- FindStat

Identifier

Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000185: Integer partitions ⟶ ℤ

Values

[1] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 1 + 9\ q + 4\ q^{2} + 9\ q^{3} + q^{6}$

$F_{5} = 1 + 16\ q + 25\ q^{2} + 36\ q^{3} + 25\ q^{4} + 16\ q^{6} + q^{10}$

$F_{6} = 1 + 25\ q + 81\ q^{2} + 125\ q^{3} + 256\ q^{4} + 125\ q^{6} + 81\ q^{7} + 25\ q^{10} + q^{15}$

$F_{7} = 1 + 36\ q + 196\ q^{2} + 421\ q^{3} + 1225\ q^{4} + 441\ q^{5} + 841\ q^{6} + 1225\ q^{7} + 196\ q^{9} + 225\ q^{10} + 196\ q^{11} + 36\ q^{15} + q^{21}$

Description

The weighted size of a partition.
Let

$\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of

$\lambda$ is

$\sum_{i=0}^m i \cdot \lambda_i.$
This is also the sum of the leg lengths of the cells in

$\lambda$ , or

$\sum_i \binom{\lambda^{\prime}_i}{2}$
where

$\lambda^{\prime}$ is the conjugate partition of

$\lambda$ .
This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2].
This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape

$\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$ , obtained uniquely by placing

$i-1$ in all the cells of the

$i$ th row of

$\lambda$ , see [2, eq.7.103].

Map

Robinson-Schensted tableau shape

Description

Sends a permutation to its Robinson-Schensted tableau shape.
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 the shape of its corresponding insertion and recording tableau.