- FindStat

Identifier

Mp00245: Signed permutations —standardize⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations

Images

[1] => [1] => [1] => [1]

[-1] => [1] => [1] => [1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 4282 entries. <<<

Map

standardize

Description

Return the standardization of the signed permutation, where 1 is the smallest and -1 the largest element.
Let $\pi\in\mathfrak H_n$ be a signed permutation. Assuming the order $1 < \dots < n < -n < \dots < -1$, this map returns the permutation in $\mathfrak S_n$ which is order isomorphic to $\pi(1),\dots,\pi(n)$.

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.).

Map

descent views to invisible inversion bottoms

Description

Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that

the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
the set of left-to-right maxima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
the set of maximal elements in the decreasing runs of $\pi$ is the set of weak deficiency positions of $\chi(\pi)$, and
the set of minimal elements in the decreasing runs of $\pi$ is the set of weak deficiency values of $\chi(\pi)$.