- FindStat

Identifier

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001461: Permutations ⟶ ℤ

Values

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of topologically connected components of the chord diagram of a permutation.
The chord diagram of a permutation $\pi\in\mathfrak S_n$ is obtained by placing labels $1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from $i$ to $\pi(i)$ for every label $i$.
This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component.
The permutation $\pi\in\mathfrak S_n$ stabilizes an interval $I=\{a,a+1,\dots,b\}$ if $\pi(I)=I$. It is stabilized-interval-free, if the only interval $\pi$ stablizes is $\{1,\dots,n\}$. Thus, this statistic is $1$ if $\pi$ is stabilized-interval-free.

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

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.