- FindStat

Identifier

Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000702: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 4

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

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

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

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

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

$F_{3} = 2\ q + 8\ q^{2} + 6\ q^{3}$

$F_{4} = 6\ q + 37\ q^{2} + 58\ q^{3} + 24\ q^{4}$

$F_{5} = 24\ q + 208\ q^{2} + 511\ q^{3} + 433\ q^{4} + 120\ q^{5}$

$F_{6} = 120\ q + 1368\ q^{2} + 4666\ q^{3} + 6483\ q^{4} + 3450\ q^{5} + 720\ q^{6}$

Description

The number of weak deficiencies of a permutation.
This is defined as

$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$
The number of weak exceedances is St000213The number of weak exceedances (also weak excedences) of a permutation., the number of deficiencies is St000703The number of deficiencies of a permutation..

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.

Map

inverse Foata bijection

Description

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