- FindStat

Identifier

Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 1000

[1,2,3,5,4] => [1,2,3,5,4] => [1,5,4,3,2] => 1000

[1,2,4,3,5] => [1,2,4,3,5] => [1,5,4,3,2] => 1000

[1,2,4,5,3] => [1,2,5,3,4] => [1,5,4,3,2] => 1000

[1,2,5,3,4] => [1,2,4,5,3] => [1,5,4,3,2] => 1000

[1,2,5,4,3] => [1,2,5,4,3] => [1,5,4,3,2] => 1000

[1,3,2,4,5] => [1,3,2,4,5] => [1,5,4,3,2] => 1000

[1,3,2,5,4] => [1,3,2,5,4] => [1,5,4,3,2] => 1000

[1,3,4,2,5] => [1,4,2,3,5] => [1,5,4,3,2] => 1000

[1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => 1000

[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => 1000

[1,3,5,4,2] => [1,5,4,2,3] => [1,5,4,3,2] => 1000

[1,4,2,3,5] => [1,3,4,2,5] => [1,5,4,3,2] => 1000

[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,3,2] => 1000

[1,4,3,2,5] => [1,4,3,2,5] => [1,5,4,3,2] => 1000

[1,4,3,5,2] => [1,5,2,4,3] => [1,5,4,3,2] => 1000

[1,4,5,2,3] => [1,3,5,2,4] => [1,5,4,3,2] => 1000

[1,4,5,3,2] => [1,5,3,2,4] => [1,5,4,3,2] => 1000

[1,5,2,3,4] => [1,3,4,5,2] => [1,5,4,3,2] => 1000

[1,5,2,4,3] => [1,4,3,5,2] => [1,5,4,3,2] => 1000

[1,5,3,2,4] => [1,4,5,3,2] => [1,5,4,3,2] => 1000

[1,5,3,4,2] => [1,4,2,5,3] => [1,5,4,3,2] => 1000

[1,5,4,2,3] => [1,3,5,4,2] => [1,5,4,3,2] => 1000

[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1000

[2,1,3,4,5] => [2,1,3,4,5] => [2,1,5,4,3] => 0100

[2,1,3,5,4] => [2,1,3,5,4] => [2,1,5,4,3] => 0100

[2,1,4,3,5] => [2,1,4,3,5] => [2,1,5,4,3] => 0100

[2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,4,3] => 0100

[2,1,5,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0100

[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 0100

[2,3,1,4,5] => [3,1,2,4,5] => [3,1,5,4,2] => 0000

[2,3,1,5,4] => [3,1,2,5,4] => [3,1,5,4,2] => 0000

[2,3,4,1,5] => [4,1,2,3,5] => [4,1,5,3,2] => 0000

[2,3,4,5,1] => [5,1,2,3,4] => [5,1,4,3,2] => 0000

[2,3,5,1,4] => [4,5,1,2,3] => [4,5,1,3,2] => 0000

[2,3,5,4,1] => [5,4,1,2,3] => [5,4,1,3,2] => 0000

[2,4,1,3,5] => [3,4,1,2,5] => [3,5,1,4,2] => 0000

[2,4,1,5,3] => [5,3,4,1,2] => [5,3,4,1,2] => 0000

[2,4,3,1,5] => [4,3,1,2,5] => [4,3,1,5,2] => 0000

[2,4,3,5,1] => [5,1,2,4,3] => [5,1,4,3,2] => 0000

[2,4,5,1,3] => [3,5,1,2,4] => [3,5,1,4,2] => 0000

[2,4,5,3,1] => [5,3,1,2,4] => [5,3,1,4,2] => 0000

[2,5,1,3,4] => [3,4,5,1,2] => [3,5,4,1,2] => 0000

[2,5,1,4,3] => [4,3,5,1,2] => [4,3,5,1,2] => 0000

[2,5,3,1,4] => [4,5,3,1,2] => [4,5,3,1,2] => 0000

[2,5,3,4,1] => [4,1,2,5,3] => [4,1,5,3,2] => 0000

[2,5,4,1,3] => [3,5,4,1,2] => [3,5,4,1,2] => 0000

[2,5,4,3,1] => [5,4,3,1,2] => [5,4,3,1,2] => 0000

[3,1,2,4,5] => [2,3,1,4,5] => [2,5,1,4,3] => 0000

[3,1,2,5,4] => [2,3,1,5,4] => [2,5,1,4,3] => 0000

[3,1,4,2,5] => [4,2,3,1,5] => [4,2,5,1,3] => 0000

[3,1,4,5,2] => [5,2,3,1,4] => [5,2,4,1,3] => 0000

[3,1,5,2,4] => [4,5,2,3,1] => [4,5,2,3,1] => 0000

[3,1,5,4,2] => [5,4,2,3,1] => [5,4,2,3,1] => 0000

[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,5,4] => 0010

[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 0010

[3,2,4,1,5] => [4,1,3,2,5] => [4,1,5,3,2] => 0000

[3,2,4,5,1] => [5,1,3,2,4] => [5,1,4,3,2] => 0000

[3,2,5,1,4] => [4,5,1,3,2] => [4,5,1,3,2] => 0000

[3,2,5,4,1] => [5,4,1,3,2] => [5,4,1,3,2] => 0000

[3,4,1,2,5] => [2,4,1,3,5] => [2,5,1,4,3] => 0000

[3,4,1,5,2] => [5,2,4,1,3] => [5,2,4,1,3] => 0000

[3,4,2,1,5] => [4,2,1,3,5] => [4,2,1,5,3] => 0000

[3,4,2,5,1] => [5,1,4,2,3] => [5,1,4,3,2] => 0000

[3,4,5,1,2] => [2,5,1,3,4] => [2,5,1,4,3] => 0000

[3,4,5,2,1] => [5,2,1,3,4] => [5,2,1,4,3] => 0000

[3,5,1,2,4] => [2,4,5,1,3] => [2,5,4,1,3] => 0000

[3,5,1,4,2] => [4,2,5,1,3] => [4,2,5,1,3] => 0000

>>> Load all 10954 entries. <<<

Map

invert Laguerre heap

Description

The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let

$\pi$ be a permutation. Following Viennot [1], we associate to

$\pi$ a heap of pieces, by considering each decreasing run

$(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of

$\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.

Map

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a

$123$ -avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between

$132$ -avoiding permutations and

$123$ -avoiding permutations, see [1, Proposition 19].

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.