- FindStat

Identifier

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000779: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]] => [4,2,1,3] => 0

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

[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]] => [4,2,1,3] => 0

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

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

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

[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]] => [4,1,2,3] => 0

[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.] => [3,1,2,4] => 0

[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]] => [4,2,1,3] => 0

[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.] => [2,1,3,4] => 0

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

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

[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]] => [4,1,2,3] => 0

[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.] => [3,1,2,4] => 0

[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]] => [4,1,2,3] => 0

[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.] => [1,2,3,4] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,2,4,1,5] => [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => 0

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

[3,2,5,1,4] => [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => 0

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

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

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

[3,4,2,1,5] => [3,5,2,1,4] => [[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => 0

[3,4,2,5,1] => [3,5,2,4,1] => [[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => 0

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

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

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

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

[3,5,2,1,4] => [3,5,2,1,4] => [[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => 0

>>> Load all 873 entries. <<<

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$F_{2} = 2$

$F_{3} = 4 + 2\ q$

$F_{4} = 10 + 14\ q$

$F_{5} = 26 + 94\ q$

$F_{6} = 76 + 604\ q + 40\ q^{2}$

Description

The tier of a permutation.
This is the number of elements

$i$ such that

$[i+1,k,i]$ is an occurrence of the pattern

$[2,3,1]$ . For example,

$[3,5,6,1,2,4]$ has tier

$2$ , with witnesses

$[3,5,2]$ (or

$[3,6,2]$ ) and

$[5,6,4]$ .
According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as OEIS:A122890 and OEIS:A158830 in the form of triangles read by rows, see [sec. 4, 1].

Map

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length

$0$ to the empty tree, and sending a permutation

$\sigma$ of length

$n \geq 1$ to a root node with two subtrees

$L$ and

$R$ by splitting

$\sigma$ at the index

$\sigma^{-1}(1)$ , normalizing both sides again to permutations and sending the permutations on the left and on the right of

$\sigma^{-1}(1)$ to the trees

$L$ and

$R$ , respectively.

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

to 132-avoiding permutation

Description

Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.