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
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
Map
descent bottoms
Description
The descent bottoms of a permutation as a binary word.