- FindStat

Identifier

Mp00052: Parking functions —to non-decreasing parking function⟶ Parking functions
Mp00301: Parking functions —leading to repeats⟶ Parking functions
St001947: Parking functions ⟶ ℤ

Values

[1] => [1] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1697 entries. <<<

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Description

The number of ties in a parking function.
This is the number of indices $i$ such that $p_i=p_{i+1}$.

Map

leading to repeats

Description

The unique parking function obtained by prepending 0 and shifting the cumulative sums of a Prüfer code, then cyclically shifting.
Pollak's map [1] sends a parking function $(p_1,\dots,p_n)$ to the Prüfer code $(c_1,\dots,c_{n-1})$ with $c_i = p_{i+1} - p_i \pmod{n+1}$.
The inverse is obtained by computing the cumulative sums, and then shifting all elements modulo $n+1$ such that the result is a parking function.
This is the inverse of Mp00298repeats to leading.
This map sends the number of occurrences of the first element to the number of repeated elements [3].

Map

to non-decreasing parking function

Description

Return the non-decreasing parking function which underlies the parking function.
Sorts the parking function into an increasing sequence.