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Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001937: Parking functions ⟶ ℤ

Values

[1,0] => [2,1] => [2,1] => [2,1] => 1
[1,0,1,0] => [3,1,2] => [1,3,2] => [1,3,2] => 2
[1,1,0,0] => [2,3,1] => [3,1,2] => [3,1,2] => 2
[1,0,1,0,1,0] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 3
[1,0,1,1,0,0] => [3,1,4,2] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,0,1,0] => [2,4,1,3] => [1,3,4,2] => [1,3,4,2] => 2
[1,1,0,1,0,0] => [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 2
[1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 3
[] => [1] => [1] => [1] => 1

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$F_{1} = q$

$F_{2} = 2\ q^{2}$

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

Description

The size of the center of a parking function.
The center of a parking function

$p_1,\dots,p_n$ is the longest subsequence

$a_1,\dots,a_k$ such that

$a_i\leq i$ .

Map

major-index to inversion-number bijection

Description

Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.

Map

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.

Map

parking function

Description

Interpret the permutation as a parking function.