- FindStat

Identifier

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000470: Permutations ⟶ ℤ (values match St000021The number of descents of a permutation., St000325The width of the tree associated to a permutation.)

Values

[1] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1041 entries. <<<

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$F_{1} = q$

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

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

$F_{4} = q + 4\ q^{2} + 12\ q^{3} + 7\ q^{4}$

$F_{5} = q + 6\ q^{2} + 40\ q^{3} + 38\ q^{4} + 35\ q^{5}$

$F_{6} = q + 9\ q^{2} + 104\ q^{3} + 208\ q^{4} + 263\ q^{5} + 135\ q^{6}$

Description

The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.

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

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