- FindStat

Identifier

Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤ

Values

[1] => [1] => ([],1) => 1

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

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

[2,1] => [2,1] => ([],2) => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of maximal elements of a poset.

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

permutation poset

Description

Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$