Identifier
Values
[1,0] => 10 => 11 => [1,1,1] => 1
[1,0,1,0] => 1010 => 1101 => [1,1,2,1] => 2
[1,1,0,0] => 1100 => 1101 => [1,1,2,1] => 2
[1,0,1,0,1,0] => 101010 => 110101 => [1,1,2,2,1] => 2
[1,0,1,1,0,0] => 101100 => 110101 => [1,1,2,2,1] => 2
[1,1,0,0,1,0] => 110010 => 110101 => [1,1,2,2,1] => 2
[1,1,0,1,0,0] => 110100 => 111001 => [1,1,1,3,1] => 3
[1,1,1,0,0,0] => 111000 => 111001 => [1,1,1,3,1] => 3
[1,0,1,0,1,0,1,0] => 10101010 => 11010101 => [1,1,2,2,2,1] => 2
[1,0,1,0,1,1,0,0] => 10101100 => 11010101 => [1,1,2,2,2,1] => 2
[1,0,1,1,0,0,1,0] => 10110010 => 11010101 => [1,1,2,2,2,1] => 2
[1,0,1,1,0,1,0,0] => 10110100 => 11011001 => [1,1,2,1,3,1] => 3
[1,0,1,1,1,0,0,0] => 10111000 => 11011001 => [1,1,2,1,3,1] => 3
[1,1,0,0,1,0,1,0] => 11001010 => 11010101 => [1,1,2,2,2,1] => 2
[1,1,0,0,1,1,0,0] => 11001100 => 11010101 => [1,1,2,2,2,1] => 2
[1,1,0,1,0,0,1,0] => 11010010 => 11100101 => [1,1,1,3,2,1] => 3
[1,1,0,1,0,1,0,0] => 11010100 => 11101001 => [1,1,1,2,3,1] => 3
[1,1,0,1,1,0,0,0] => 11011000 => 11101001 => [1,1,1,2,3,1] => 3
[1,1,1,0,0,0,1,0] => 11100010 => 11100101 => [1,1,1,3,2,1] => 3
[1,1,1,0,0,1,0,0] => 11100100 => 11101001 => [1,1,1,2,3,1] => 3
[1,1,1,0,1,0,0,0] => 11101000 => 11110001 => [1,1,1,1,4,1] => 4
[1,1,1,1,0,0,0,0] => 11110000 => 11110001 => [1,1,1,1,4,1] => 4
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Description
The largest part of an integer composition.
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
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.