- FindStat

Identifier

Mp00079: Set partitions —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

{{1},{2}} => [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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 116 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q$

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

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

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

shape

Description

Sends a set partition to the integer partition obtained by the sizes of the blocks.

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.