- FindStat

Identifier

Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001811: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 3$

$F_{3} = 14 + 2\ q$

$F_{4} = 88 + 26\ q + 11\ q^{2}$

$F_{5} = 680 + 307\ q + 200\ q^{2} + 92\ q^{3} + 17\ q^{4}$

Description

The Castelnuovo-Mumford regularity of a permutation.
The Castelnuovo-Mumford regularity of a permutation

$\sigma$ is the Castelnuovo-Mumford regularity of the matrix Schubert variety

$X_\sigma$ .
Equivalently, it is the difference between the degrees of the Grothendieck polynomial and the Schubert polynomial for

$\sigma$ . It can be computed by subtracting the Coxeter length St000018The number of inversions of a permutation. from the Rajchgot index St001759The Rajchgot index of a permutation..

Map

inverse Foata bijection

Description

The inverse of Foata's bijection.
See Mp00067Foata bijection.

Map

to car permutation

Description

Maps a parking function to the permutation of cars that correspond to that parking function.
For example, the image

$[2, 4, 5, 6, 3, 1, 7]$ means that car 2 takes spots 1, car 4 takes spot 2, and so on.