- FindStat

Identifier

Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001207: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

Description

The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ .

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.

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.