- FindStat

Identifier

Mp00163: Signed permutations —permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤ

Values

[1,2] => [1,2] => [1,2] => 1 => 1

[1,-2] => [1,2] => [1,2] => 1 => 1

[-1,2] => [1,2] => [1,2] => 1 => 1

[-1,-2] => [1,2] => [1,2] => 1 => 1

[1,2,3] => [1,2,3] => [1,2,3] => 11 => 2

[1,2,-3] => [1,2,3] => [1,2,3] => 11 => 2

[1,-2,3] => [1,2,3] => [1,2,3] => 11 => 2

[1,-2,-3] => [1,2,3] => [1,2,3] => 11 => 2

[-1,2,3] => [1,2,3] => [1,2,3] => 11 => 2

[-1,2,-3] => [1,2,3] => [1,2,3] => 11 => 2

[-1,-2,3] => [1,2,3] => [1,2,3] => 11 => 2

[-1,-2,-3] => [1,2,3] => [1,2,3] => 11 => 2

[2,1,3] => [2,1,3] => [2,1,3] => 01 => 1

[2,1,-3] => [2,1,3] => [2,1,3] => 01 => 1

[2,-1,3] => [2,1,3] => [2,1,3] => 01 => 1

[2,-1,-3] => [2,1,3] => [2,1,3] => 01 => 1

[-2,1,3] => [2,1,3] => [2,1,3] => 01 => 1

[-2,1,-3] => [2,1,3] => [2,1,3] => 01 => 1

[-2,-1,3] => [2,1,3] => [2,1,3] => 01 => 1

[-2,-1,-3] => [2,1,3] => [2,1,3] => 01 => 1

[3,1,2] => [3,1,2] => [1,3,2] => 10 => 1

[3,1,-2] => [3,1,2] => [1,3,2] => 10 => 1

[3,-1,2] => [3,1,2] => [1,3,2] => 10 => 1

[3,-1,-2] => [3,1,2] => [1,3,2] => 10 => 1

[-3,1,2] => [3,1,2] => [1,3,2] => 10 => 1

[-3,1,-2] => [3,1,2] => [1,3,2] => 10 => 1

[-3,-1,2] => [3,1,2] => [1,3,2] => 10 => 1

[-3,-1,-2] => [3,1,2] => [1,3,2] => 10 => 1

[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[-1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[-1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => 111 => 3

[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[1,3,2,-4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[1,3,-2,4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[1,3,-2,-4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[1,-3,2,4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[1,-3,2,-4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[1,-3,-2,4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[1,-3,-2,-4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[-1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[-1,3,2,-4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[-1,3,-2,4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[-1,3,-2,-4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[-1,-3,2,4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[-1,-3,2,-4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[-1,-3,-2,4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[-1,-3,-2,-4] => [1,3,2,4] => [3,1,2,4] => 001 => 1

[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[1,4,2,-3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[1,4,-2,3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[1,4,-2,-3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[1,-4,2,3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[1,-4,2,-3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[1,-4,-2,3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[1,-4,-2,-3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[-1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[-1,4,2,-3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[-1,4,-2,3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[-1,4,-2,-3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[-1,-4,2,3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[-1,-4,2,-3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[-1,-4,-2,3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[-1,-4,-2,-3] => [1,4,2,3] => [1,4,2,3] => 100 => 1

[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[2,1,3,-4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[2,1,-3,4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[2,1,-3,-4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[2,-1,3,4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[2,-1,3,-4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[2,-1,-3,4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[2,-1,-3,-4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[-2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[-2,1,3,-4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[-2,1,-3,4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[-2,1,-3,-4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[-2,-1,3,4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[-2,-1,3,-4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[-2,-1,-3,4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[-2,-1,-3,-4] => [2,1,3,4] => [2,1,3,4] => 011 => 1

[2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 001 => 1

[2,3,1,-4] => [2,3,1,4] => [2,3,1,4] => 001 => 1

[2,3,-1,4] => [2,3,1,4] => [2,3,1,4] => 001 => 1

[2,3,-1,-4] => [2,3,1,4] => [2,3,1,4] => 001 => 1

[2,-3,1,4] => [2,3,1,4] => [2,3,1,4] => 001 => 1

[2,-3,1,-4] => [2,3,1,4] => [2,3,1,4] => 001 => 1

[2,-3,-1,4] => [2,3,1,4] => [2,3,1,4] => 001 => 1

[2,-3,-1,-4] => [2,3,1,4] => [2,3,1,4] => 001 => 1

[-2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 001 => 1

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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

permutation

Description

The permutation obtained by forgetting the colours.

Map

Foata bijection

Description

Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.

If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.

In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is

$1$
$|1|4 \to 14$
$|14|2 \to 412$
$|4|1|2|5 \to 4125$
$|4|125|3 \to 45123.$

In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).