Identifier
Values
[1] => [1,0] => 10 => [1,2] => 1
[2] => [1,0,1,0] => 1010 => [1,2,2] => 1
[1,1] => [1,1,0,0] => 1100 => [1,1,3] => 1
[3] => [1,0,1,0,1,0] => 101010 => [1,2,2,2] => 1
[2,1] => [1,0,1,1,0,0] => 101100 => [1,2,1,3] => 2
[1,1,1] => [1,1,0,1,0,0] => 110100 => [1,1,2,3] => 2
[4] => [1,0,1,0,1,0,1,0] => 10101010 => [1,2,2,2,2] => 1
[3,1] => [1,0,1,0,1,1,0,0] => 10101100 => [1,2,2,1,3] => 2
[2,2] => [1,1,1,0,0,0] => 111000 => [1,1,1,4] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => [1,2,1,2,3] => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => 11010100 => [1,1,2,2,3] => 2
[3,2] => [1,0,1,1,1,0,0,0] => 10111000 => [1,2,1,1,4] => 2
[2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => [1,1,1,3,3] => 2
[3,3] => [1,1,1,0,1,0,0,0] => 11101000 => [1,1,1,2,4] => 2
[2,2,2] => [1,1,1,1,0,0,0,0] => 11110000 => [1,1,1,1,5] => 1
[] => [] => => [1] => 0
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Description
The degree of asymmetry of an integer composition.
This is the number of pairs of symmetrically positioned distinct entries.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
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.