- FindStat

Identifier

Mp00069: Permutations —complement⟶ Permutations
St000246: Permutations ⟶ ℤ (values match St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.)

Values

[1] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,5,4,3,1] => [4,1,2,3,5] => 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,4,5,2,1] => [3,2,1,4,5] => 7

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

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

$F_{5} = 1 + 4\ q + 9\ q^{2} + 15\ q^{3} + 20\ q^{4} + 22\ q^{5} + 20\ q^{6} + 15\ q^{7} + 9\ q^{8} + 4\ q^{9} + q^{10}$

$F_{6} = 1 + 5\ q + 14\ q^{2} + 29\ q^{3} + 49\ q^{4} + 71\ q^{5} + 90\ q^{6} + 101\ q^{7} + 101\ q^{8} + 90\ q^{9} + 71\ q^{10} + 49\ q^{11} + 29\ q^{12} + 14\ q^{13} + 5\ q^{14} + q^{15}$

Description

The number of non-inversions of a permutation.
For a permutation of

$\{1,\ldots,n\}$ , this is given by

$\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$ .

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation

$\sigma$ of length

$n$ is the permutation

$\tau$ with

$\tau(i) = n+1-\sigma(i)$