- FindStat

Identifier

Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000054: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

complement

Description

Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.

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.