Identifier
-
Mp00230:
Integer partitions
—parallelogram polyomino⟶
Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001524: Binary words ⟶ ℤ
Values
[1] => [1,0] => [1,1,0,0] => 1100 => 0
[2] => [1,0,1,0] => [1,1,0,1,0,0] => 110100 => 0
[1,1] => [1,1,0,0] => [1,1,1,0,0,0] => 111000 => 0
[3] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 11010100 => 0
[2,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 2
[1,1,1] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => 0
[2,2] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => 11110000 => 0
[] => [] => [1,0] => 10 => 0
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Description
The degree of symmetry of a binary word.
For a binary word $w$ of length $n$, this is the number of positions $i\leq n/2$ such that $w_i = w_{n+1-i}$.
For a binary word $w$ of length $n$, this is the number of positions $i\leq n/2$ such that $w_i = w_{n+1-i}$.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
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.
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 binary word
Description
Return the Dyck word as binary word.
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