- FindStat

Identifier

Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000223: Permutations ⟶ ℤ

Values

[1] => [[1]] => [1] => [1] => 0
[2] => [[1,2]] => [1,2] => [1,2] => 0
[1,1] => [[1],[2]] => [2,1] => [1,2] => 0
[3] => [[1,2,3]] => [1,2,3] => [1,2,3] => 0
[2,1] => [[1,3],[2]] => [2,1,3] => [1,3,2] => 0
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [1,2,3] => 0
[4] => [[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 0
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => [1,3,4,2] => 0
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [1,2,3,4] => 0
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => [1,4,2,3] => 0
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => 0
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [1,3,4,5,2] => 0
[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [1,2,5,3,4] => 0
[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [1,4,5,2,3] => 0
[2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [1,3,2,5,4] => 0
[2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [1,5,2,3,4] => 0
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [1,3,4,5,6,2] => 0
[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [1,2,5,6,3,4] => 0
[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [1,4,5,6,2,3] => 0
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [1,2,3,4,5,6] => 0
[3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [1,3,6,2,5,4] => 1
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [1,5,6,2,3,4] => 0
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [1,2,3,4,5,6] => 0
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [1,4,2,6,3,5] => 0
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 0
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [4,3,2,1,5,6,7] => [1,5,6,7,2,3,4] => 0
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [5,4,3,2,1,6,7] => [1,6,7,2,3,4,5] => 0
[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [6,4,3,2,7,1,5] => [1,5,2,7,3,4,6] => 0
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7] => [1,7,2,3,4,5,6] => 0
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0
[8] => [[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => [1,2,3,4,5,6,7,8] => 0
[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8] => [1,6,7,8,2,3,4,5] => 0
[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8] => [1,7,8,2,3,4,5,6] => 0
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8] => 0
[2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => [7,5,3,8,2,6,1,4] => [1,4,2,6,3,8,5,7] => 0
[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => [7,5,4,3,2,8,1,6] => [1,6,2,8,3,4,5,7] => 0
[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8] => [1,8,2,3,4,5,6,7] => 0
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 0
[9] => [[1,2,3,4,5,6,7,8,9]] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 0
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [7,8,9,4,5,6,1,2,3] => [1,2,3,4,5,6,7,8,9] => 0
[3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8,9] => [1,8,9,2,3,4,5,6,7] => 0
[2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1,9] => [1,9,2,3,4,5,6,7,8] => 0
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => 0
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 0
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [6,7,8,9,10,1,2,3,4,5] => [1,2,3,4,5,6,7,8,9,10] => 0
[3,1,1,1,1,1,1,1] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1,9,10] => [1,9,10,2,3,4,5,6,7,8] => 0
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [9,10,7,8,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8,9,10] => 0
[2,1,1,1,1,1,1,1,1] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1,10] => [1,10,2,3,4,5,6,7,8,9] => 0
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 3$

$F_{4} = 5$

$F_{5} = 7$

$F_{6} = 10 + q$

Description

The number of nestings in the permutation.

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

reading tableau

Description

Return the RSK recording tableau of the reading word of the (standard) tableau

$T$ labeled down (in English convention) each column to the shape of a partition.

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.