- FindStat

Identifier

Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001214: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,3,4] => [1,1,3,4] => [4,3,1,1] => 5

[1,1,4,3] => [1,1,4,3] => [4,3,1,1] => 5

[1,3,1,4] => [1,3,1,4] => [4,3,1,1] => 5

[1,3,4,1] => [1,3,4,1] => [4,3,1,1] => 5

[1,4,1,3] => [1,4,1,3] => [4,3,1,1] => 5

[1,4,3,1] => [1,4,3,1] => [4,3,1,1] => 5

[3,1,1,4] => [3,1,1,4] => [4,3,1,1] => 5

[3,1,4,1] => [3,1,4,1] => [4,3,1,1] => 5

[3,4,1,1] => [3,4,1,1] => [4,3,1,1] => 5

[4,1,1,3] => [4,1,1,3] => [4,3,1,1] => 5

[4,1,3,1] => [4,1,3,1] => [4,3,1,1] => 5

[4,3,1,1] => [4,3,1,1] => [4,3,1,1] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,2,4] => [1,2,2,4] => [4,2,2,1] => 5

[1,2,4,2] => [1,2,4,2] => [4,2,2,1] => 5

[1,4,2,2] => [1,4,2,2] => [4,2,2,1] => 5

[2,1,2,4] => [2,1,2,4] => [4,2,2,1] => 5

>>> Load all 704 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + 2\ q$

$F_{3} = 1 + 3\ q + 6\ q^{2} + 6\ q^{3}$

$F_{4} = 1 + 4\ q + 10\ q^{2} + 20\ q^{3} + 30\ q^{4} + 36\ q^{5} + 24\ q^{6}$

Description

The aft of an integer partition.
The aft is the size of the partition minus the length of the first row or column, whichever is larger.
See also St000784The maximum of the length and the largest part of the integer partition..

Map

to composition

Description

Return the parking function interpreted as an integer composition.

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.