- FindStat

Identifier

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
St001870: Signed permutations ⟶ ℤ

Values

{{1}} => [1] => [1] => [-1] => 1
{{1,2}} => [2,1] => [2,1] => [1,-2] => 2
{{1},{2}} => [1,2] => [1,2] => [2,-1] => 2
{{1,2,3}} => [2,3,1] => [2,3,1] => [1,2,-3] => 2
{{1,2},{3}} => [2,1,3] => [2,1,3] => [1,3,-2] => 2
{{1,3},{2}} => [3,2,1] => [3,2,1] => [2,1,-3] => 2
{{1},{2,3}} => [1,3,2] => [1,3,2] => [3,2,-1] => 2
{{1},{2},{3}} => [1,2,3] => [1,2,3] => [2,3,-1] => 2
{{1,2,3,4}} => [2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => 2
{{1,2,3},{4}} => [2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => 2
{{1,2,4},{3}} => [2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => 2
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => 2
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => 2
{{1,3,4},{2}} => [3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => 2
{{1,3},{2,4}} => [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2
{{1,3},{2},{4}} => [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 2
{{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => 2
{{1},{2,3,4}} => [1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => 2
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => 2
{{1,4},{2},{3}} => [4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => 2
{{1},{2,4},{3}} => [1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => 2
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => 2
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 2
{{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => 2
{{1,2,3,4},{5}} => [2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => 2
{{1,2,3,5},{4}} => [2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,-5] => 2
{{1,2,3},{4,5}} => [2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => 2
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => 2
{{1,2,4,5},{3}} => [2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,-5] => 2
{{1,2,4},{3,5}} => [2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => 2
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,-4] => 2
{{1,2,5},{3,4}} => [2,5,4,3,1] => [2,5,4,3,1] => [1,4,3,2,-5] => 2
{{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => 2
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => 2
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [2,5,3,4,1] => [1,3,4,2,-5] => 2
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => 2
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => 2
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => 2

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

$F_{1} = q$

$F_{2} = 2\ q^{2}$

$F_{3} = 5\ q^{2}$

$F_{4} = 15\ q^{2}$

Description

The number of positive entries followed by a negative entry in a signed permutation.
For a signed permutation

$\pi\in\mathfrak H_n$ , this is the number of positive entries followed by a negative entry in

$\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$ .

Map

to permutation

Description

Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.

Map

inverse Kreweras complement

Description

The inverse Kreweras complement of a signed permutation.
This is the signed permutation

$c \pi^{-1}$ where

$c = (1,\ldots,n,-1,-2,\dots,-n)$ is the long cycle.
The order of the inverse Kreweras complement on signed permutations of

$\{\pm 1,\dots, \pm n\}$ is

$2n$ .

Map

to signed permutation

Description

The signed permutation with all signs positive.