- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤ

Values

[[1]] => [1] => [[1]] => 0

[[1,2]] => [1,2] => [[1,2]] => 0

[[1],[2]] => [2,1] => [[1],[2]] => 0

[[1,2,3]] => [1,2,3] => [[1,2,3]] => 0

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

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

[[1],[2],[3]] => [3,2,1] => [[1],[2],[3]] => 0

[[1,2,3,4]] => [1,2,3,4] => [[1,2,3,4]] => 0

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

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

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

[[1,3],[2,4]] => [2,4,1,3] => [[1,2],[3,4]] => 0

[[1,2],[3,4]] => [3,4,1,2] => [[1,2],[3,4]] => 0

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

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

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

[[1],[2],[3],[4]] => [4,3,2,1] => [[1],[2],[3],[4]] => 0

[[1,2,3,4,5]] => [1,2,3,4,5] => [[1,2,3,4,5]] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]] => 0

[[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [[1,2,3,4,5,6]] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [[1,2,3],[4,5,6]] => 0

[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [[1,2,3],[4,5,6]] => 0

[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [[1,2,3],[4,5,6]] => 0

[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [[1,2,3],[4,5,6]] => 0

[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [[1,2,3],[4,5,6]] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [[1,2],[3,4],[5,6]] => 0

[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [[1,2],[3,4],[5,6]] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 2 + 2\ q$

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

$F_{5} = 2 + 5\ q + 5\ q^{2} + 8\ q^{3} + 6\ q^{4}$

$F_{6} = 12 + 9\ q^{2} + 35\ q^{4} + 20\ q^{6}$

$F_{7} = 2 + 14\ q + 21\ q^{2} + 28\ q^{3} + 21\ q^{4} + 12\ q^{5} + 49\ q^{6} + 35\ q^{7} + 30\ q^{8} + 20\ q^{9}$

Description

The comajor index of a standard tableau minus the weighted size of its 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

Robinson-Schensted recording tableau

Description

Sends a permutation to its Robinson-Schensted recording tableau.
The Robinson-Schensted corrspondence is a bijection between permutations of length

$n$ and pairs of standard Young tableaux of the same shape and of size

$n$ , see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to its corresponding recording tableau.