- FindStat

Identifier

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000830: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,3,4,1] => [4,2,3,1] => [4,3,1,2] => [4,3,1,2] => 8

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

[2,4,3,1] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 6

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

[3,1,4,2] => [4,2,3,1] => [4,3,1,2] => [4,3,1,2] => 8

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

[3,2,4,1] => [4,2,3,1] => [4,3,1,2] => [4,3,1,2] => 8

[3,4,1,2] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 6

[3,4,2,1] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 6

[4,1,2,3] => [4,2,3,1] => [4,3,1,2] => [4,3,1,2] => 8

[4,1,3,2] => [4,2,3,1] => [4,3,1,2] => [4,3,1,2] => 8

[4,2,1,3] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 6

[4,2,3,1] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 6

[4,3,1,2] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 6

[4,3,2,1] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 6

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

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

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

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

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

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

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

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

[1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,2,3] => [1,5,4,2,3] => 8

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

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

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

[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,2,3] => [1,5,4,2,3] => 8

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

[1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,2,3] => [1,5,4,2,3] => 8

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

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

[1,5,2,3,4] => [1,5,3,4,2] => [1,5,4,2,3] => [1,5,4,2,3] => 8

[1,5,2,4,3] => [1,5,3,4,2] => [1,5,4,2,3] => [1,5,4,2,3] => 8

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

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

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

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

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

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

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

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

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

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

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

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

[2,3,4,1,5] => [4,2,3,1,5] => [4,3,1,2,5] => [4,3,1,2,5] => 8

[2,3,4,5,1] => [5,2,3,4,1] => [5,3,4,1,2] => [5,2,4,1,3] => 10

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

[2,3,5,4,1] => [5,2,4,3,1] => [5,4,1,3,2] => [5,4,1,3,2] => 12

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

[2,4,1,5,3] => [3,5,1,4,2] => [4,5,3,2,1] => [1,5,4,3,2] => 8

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

[2,4,3,5,1] => [5,3,2,4,1] => [5,4,2,1,3] => [5,4,2,1,3] => 12

[2,4,5,1,3] => [4,5,3,1,2] => [2,5,1,4,3] => [2,5,1,4,3] => 8

[2,4,5,3,1] => [5,4,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => 8

[2,5,1,3,4] => [3,5,1,4,2] => [4,5,3,2,1] => [1,5,4,3,2] => 8

[2,5,1,4,3] => [3,5,1,4,2] => [4,5,3,2,1] => [1,5,4,3,2] => 8

[2,5,3,1,4] => [4,5,3,1,2] => [2,5,1,4,3] => [2,5,1,4,3] => 8

[2,5,3,4,1] => [5,4,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => 8

[2,5,4,1,3] => [4,5,3,1,2] => [2,5,1,4,3] => [2,5,1,4,3] => 8

[2,5,4,3,1] => [5,4,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => 8

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

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

[3,1,4,2,5] => [4,2,3,1,5] => [4,3,1,2,5] => [4,3,1,2,5] => 8

[3,1,4,5,2] => [5,2,3,4,1] => [5,3,4,1,2] => [5,2,4,1,3] => 10

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

[3,1,5,4,2] => [5,2,4,3,1] => [5,4,1,3,2] => [5,4,1,3,2] => 12

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

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

[3,2,4,1,5] => [4,2,3,1,5] => [4,3,1,2,5] => [4,3,1,2,5] => 8

[3,2,4,5,1] => [5,2,3,4,1] => [5,3,4,1,2] => [5,2,4,1,3] => 10

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

[3,2,5,4,1] => [5,2,4,3,1] => [5,4,1,3,2] => [5,4,1,3,2] => 12

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

[3,4,1,5,2] => [5,3,2,4,1] => [5,4,2,1,3] => [5,4,2,1,3] => 12

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

[3,4,2,5,1] => [5,3,2,4,1] => [5,4,2,1,3] => [5,4,2,1,3] => 12

[3,4,5,1,2] => [5,4,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => 8

[3,4,5,2,1] => [5,4,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => 8

[3,5,1,2,4] => [4,5,3,1,2] => [2,5,1,4,3] => [2,5,1,4,3] => 8

[3,5,1,4,2] => [5,4,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => 8

[3,5,2,1,4] => [4,5,3,1,2] => [2,5,1,4,3] => [2,5,1,4,3] => 8

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$F_{2} = 1 + q^{2}$

$F_{3} = 1 + 2\ q^{2} + 3\ q^{4}$

$F_{4} = 1 + 3\ q^{2} + 8\ q^{4} + 7\ q^{6} + 5\ q^{8}$

$F_{5} = 1 + 4\ q^{2} + 17\ q^{4} + 20\ q^{6} + 53\ q^{8} + 7\ q^{10} + 18\ q^{12}$

$F_{6} = 1 + 5\ q^{2} + 27\ q^{4} + 55\ q^{6} + 137\ q^{8} + 199\ q^{10} + 144\ q^{12} + 57\ q^{14} + 95\ q^{16}$

Description

The total displacement of a permutation.
This is, for a permutation

$\pi$ of

$n$ , given by

$\sum_{i = 1}^n | \pi(i) - i |.$
This is twice the statistic St000029The depth of a permutation. and can be found in [3, Problem 5.1.1.28] and also in [1, 2].

Map

Inverse fireworks map

Description

Sends a permutation to an inverse fireworks permutation.
A permutation

$\sigma$ is inverse fireworks if its inverse avoids the vincular pattern

$3-12$ . The inverse fireworks map sends any permutation

$\sigma$ to an inverse fireworks permutation that is below

$\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..

Map

first fundamental transformation

Description

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

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

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