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Identifier

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000109: Permutations ⟶ ℤ

Values

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

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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

initial tableau

Description

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

$1$ through

$n$ row by row.

Map

major-index to inversion-number bijection

Description

Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.

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.