- FindStat

Identifier

Mp00223: Permutations —runsort⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St001549: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of restricted non-inversions between exceedances.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$$

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.

Map

inverse Foata bijection

Description

The inverse of Foata's bijection.
See Mp00067Foata bijection.

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 $\pi = \pi^{-1}$.