- FindStat

Identifier

Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St000433: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1352 entries. <<<

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$F_{1} = 1$

$F_{2} = 2$

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

$F_{4} = 7 + 6\ q + 9\ q^{2} + 2\ q^{4}$

$F_{5} = 11 + 10\ q + 24\ q^{2} + 22\ q^{3} + 26\ q^{4} + 9\ q^{5} + 5\ q^{6} + 11\ q^{7} + 2\ q^{10}$

$F_{6} = 16 + 14\ q + 38\ q^{2} + 44\ q^{3} + 94\ q^{4} + 59\ q^{5} + 120\ q^{6} + 82\ q^{7} + 61\ q^{8} + 45\ q^{9} + 61\ q^{10} + 25\ q^{11} + 23\ q^{12} + 11\ q^{13} + 12\ q^{14} + 13\ q^{16} + 2\ q^{20}$

Description

The number of occurrences of the pattern 132 or of the pattern 321 in a permutation.

Map

complement

Description

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

$\sigma$ of length

$n$ is the permutation

$\tau$ with

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

Map

Kreweras complement

Description

Sends the permutation

$\pi \in \mathfrak{S}_n$ to the permutation

$\pi^{-1}c$ where

$c = (1,\ldots,n)$ is the long cycle.

Map

Tanimoto

Description

Add 1 to every entry of the permutation (n becomes 1 instead of n+1), except that when n appears at the front or the back of the permutation, instead remove it and place 1 at the other end of the permutation.