0,1,1,0,0] => 11001100 => 11010101 => 00101010 => 1
[1,1,0,1,0,0,1,0] => 11010010 => 11100101 => 00011010 => 2
[1,1,0,1,0,1,0,0] => 11010100 => 11101001 => 00010110 => 2
[1,1,0,1,1,0,0,0] => 11011000 => 11101001 => 00010110 => 2
[1,1,1,0,0,0,1,0] => 11100010 => 11100101 => 00011010 => 2
[1,1,1,0,0,1,0,0] => 11100100 => 11101001 => 00010110 => 2
[1,1,1,0,1,0,0,0] => 11101000 => 11110001 => 00001110 => 3
[1,1,1,1,0,0,0,0] => 11110000 => 11110001 => 00001110 => 3
[1,0,1,0,1,0,1,0,1,0] => 1010101010 => 1101010101 => 0010101010 => 1
[1,0,1,0,1,0,1,1,0,0] => 1010101100 => 1101010101 => 0010101010 => 1
[1,0,1,0,1,1,0,0,1,0] => 1010110010 => 1101010101 => 0010101010 => 1
[1,0,1,0,1,1,0,1,0,0] => 1010110100 => 1101011001 => 0010100110 => 2
[1,0,1,0,1,1,1,0,0,0] => 1010111000 => 1101011001 => 0010100110 => 2
[1,0,1,1,0,0,1,0,1,0] => 1011001010 => 1101010101 => 0010101010 => 1
[1,0,1,1,0,0,1,1,0,0] => 1011001100 => 1101010101 => 0010101010 => 1
[1,0,1,1,0,1,0,0,1,0] => 1011010010 => 1101100101 => 0010011010 => 2
[1,0,1,1,0,1,0,1,0,0] => 1011010100 => 1101101001 => 0010010110 => 2
[1,0,1,1,0,1,1,0,0,0] => 1011011000 => 1101101001 => 0010010110 => 2
[1,0,1,1,1,0,0,0,1,0] => 1011100010 => 1101100101 => 0010011010 => 2
[1,0,1,1,1,0,0,1,0,0] => 1011100100 => 1101101001 => 0010010110 => 2
[1,0,1,1,1,0,1,0,0,0] => 1011101000 => 1101110001 => 0010001110 => 3
[1,0,1,1,1,1,0,0,0,0] => 1011110000 => 1101110001 => 0010001110 => 3
[1,1,0,0,1,0,1,0,1,0] => 1100101010 => 1101010101 => 0010101010 => 1
[1,1,0,0,1,0,1,1,0,0] => 1100101100 => 1101010101 => 0010101010 => 1
[1,1,0,0,1,1,0,0,1,0] => 1100110010 => 1101010101 => 0010101010 => 1
[1,1,0,0,1,1,0,1,0,0] => 1100110100 => 1101011001 => 0010100110 => 2
[1,1,0,0,1,1,1,0,0,0] => 1100111000 => 1101011001 => 0010100110 => 2
[1,1,0,1,0,0,1,0,1,0] => 1101001010 => 1110010101 => 0001101010 => 2
[1,1,0,1,0,0,1,1,0,0] => 1101001100 => 1110010101 => 0001101010 => 2
[1,1,0,1,0,1,0,0,1,0] => 1101010010 => 1110100101 => 0001011010 => 2
[1,1,0,1,0,1,0,1,0,0] => 1101010100 => 1110101001 => 0001010110 => 2
[1,1,0,1,0,1,1,0,0,0] => 1101011000 => 1110101001 => 0001010110 => 2
[1,1,0,1,1,0,0,0,1,0] => 1101100010 => 1110100101 => 0001011010 => 2
[1,1,0,1,1,0,0,1,0,0] => 1101100100 => 1110101001 => 0001010110 => 2
[1,1,0,1,1,0,1,0,0,0] => 1101101000 => 1110110001 => 0001001110 => 3
[1,1,0,1,1,1,0,0,0,0] => 1101110000 => 1110110001 => 0001001110 => 3
[1,1,1,0,0,0,1,0,1,0] => 1110001010 => 1110010101 => 0001101010 => 2
[1,1,1,0,0,0,1,1,0,0] => 1110001100 => 1110010101 => 0001101010 => 2
[1,1,1,0,0,1,0,0,1,0] => 1110010010 => 1110100101 => 0001011010 => 2
[1,1,1,0,0,1,0,1,0,0] => 1110010100 => 1110101001 => 0001010110 => 2
[1,1,1,0,0,1,1,0,0,0] => 1110011000 => 1110101001 => 0001010110 => 2
[1,1,1,0,1,0,0,0,1,0] => 1110100010 => 1111000101 => 0000111010 => 3
[1,1,1,0,1,0,0,1,0,0] => 1110100100 => 1111001001 => 0000110110 => 3
[1,1,1,0,1,0,1,0,0,0] => 1110101000 => 1111010001 => 0000101110 => 3
[1,1,1,0,1,1,0,0,0,0] => 1110110000 => 1111010001 => 0000101110 => 3
[1,1,1,1,0,0,0,0,1,0] => 1111000010 => 1111000101 => 0000111010 => 3
[1,1,1,1,0,0,0,1,0,0] => 1111000100 => 1111001001 => 0000110110 => 3
[1,1,1,1,0,0,1,0,0,0] => 1111001000 => 1111010001 => 0000101110 => 3
[1,1,1,1,0,1,0,0,0,0] => 1111010000 => 1111100001 => 0000011110 => 4
[1,1,1,1,1,0,0,0,0,0] => 1111100000 => 1111100001 => 0000011110 => 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       
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 $q$.
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
complement
Description
Send a binary word to the word obtained by interchanging the two letters.