- FindStat

Identifier

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000702: Permutations ⟶ ℤ

Values

{{1,2}} => [2,1] => [1,2] => [1,2] => 2

{{1},{2}} => [1,2] => [2,1] => [2,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 5

{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 5

{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => 5

{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [1,2,3,5,6,4] => [1,2,3,6,4,5] => 5

{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 5

{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [1,2,6,3,4,5] => [1,2,4,5,6,3] => 3

{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 4

{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => 5

{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => 5

{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [1,2,5,4,6,3] => [1,2,6,4,3,5] => 5

{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [1,2,4,5,3,6] => [1,2,5,3,4,6] => 5

{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 4

{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [1,2,4,6,5,3] => [1,2,6,3,5,4] => 5

{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 5

{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 5

{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [1,6,2,4,3,5] => [1,3,5,4,6,2] => 3

{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 4

{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [1,5,2,3,4,6] => [1,3,4,5,2,6] => 3

{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [1,6,2,3,5,4] => [1,3,4,6,5,2] => 3

{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [1,5,2,3,6,4] => [1,3,4,6,2,5] => 3

{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 4

{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [1,6,2,5,3,4] => [1,3,5,6,4,2] => 3

{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => 4

{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [1,3,2,5,6,4] => [1,3,2,6,4,5] => 4

{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => 5

{{1,2,5},{3,4,6}} => [2,5,4,6,1,3] => [1,6,3,2,4,5] => [1,4,3,5,6,2] => 3

>>> Load all 392 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

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

$F_{3} = 4\ q^{2} + q^{3}$

$F_{4} = q + 2\ q^{2} + 11\ q^{3} + q^{4}$

$F_{5} = 10\ q^{2} + 15\ q^{3} + 26\ q^{4} + q^{5}$

$F_{6} = q + 10\ q^{2} + 65\ q^{3} + 69\ q^{4} + 57\ q^{5} + q^{6}$

Description

The number of weak deficiencies of a permutation.
This is defined as

$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$
The number of weak exceedances is St000213The number of weak exceedances (also weak excedences) of a permutation., the number of deficiencies is St000703The number of deficiencies of a permutation..

Map

to permutation

Description

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

Map

inverse

Description

Sends a permutation to its inverse.

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.