- FindStat

Identifier

Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[+,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5]

[-,+,+,+,+] => [2,3,4,5,1] => [2,3,4,5,1] => [5,2,3,4,1]

[+,-,+,+,+] => [1,3,4,5,2] => [3,1,4,5,2] => [3,5,1,4,2]

[+,+,-,+,+] => [1,2,4,5,3] => [4,1,2,5,3] => [4,1,5,2,3]

[+,+,+,-,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4]

[+,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5]

[-,-,+,+,+] => [3,4,5,1,2] => [1,3,4,5,2] => [1,5,3,4,2]

[-,+,-,+,+] => [2,4,5,1,3] => [2,1,4,5,3] => [2,1,5,4,3]

[-,+,+,-,+] => [2,3,5,1,4] => [2,3,1,5,4] => [3,2,1,5,4]

[-,+,+,+,-] => [2,3,4,1,5] => [2,3,4,1,5] => [4,2,3,1,5]

[+,-,-,+,+] => [1,4,5,2,3] => [1,4,2,5,3] => [1,4,5,2,3]

[+,-,+,-,+] => [1,3,5,2,4] => [3,1,2,5,4] => [3,1,2,5,4]

[+,-,+,+,-] => [1,3,4,2,5] => [3,1,4,2,5] => [3,4,1,2,5]

>>> Load all 2371 entries. <<<

Map

upper permutation

Description

The upper bound in the Grassmann interval corresponding to the decorated permutation.
Let

$I$ be the anti-exceedance set of a decorated permutation

$w$ . Let

$v$ be the

$k$ -Grassmannian permutation determined by

$v[k] = w^{-1}(I)$ and let

$u$ be the permutation satisfying

$u = wv$ . Then

$[u, v]$ is the Grassmann interval corresponding to

$w$ .
This map returns

$v$ .

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.
This map is similar to Mp00235descent views to invisible inversion bottoms, but different beginning with permutations of six elements.
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 maximima 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 deficiency positions of $\chi(\pi)$ , and
the set of minimal elements in the decreasing runs of $\pi$ is the set of deficiency values of $\chi(\pi)$ .