- FindStat

Identifier

Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00064: Permutations —reverse⟶ 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] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.

Map

major-index to inversion-number bijection

Description

Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.