- FindStat

Identifier

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St001558: Permutations ⟶ ℤ

Values

[1] => [1] => 0

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

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

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

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

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

[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] => 0

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 6

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

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

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

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

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

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

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

[2,4,5,1,3] => [4,5,3,1,2] => 9

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

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

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

[2,5,3,1,4] => [4,5,3,1,2] => 9

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

[2,5,4,1,3] => [4,5,3,1,2] => 9

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

[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] => 6

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

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

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

[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] => 6

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

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

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

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

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

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

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

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

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

[3,5,1,2,4] => [4,5,3,1,2] => 9

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

>>> Load all 1041 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 1 + 3\ q + q^{2} + 6\ q^{3} + q^{5} + 12\ q^{6}$

$F_{5} = 1 + 4\ q + 3\ q^{2} + 9\ q^{3} + 6\ q^{4} + 2\ q^{5} + 24\ q^{6} + 6\ q^{8} + 5\ q^{9} + 60\ q^{10}$

$F_{6} = 1 + 5\ q + 6\ q^{2} + 13\ q^{3} + 18\ q^{4} + 3\ q^{5} + 47\ q^{6} + 24\ q^{7} + 12\ q^{8} + 10\ q^{9} + 121\ q^{10} + 9\ q^{11} + 25\ q^{12} + 36\ q^{13} + 30\ q^{14} + 360\ q^{15}$

Description

The number of transpositions that are smaller or equal to a permutation in Bruhat order.
A statistic is known to be smooth if and only if this number coincides with the number of inversions. This is also equivalent for a permutation to avoid the two pattern

$4231$ and

$3412$ .

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

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