- FindStat

Identifier

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000022: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of fixed points of a permutation.

Map

Corteel

Description

Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.

Map

Demazure product with inverse

Description

This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.