Identifier
Values
[1,3,2] => [1,3,2] => [3,1,2] => 10 => 1
[2,1,3] => [1,3,2] => [3,1,2] => 10 => 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 100 => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[1,3,4,2] => [1,3,4,2] => [3,1,2,4] => 100 => 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 010 => 1
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 010 => 1
[2,1,3,4] => [1,3,4,2] => [3,1,2,4] => 100 => 1
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 010 => 1
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 010 => 1
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 100 => 1
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 100 => 1
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => 010 => 1
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 010 => 1
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => 100 => 1
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => 0100 => 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,1,2,3,5] => 1000 => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => 0100 => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => 0100 => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,4,2,5] => 0100 => 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => 1100 => 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,3,2,4,5] => 0100 => 1
[1,3,4,5,2] => [1,3,4,5,2] => [3,1,2,4,5] => 1000 => 1
[1,3,5,2,4] => [1,3,5,2,4] => [3,5,1,2,4] => 1000 => 1
[1,3,5,4,2] => [1,3,5,2,4] => [3,5,1,2,4] => 1000 => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,2,5,3] => [1,4,2,5,3] => [4,1,5,2,3] => 1100 => 1
[1,4,3,2,5] => [1,4,2,5,3] => [4,1,5,2,3] => 1100 => 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => 1000 => 1
[1,4,5,3,2] => [1,4,5,2,3] => [4,5,1,2,3] => 1000 => 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => 1100 => 1
[1,5,3,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => 1100 => 1
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[2,1,3,4,5] => [1,3,4,5,2] => [3,1,2,4,5] => 1000 => 1
[2,1,3,5,4] => [1,3,5,2,4] => [3,5,1,2,4] => 1000 => 1
[2,1,4,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[2,1,4,5,3] => [1,4,5,2,3] => [4,5,1,2,3] => 1000 => 1
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[2,3,1,4,5] => [1,4,5,2,3] => [4,5,1,2,3] => 1000 => 1
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[2,3,5,1,4] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1
[2,4,1,3,5] => [1,3,5,2,4] => [3,5,1,2,4] => 1000 => 1
[2,4,1,5,3] => [1,5,2,4,3] => [5,1,4,2,3] => 1100 => 1
[2,4,3,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => 1100 => 1
[2,4,3,5,1] => [1,2,4,3,5] => [1,4,2,3,5] => 0100 => 1
[2,4,5,1,3] => [1,3,2,4,5] => [1,3,4,2,5] => 0100 => 1
[2,4,5,3,1] => [1,2,4,5,3] => [4,1,2,3,5] => 1000 => 1
[2,5,1,3,4] => [1,3,4,2,5] => [1,3,2,4,5] => 0100 => 1
[2,5,1,4,3] => [1,4,2,5,3] => [4,1,5,2,3] => 1100 => 1
[2,5,3,1,4] => [1,4,2,5,3] => [4,1,5,2,3] => 1100 => 1
[2,5,3,4,1] => [1,2,5,3,4] => [1,5,2,3,4] => 0100 => 1
[2,5,4,1,3] => [1,3,2,5,4] => [3,1,5,2,4] => 1100 => 1
[2,5,4,3,1] => [1,2,5,3,4] => [1,5,2,3,4] => 0100 => 1
[3,1,2,4,5] => [1,2,4,5,3] => [4,1,2,3,5] => 1000 => 1
[3,1,2,5,4] => [1,2,5,3,4] => [1,5,2,3,4] => 0100 => 1
[3,1,4,2,5] => [1,4,2,5,3] => [4,1,5,2,3] => 1100 => 1
[3,1,4,5,2] => [1,4,5,2,3] => [4,5,1,2,3] => 1000 => 1
[3,1,5,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => 1100 => 1
[3,1,5,4,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[3,2,1,4,5] => [1,4,5,2,3] => [4,5,1,2,3] => 1000 => 1
[3,2,1,5,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[3,2,4,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => 1100 => 1
[3,2,4,5,1] => [1,2,4,5,3] => [4,1,2,3,5] => 1000 => 1
[3,2,5,1,4] => [1,4,2,5,3] => [4,1,5,2,3] => 1100 => 1
[3,2,5,4,1] => [1,2,5,3,4] => [1,5,2,3,4] => 0100 => 1
[3,4,1,2,5] => [1,2,5,3,4] => [1,5,2,3,4] => 0100 => 1
[3,4,1,5,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[3,4,2,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[3,4,2,5,1] => [1,2,5,3,4] => [1,5,2,3,4] => 0100 => 1
[3,5,1,2,4] => [1,2,4,3,5] => [1,4,2,3,5] => 0100 => 1
[3,5,1,4,2] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[3,5,2,1,4] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[3,5,2,4,1] => [1,2,4,3,5] => [1,4,2,3,5] => 0100 => 1
[3,5,4,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1
[3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1
[4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1
[4,1,2,5,3] => [1,2,5,3,4] => [1,5,2,3,4] => 0100 => 1
[4,1,3,2,5] => [1,3,2,5,4] => [3,1,5,2,4] => 1100 => 1
[4,1,3,5,2] => [1,3,5,2,4] => [3,5,1,2,4] => 1000 => 1
[4,1,5,2,3] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[4,1,5,3,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[4,2,1,3,5] => [1,3,5,2,4] => [3,5,1,2,4] => 1000 => 1
[4,2,1,5,3] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[4,2,3,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1
[4,2,5,1,3] => [1,3,2,5,4] => [3,1,5,2,4] => 1100 => 1
[4,2,5,3,1] => [1,2,5,3,4] => [1,5,2,3,4] => 0100 => 1
[4,3,1,2,5] => [1,2,5,3,4] => [1,5,2,3,4] => 0100 => 1
[4,3,1,5,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
[4,3,2,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1
>>> Load all 122 entries. <<<
[4,3,2,5,1] => [1,2,5,3,4] => [1,5,2,3,4] => 0100 => 1
[4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1
[4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1
[4,5,1,3,2] => [1,3,2,4,5] => [1,3,4,2,5] => 0100 => 1
[4,5,2,1,3] => [1,3,2,4,5] => [1,3,4,2,5] => 0100 => 1
[5,1,2,4,3] => [1,2,4,3,5] => [1,4,2,3,5] => 0100 => 1
[5,1,3,2,4] => [1,3,2,4,5] => [1,3,4,2,5] => 0100 => 1
[5,1,3,4,2] => [1,3,4,2,5] => [1,3,2,4,5] => 0100 => 1
[5,1,4,2,3] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[5,1,4,3,2] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[5,2,1,3,4] => [1,3,4,2,5] => [1,3,2,4,5] => 0100 => 1
[5,2,1,4,3] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[5,2,3,1,4] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[5,2,4,1,3] => [1,3,2,4,5] => [1,3,4,2,5] => 0100 => 1
[5,2,4,3,1] => [1,2,4,3,5] => [1,4,2,3,5] => 0100 => 1
[5,3,1,2,4] => [1,2,4,3,5] => [1,4,2,3,5] => 0100 => 1
[5,3,1,4,2] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[5,3,2,1,4] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[5,3,2,4,1] => [1,2,4,3,5] => [1,4,2,3,5] => 0100 => 1
[5,4,1,3,2] => [1,3,2,4,5] => [1,3,4,2,5] => 0100 => 1
[5,4,2,1,3] => [1,3,2,4,5] => [1,3,4,2,5] => 0100 => 1
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
cactus evacuation
Description
The cactus evacuation of a permutation.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.
Map
descent bottoms
Description
The descent bottoms of a permutation as a binary word.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.