- FindStat

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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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 2 + q^{2}$

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

$w_i = w_{n+1-i}$ .

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

prime Dyck path

Description

Return the Dyck path obtained by adding an initial up and a final down step.