- FindStat

Identifier

Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001697: Standard tableaux ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + 2\ q^{2} + q^{3}$

$F_{4} = 1 + 3\ q + 5\ q^{2} + 6\ q^{3} + 5\ q^{4} + 3\ q^{5} + q^{6}$

$F_{5} = 1 + 4\ q + 9\ q^{2} + 15\ q^{3} + 20\ q^{4} + 22\ q^{5} + 20\ q^{6} + 15\ q^{7} + 9\ q^{8} + 4\ q^{9} + q^{10}$

$F_{6} = 1 + 5\ q + 14\ q^{2} + 29\ q^{3} + 49\ q^{4} + 71\ q^{5} + 90\ q^{6} + 101\ q^{7} + 101\ q^{8} + 90\ q^{9} + 71\ q^{10} + 49\ q^{11} + 29\ q^{12} + 14\ q^{13} + 5\ q^{14} + q^{15}$

$F_{7} = 1 + 6\ q + 20\ q^{2} + 49\ q^{3} + 98\ q^{4} + 169\ q^{5} + 259\ q^{6} + 359\ q^{7} + 455\ q^{8} + 531\ q^{9} + 573\ q^{10} + 573\ q^{11} + 531\ q^{12} + 455\ q^{13} + 359\ q^{14} + 259\ q^{15} + 169\ q^{16} + 98\ q^{17} + 49\ q^{18} + 20\ q^{19} + 6\ q^{20} + q^{21}$

Description

The shifted natural comajor index of a standard Young tableau.
A natural descent of a standard tableau

$T$ is an entry

$i$ such that

$i+1$ appears in a higher row than

$i$ in English notation.
The natural comajor index of a tableau of shape

$\lambda$ , size

$n$ with natural descent set

$D$ is then

$b(\lambda)+\sum_{d\in D} n-d$ , where

$b(\lambda) = \sum_i (i-1)\lambda_i$ .

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.