- FindStat

Identifier

Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words

Images

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

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

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

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

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

[2,1,3] => [2,1,3] => [2,1] => 0

[2,3,1] => [3,2,1] => [2,1] => 0

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

[3,2,1] => [2,3,1] => [2,1] => 0

[1,2,3,4] => [1,2,3,4] => [1,2,3] => 11

[1,2,4,3] => [1,2,4,3] => [1,2,3] => 11

[1,3,2,4] => [1,3,2,4] => [1,3,2] => 10

[1,3,4,2] => [1,4,3,2] => [1,3,2] => 10

[1,4,2,3] => [1,4,2,3] => [1,2,3] => 11

[1,4,3,2] => [1,3,4,2] => [1,3,2] => 10

[2,1,3,4] => [2,1,3,4] => [2,1,3] => 01

[2,1,4,3] => [2,1,4,3] => [2,1,3] => 01

[2,3,1,4] => [3,2,1,4] => [3,2,1] => 00

[2,3,4,1] => [4,2,3,1] => [2,3,1] => 00

[2,4,1,3] => [4,2,1,3] => [2,1,3] => 01

[2,4,3,1] => [3,2,4,1] => [3,2,1] => 00

[3,1,2,4] => [3,1,2,4] => [3,1,2] => 00

[3,1,4,2] => [3,4,1,2] => [3,1,2] => 00

[3,2,1,4] => [2,3,1,4] => [2,3,1] => 00

[3,2,4,1] => [4,3,2,1] => [3,2,1] => 00

[3,4,1,2] => [4,1,3,2] => [1,3,2] => 10

[3,4,2,1] => [2,4,3,1] => [2,3,1] => 00

[4,1,2,3] => [4,1,2,3] => [1,2,3] => 11

[4,1,3,2] => [4,3,1,2] => [3,1,2] => 00

[4,2,1,3] => [2,4,1,3] => [2,1,3] => 01

[4,2,3,1] => [3,4,2,1] => [3,2,1] => 00

[4,3,1,2] => [3,1,4,2] => [3,1,2] => 00

[4,3,2,1] => [2,3,4,1] => [2,3,1] => 00

[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4] => 111

[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4] => 111

[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3] => 110

[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,3] => 110

[1,2,5,3,4] => [1,2,5,3,4] => [1,2,3,4] => 111

[1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,3] => 110

[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4] => 101

[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,4] => 101

[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2] => 100

[1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,2] => 100

[1,3,5,2,4] => [1,5,3,2,4] => [1,3,2,4] => 101

[1,3,5,4,2] => [1,4,3,5,2] => [1,4,3,2] => 100

[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3] => 100

[1,4,2,5,3] => [1,4,5,2,3] => [1,4,2,3] => 100

[1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2] => 100

[1,4,3,5,2] => [1,5,4,3,2] => [1,4,3,2] => 100

[1,4,5,2,3] => [1,5,2,4,3] => [1,2,4,3] => 110

[1,4,5,3,2] => [1,3,5,4,2] => [1,3,4,2] => 100

[1,5,2,3,4] => [1,5,2,3,4] => [1,2,3,4] => 111

[1,5,2,4,3] => [1,5,4,2,3] => [1,4,2,3] => 100

[1,5,3,2,4] => [1,3,5,2,4] => [1,3,2,4] => 101

[1,5,3,4,2] => [1,4,5,3,2] => [1,4,3,2] => 100

[1,5,4,2,3] => [1,4,2,5,3] => [1,4,2,3] => 100

[1,5,4,3,2] => [1,3,4,5,2] => [1,3,4,2] => 100

[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4] => 011

[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => 011

[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3] => 010

[2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,3] => 010

[2,1,5,3,4] => [2,1,5,3,4] => [2,1,3,4] => 011

[2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,3] => 010

[2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4] => 001

[2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => 001

[2,3,4,1,5] => [4,2,3,1,5] => [4,2,3,1] => 000

[2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,1] => 000

[2,3,5,1,4] => [5,2,3,1,4] => [2,3,1,4] => 001

[2,3,5,4,1] => [4,2,3,5,1] => [4,2,3,1] => 000

[2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3] => 000

[2,4,1,5,3] => [4,2,5,1,3] => [4,2,1,3] => 000

[2,4,3,1,5] => [3,2,4,1,5] => [3,2,4,1] => 000

[2,4,3,5,1] => [5,2,4,3,1] => [2,4,3,1] => 000

[2,4,5,1,3] => [5,2,1,4,3] => [2,1,4,3] => 010

[2,4,5,3,1] => [3,2,5,4,1] => [3,2,4,1] => 000

[2,5,1,3,4] => [5,2,1,3,4] => [2,1,3,4] => 011

[2,5,1,4,3] => [5,2,4,1,3] => [2,4,1,3] => 000

[2,5,3,1,4] => [3,2,5,1,4] => [3,2,1,4] => 001

[2,5,3,4,1] => [4,2,5,3,1] => [4,2,3,1] => 000

[2,5,4,1,3] => [4,2,1,5,3] => [4,2,1,3] => 000

[2,5,4,3,1] => [3,2,4,5,1] => [3,2,4,1] => 000

[3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4] => 001

[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,4] => 001

[3,1,4,2,5] => [3,4,1,2,5] => [3,4,1,2] => 000

[3,1,4,5,2] => [3,5,1,4,2] => [3,1,4,2] => 000

[3,1,5,2,4] => [3,5,1,2,4] => [3,1,2,4] => 001

[3,1,5,4,2] => [3,4,1,5,2] => [3,4,1,2] => 000

[3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4] => 001

[3,2,1,5,4] => [2,3,1,5,4] => [2,3,1,4] => 001

[3,2,4,1,5] => [4,3,2,1,5] => [4,3,2,1] => 000

[3,2,4,5,1] => [5,3,2,4,1] => [3,2,4,1] => 000

[3,2,5,1,4] => [5,3,2,1,4] => [3,2,1,4] => 001

[3,2,5,4,1] => [4,3,2,5,1] => [4,3,2,1] => 000

[3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2] => 000

[3,4,1,5,2] => [4,5,3,1,2] => [4,3,1,2] => 000

[3,4,2,1,5] => [2,4,3,1,5] => [2,4,3,1] => 000

[3,4,2,5,1] => [5,4,3,2,1] => [4,3,2,1] => 000

[3,4,5,1,2] => [5,1,3,4,2] => [1,3,4,2] => 100

[3,4,5,2,1] => [2,5,3,4,1] => [2,3,4,1] => 000

[3,5,1,2,4] => [5,1,3,2,4] => [1,3,2,4] => 101

[3,5,1,4,2] => [5,4,3,1,2] => [4,3,1,2] => 000

>>> Load all 12675 entries. <<<

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 $\chi(\pi)$ .

Map

restriction

Description

The permutation obtained by removing the largest letter.
This map is undefined for the empty 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.