- FindStat

Identifier

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000054: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies St000703The number of deficiencies of a permutation. as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

Map

reverse

Description

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