Identifier
Values
[2,1] => [2,1] => [1,2] => 1 => 1
[+,3,2] => [1,3,2] => [2,1,3] => 01 => 1
[-,3,2] => [1,3,2] => [2,1,3] => 01 => 1
[3,1,2] => [3,1,2] => [1,3,2] => 10 => 1
[3,+,1] => [3,2,1] => [1,2,3] => 11 => 2
[3,-,1] => [3,2,1] => [1,2,3] => 11 => 2
[+,+,4,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[-,+,4,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[+,-,4,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[-,-,4,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[+,4,2,3] => [1,4,2,3] => [2,1,4,3] => 010 => 1
[-,4,2,3] => [1,4,2,3] => [2,1,4,3] => 010 => 1
[+,4,+,2] => [1,4,3,2] => [2,1,3,4] => 011 => 1
[-,4,+,2] => [1,4,3,2] => [2,1,3,4] => 011 => 1
[+,4,-,2] => [1,4,3,2] => [2,1,3,4] => 011 => 1
[-,4,-,2] => [1,4,3,2] => [2,1,3,4] => 011 => 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 001 => 1
[2,4,+,1] => [2,4,3,1] => [3,1,2,4] => 001 => 1
[2,4,-,1] => [2,4,3,1] => [3,1,2,4] => 001 => 1
[4,1,2,3] => [4,1,2,3] => [1,3,4,2] => 100 => 1
[4,1,+,2] => [4,1,3,2] => [1,3,2,4] => 101 => 2
[4,1,-,2] => [4,1,3,2] => [1,3,2,4] => 101 => 2
[4,+,1,3] => [4,2,1,3] => [1,4,3,2] => 100 => 1
[4,-,1,3] => [4,2,1,3] => [1,4,3,2] => 100 => 1
[4,+,+,1] => [4,2,3,1] => [1,4,2,3] => 100 => 1
[4,-,+,1] => [4,2,3,1] => [1,4,2,3] => 100 => 1
[4,+,-,1] => [4,2,3,1] => [1,4,2,3] => 100 => 1
[4,-,-,1] => [4,2,3,1] => [1,4,2,3] => 100 => 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 110 => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 111 => 3
[+,+,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[-,+,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[+,-,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[+,+,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[-,-,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[-,+,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[+,-,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[-,-,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[+,+,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[-,+,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[+,-,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[-,-,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[+,+,5,+,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[-,+,5,+,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[+,-,5,+,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[+,+,5,-,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[-,-,5,+,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[-,+,5,-,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[+,-,5,-,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[-,-,5,-,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[+,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1
[-,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1
[+,3,5,+,2] => [1,3,5,4,2] => [2,4,1,3,5] => 0001 => 1
[-,3,5,+,2] => [1,3,5,4,2] => [2,4,1,3,5] => 0001 => 1
[+,3,5,-,2] => [1,3,5,4,2] => [2,4,1,3,5] => 0001 => 1
[-,3,5,-,2] => [1,3,5,4,2] => [2,4,1,3,5] => 0001 => 1
[+,5,2,3,4] => [1,5,2,3,4] => [2,1,4,5,3] => 0100 => 1
[-,5,2,3,4] => [1,5,2,3,4] => [2,1,4,5,3] => 0100 => 1
[+,5,2,+,3] => [1,5,2,4,3] => [2,1,4,3,5] => 0101 => 0
[-,5,2,+,3] => [1,5,2,4,3] => [2,1,4,3,5] => 0101 => 0
[+,5,2,-,3] => [1,5,2,4,3] => [2,1,4,3,5] => 0101 => 0
[-,5,2,-,3] => [1,5,2,4,3] => [2,1,4,3,5] => 0101 => 0
[+,5,+,2,4] => [1,5,3,2,4] => [2,1,5,4,3] => 0100 => 1
[-,5,+,2,4] => [1,5,3,2,4] => [2,1,5,4,3] => 0100 => 1
[+,5,-,2,4] => [1,5,3,2,4] => [2,1,5,4,3] => 0100 => 1
[-,5,-,2,4] => [1,5,3,2,4] => [2,1,5,4,3] => 0100 => 1
[+,5,+,+,2] => [1,5,3,4,2] => [2,1,5,3,4] => 0100 => 1
[-,5,+,+,2] => [1,5,3,4,2] => [2,1,5,3,4] => 0100 => 1
[+,5,-,+,2] => [1,5,3,4,2] => [2,1,5,3,4] => 0100 => 1
[+,5,+,-,2] => [1,5,3,4,2] => [2,1,5,3,4] => 0100 => 1
[-,5,-,+,2] => [1,5,3,4,2] => [2,1,5,3,4] => 0100 => 1
[-,5,+,-,2] => [1,5,3,4,2] => [2,1,5,3,4] => 0100 => 1
[+,5,-,-,2] => [1,5,3,4,2] => [2,1,5,3,4] => 0100 => 1
[-,5,-,-,2] => [1,5,3,4,2] => [2,1,5,3,4] => 0100 => 1
[+,5,4,2,3] => [1,5,4,2,3] => [2,1,3,5,4] => 0110 => 2
[-,5,4,2,3] => [1,5,4,2,3] => [2,1,3,5,4] => 0110 => 2
[+,5,4,3,2] => [1,5,4,3,2] => [2,1,3,4,5] => 0111 => 2
[-,5,4,3,2] => [1,5,4,3,2] => [2,1,3,4,5] => 0111 => 2
[2,1,+,5,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0001 => 1
[2,1,-,5,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0001 => 1
[2,1,5,3,4] => [2,1,5,3,4] => [3,2,1,5,4] => 0010 => 1
[2,1,5,+,3] => [2,1,5,4,3] => [3,2,1,4,5] => 0011 => 1
[2,1,5,-,3] => [2,1,5,4,3] => [3,2,1,4,5] => 0011 => 1
[2,3,1,5,4] => [2,3,1,5,4] => [3,4,2,1,5] => 0001 => 1
[2,3,5,+,1] => [2,3,5,4,1] => [3,4,1,2,5] => 0001 => 1
[2,3,5,-,1] => [2,3,5,4,1] => [3,4,1,2,5] => 0001 => 1
[2,5,1,+,3] => [2,5,1,4,3] => [3,1,4,2,5] => 0001 => 1
[2,5,1,-,3] => [2,5,1,4,3] => [3,1,4,2,5] => 0001 => 1
[2,5,4,1,3] => [2,5,4,1,3] => [3,1,2,5,4] => 0010 => 1
[2,5,4,3,1] => [2,5,4,3,1] => [3,1,2,4,5] => 0011 => 1
[3,1,2,5,4] => [3,1,2,5,4] => [4,2,3,1,5] => 0001 => 1
[3,1,5,+,2] => [3,1,5,4,2] => [4,2,1,3,5] => 0001 => 1
[3,1,5,-,2] => [3,1,5,4,2] => [4,2,1,3,5] => 0001 => 1
[3,+,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => 0001 => 1
[3,-,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => 0001 => 1
[3,+,5,+,1] => [3,2,5,4,1] => [4,3,1,2,5] => 0001 => 1
[3,-,5,+,1] => [3,2,5,4,1] => [4,3,1,2,5] => 0001 => 1
[3,+,5,-,1] => [3,2,5,4,1] => [4,3,1,2,5] => 0001 => 1
[3,-,5,-,1] => [3,2,5,4,1] => [4,3,1,2,5] => 0001 => 1
[3,5,1,+,2] => [3,5,1,4,2] => [4,1,3,2,5] => 0001 => 1
[3,5,1,-,2] => [3,5,1,4,2] => [4,1,3,2,5] => 0001 => 1
>>> Load all 151 entries. <<<
[3,5,4,2,1] => [3,5,4,2,1] => [4,1,2,3,5] => 0001 => 1
[5,1,2,3,4] => [5,1,2,3,4] => [1,3,4,5,2] => 1000 => 1
[5,1,2,+,3] => [5,1,2,4,3] => [1,3,4,2,5] => 1001 => 2
[5,1,2,-,3] => [5,1,2,4,3] => [1,3,4,2,5] => 1001 => 2
[5,1,+,2,4] => [5,1,3,2,4] => [1,3,5,4,2] => 1000 => 1
[5,1,-,2,4] => [5,1,3,2,4] => [1,3,5,4,2] => 1000 => 1
[5,1,+,+,2] => [5,1,3,4,2] => [1,3,5,2,4] => 1000 => 1
[5,1,-,+,2] => [5,1,3,4,2] => [1,3,5,2,4] => 1000 => 1
[5,1,+,-,2] => [5,1,3,4,2] => [1,3,5,2,4] => 1000 => 1
[5,1,-,-,2] => [5,1,3,4,2] => [1,3,5,2,4] => 1000 => 1
[5,1,4,2,3] => [5,1,4,2,3] => [1,3,2,5,4] => 1010 => 0
[5,1,4,3,2] => [5,1,4,3,2] => [1,3,2,4,5] => 1011 => 2
[5,+,1,3,4] => [5,2,1,3,4] => [1,4,3,5,2] => 1000 => 1
[5,-,1,3,4] => [5,2,1,3,4] => [1,4,3,5,2] => 1000 => 1
[5,+,1,+,3] => [5,2,1,4,3] => [1,4,3,2,5] => 1001 => 2
[5,-,1,+,3] => [5,2,1,4,3] => [1,4,3,2,5] => 1001 => 2
[5,+,1,-,3] => [5,2,1,4,3] => [1,4,3,2,5] => 1001 => 2
[5,-,1,-,3] => [5,2,1,4,3] => [1,4,3,2,5] => 1001 => 2
[5,+,+,1,4] => [5,2,3,1,4] => [1,4,5,3,2] => 1000 => 1
[5,-,+,1,4] => [5,2,3,1,4] => [1,4,5,3,2] => 1000 => 1
[5,+,-,1,4] => [5,2,3,1,4] => [1,4,5,3,2] => 1000 => 1
[5,-,-,1,4] => [5,2,3,1,4] => [1,4,5,3,2] => 1000 => 1
[5,+,+,+,1] => [5,2,3,4,1] => [1,4,5,2,3] => 1000 => 1
[5,-,+,+,1] => [5,2,3,4,1] => [1,4,5,2,3] => 1000 => 1
[5,+,-,+,1] => [5,2,3,4,1] => [1,4,5,2,3] => 1000 => 1
[5,+,+,-,1] => [5,2,3,4,1] => [1,4,5,2,3] => 1000 => 1
[5,-,-,+,1] => [5,2,3,4,1] => [1,4,5,2,3] => 1000 => 1
[5,-,+,-,1] => [5,2,3,4,1] => [1,4,5,2,3] => 1000 => 1
[5,+,-,-,1] => [5,2,3,4,1] => [1,4,5,2,3] => 1000 => 1
[5,-,-,-,1] => [5,2,3,4,1] => [1,4,5,2,3] => 1000 => 1
[5,+,4,1,3] => [5,2,4,1,3] => [1,4,2,5,3] => 1000 => 1
[5,-,4,1,3] => [5,2,4,1,3] => [1,4,2,5,3] => 1000 => 1
[5,+,4,3,1] => [5,2,4,3,1] => [1,4,2,3,5] => 1001 => 2
[5,-,4,3,1] => [5,2,4,3,1] => [1,4,2,3,5] => 1001 => 2
[5,3,1,2,4] => [5,3,1,2,4] => [1,5,3,4,2] => 1000 => 1
[5,3,1,+,2] => [5,3,1,4,2] => [1,5,3,2,4] => 1000 => 1
[5,3,1,-,2] => [5,3,1,4,2] => [1,5,3,2,4] => 1000 => 1
[5,3,2,1,4] => [5,3,2,1,4] => [1,5,4,3,2] => 1000 => 1
[5,3,2,+,1] => [5,3,2,4,1] => [1,5,4,2,3] => 1000 => 1
[5,3,2,-,1] => [5,3,2,4,1] => [1,5,4,2,3] => 1000 => 1
[5,3,4,1,2] => [5,3,4,1,2] => [1,5,2,4,3] => 1000 => 1
[5,3,4,2,1] => [5,3,4,2,1] => [1,5,2,3,4] => 1000 => 1
[5,4,1,2,3] => [5,4,1,2,3] => [1,2,4,5,3] => 1100 => 1
[5,4,1,3,2] => [5,4,1,3,2] => [1,2,4,3,5] => 1101 => 2
[5,4,2,1,3] => [5,4,2,1,3] => [1,2,5,4,3] => 1100 => 1
[5,4,2,3,1] => [5,4,2,3,1] => [1,2,5,3,4] => 1100 => 1
[5,4,+,1,2] => [5,4,3,1,2] => [1,2,3,5,4] => 1110 => 2
[5,4,-,1,2] => [5,4,3,1,2] => [1,2,3,5,4] => 1110 => 2
[5,4,+,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4
[5,4,-,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4
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
permutation
Description
The underlying permutation of the decorated permutation.
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 $\pi$.