- FindStat

Identifier

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000109: Permutations ⟶ ℤ

Values

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

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$F_{1} = q$

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

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

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

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

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

Description

The number of elements less than or equal to the given element in Bruhat order.

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation

$\sigma$ of length

$n$ is the permutation

$\tau$ with

$\tau(i) = n+1-\sigma(i)$

Map

initial tableau

Description

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

$1$ through

$n$ row by row.

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.