- FindStat

Identifier

Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000237: Permutations ⟶ ℤ

Values

[1] => [1] => [1] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1321 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.

Map

first fundamental transformation

Description

Return the permutation whose cycles are the subsequences between successive left to right maxima.

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

Inverse Kreweras complement

Description

Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $c\pi^{-1}$ where $c = (1,\ldots,n)$ is the long cycle.