- FindStat

Identifier

Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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]] => 1

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

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

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

[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]] => 1

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

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

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

[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]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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]] => 1

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

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

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

[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]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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]] => 1

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

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$F_{1} = q$

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

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

$F_{4} = 9 + 8\ q + 6\ q^{2} + q^{4}$

$F_{5} = 44 + 45\ q + 20\ q^{2} + 10\ q^{3} + q^{5}$

$F_{6} = 265 + 264\ q + 135\ q^{2} + 40\ q^{3} + 15\ q^{4} + q^{6}$

$F_{7} = 1854 + 1855\ q + 924\ q^{2} + 315\ q^{3} + 70\ q^{4} + 21\ q^{5} + q^{7}$

Description

Eigenvalues of the top-to-random operator acting on a simple module.
These eigenvalues are given in [1] and [3].
The simple module of the symmetric group indexed by a partition

$\lambda$ has dimension equal to the number of standard tableaux of shape

$\lambda$ . Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape

$\lambda$ ; this statistic gives all the eigenvalues of the operator acting on the module.
This statistic bears different names, such as the type in [2] or eig in [3].
Similarly, the eigenvalues of the random-to-random operator acting on a simple module is St000508Eigenvalues of the random-to-random operator acting on a simple module..

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.