Identifier
Values
[3,1,2] => [1,3,2] => [2,1,3] => 01 => 1
[3,2,1] => [1,3,2] => [2,1,3] => 01 => 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[4,1,2,3] => [1,4,3,2] => [2,1,3,4] => 011 => 1
[4,1,3,2] => [1,4,2,3] => [2,1,4,3] => 010 => 1
[4,2,1,3] => [1,4,3,2] => [2,1,3,4] => 011 => 1
[4,2,3,1] => [1,4,2,3] => [2,1,4,3] => 010 => 1
[4,3,1,2] => [1,4,2,3] => [2,1,4,3] => 010 => 1
[4,3,2,1] => [1,4,2,3] => [2,1,4,3] => 010 => 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[1,5,2,3,4] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[1,5,2,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[1,5,3,2,4] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[1,5,3,4,2] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[1,5,4,2,3] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[1,5,4,3,2] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[2,1,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[2,1,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[2,3,5,1,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[2,5,1,3,4] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[2,5,1,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[2,5,3,1,4] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[2,5,3,4,1] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[2,5,4,1,3] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[2,5,4,3,1] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[3,1,5,2,4] => [1,3,5,4,2] => [2,4,1,3,5] => 0001 => 1
[3,2,5,1,4] => [1,3,5,4,2] => [2,4,1,3,5] => 0001 => 1
[3,5,1,2,4] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1
[3,5,1,4,2] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1
[3,5,2,1,4] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1
[3,5,2,4,1] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1
[5,1,2,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => 0111 => 2
[5,1,2,4,3] => [1,5,3,2,4] => [2,1,5,4,3] => 0100 => 1
[5,1,3,2,4] => [1,5,4,2,3] => [2,1,3,5,4] => 0110 => 2
[5,1,3,4,2] => [1,5,2,3,4] => [2,1,4,5,3] => 0100 => 1
[5,1,4,2,3] => [1,5,3,4,2] => [2,1,5,3,4] => 0100 => 1
[5,1,4,3,2] => [1,5,2,3,4] => [2,1,4,5,3] => 0100 => 1
[5,2,1,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => 0111 => 2
[5,2,1,4,3] => [1,5,3,2,4] => [2,1,5,4,3] => 0100 => 1
[5,2,3,1,4] => [1,5,4,2,3] => [2,1,3,5,4] => 0110 => 2
[5,2,3,4,1] => [1,5,2,3,4] => [2,1,4,5,3] => 0100 => 1
[5,2,4,1,3] => [1,5,3,4,2] => [2,1,5,3,4] => 0100 => 1
[5,2,4,3,1] => [1,5,2,3,4] => [2,1,4,5,3] => 0100 => 1
[5,3,1,2,4] => [1,5,4,2,3] => [2,1,3,5,4] => 0110 => 2
[5,3,1,4,2] => [1,5,2,3,4] => [2,1,4,5,3] => 0100 => 1
[5,3,2,1,4] => [1,5,4,2,3] => [2,1,3,5,4] => 0110 => 2
[5,3,2,4,1] => [1,5,2,3,4] => [2,1,4,5,3] => 0100 => 1
[5,3,4,1,2] => [1,5,2,3,4] => [2,1,4,5,3] => 0100 => 1
[5,3,4,2,1] => [1,5,2,3,4] => [2,1,4,5,3] => 0100 => 1
[5,4,1,2,3] => [1,5,3,2,4] => [2,1,5,4,3] => 0100 => 1
[5,4,1,3,2] => [1,5,2,4,3] => [2,1,4,3,5] => 0101 => 0
[5,4,2,1,3] => [1,5,3,2,4] => [2,1,5,4,3] => 0100 => 1
[5,4,2,3,1] => [1,5,2,4,3] => [2,1,4,3,5] => 0101 => 0
[5,4,3,1,2] => [1,5,2,4,3] => [2,1,4,3,5] => 0101 => 0
[5,4,3,2,1] => [1,5,2,4,3] => [2,1,4,3,5] => 0101 => 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
connectivity set
Description
The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is
$$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$$
For $n > 1$ it can also be described as the set of occurrences of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
The permutation is connected, when the connectivity set is empty.
Map
Lehmer code rotation
Description
Sends a permutation $\pi$ to the unique permutation $\tau$ (of the same length) such that every entry in the Lehmer code of $\tau$ is cyclically one larger than the Lehmer code of $\pi$.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.