Identifier
Values
{{1},{2}} => [1,2] => [1,2] => 1 => 1
{{1,2},{3}} => [2,1,3] => [2,1,3] => 01 => 1
{{1},{2,3}} => [1,3,2] => [1,3,2] => 10 => 1
{{1},{2},{3}} => [1,2,3] => [1,3,2] => 10 => 1
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => 010 => 1
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,4,3] => 010 => 1
{{1,3},{2},{4}} => [3,2,1,4] => [3,2,1,4] => 001 => 1
{{1},{2,3,4}} => [1,3,4,2] => [1,4,3,2] => 100 => 1
{{1},{2,3},{4}} => [1,3,2,4] => [1,4,3,2] => 100 => 1
{{1},{2,4},{3}} => [1,4,3,2] => [1,4,3,2] => 100 => 1
{{1},{2},{3,4}} => [1,2,4,3] => [1,4,3,2] => 100 => 1
{{1},{2},{3},{4}} => [1,2,3,4] => [1,4,3,2] => 100 => 1
{{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,5,4,3] => 0100 => 1
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,5,4,3] => 0100 => 1
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [2,1,5,4,3] => 0100 => 1
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,5,4,3] => 0100 => 1
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,1,5,4,3] => 0100 => 1
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [3,2,1,5,4] => 0010 => 1
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [3,2,1,5,4] => 0010 => 1
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [4,3,2,1,5] => 0001 => 1
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,5,4,3,2] => 1000 => 1
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,5,4,3,2] => 1000 => 1
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,5,4,3,2] => 1000 => 1
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,5,4,3,2] => 1000 => 1
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,5,4,3,2] => 1000 => 1
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,5,4,3,2] => 1000 => 1
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,5,4,3,2] => 1000 => 1
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,5,4,3,2] => 1000 => 1
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,5,4,3,2] => 1000 => 1
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,5,4,3,2] => 1000 => 1
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,5,4,3,2] => 1000 => 1
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,5,4,3,2] => 1000 => 1
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,5,4,3,2] => 1000 => 1
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,5,4,3,2] => 1000 => 1
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,5,4,3,2] => 1000 => 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
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
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].