- FindStat

Identifier

Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000862: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 1

[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 1

[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 2

[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 1

[2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 2

[2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 1

[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] => 1

[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => 1

[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => 2

[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => 1

[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 2

[3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => 2

[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => 1

[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => 2

[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => 2

[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => 1

[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => 1

[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => 1

[6,1] => [[1,3,4,5,6,7],[2]] => [2,1,3,4,5,6,7] => 1

[5,2] => [[1,2,5,6,7],[3,4]] => [3,4,1,2,5,6,7] => 2

[5,1,1] => [[1,4,5,6,7],[2],[3]] => [3,2,1,4,5,6,7] => 1

[4,3] => [[1,2,3,7],[4,5,6]] => [4,5,6,1,2,3,7] => 2

[4,2,1] => [[1,3,6,7],[2,5],[4]] => [4,2,5,1,3,6,7] => 2

[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [4,3,2,1,5,6,7] => 1

[3,3,1] => [[1,3,4],[2,6,7],[5]] => [5,2,6,7,1,3,4] => 2

[3,2,2] => [[1,2,7],[3,4],[5,6]] => [5,6,3,4,1,2,7] => 2

[3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [5,3,2,6,1,4,7] => 2

[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [5,4,3,2,1,6,7] => 1

[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [6,4,7,2,5,1,3] => 2

[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [6,4,3,2,7,1,5] => 2

[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7] => 1

[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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

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

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

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

[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

[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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 3\ q$

$F_{4} = 4\ q + q^{2}$

$F_{5} = 5\ q + 2\ q^{2}$

$F_{6} = 6\ q + 5\ q^{2}$

$F_{7} = 7\ q + 8\ q^{2}$

$F_{8} = 8\ q + 14\ q^{2}$

$F_{9} = 9\ q + 20\ q^{2} + q^{3}$

$F_{10} = 10\ q + 30\ q^{2} + 2\ q^{3}$

Description

The number of parts of the shifted shape of a permutation.
The diagram of a strict partition

$\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of

$n$ is a tableau with

$\ell$ rows, the

$i$ -th row being indented by

$i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair

$(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in

$Q$ may be circled.
This statistic records the number of parts of the shifted shape.

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.