- FindStat

Identifier

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤ

Values

[1,1] => [1] => [1,0,1,0] => 1010 => 0
[2,1] => [1] => [1,0,1,0] => 1010 => 0
[3,1] => [1] => [1,0,1,0] => 1010 => 0
[4,1] => [1] => [1,0,1,0] => 1010 => 0
[5,1] => [1] => [1,0,1,0] => 1010 => 0
[6,1] => [1] => [1,0,1,0] => 1010 => 0
[7,1] => [1] => [1,0,1,0] => 1010 => 0
[8,1] => [1] => [1,0,1,0] => 1010 => 0
[9,1] => [1] => [1,0,1,0] => 1010 => 0
[10,1] => [1] => [1,0,1,0] => 1010 => 0
[11,1] => [1] => [1,0,1,0] => 1010 => 0
[12,1] => [1] => [1,0,1,0] => 1010 => 0
[13,1] => [1] => [1,0,1,0] => 1010 => 0
[14,1] => [1] => [1,0,1,0] => 1010 => 0
[15,1] => [1] => [1,0,1,0] => 1010 => 0
[16,1] => [1] => [1,0,1,0] => 1010 => 0

search for individual values

searching the database for the individual values of this statistic

Description

The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let

$A_n=K[x]/(x^n)$ .
We associate to a nonempty subset S of an (n-1)-set the module

$M_S$ , which is the direct sum of

$A_n$ -modules with indecomposable non-projective direct summands of dimension

$i$ when

$i$ is in

$S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of

$M_S$ . We decode the subset as a binary word so that for example the subset

$S=\{1,3 \}$ of

$\{1,2,3 \}$ is decoded as 101.

Map

first row removal

Description

Removes the first entry of an integer partition

Map

to binary word

Description

Return the Dyck word as binary word.

Map

to Dyck path

Description

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