- FindStat

Identifier

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001346: Permutations ⟶ ℤ

Values

[2] => [[1,2]] => [1,2] => 2
[1,1] => [[1],[2]] => [2,1] => 1
[3] => [[1,2,3]] => [1,2,3] => 6
[2,1] => [[1,2],[3]] => [3,1,2] => 2
[1,1,1] => [[1],[2],[3]] => [3,2,1] => 1
[4] => [[1,2,3,4]] => [1,2,3,4] => 24
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => 6
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => 4
[2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => 2
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 1
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 120
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 24
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 12
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 6
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 4
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 2
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 1
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 720
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => 120
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => 48
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 24
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 36
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 12
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => 6
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => 8
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => 4
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => 2
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => 1

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

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

$F_{4} = q + q^{2} + q^{4} + q^{6} + q^{24}$

$F_{5} = q + q^{2} + q^{4} + q^{6} + q^{12} + q^{24} + q^{120}$

$F_{6} = q + q^{2} + q^{4} + q^{6} + q^{8} + q^{12} + q^{24} + q^{36} + q^{48} + q^{120} + q^{720}$

Description

The number of parking functions that give the same permutation.
A parking function

$(a_1,\dots,a_n)$ is a list of preferred parking spots of

$n$ cars entering a one-way street. Once the cars have parked, the order of the cars gives a permutation of

$\{1,\dots,n\}$ . This statistic records the number of parking functions that yield the same permutation of cars.

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.

Map

initial tableau

Description

Sends an integer partition to the standard tableau obtained by filling the numbers

$1$ through

$n$ row by row.