Identifier
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Images
=>
Cc0005;cc-rep-0
[1,0]=>10=>[1,2]=>[2,1] [1,0,1,0]=>1010=>[1,2,2]=>[2,2,1] [1,1,0,0]=>1100=>[1,1,3]=>[1,3,1] [1,0,1,0,1,0]=>101010=>[1,2,2,2]=>[2,2,2,1] [1,0,1,1,0,0]=>101100=>[1,2,1,3]=>[2,1,3,1] [1,1,0,0,1,0]=>110010=>[1,1,3,2]=>[1,3,2,1] [1,1,0,1,0,0]=>110100=>[1,1,2,3]=>[1,2,3,1] [1,1,1,0,0,0]=>111000=>[1,1,1,4]=>[1,1,4,1] [1,0,1,0,1,0,1,0]=>10101010=>[1,2,2,2,2]=>[2,2,2,2,1] [1,0,1,0,1,1,0,0]=>10101100=>[1,2,2,1,3]=>[2,2,1,3,1] [1,0,1,1,0,0,1,0]=>10110010=>[1,2,1,3,2]=>[2,1,3,2,1] [1,0,1,1,0,1,0,0]=>10110100=>[1,2,1,2,3]=>[2,1,2,3,1] [1,0,1,1,1,0,0,0]=>10111000=>[1,2,1,1,4]=>[2,1,1,4,1] [1,1,0,0,1,0,1,0]=>11001010=>[1,1,3,2,2]=>[1,3,2,2,1] [1,1,0,0,1,1,0,0]=>11001100=>[1,1,3,1,3]=>[1,3,1,3,1] [1,1,0,1,0,0,1,0]=>11010010=>[1,1,2,3,2]=>[1,2,3,2,1] [1,1,0,1,0,1,0,0]=>11010100=>[1,1,2,2,3]=>[1,2,2,3,1] [1,1,0,1,1,0,0,0]=>11011000=>[1,1,2,1,4]=>[1,2,1,4,1] [1,1,1,0,0,0,1,0]=>11100010=>[1,1,1,4,2]=>[1,1,4,2,1] [1,1,1,0,0,1,0,0]=>11100100=>[1,1,1,3,3]=>[1,1,3,3,1] [1,1,1,0,1,0,0,0]=>11101000=>[1,1,1,2,4]=>[1,1,2,4,1] [1,1,1,1,0,0,0,0]=>11110000=>[1,1,1,1,5]=>[1,1,1,5,1]
Map
to binary word
Description
Return the Dyck word as binary word.
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.
Map
rotate front to back
Description
The front to back rotation of the entries of an integer composition.