- FindStat

Your data matches 20 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000018
(load all 5 compositions to match this statistic)

Mapping

Mp00310: Permutations —toric promotion⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [3,2,1] => 3
[1,3,2] => [2,3,1] => 2
[2,1,3] => [3,1,2] => 2
[2,3,1] => [1,3,2] => 1
[3,1,2] => [2,1,3] => 1
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,2,3,1] => 5
[1,2,4,3] => [4,3,1,2] => 5
[1,3,2,4] => [2,4,1,3] => 3
[1,3,4,2] => [2,3,4,1] => 3
[1,4,2,3] => [3,4,1,2] => 4
[1,4,3,2] => [3,1,2,4] => 2
[2,1,3,4] => [4,1,2,3] => 3
[2,1,4,3] => [4,1,3,2] => 4
[2,3,1,4] => [1,4,2,3] => 2
[2,3,4,1] => [4,2,1,3] => 4
[2,4,1,3] => [4,3,2,1] => 6
[2,4,3,1] => [1,4,3,2] => 3
[3,1,2,4] => [2,4,3,1] => 4
[3,1,4,2] => [1,2,3,4] => 0
[3,2,1,4] => [1,2,4,3] => 1
[3,2,4,1] => [2,1,4,3] => 2
[3,4,1,2] => [2,3,1,4] => 2
[3,4,2,1] => [2,1,3,4] => 1
[4,1,2,3] => [3,4,2,1] => 5
[4,1,3,2] => [3,2,4,1] => 4
[4,2,1,3] => [1,3,4,2] => 2
[4,2,3,1] => [3,1,4,2] => 3
[4,3,1,2] => [3,2,1,4] => 3
[4,3,2,1] => [1,3,2,4] => 1
[1,2,3,4,5] => [5,2,3,4,1] => 7
[1,2,3,5,4] => [5,2,4,1,3] => 7
[1,2,4,3,5] => [5,3,1,2,4] => 6
[1,2,4,5,3] => [5,3,4,1,2] => 8
[1,2,5,3,4] => [5,4,1,2,3] => 7
[1,2,5,4,3] => [5,4,1,3,2] => 8
[1,3,2,4,5] => [2,5,1,3,4] => 4
[1,3,2,5,4] => [2,5,1,4,3] => 5
[1,3,4,2,5] => [2,3,5,1,4] => 4
[1,3,4,5,2] => [2,3,4,5,1] => 4
[1,3,5,2,4] => [2,4,5,1,3] => 5
[1,3,5,4,2] => [2,4,1,3,5] => 3
[1,4,2,3,5] => [3,5,1,2,4] => 5
[1,4,2,5,3] => [3,5,1,4,2] => 6
[1,4,3,2,5] => [3,1,2,5,4] => 3
[1,4,3,5,2] => [3,1,2,4,5] => 2
[1,4,5,2,3] => [3,4,5,1,2] => 6

Description

The number of inversions of a permutation. This equals the minimal number of simple transpositions

$(i,i+1)$ needed to write

$\pi$ . Thus, it is also the Coxeter length of

$\pi$ .

Matching statistic: St000391
(load all 2 compositions to match this statistic)

Mapping

Mp00310: Permutations —toric promotion⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] =>  => ? = 0
[1,2] => [1,2] => [1,2] => 0 => 0
[2,1] => [2,1] => [2,1] => 1 => 1
[1,2,3] => [3,2,1] => [3,2,1] => 11 => 3
[1,3,2] => [2,3,1] => [2,3,1] => 01 => 2
[2,1,3] => [3,1,2] => [1,3,2] => 01 => 2
[2,3,1] => [1,3,2] => [3,1,2] => 10 => 1
[3,1,2] => [2,1,3] => [2,1,3] => 10 => 1
[3,2,1] => [1,2,3] => [1,2,3] => 00 => 0
[1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 011 => 5
[1,2,4,3] => [4,3,1,2] => [1,4,3,2] => 011 => 5
[1,3,2,4] => [2,4,1,3] => [4,2,1,3] => 110 => 3
[1,3,4,2] => [2,3,4,1] => [2,3,4,1] => 001 => 3
[1,4,2,3] => [3,4,1,2] => [3,1,4,2] => 101 => 4
[1,4,3,2] => [3,1,2,4] => [1,3,2,4] => 010 => 2
[2,1,3,4] => [4,1,2,3] => [1,2,4,3] => 001 => 3
[2,1,4,3] => [4,1,3,2] => [4,1,3,2] => 101 => 4
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 010 => 2
[2,3,4,1] => [4,2,1,3] => [2,1,4,3] => 101 => 4
[2,4,1,3] => [4,3,2,1] => [4,3,2,1] => 111 => 6
[2,4,3,1] => [1,4,3,2] => [4,3,1,2] => 110 => 3
[3,1,2,4] => [2,4,3,1] => [4,2,3,1] => 101 => 4
[3,1,4,2] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[3,2,1,4] => [1,2,4,3] => [4,1,2,3] => 100 => 1
[3,2,4,1] => [2,1,4,3] => [2,4,1,3] => 010 => 2
[3,4,1,2] => [2,3,1,4] => [2,3,1,4] => 010 => 2
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => 100 => 1
[4,1,2,3] => [3,4,2,1] => [3,4,2,1] => 011 => 5
[4,1,3,2] => [3,2,4,1] => [3,2,4,1] => 101 => 4
[4,2,1,3] => [1,3,4,2] => [3,4,1,2] => 010 => 2
[4,2,3,1] => [3,1,4,2] => [1,3,4,2] => 001 => 3
[4,3,1,2] => [3,2,1,4] => [3,2,1,4] => 110 => 3
[4,3,2,1] => [1,3,2,4] => [3,1,2,4] => 100 => 1
[1,2,3,4,5] => [5,2,3,4,1] => [2,3,5,4,1] => 0011 => 7
[1,2,3,5,4] => [5,2,4,1,3] => [5,2,1,4,3] => 1101 => 7
[1,2,4,3,5] => [5,3,1,2,4] => [1,3,2,5,4] => 0101 => 6
[1,2,4,5,3] => [5,3,4,1,2] => [3,1,5,4,2] => 1011 => 8
[1,2,5,3,4] => [5,4,1,2,3] => [1,2,5,4,3] => 0011 => 7
[1,2,5,4,3] => [5,4,1,3,2] => [5,1,4,3,2] => 1011 => 8
[1,3,2,4,5] => [2,5,1,3,4] => [2,1,5,3,4] => 1010 => 4
[1,3,2,5,4] => [2,5,1,4,3] => [2,5,4,1,3] => 0110 => 5
[1,3,4,2,5] => [2,3,5,1,4] => [5,2,3,1,4] => 1010 => 4
[1,3,4,5,2] => [2,3,4,5,1] => [2,3,4,5,1] => 0001 => 4
[1,3,5,2,4] => [2,4,5,1,3] => [4,5,2,1,3] => 0110 => 5
[1,3,5,4,2] => [2,4,1,3,5] => [4,2,1,3,5] => 1100 => 3
[1,4,2,3,5] => [3,5,1,2,4] => [1,5,3,2,4] => 0110 => 5
[1,4,2,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => 0101 => 6
[1,4,3,2,5] => [3,1,2,5,4] => [1,3,5,2,4] => 0010 => 3
[1,4,3,5,2] => [3,1,2,4,5] => [1,3,2,4,5] => 0100 => 2
[1,4,5,2,3] => [3,4,5,1,2] => [3,4,1,5,2] => 0101 => 6
[1,4,5,3,2] => [3,4,1,2,5] => [3,1,4,2,5] => 1010 => 4

Description

The sum of the positions of the ones in a binary word.

Matching statistic: St000330
(load all 2 compositions to match this statistic)

Mapping

Mp00310: Permutations —toric promotion⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => [[1]]
 => 0
[1,2] => [1,2] => [1,2] => [[1,2]]
 => 0
[2,1] => [2,1] => [2,1] => [[1],[2]]
 => 1
[1,2,3] => [3,2,1] => [3,2,1] => [[1],[2],[3]]
 => 3
[1,3,2] => [2,3,1] => [2,3,1] => [[1,2],[3]]
 => 2
[2,1,3] => [3,1,2] => [1,3,2] => [[1,2],[3]]
 => 2
[2,3,1] => [1,3,2] => [3,1,2] => [[1,3],[2]]
 => 1
[3,1,2] => [2,1,3] => [2,1,3] => [[1,3],[2]]
 => 1
[3,2,1] => [1,2,3] => [1,2,3] => [[1,2,3]]
 => 0
[1,2,3,4] => [4,2,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
 => 5
[1,2,4,3] => [4,3,1,2] => [1,4,3,2] => [[1,2],[3],[4]]
 => 5
[1,3,2,4] => [2,4,1,3] => [4,2,1,3] => [[1,4],[2],[3]]
 => 3
[1,3,4,2] => [2,3,4,1] => [2,3,4,1] => [[1,2,3],[4]]
 => 3
[1,4,2,3] => [3,4,1,2] => [3,1,4,2] => [[1,3],[2,4]]
 => 4
[1,4,3,2] => [3,1,2,4] => [1,3,2,4] => [[1,2,4],[3]]
 => 2
[2,1,3,4] => [4,1,2,3] => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[2,1,4,3] => [4,1,3,2] => [4,1,3,2] => [[1,3],[2],[4]]
 => 4
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [[1,2,4],[3]]
 => 2
[2,3,4,1] => [4,2,1,3] => [2,1,4,3] => [[1,3],[2,4]]
 => 4
[2,4,1,3] => [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 6
[2,4,3,1] => [1,4,3,2] => [4,3,1,2] => [[1,4],[2],[3]]
 => 3
[3,1,2,4] => [2,4,3,1] => [4,2,3,1] => [[1,3],[2],[4]]
 => 4
[3,1,4,2] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[3,2,1,4] => [1,2,4,3] => [4,1,2,3] => [[1,3,4],[2]]
 => 1
[3,2,4,1] => [2,1,4,3] => [2,4,1,3] => [[1,2],[3,4]]
 => 2
[3,4,1,2] => [2,3,1,4] => [2,3,1,4] => [[1,2,4],[3]]
 => 2
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
 => 1
[4,1,2,3] => [3,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
 => 5
[4,1,3,2] => [3,2,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
 => 4
[4,2,1,3] => [1,3,4,2] => [3,4,1,2] => [[1,2],[3,4]]
 => 2
[4,2,3,1] => [3,1,4,2] => [1,3,4,2] => [[1,2,3],[4]]
 => 3
[4,3,1,2] => [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
 => 3
[4,3,2,1] => [1,3,2,4] => [3,1,2,4] => [[1,3,4],[2]]
 => 1
[1,2,3,4,5] => [5,2,3,4,1] => [2,3,5,4,1] => [[1,2,3],[4],[5]]
 => 7
[1,2,3,5,4] => [5,2,4,1,3] => [5,2,1,4,3] => [[1,4],[2,5],[3]]
 => 7
[1,2,4,3,5] => [5,3,1,2,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 6
[1,2,4,5,3] => [5,3,4,1,2] => [3,1,5,4,2] => [[1,3],[2,4],[5]]
 => 8
[1,2,5,3,4] => [5,4,1,2,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => 7
[1,2,5,4,3] => [5,4,1,3,2] => [5,1,4,3,2] => [[1,3],[2],[4],[5]]
 => 8
[1,3,2,4,5] => [2,5,1,3,4] => [2,1,5,3,4] => [[1,3,5],[2,4]]
 => 4
[1,3,2,5,4] => [2,5,1,4,3] => [2,5,4,1,3] => [[1,2],[3,5],[4]]
 => 5
[1,3,4,2,5] => [2,3,5,1,4] => [5,2,3,1,4] => [[1,3,5],[2],[4]]
 => 4
[1,3,4,5,2] => [2,3,4,5,1] => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => 4
[1,3,5,2,4] => [2,4,5,1,3] => [4,5,2,1,3] => [[1,2],[3,5],[4]]
 => 5
[1,3,5,4,2] => [2,4,1,3,5] => [4,2,1,3,5] => [[1,4,5],[2],[3]]
 => 3
[1,4,2,3,5] => [3,5,1,2,4] => [1,5,3,2,4] => [[1,2,5],[3],[4]]
 => 5
[1,4,2,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => [[1,2],[3,4],[5]]
 => 6
[1,4,3,2,5] => [3,1,2,5,4] => [1,3,5,2,4] => [[1,2,3],[4,5]]
 => 3
[1,4,3,5,2] => [3,1,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => 2
[1,4,5,2,3] => [3,4,5,1,2] => [3,4,1,5,2] => [[1,2,4],[3,5]]
 => 6
[3,4,5,6,7,8,9,1,2] => [2,3,4,5,6,7,8,1,9] => [2,3,4,5,6,7,8,1,9] => [[1,2,3,4,5,6,7,9],[8]]
 => ? = 7
[2,1,3,4,5,6,7,8,9,10,11] => [11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
 => ? = 10
[9,2,1,3,4,5,6,7,8] => [1,8,9,2,3,4,5,6,7] => [1,2,3,4,8,5,9,6,7] => [[1,2,3,4,5,7,9],[6,8]]
 => ? = 12

Description

The (standard) major index of a standard tableau. A descent of a standard tableau

$T$ is an index

$i$ such that

$i+1$ appears in a row strictly below the row of

$i$ . The (standard) major index is the the sum of the descents.

Matching statistic: St000008
(load all 2 compositions to match this statistic)

Mapping

Mp00310: Permutations —toric promotion⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 97% ●values known / values provided: 99%●distinct values known / distinct values provided: 97%

Values

[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [2] => 0
[2,1] => [2,1] => [2,1] => [1,1] => 1
[1,2,3] => [3,2,1] => [3,2,1] => [1,1,1] => 3
[1,3,2] => [2,3,1] => [2,3,1] => [2,1] => 2
[2,1,3] => [3,1,2] => [1,3,2] => [2,1] => 2
[2,3,1] => [1,3,2] => [3,1,2] => [1,2] => 1
[3,1,2] => [2,1,3] => [2,1,3] => [1,2] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [3] => 0
[1,2,3,4] => [4,2,3,1] => [2,4,3,1] => [2,1,1] => 5
[1,2,4,3] => [4,3,1,2] => [1,4,3,2] => [2,1,1] => 5
[1,3,2,4] => [2,4,1,3] => [4,2,1,3] => [1,1,2] => 3
[1,3,4,2] => [2,3,4,1] => [2,3,4,1] => [3,1] => 3
[1,4,2,3] => [3,4,1,2] => [3,1,4,2] => [1,2,1] => 4
[1,4,3,2] => [3,1,2,4] => [1,3,2,4] => [2,2] => 2
[2,1,3,4] => [4,1,2,3] => [1,2,4,3] => [3,1] => 3
[2,1,4,3] => [4,1,3,2] => [4,1,3,2] => [1,2,1] => 4
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [2,2] => 2
[2,3,4,1] => [4,2,1,3] => [2,1,4,3] => [1,2,1] => 4
[2,4,1,3] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 6
[2,4,3,1] => [1,4,3,2] => [4,3,1,2] => [1,1,2] => 3
[3,1,2,4] => [2,4,3,1] => [4,2,3,1] => [1,2,1] => 4
[3,1,4,2] => [1,2,3,4] => [1,2,3,4] => [4] => 0
[3,2,1,4] => [1,2,4,3] => [4,1,2,3] => [1,3] => 1
[3,2,4,1] => [2,1,4,3] => [2,4,1,3] => [2,2] => 2
[3,4,1,2] => [2,3,1,4] => [2,3,1,4] => [2,2] => 2
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [1,3] => 1
[4,1,2,3] => [3,4,2,1] => [3,4,2,1] => [2,1,1] => 5
[4,1,3,2] => [3,2,4,1] => [3,2,4,1] => [1,2,1] => 4
[4,2,1,3] => [1,3,4,2] => [3,4,1,2] => [2,2] => 2
[4,2,3,1] => [3,1,4,2] => [1,3,4,2] => [3,1] => 3
[4,3,1,2] => [3,2,1,4] => [3,2,1,4] => [1,1,2] => 3
[4,3,2,1] => [1,3,2,4] => [3,1,2,4] => [1,3] => 1
[1,2,3,4,5] => [5,2,3,4,1] => [2,3,5,4,1] => [3,1,1] => 7
[1,2,3,5,4] => [5,2,4,1,3] => [5,2,1,4,3] => [1,1,2,1] => 7
[1,2,4,3,5] => [5,3,1,2,4] => [1,3,2,5,4] => [2,2,1] => 6
[1,2,4,5,3] => [5,3,4,1,2] => [3,1,5,4,2] => [1,2,1,1] => 8
[1,2,5,3,4] => [5,4,1,2,3] => [1,2,5,4,3] => [3,1,1] => 7
[1,2,5,4,3] => [5,4,1,3,2] => [5,1,4,3,2] => [1,2,1,1] => 8
[1,3,2,4,5] => [2,5,1,3,4] => [2,1,5,3,4] => [1,2,2] => 4
[1,3,2,5,4] => [2,5,1,4,3] => [2,5,4,1,3] => [2,1,2] => 5
[1,3,4,2,5] => [2,3,5,1,4] => [5,2,3,1,4] => [1,2,2] => 4
[1,3,4,5,2] => [2,3,4,5,1] => [2,3,4,5,1] => [4,1] => 4
[1,3,5,2,4] => [2,4,5,1,3] => [4,5,2,1,3] => [2,1,2] => 5
[1,3,5,4,2] => [2,4,1,3,5] => [4,2,1,3,5] => [1,1,3] => 3
[1,4,2,3,5] => [3,5,1,2,4] => [1,5,3,2,4] => [2,1,2] => 5
[1,4,2,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => [2,2,1] => 6
[1,4,3,2,5] => [3,1,2,5,4] => [1,3,5,2,4] => [3,2] => 3
[1,4,3,5,2] => [3,1,2,4,5] => [1,3,2,4,5] => [2,3] => 2
[1,4,5,2,3] => [3,4,5,1,2] => [3,4,1,5,2] => [2,2,1] => 6
[3,1,4,5,6,7,8,9,2] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => [9] => ? = 0
[3,4,5,6,7,8,9,1,2] => [2,3,4,5,6,7,8,1,9] => [2,3,4,5,6,7,8,1,9] => [7,2] => ? = 7
[3,4,5,6,7,8,9,10,1,2] => [2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1,10] => [8,2] => ? = 8
[8,7,6,5,4,3,9,2,1] => [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8,9] => [1,1,1,1,1,1,3] => ? = 21
[9,8,7,6,5,4,3,10,2,1] => [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9,10] => [1,1,1,1,1,1,1,3] => ? = 28
[9,2,1,3,4,5,6,7,8] => [1,8,9,2,3,4,5,6,7] => [1,2,3,4,8,5,9,6,7] => [5,2,2] => ? = 12
[10,2,1,3,4,5,6,7,8,9] => [1,9,10,2,3,4,5,6,7,8] => [1,2,3,4,5,9,6,10,7,8] => [6,2,2] => ? = 14
[3,1,4,5,6,7,8,9,10,2] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => [10] => ? = 0
[9,8,7,6,5,4,3,1,2] => [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,2] => ? = 28
[10,9,8,7,6,5,4,3,1,2] => [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1,10] => [1,1,1,1,1,1,1,1,2] => ? = 36
[1,8,9,2,3,4,5,6,7] => [7,8,9,1,2,3,4,5,6] => [1,2,3,7,4,8,5,9,6] => [4,2,2,1] => ? = 18
[1,10,2,3,4,5,6,7,8,9] => [9,10,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,9,7,10,8] => [7,2,1] => ? = 16
[1,9,10,2,3,4,5,6,7,8] => [8,9,10,1,2,3,4,5,6,7] => [1,2,3,4,8,5,9,6,10,7] => [5,2,2,1] => ? = 21

Description

The major index of the composition. The descents of a composition

$[c_1,c_2,\dots,c_k]$ are the partial sums

$c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$ , excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].

Matching statistic: St000246
(load all 4 compositions to match this statistic)

Mapping

Mp00310: Permutations —toric promotion⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 86%

Values

[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [1,2] => 1
[1,2,3] => [3,2,1] => [1,2,3] => 3
[1,3,2] => [2,3,1] => [2,1,3] => 2
[2,1,3] => [3,1,2] => [1,3,2] => 2
[2,3,1] => [1,3,2] => [3,1,2] => 1
[3,1,2] => [2,1,3] => [2,3,1] => 1
[3,2,1] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [4,2,3,1] => [1,3,2,4] => 5
[1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 5
[1,3,2,4] => [2,4,1,3] => [3,1,4,2] => 3
[1,3,4,2] => [2,3,4,1] => [3,2,1,4] => 3
[1,4,2,3] => [3,4,1,2] => [2,1,4,3] => 4
[1,4,3,2] => [3,1,2,4] => [2,4,3,1] => 2
[2,1,3,4] => [4,1,2,3] => [1,4,3,2] => 3
[2,1,4,3] => [4,1,3,2] => [1,4,2,3] => 4
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => 2
[2,3,4,1] => [4,2,1,3] => [1,3,4,2] => 4
[2,4,1,3] => [4,3,2,1] => [1,2,3,4] => 6
[2,4,3,1] => [1,4,3,2] => [4,1,2,3] => 3
[3,1,2,4] => [2,4,3,1] => [3,1,2,4] => 4
[3,1,4,2] => [1,2,3,4] => [4,3,2,1] => 0
[3,2,1,4] => [1,2,4,3] => [4,3,1,2] => 1
[3,2,4,1] => [2,1,4,3] => [3,4,1,2] => 2
[3,4,1,2] => [2,3,1,4] => [3,2,4,1] => 2
[3,4,2,1] => [2,1,3,4] => [3,4,2,1] => 1
[4,1,2,3] => [3,4,2,1] => [2,1,3,4] => 5
[4,1,3,2] => [3,2,4,1] => [2,3,1,4] => 4
[4,2,1,3] => [1,3,4,2] => [4,2,1,3] => 2
[4,2,3,1] => [3,1,4,2] => [2,4,1,3] => 3
[4,3,1,2] => [3,2,1,4] => [2,3,4,1] => 3
[4,3,2,1] => [1,3,2,4] => [4,2,3,1] => 1
[1,2,3,4,5] => [5,2,3,4,1] => [1,4,3,2,5] => 7
[1,2,3,5,4] => [5,2,4,1,3] => [1,4,2,5,3] => 7
[1,2,4,3,5] => [5,3,1,2,4] => [1,3,5,4,2] => 6
[1,2,4,5,3] => [5,3,4,1,2] => [1,3,2,5,4] => 8
[1,2,5,3,4] => [5,4,1,2,3] => [1,2,5,4,3] => 7
[1,2,5,4,3] => [5,4,1,3,2] => [1,2,5,3,4] => 8
[1,3,2,4,5] => [2,5,1,3,4] => [4,1,5,3,2] => 4
[1,3,2,5,4] => [2,5,1,4,3] => [4,1,5,2,3] => 5
[1,3,4,2,5] => [2,3,5,1,4] => [4,3,1,5,2] => 4
[1,3,4,5,2] => [2,3,4,5,1] => [4,3,2,1,5] => 4
[1,3,5,2,4] => [2,4,5,1,3] => [4,2,1,5,3] => 5
[1,3,5,4,2] => [2,4,1,3,5] => [4,2,5,3,1] => 3
[1,4,2,3,5] => [3,5,1,2,4] => [3,1,5,4,2] => 5
[1,4,2,5,3] => [3,5,1,4,2] => [3,1,5,2,4] => 6
[1,4,3,2,5] => [3,1,2,5,4] => [3,5,4,1,2] => 3
[1,4,3,5,2] => [3,1,2,4,5] => [3,5,4,2,1] => 2
[1,4,5,2,3] => [3,4,5,1,2] => [3,2,1,5,4] => 6
[1,2,5,6,7,3,4] => [7,4,5,6,1,2,3] => [1,4,3,2,7,6,5] => ? = 15
[1,2,6,7,3,4,5] => [7,5,6,1,2,3,4] => [1,3,2,7,6,5,4] => ? = 14
[1,2,7,3,4,5,6] => [7,6,1,2,3,4,5] => [1,2,7,6,5,4,3] => ? = 11
[1,3,2,4,5,6,7] => [2,7,1,3,4,5,6] => [6,1,7,5,4,3,2] => ? = 6
[1,3,4,5,2,6,7] => [2,3,4,7,1,5,6] => [6,5,4,1,7,3,2] => ? = 6
[1,3,4,5,6,2,7] => [2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => ? = 6
[1,3,4,5,7,2,6] => [2,3,4,6,7,1,5] => [6,5,4,2,1,7,3] => ? = 7
[1,3,4,6,7,2,5] => [2,3,5,6,7,1,4] => [6,5,3,2,1,7,4] => ? = 8
[1,3,5,4,6,7,2] => [2,4,1,3,5,6,7] => [6,4,7,5,3,2,1] => ? = 3
[1,3,6,2,4,5,7] => [2,5,7,1,3,4,6] => [6,3,1,7,5,4,2] => ? = 8
[1,3,6,4,5,7,2] => [2,5,1,3,4,6,7] => [6,3,7,5,4,2,1] => ? = 4
[1,3,7,2,4,5,6] => [2,6,7,1,3,4,5] => [6,2,1,7,5,4,3] => ? = 9
[1,3,7,4,5,6,2] => [2,6,1,3,4,5,7] => [6,2,7,5,4,3,1] => ? = 5
[1,4,3,5,6,2,7] => [3,1,2,4,5,7,6] => [5,7,6,4,3,1,2] => ? = 3
[1,4,3,5,6,7,2] => [3,1,2,4,5,6,7] => [5,7,6,4,3,2,1] => ? = 2
[1,4,5,3,6,7,2] => [3,4,1,2,5,6,7] => [5,4,7,6,3,2,1] => ? = 4
[1,4,6,7,3,5,2] => [3,5,6,1,2,4,7] => [5,3,2,7,6,4,1] => ? = 8
[1,4,7,3,5,6,2] => [3,6,1,2,4,5,7] => [5,2,7,6,4,3,1] => ? = 6
[1,5,3,4,6,2,7] => [4,1,2,3,5,7,6] => [4,7,6,5,3,1,2] => ? = 4
[1,5,3,4,6,7,2] => [4,1,2,3,5,6,7] => [4,7,6,5,3,2,1] => ? = 3
[1,5,6,7,3,4,2] => [4,5,6,1,2,3,7] => [4,3,2,7,6,5,1] => ? = 9
[1,5,7,3,4,2,6] => [4,6,1,2,3,7,5] => [4,2,7,6,5,1,3] => ? = 8
[1,5,7,3,4,6,2] => [4,6,1,2,3,5,7] => [4,2,7,6,5,3,1] => ? = 7
[1,6,3,4,2,5,7] => [5,1,2,3,7,4,6] => [3,7,6,5,1,4,2] => ? = 6
[1,6,3,4,5,2,7] => [5,1,2,3,4,7,6] => [3,7,6,5,4,1,2] => ? = 5
[1,6,3,4,5,7,2] => [5,1,2,3,4,6,7] => [3,7,6,5,4,2,1] => ? = 4
[1,6,3,4,7,2,5] => [5,1,2,3,6,7,4] => [3,7,6,5,2,1,4] => ? = 6
[1,6,3,4,7,5,2] => [5,1,2,3,6,4,7] => [3,7,6,5,2,4,1] => ? = 5
[1,6,3,7,4,5,2] => [5,1,2,6,3,4,7] => [3,7,6,2,5,4,1] => ? = 6
[1,6,7,2,3,4,5] => [5,6,7,1,2,3,4] => [3,2,1,7,6,5,4] => ? = 12
[1,6,7,3,4,2,5] => [5,6,1,2,3,7,4] => [3,2,7,6,5,1,4] => ? = 9
[1,6,7,3,4,5,2] => [5,6,1,2,3,4,7] => [3,2,7,6,5,4,1] => ? = 8
[1,7,3,2,4,5,6] => [6,1,2,7,3,4,5] => [2,7,6,1,5,4,3] => ? = 8
[1,7,3,4,2,5,6] => [6,1,2,3,7,4,5] => [2,7,6,5,1,4,3] => ? = 7
[1,7,3,4,5,2,6] => [6,1,2,3,4,7,5] => [2,7,6,5,4,1,3] => ? = 6
[1,7,3,4,5,6,2] => [6,1,2,3,4,5,7] => [2,7,6,5,4,3,1] => ? = 5
[2,3,1,4,5,6,7] => [1,7,2,3,4,5,6] => [7,1,6,5,4,3,2] => ? = 5
[2,4,1,6,7,5,3] => [1,7,3,5,6,4,2] => [7,1,5,3,2,4,6] => ? = 11
[2,4,1,7,6,5,3] => [1,7,3,6,5,4,2] => [7,1,5,2,3,4,6] => ? = 12
[2,5,1,6,4,7,3] => [1,7,4,5,3,6,2] => [7,1,4,3,5,2,6] => ? = 11
[2,5,1,6,7,4,3] => [1,7,4,5,6,3,2] => [7,1,4,3,2,5,6] => ? = 12
[2,5,1,7,6,4,3] => [1,7,4,6,5,3,2] => [7,1,4,2,3,5,6] => ? = 13
[2,5,4,1,7,6,3] => [1,7,4,3,6,5,2] => [7,1,4,5,2,3,6] => ? = 11
[2,6,5,1,7,4,3] => [1,7,5,4,6,3,2] => [7,1,3,4,2,5,6] => ? = 13
[2,6,5,4,1,7,3] => [1,7,5,4,3,6,2] => [7,1,3,4,5,2,6] => ? = 12
[2,6,7,1,4,5,3] => [7,5,6,3,4,1,2] => [1,3,2,5,4,7,6] => ? = 18
[2,7,1,5,6,3,4] => [7,6,4,5,1,2,3] => [1,2,4,3,7,6,5] => ? = 17
[2,7,1,6,3,4,5] => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => ? = 15
[2,7,5,1,6,4,3] => [1,7,6,4,5,3,2] => [7,1,2,4,3,5,6] => ? = 14
[2,7,6,1,4,5,3] => [7,6,5,3,4,1,2] => [1,2,3,5,4,7,6] => ? = 19

Description

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

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

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

Matching statistic: St000081

Mapping

Mp00310: Permutations —toric promotion⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 86%

Values

[1] => [1] => ([],1)
 => 0
[1,2] => [1,2] => ([],2)
 => 0
[2,1] => [2,1] => ([(0,1)],2)
 => 1
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[2,3,1] => [1,3,2] => ([(1,2)],3)
 => 1
[3,1,2] => [2,1,3] => ([(1,2)],3)
 => 1
[3,2,1] => [1,2,3] => ([],3)
 => 0
[1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 5
[1,2,4,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 5
[1,3,2,4] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 3
[1,3,4,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[1,4,3,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
[2,1,3,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,1,4,3] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 2
[2,3,4,1] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[2,4,3,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,4,2] => [1,2,3,4] => ([],4)
 => 0
[3,2,1,4] => [1,2,4,3] => ([(2,3)],4)
 => 1
[3,2,4,1] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
[3,4,1,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
[3,4,2,1] => [2,1,3,4] => ([(2,3)],4)
 => 1
[4,1,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 5
[4,1,3,2] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 2
[4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 3
[4,3,1,2] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => [1,3,2,4] => ([(2,3)],4)
 => 1
[1,2,3,4,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
[1,2,3,5,4] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
[1,2,4,3,5] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[1,2,4,5,3] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 8
[1,2,5,3,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
[1,2,5,4,3] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8
[1,3,2,4,5] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 4
[1,3,2,5,4] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 5
[1,3,4,2,5] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 4
[1,3,4,5,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 5
[1,3,5,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,4,2,3,5] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 5
[1,4,2,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 6
[1,4,3,2,5] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 2
[1,4,5,2,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 6
[1,2,4,5,3,6,7] => [7,3,4,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 10
[1,2,5,6,7,3,4] => [7,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 15
[1,2,6,7,3,4,5] => [7,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 14
[1,3,4,5,7,2,6] => [2,3,4,6,7,1,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 7
[1,3,4,6,7,2,5] => [2,3,5,6,7,1,4] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 8
[1,3,5,4,6,7,2] => [2,4,1,3,5,6,7] => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,3,6,2,4,5,7] => [2,5,7,1,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 8
[1,3,6,4,5,7,2] => [2,5,1,3,4,6,7] => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
[1,3,7,2,4,5,6] => [2,6,7,1,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 9
[1,3,7,4,5,6,2] => [2,6,1,3,4,5,7] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 5
[1,4,3,5,6,2,7] => [3,1,2,4,5,7,6] => ([(2,3),(4,6),(5,6)],7)
 => ? = 3
[1,4,5,3,6,7,2] => [3,4,1,2,5,6,7] => ([(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 4
[1,4,5,6,7,2,3] => [3,4,5,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 10
[1,4,6,7,3,5,2] => [3,5,6,1,2,4,7] => ([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 8
[1,4,7,3,5,6,2] => [3,6,1,2,4,5,7] => ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 6
[1,5,3,4,6,2,7] => [4,1,2,3,5,7,6] => ([(1,2),(3,6),(4,6),(5,6)],7)
 => ? = 4
[1,5,6,7,3,4,2] => [4,5,6,1,2,3,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 9
[1,5,7,3,4,2,6] => [4,6,1,2,3,7,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => ? = 8
[1,5,7,3,4,6,2] => [4,6,1,2,3,5,7] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 7
[1,6,3,4,5,2,7] => [5,1,2,3,4,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 5
[1,6,3,4,7,5,2] => [5,1,2,3,6,4,7] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 5
[1,6,3,7,4,5,2] => [5,1,2,6,3,4,7] => ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 6
[1,6,7,2,3,4,5] => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 12
[1,6,7,3,4,2,5] => [5,6,1,2,3,7,4] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 9
[1,6,7,3,4,5,2] => [5,6,1,2,3,4,7] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 8
[1,7,2,3,4,5,6] => [6,7,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 10
[1,7,3,2,4,5,6] => [6,1,2,7,3,4,5] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 8
[1,7,3,4,2,5,6] => [6,1,2,3,7,4,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 7
[2,5,4,1,7,6,3] => [1,7,4,3,6,5,2] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 11
[2,6,7,1,4,5,3] => [7,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 18
[2,7,1,5,6,3,4] => [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 17
[3,1,4,7,5,2,6] => [1,2,3,6,4,7,5] => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[3,1,7,4,5,2,6] => [1,2,6,3,4,7,5] => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
[3,2,4,1,5,6,7] => [2,1,7,3,4,5,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 5
[3,2,7,6,5,4,1] => [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 11
[3,4,1,5,2,6,7] => [2,1,3,4,7,5,6] => ([(2,3),(4,6),(5,6)],7)
 => ? = 3
[3,4,2,1,5,6,7] => [2,1,3,7,4,5,6] => ([(1,2),(3,6),(4,6),(5,6)],7)
 => ? = 4
[3,4,5,6,2,1,7] => [2,3,4,1,5,7,6] => ([(1,2),(3,6),(4,6),(5,6)],7)
 => ? = 4
[3,4,5,6,2,7,1] => [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 5
[3,7,2,4,1,5,6] => [2,6,1,7,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => ? = 8
[3,7,4,1,5,6,2] => [2,1,6,3,4,5,7] => ([(1,2),(3,6),(4,6),(5,6)],7)
 => ? = 4
[4,2,1,3,5,6,7] => [1,3,7,2,4,5,6] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 5
[5,1,3,6,4,7,2] => [4,2,5,1,3,6,7] => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 6
[5,1,7,2,3,4,6] => [1,4,6,7,2,3,5] => ([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 8
[5,2,1,3,4,6,7] => [1,4,7,2,3,5,6] => ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 6
[5,4,7,6,3,1,2] => [4,3,6,5,2,1,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 11
[5,6,1,3,4,7,2] => [4,5,2,3,6,7,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 10
[5,6,7,3,1,4,2] => [4,5,1,6,2,3,7] => ([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 8
[6,1,7,2,3,4,5] => [1,5,6,7,2,3,4] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 9
[6,1,7,3,2,4,5] => [1,5,6,2,7,3,4] => ([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 8

Description

The number of edges of a graph.

Matching statistic: St001397
(load all 2 compositions to match this statistic)

Mapping

Mp00310: Permutations —toric promotion⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 69% ●values known / values provided: 76%●distinct values known / distinct values provided: 69%

Values

[1] => [1] => ([],1)
 => 0
[1,2] => [1,2] => ([(0,1)],2)
 => 0
[2,1] => [2,1] => ([],2)
 => 1
[1,2,3] => [3,2,1] => ([],3)
 => 3
[1,3,2] => [2,3,1] => ([(1,2)],3)
 => 2
[2,1,3] => [3,1,2] => ([(1,2)],3)
 => 2
[2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => 1
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 0
[1,2,3,4] => [4,2,3,1] => ([(2,3)],4)
 => 5
[1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
 => 5
[1,3,2,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 3
[1,3,4,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 3
[1,4,2,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 4
[1,4,3,2] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[2,1,3,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 3
[2,1,4,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => 4
[2,3,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[2,3,4,1] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 4
[2,4,1,3] => [4,3,2,1] => ([],4)
 => 6
[2,4,3,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[3,1,2,4] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 4
[3,1,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0
[3,2,1,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 1
[3,2,4,1] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,4,1,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[3,4,2,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 1
[4,1,2,3] => [3,4,2,1] => ([(2,3)],4)
 => 5
[4,1,3,2] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 4
[4,2,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 2
[4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 3
[4,3,1,2] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,2,3,4,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 7
[1,2,3,5,4] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => 7
[1,2,4,3,5] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 6
[1,2,4,5,3] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 8
[1,2,5,3,4] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 7
[1,2,5,4,3] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
 => 8
[1,3,2,4,5] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => 4
[1,3,2,5,4] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 5
[1,3,4,2,5] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => 4
[1,3,4,5,2] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 4
[1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 5
[1,3,5,4,2] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[1,4,2,3,5] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 5
[1,4,2,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 6
[1,4,3,2,5] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => 3
[1,4,3,5,2] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[1,4,5,2,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => 6
[1,2,4,3,5,6,7] => [7,3,1,2,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => ? = 8
[1,2,4,5,3,6,7] => [7,3,4,1,2,5,6] => ([(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ? = 10
[1,2,5,3,4,6,7] => [7,4,1,2,3,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
 => ? = 9
[1,2,5,6,7,3,4] => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
 => ? = 15
[1,2,6,7,3,4,5] => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 14
[1,2,7,3,4,5,6] => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ? = 11
[1,3,2,4,5,6,7] => [2,7,1,3,4,5,6] => ([(0,6),(1,3),(1,6),(4,2),(5,4),(6,5)],7)
 => ? = 6
[1,3,4,5,2,6,7] => [2,3,4,7,1,5,6] => ([(0,6),(1,4),(4,5),(5,2),(5,6),(6,3)],7)
 => ? = 6
[1,3,4,5,6,2,7] => [2,3,4,5,7,1,6] => ([(0,6),(1,4),(3,5),(4,3),(5,2),(5,6)],7)
 => ? = 6
[1,3,4,5,6,7,2] => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 6
[1,3,4,5,7,2,6] => [2,3,4,6,7,1,5] => ([(0,6),(1,4),(3,2),(4,5),(5,3),(5,6)],7)
 => ? = 7
[1,3,4,6,7,2,5] => [2,3,5,6,7,1,4] => ([(0,6),(1,5),(3,4),(4,2),(5,3),(5,6)],7)
 => ? = 8
[1,3,5,4,6,7,2] => [2,4,1,3,5,6,7] => ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => ? = 3
[1,3,6,2,4,5,7] => [2,5,7,1,3,4,6] => ([(0,6),(1,4),(1,6),(3,5),(4,2),(4,5),(6,3)],7)
 => ? = 8
[1,3,6,4,5,7,2] => [2,5,1,3,4,6,7] => ([(0,6),(1,3),(1,6),(2,5),(3,5),(5,4),(6,2)],7)
 => ? = 4
[1,3,7,2,4,5,6] => [2,6,7,1,3,4,5] => ([(0,6),(1,4),(1,6),(4,3),(5,2),(6,5)],7)
 => ? = 9
[1,3,7,4,5,6,2] => [2,6,1,3,4,5,7] => ([(0,6),(1,3),(1,6),(2,5),(3,5),(4,2),(6,4)],7)
 => ? = 5
[1,4,3,5,6,2,7] => [3,1,2,4,5,7,6] => ([(0,6),(1,4),(4,6),(5,2),(5,3),(6,5)],7)
 => ? = 3
[1,4,3,5,6,7,2] => [3,1,2,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => ? = 2
[1,4,5,3,6,7,2] => [3,4,1,2,5,6,7] => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
 => ? = 4
[1,4,5,6,7,2,3] => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 10
[1,4,6,7,3,5,2] => [3,5,6,1,2,4,7] => ([(0,3),(1,4),(1,6),(2,5),(3,6),(4,2),(6,5)],7)
 => ? = 8
[1,4,7,3,5,6,2] => [3,6,1,2,4,5,7] => ([(0,3),(1,4),(1,6),(2,5),(3,6),(4,5),(6,2)],7)
 => ? = 6
[1,5,3,4,6,2,7] => [4,1,2,3,5,7,6] => ([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
 => ? = 4
[1,5,3,4,6,7,2] => [4,1,2,3,5,6,7] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
 => ? = 3
[1,5,6,7,3,4,2] => [4,5,6,1,2,3,7] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
 => ? = 9
[1,5,7,3,4,2,6] => [4,6,1,2,3,7,5] => ([(0,3),(0,6),(1,4),(2,5),(2,6),(3,5),(4,2)],7)
 => ? = 8
[1,5,7,3,4,6,2] => [4,6,1,2,3,5,7] => ([(0,3),(0,6),(1,4),(2,6),(3,5),(4,2),(6,5)],7)
 => ? = 7
[1,6,3,4,2,5,7] => [5,1,2,3,7,4,6] => ([(0,5),(0,6),(1,3),(2,6),(3,4),(4,2),(4,5)],7)
 => ? = 6
[1,6,3,4,5,2,7] => [5,1,2,3,4,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
 => ? = 5
[1,6,3,4,7,2,5] => [5,1,2,3,6,7,4] => ([(0,6),(1,4),(4,5),(5,2),(5,6),(6,3)],7)
 => ? = 6
[1,6,3,4,7,5,2] => [5,1,2,3,6,4,7] => ([(0,6),(1,4),(2,5),(3,2),(3,6),(4,3),(6,5)],7)
 => ? = 5
[1,6,3,7,4,5,2] => [5,1,2,6,3,4,7] => ([(0,6),(1,4),(2,5),(3,2),(4,3),(4,6),(6,5)],7)
 => ? = 6
[1,6,7,2,3,4,5] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ? = 12
[1,6,7,3,4,2,5] => [5,6,1,2,3,7,4] => ([(0,5),(1,3),(3,6),(4,2),(4,6),(5,4)],7)
 => ? = 9
[1,6,7,3,4,5,2] => [5,6,1,2,3,4,7] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
 => ? = 8
[1,7,2,3,4,5,6] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 10
[1,7,3,2,4,5,6] => [6,1,2,7,3,4,5] => ([(0,6),(1,5),(3,4),(4,2),(5,3),(5,6)],7)
 => ? = 8
[1,7,3,4,2,5,6] => [6,1,2,3,7,4,5] => ([(0,6),(1,4),(3,2),(4,5),(5,3),(5,6)],7)
 => ? = 7
[1,7,3,4,5,2,6] => [6,1,2,3,4,7,5] => ([(0,6),(1,4),(3,5),(4,3),(5,2),(5,6)],7)
 => ? = 6
[1,7,3,4,5,6,2] => [6,1,2,3,4,5,7] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ? = 5
[2,1,3,4,5,6,7] => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 6
[2,3,1,4,5,6,7] => [1,7,2,3,4,5,6] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => ? = 5
[2,3,4,5,1,6,7] => [7,2,3,1,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => ? = 8
[2,3,4,5,6,1,7] => [7,2,3,4,1,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
 => ? = 9
[2,4,1,6,7,5,3] => [1,7,3,5,6,4,2] => ([(0,3),(0,4),(0,6),(5,2),(6,1),(6,5)],7)
 => ? = 11
[2,6,7,1,4,5,3] => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
 => ? = 18
[2,7,1,5,6,3,4] => [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
 => ? = 17
[3,1,4,5,6,2,7] => [1,2,3,4,5,7,6] => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
 => ? = 1
[3,1,4,5,6,7,2] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 0

Description

Number of pairs of incomparable elements in a finite poset. For a finite poset

$(P,\leq)$ , this is the number of unordered pairs

$\{x,y\} \in \binom{P}{2}$ with

$x \not\leq y$ and

$y \not\leq x$ .

Matching statistic: St000833
(load all 2 compositions to match this statistic)

Mapping

Mp00310: Permutations —toric promotion⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000833: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 71%●distinct values known / distinct values provided: 55%

Values

[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3] => [3,2,1] => [3,2,1] => [3,2,1] => 3
[1,3,2] => [2,3,1] => [2,3,1] => [2,1,3] => 2
[2,1,3] => [3,1,2] => [1,3,2] => [3,1,2] => 2
[2,3,1] => [1,3,2] => [3,1,2] => [1,3,2] => 1
[3,1,2] => [2,1,3] => [2,1,3] => [2,3,1] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,2,3,1] => [2,4,3,1] => [4,2,1,3] => 5
[1,2,4,3] => [4,3,1,2] => [1,4,3,2] => [4,3,1,2] => 5
[1,3,2,4] => [2,4,1,3] => [4,2,1,3] => [2,4,3,1] => 3
[1,3,4,2] => [2,3,4,1] => [2,3,4,1] => [2,1,3,4] => 3
[1,4,2,3] => [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 4
[1,4,3,2] => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[2,1,3,4] => [4,1,2,3] => [1,2,4,3] => [4,1,2,3] => 3
[2,1,4,3] => [4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 4
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[2,3,4,1] => [4,2,1,3] => [2,1,4,3] => [2,1,4,3] => 4
[2,4,1,3] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[2,4,3,1] => [1,4,3,2] => [4,3,1,2] => [1,4,3,2] => 3
[3,1,2,4] => [2,4,3,1] => [4,2,3,1] => [4,2,3,1] => 4
[3,1,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,2,1,4] => [1,2,4,3] => [4,1,2,3] => [1,2,4,3] => 1
[3,2,4,1] => [2,1,4,3] => [2,4,1,3] => [2,4,1,3] => 2
[3,4,1,2] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [2,3,4,1] => 1
[4,1,2,3] => [3,4,2,1] => [3,4,2,1] => [3,2,1,4] => 5
[4,1,3,2] => [3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 4
[4,2,1,3] => [1,3,4,2] => [3,4,1,2] => [3,4,1,2] => 2
[4,2,3,1] => [3,1,4,2] => [1,3,4,2] => [3,1,2,4] => 3
[4,3,1,2] => [3,2,1,4] => [3,2,1,4] => [3,4,2,1] => 3
[4,3,2,1] => [1,3,2,4] => [3,1,2,4] => [1,3,4,2] => 1
[1,2,3,4,5] => [5,2,3,4,1] => [2,3,5,4,1] => [5,2,1,3,4] => 7
[1,2,3,5,4] => [5,2,4,1,3] => [5,2,1,4,3] => [2,1,5,4,3] => 7
[1,2,4,3,5] => [5,3,1,2,4] => [1,3,2,5,4] => [3,1,5,2,4] => 6
[1,2,4,5,3] => [5,3,4,1,2] => [3,1,5,4,2] => [5,3,1,4,2] => 8
[1,2,5,3,4] => [5,4,1,2,3] => [1,2,5,4,3] => [5,4,1,2,3] => 7
[1,2,5,4,3] => [5,4,1,3,2] => [5,1,4,3,2] => [5,4,1,3,2] => 8
[1,3,2,4,5] => [2,5,1,3,4] => [2,1,5,3,4] => [2,3,1,5,4] => 4
[1,3,2,5,4] => [2,5,1,4,3] => [2,5,4,1,3] => [2,5,4,1,3] => 5
[1,3,4,2,5] => [2,3,5,1,4] => [5,2,3,1,4] => [2,5,3,4,1] => 4
[1,3,4,5,2] => [2,3,4,5,1] => [2,3,4,5,1] => [2,1,3,4,5] => 4
[1,3,5,2,4] => [2,4,5,1,3] => [4,5,2,1,3] => [4,5,2,1,3] => 5
[1,3,5,4,2] => [2,4,1,3,5] => [4,2,1,3,5] => [2,4,5,3,1] => 3
[1,4,2,3,5] => [3,5,1,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 5
[1,4,2,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => [5,3,4,1,2] => 6
[1,4,3,2,5] => [3,1,2,5,4] => [1,3,5,2,4] => [3,5,1,2,4] => 3
[1,4,3,5,2] => [3,1,2,4,5] => [1,3,2,4,5] => [1,3,4,2,5] => 2
[1,4,5,2,3] => [3,4,5,1,2] => [3,4,1,5,2] => [3,1,4,2,5] => 6
[1,4,5,3,2] => [3,4,1,2,5] => [3,1,4,2,5] => [3,4,1,5,2] => 4
[1,2,4,3,5,6,7] => [7,3,1,2,4,5,6] => [1,3,2,4,5,7,6] => [3,1,2,4,7,5,6] => ? = 8
[1,2,4,5,3,6,7] => [7,3,4,1,2,5,6] => [3,1,4,2,5,7,6] => [3,1,4,5,2,7,6] => ? = 10
[1,2,5,3,4,6,7] => [7,4,1,2,3,5,6] => [1,2,4,3,5,7,6] => [4,1,2,7,3,5,6] => ? = 9
[1,2,5,6,7,3,4] => [7,4,5,6,1,2,3] => [4,1,5,2,7,6,3] => [7,4,1,5,2,6,3] => ? = 15
[1,2,6,3,4,5,7] => [7,5,1,2,3,4,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 10
[1,2,6,7,3,4,5] => [7,5,6,1,2,3,4] => [1,2,5,3,7,6,4] => [7,5,1,6,2,3,4] => ? = 14
[1,2,7,3,4,5,6] => [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 11
[1,3,2,4,5,6,7] => [2,7,1,3,4,5,6] => [2,1,3,4,7,5,6] => [2,3,1,4,5,7,6] => ? = 6
[1,3,4,5,2,6,7] => [2,3,4,7,1,5,6] => [2,7,3,4,1,5,6] => [2,3,7,4,5,1,6] => ? = 6
[1,3,4,5,6,2,7] => [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => [2,7,3,4,5,6,1] => ? = 6
[1,3,4,5,6,7,2] => [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => [2,1,3,4,5,6,7] => ? = 6
[1,3,4,5,7,2,6] => [2,3,4,6,7,1,5] => [6,7,2,3,4,1,5] => [2,6,3,4,7,1,5] => ? = 7
[1,3,4,6,7,2,5] => [2,3,5,6,7,1,4] => [5,6,7,2,3,1,4] => [5,6,2,7,1,3,4] => ? = 8
[1,3,5,4,6,7,2] => [2,4,1,3,5,6,7] => [4,2,1,3,5,6,7] => [2,4,5,6,7,3,1] => ? = 3
[1,3,6,2,4,5,7] => [2,5,7,1,3,4,6] => [2,1,7,5,3,4,6] => [2,3,7,5,1,6,4] => ? = 8
[1,3,6,4,5,7,2] => [2,5,1,3,4,6,7] => [2,1,5,3,4,6,7] => [2,3,5,6,1,7,4] => ? = 4
[1,3,7,2,4,5,6] => [2,6,7,1,3,4,5] => [2,1,6,3,7,4,5] => [2,3,1,6,4,7,5] => ? = 9
[1,3,7,4,5,6,2] => [2,6,1,3,4,5,7] => [2,1,3,6,4,5,7] => [2,3,4,1,6,7,5] => ? = 5
[1,4,5,3,6,7,2] => [3,4,1,2,5,6,7] => [3,1,4,2,5,6,7] => [3,4,5,6,1,7,2] => ? = 4
[1,4,5,6,7,2,3] => [3,4,5,6,7,1,2] => [3,4,5,6,1,7,2] => [3,1,4,2,5,6,7] => ? = 10
[1,4,6,7,3,5,2] => [3,5,6,1,2,4,7] => [5,1,6,3,2,4,7] => [1,5,6,3,2,7,4] => ? = 8
[1,5,3,4,6,2,7] => [4,1,2,3,5,7,6] => [1,2,4,7,3,5,6] => [1,4,7,2,3,5,6] => ? = 4
[1,5,6,7,3,4,2] => [4,5,6,1,2,3,7] => [4,1,5,2,6,3,7] => [4,5,1,6,2,7,3] => ? = 9
[1,5,7,3,4,2,6] => [4,6,1,2,3,7,5] => [1,2,6,4,7,3,5] => [6,7,1,4,2,3,5] => ? = 8
[1,6,3,4,2,5,7] => [5,1,2,3,7,4,6] => [5,1,2,3,7,4,6] => [1,5,2,3,4,7,6] => ? = 6
[1,6,3,4,5,2,7] => [5,1,2,3,4,7,6] => [1,2,3,5,7,4,6] => [5,7,1,2,3,4,6] => ? = 5
[1,6,3,4,7,2,5] => [5,1,2,3,6,7,4] => [1,2,3,5,6,7,4] => [5,1,2,3,4,6,7] => ? = 6
[1,6,3,4,7,5,2] => [5,1,2,3,6,4,7] => [1,2,3,5,6,4,7] => [1,5,2,3,4,6,7] => ? = 5
[1,6,3,7,4,5,2] => [5,1,2,6,3,4,7] => [5,1,2,3,6,4,7] => [1,5,2,3,6,7,4] => ? = 6
[1,6,7,2,3,4,5] => [5,6,7,1,2,3,4] => [1,5,2,6,3,7,4] => [5,1,6,2,7,3,4] => ? = 12
[1,6,7,3,4,2,5] => [5,6,1,2,3,7,4] => [1,2,5,3,6,7,4] => [5,1,2,6,3,4,7] => ? = 9
[1,6,7,3,4,5,2] => [5,6,1,2,3,4,7] => [1,2,5,3,6,4,7] => [1,5,2,6,3,4,7] => ? = 8
[1,7,2,3,4,5,6] => [6,7,1,2,3,4,5] => [1,2,3,6,4,7,5] => [6,1,7,2,3,4,5] => ? = 10
[1,7,3,2,4,5,6] => [6,1,2,7,3,4,5] => [1,6,2,3,4,7,5] => [6,1,2,3,7,4,5] => ? = 8
[1,7,3,4,2,5,6] => [6,1,2,3,7,4,5] => [6,1,2,3,4,7,5] => [6,1,2,3,4,7,5] => ? = 7
[1,7,3,4,5,2,6] => [6,1,2,3,4,7,5] => [1,2,3,4,6,7,5] => [6,1,2,3,4,5,7] => ? = 6
[1,7,3,4,5,6,2] => [6,1,2,3,4,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 5
[2,1,3,4,5,6,7] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 6
[2,3,1,4,5,6,7] => [1,7,2,3,4,5,6] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 5
[2,3,4,1,5,6,7] => [7,2,1,3,4,5,6] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 7
[2,3,4,5,1,6,7] => [7,2,3,1,4,5,6] => [2,3,1,4,5,7,6] => [2,1,3,4,7,5,6] => ? = 8
[2,3,4,5,6,1,7] => [7,2,3,4,1,5,6] => [2,3,4,1,5,7,6] => [2,1,3,7,4,5,6] => ? = 9
[2,3,4,5,6,7,1] => [7,2,3,4,5,1,6] => [2,3,4,5,1,7,6] => [2,1,7,3,4,5,6] => ? = 10
[2,4,1,6,7,5,3] => [1,7,3,5,6,4,2] => [3,7,5,6,4,1,2] => [3,7,5,1,4,2,6] => ? = 11
[2,4,1,7,6,5,3] => [1,7,3,6,5,4,2] => [7,6,3,5,4,1,2] => [3,7,6,1,5,4,2] => ? = 12
[2,4,3,5,1,6,7] => [7,3,2,1,4,5,6] => [3,2,1,4,5,7,6] => [3,2,4,5,7,6,1] => ? = 9
[2,4,3,5,6,1,7] => [7,3,2,4,1,5,6] => [3,2,4,1,5,7,6] => [3,2,4,7,5,6,1] => ? = 10
[2,5,1,6,4,7,3] => [1,7,4,5,3,6,2] => [4,5,3,7,6,1,2] => [4,7,3,1,5,2,6] => ? = 11
[2,5,1,6,7,4,3] => [1,7,4,5,6,3,2] => [4,5,7,6,3,1,2] => [4,7,5,3,1,2,6] => ? = 12

Description

The comajor index of a permutation. This is,

$\operatorname{comaj}(\pi) = \sum_{i \in \operatorname{Des}(\pi)} (n-i)$ for a permutation

$\pi$ of length

$n$ .

Matching statistic: St000795
(load all 2 compositions to match this statistic)

Mapping

Mp00310: Permutations —toric promotion⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000795: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 71%●distinct values known / distinct values provided: 55%

Values

[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3] => [3,2,1] => [3,2,1] => [2,3,1] => 3
[1,3,2] => [2,3,1] => [3,1,2] => [3,1,2] => 2
[2,1,3] => [3,1,2] => [2,3,1] => [3,2,1] => 2
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 5
[1,2,4,3] => [4,3,1,2] => [3,4,2,1] => [4,2,3,1] => 5
[1,3,2,4] => [2,4,1,3] => [3,1,4,2] => [4,3,1,2] => 3
[1,3,4,2] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 3
[1,4,2,3] => [3,4,1,2] => [3,4,1,2] => [4,1,3,2] => 4
[1,4,3,2] => [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 2
[2,1,3,4] => [4,1,2,3] => [2,3,4,1] => [4,3,2,1] => 3
[2,1,4,3] => [4,1,3,2] => [2,4,3,1] => [3,4,2,1] => 4
[2,3,1,4] => [1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 2
[2,3,4,1] => [4,2,1,3] => [3,2,4,1] => [2,4,3,1] => 4
[2,4,1,3] => [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 6
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 3
[3,1,2,4] => [2,4,3,1] => [4,1,3,2] => [3,4,1,2] => 4
[3,1,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,2,1,4] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,1,2] => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[4,1,2,3] => [3,4,2,1] => [4,3,1,2] => [3,1,4,2] => 5
[4,1,3,2] => [3,2,4,1] => [4,2,1,3] => [2,4,1,3] => 4
[4,2,1,3] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 2
[4,2,3,1] => [3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 3
[4,3,1,2] => [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 3
[4,3,2,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,2,3,4,5] => [5,2,3,4,1] => [5,2,3,4,1] => [2,3,4,5,1] => 7
[1,2,3,5,4] => [5,2,4,1,3] => [4,2,5,3,1] => [2,5,3,4,1] => 7
[1,2,4,3,5] => [5,3,1,2,4] => [3,4,2,5,1] => [5,4,2,3,1] => 6
[1,2,4,5,3] => [5,3,4,1,2] => [4,5,2,3,1] => [5,2,3,4,1] => 8
[1,2,5,3,4] => [5,4,1,2,3] => [3,4,5,2,1] => [5,2,4,3,1] => 7
[1,2,5,4,3] => [5,4,1,3,2] => [3,5,4,2,1] => [4,2,5,3,1] => 8
[1,3,2,4,5] => [2,5,1,3,4] => [3,1,4,5,2] => [5,4,3,1,2] => 4
[1,3,2,5,4] => [2,5,1,4,3] => [3,1,5,4,2] => [4,5,3,1,2] => 5
[1,3,4,2,5] => [2,3,5,1,4] => [4,1,2,5,3] => [5,4,1,2,3] => 4
[1,3,4,5,2] => [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 4
[1,3,5,2,4] => [2,4,5,1,3] => [4,1,5,2,3] => [5,1,2,4,3] => 5
[1,3,5,4,2] => [2,4,1,3,5] => [3,1,4,2,5] => [4,3,1,2,5] => 3
[1,4,2,3,5] => [3,5,1,2,4] => [3,4,1,5,2] => [5,4,1,3,2] => 5
[1,4,2,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => [4,5,1,3,2] => 6
[1,4,3,2,5] => [3,1,2,5,4] => [2,3,1,5,4] => [3,2,1,5,4] => 3
[1,4,3,5,2] => [3,1,2,4,5] => [2,3,1,4,5] => [3,2,1,4,5] => 2
[1,4,5,2,3] => [3,4,5,1,2] => [4,5,1,2,3] => [5,1,4,2,3] => 6
[1,4,5,3,2] => [3,4,1,2,5] => [3,4,1,2,5] => [4,1,3,2,5] => 4
[1,2,4,3,5,6,7] => [7,3,1,2,4,5,6] => [3,4,2,5,6,7,1] => [7,6,5,4,2,3,1] => ? = 8
[1,2,4,5,3,6,7] => [7,3,4,1,2,5,6] => [4,5,2,3,6,7,1] => [7,6,5,2,3,4,1] => ? = 10
[1,2,5,3,4,6,7] => [7,4,1,2,3,5,6] => [3,4,5,2,6,7,1] => [7,6,5,2,4,3,1] => ? = 9
[1,2,5,6,7,3,4] => [7,4,5,6,1,2,3] => [5,6,7,2,3,4,1] => [7,2,6,3,4,5,1] => ? = 15
[1,2,6,3,4,5,7] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => [7,6,2,5,4,3,1] => ? = 10
[1,2,6,7,3,4,5] => [7,5,6,1,2,3,4] => [4,5,6,7,2,3,1] => [7,2,6,3,5,4,1] => ? = 14
[1,2,7,3,4,5,6] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => [7,2,6,5,4,3,1] => ? = 11
[1,3,2,4,5,6,7] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => [7,6,5,4,3,1,2] => ? = 6
[1,3,4,5,2,6,7] => [2,3,4,7,1,5,6] => [5,1,2,3,6,7,4] => [7,6,5,1,2,3,4] => ? = 6
[1,3,4,5,6,2,7] => [2,3,4,5,7,1,6] => [6,1,2,3,4,7,5] => [7,6,1,2,3,4,5] => ? = 6
[1,3,4,5,6,7,2] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 6
[1,3,4,5,7,2,6] => [2,3,4,6,7,1,5] => [6,1,2,3,7,4,5] => [7,1,2,3,4,6,5] => ? = 7
[1,3,4,6,7,2,5] => [2,3,5,6,7,1,4] => [6,1,2,7,3,4,5] => [7,1,2,3,6,4,5] => ? = 8
[1,3,5,4,6,7,2] => [2,4,1,3,5,6,7] => [3,1,4,2,5,6,7] => [4,3,1,2,5,6,7] => ? = 3
[1,3,6,2,4,5,7] => [2,5,7,1,3,4,6] => [4,1,5,6,2,7,3] => [7,6,1,2,5,4,3] => ? = 8
[1,3,6,4,5,7,2] => [2,5,1,3,4,6,7] => [3,1,4,5,2,6,7] => [5,4,3,1,2,6,7] => ? = 4
[1,3,7,2,4,5,6] => [2,6,7,1,3,4,5] => [4,1,5,6,7,2,3] => [7,1,2,6,5,4,3] => ? = 9
[1,3,7,4,5,6,2] => [2,6,1,3,4,5,7] => [3,1,4,5,6,2,7] => [6,5,4,3,1,2,7] => ? = 5
[1,4,3,5,6,2,7] => [3,1,2,4,5,7,6] => [2,3,1,4,5,7,6] => [3,2,1,4,5,7,6] => ? = 3
[1,4,3,5,6,7,2] => [3,1,2,4,5,6,7] => [2,3,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 2
[1,4,5,3,6,7,2] => [3,4,1,2,5,6,7] => [3,4,1,2,5,6,7] => [4,1,3,2,5,6,7] => ? = 4
[1,4,5,6,7,2,3] => [3,4,5,6,7,1,2] => [6,7,1,2,3,4,5] => [7,1,6,2,3,4,5] => ? = 10
[1,4,6,7,3,5,2] => [3,5,6,1,2,4,7] => [4,5,1,6,2,3,7] => [6,1,5,4,2,3,7] => ? = 8
[1,4,7,3,5,6,2] => [3,6,1,2,4,5,7] => [3,4,1,5,6,2,7] => [6,5,4,1,3,2,7] => ? = 6
[1,5,3,4,6,2,7] => [4,1,2,3,5,7,6] => [2,3,4,1,5,7,6] => [4,3,2,1,5,7,6] => ? = 4
[1,5,3,4,6,7,2] => [4,1,2,3,5,6,7] => [2,3,4,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 3
[1,5,6,7,3,4,2] => [4,5,6,1,2,3,7] => [4,5,6,1,2,3,7] => [6,1,5,2,4,3,7] => ? = 9
[1,5,7,3,4,2,6] => [4,6,1,2,3,7,5] => [3,4,5,1,7,2,6] => [7,5,1,4,3,2,6] => ? = 8
[1,5,7,3,4,6,2] => [4,6,1,2,3,5,7] => [3,4,5,1,6,2,7] => [6,5,1,4,3,2,7] => ? = 7
[1,6,3,4,2,5,7] => [5,1,2,3,7,4,6] => [2,3,4,6,1,7,5] => [7,6,4,3,2,1,5] => ? = 6
[1,6,3,4,5,2,7] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => [5,4,3,2,1,7,6] => ? = 5
[1,6,3,4,5,7,2] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => [5,4,3,2,1,6,7] => ? = 4
[1,6,3,4,7,2,5] => [5,1,2,3,6,7,4] => [2,3,4,7,1,5,6] => [7,4,3,2,1,5,6] => ? = 6
[1,6,3,4,7,5,2] => [5,1,2,3,6,4,7] => [2,3,4,6,1,5,7] => [6,4,3,2,1,5,7] => ? = 5
[1,6,3,7,4,5,2] => [5,1,2,6,3,4,7] => [2,3,5,6,1,4,7] => [6,3,2,1,5,4,7] => ? = 6
[1,6,7,2,3,4,5] => [5,6,7,1,2,3,4] => [4,5,6,7,1,2,3] => [7,1,6,2,5,4,3] => ? = 12
[1,6,7,3,4,2,5] => [5,6,1,2,3,7,4] => [3,4,5,7,1,2,6] => [7,1,5,4,3,2,6] => ? = 9
[1,6,7,3,4,5,2] => [5,6,1,2,3,4,7] => [3,4,5,6,1,2,7] => [6,1,5,4,3,2,7] => ? = 8
[1,7,2,3,4,5,6] => [6,7,1,2,3,4,5] => [3,4,5,6,7,1,2] => [7,1,6,5,4,3,2] => ? = 10
[1,7,3,2,4,5,6] => [6,1,2,7,3,4,5] => [2,3,5,6,7,1,4] => [7,3,2,1,6,5,4] => ? = 8
[1,7,3,4,2,5,6] => [6,1,2,3,7,4,5] => [2,3,4,6,7,1,5] => [7,4,3,2,1,6,5] => ? = 7
[1,7,3,4,5,2,6] => [6,1,2,3,4,7,5] => [2,3,4,5,7,1,6] => [7,5,4,3,2,1,6] => ? = 6
[1,7,3,4,5,6,2] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => [6,5,4,3,2,1,7] => ? = 5
[2,1,3,4,5,6,7] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => [7,6,5,4,3,2,1] => ? = 6
[2,3,1,4,5,6,7] => [1,7,2,3,4,5,6] => [1,3,4,5,6,7,2] => [1,7,6,5,4,3,2] => ? = 5
[2,3,4,1,5,6,7] => [7,2,1,3,4,5,6] => [3,2,4,5,6,7,1] => [2,7,6,5,4,3,1] => ? = 7
[2,3,4,5,1,6,7] => [7,2,3,1,4,5,6] => [4,2,3,5,6,7,1] => [2,3,7,6,5,4,1] => ? = 8
[2,3,4,5,6,1,7] => [7,2,3,4,1,5,6] => [5,2,3,4,6,7,1] => [2,3,4,7,6,5,1] => ? = 9
[2,3,4,5,6,7,1] => [7,2,3,4,5,1,6] => [6,2,3,4,5,7,1] => [2,3,4,5,7,6,1] => ? = 10

Description

The mad of a permutation. According to [1], this is the sum of twice the number of occurrences of the vincular pattern of

$(2\underline{31})$ plus the number of occurrences of the vincular patterns

$(\underline{31}2)$ and

$(\underline{21})$ , where matches of the underlined letters must be adjacent.

Matching statistic: St000004
(load all 4 compositions to match this statistic)

Mapping

Mp00310: Permutations —toric promotion⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 70%●distinct values known / distinct values provided: 55%

Values

[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [3,2,1] => [3,2,1] => 3
[1,3,2] => [2,3,1] => [2,3,1] => 2
[2,1,3] => [3,1,2] => [1,3,2] => 2
[2,3,1] => [1,3,2] => [3,1,2] => 1
[3,1,2] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 5
[1,2,4,3] => [4,3,1,2] => [1,4,3,2] => 5
[1,3,2,4] => [2,4,1,3] => [4,2,1,3] => 3
[1,3,4,2] => [2,3,4,1] => [2,3,4,1] => 3
[1,4,2,3] => [3,4,1,2] => [3,1,4,2] => 4
[1,4,3,2] => [3,1,2,4] => [1,3,2,4] => 2
[2,1,3,4] => [4,1,2,3] => [1,2,4,3] => 3
[2,1,4,3] => [4,1,3,2] => [4,1,3,2] => 4
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 2
[2,3,4,1] => [4,2,1,3] => [2,1,4,3] => 4
[2,4,1,3] => [4,3,2,1] => [4,3,2,1] => 6
[2,4,3,1] => [1,4,3,2] => [4,3,1,2] => 3
[3,1,2,4] => [2,4,3,1] => [4,2,3,1] => 4
[3,1,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,2,1,4] => [1,2,4,3] => [4,1,2,3] => 1
[3,2,4,1] => [2,1,4,3] => [2,4,1,3] => 2
[3,4,1,2] => [2,3,1,4] => [2,3,1,4] => 2
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => 1
[4,1,2,3] => [3,4,2,1] => [3,4,2,1] => 5
[4,1,3,2] => [3,2,4,1] => [3,2,4,1] => 4
[4,2,1,3] => [1,3,4,2] => [3,4,1,2] => 2
[4,2,3,1] => [3,1,4,2] => [1,3,4,2] => 3
[4,3,1,2] => [3,2,1,4] => [3,2,1,4] => 3
[4,3,2,1] => [1,3,2,4] => [3,1,2,4] => 1
[1,2,3,4,5] => [5,2,3,4,1] => [2,3,5,4,1] => 7
[1,2,3,5,4] => [5,2,4,1,3] => [5,2,1,4,3] => 7
[1,2,4,3,5] => [5,3,1,2,4] => [1,3,2,5,4] => 6
[1,2,4,5,3] => [5,3,4,1,2] => [3,1,5,4,2] => 8
[1,2,5,3,4] => [5,4,1,2,3] => [1,2,5,4,3] => 7
[1,2,5,4,3] => [5,4,1,3,2] => [5,1,4,3,2] => 8
[1,3,2,4,5] => [2,5,1,3,4] => [2,1,5,3,4] => 4
[1,3,2,5,4] => [2,5,1,4,3] => [2,5,4,1,3] => 5
[1,3,4,2,5] => [2,3,5,1,4] => [5,2,3,1,4] => 4
[1,3,4,5,2] => [2,3,4,5,1] => [2,3,4,5,1] => 4
[1,3,5,2,4] => [2,4,5,1,3] => [4,5,2,1,3] => 5
[1,3,5,4,2] => [2,4,1,3,5] => [4,2,1,3,5] => 3
[1,4,2,3,5] => [3,5,1,2,4] => [1,5,3,2,4] => 5
[1,4,2,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => 6
[1,4,3,2,5] => [3,1,2,5,4] => [1,3,5,2,4] => 3
[1,4,3,5,2] => [3,1,2,4,5] => [1,3,2,4,5] => 2
[1,4,5,2,3] => [3,4,5,1,2] => [3,4,1,5,2] => 6
[1,2,4,3,5,6,7] => [7,3,1,2,4,5,6] => [1,3,2,4,5,7,6] => ? = 8
[1,2,4,5,3,6,7] => [7,3,4,1,2,5,6] => [3,1,4,2,5,7,6] => ? = 10
[1,2,5,3,4,6,7] => [7,4,1,2,3,5,6] => [1,2,4,3,5,7,6] => ? = 9
[1,2,5,6,7,3,4] => [7,4,5,6,1,2,3] => [4,1,5,2,7,6,3] => ? = 15
[1,2,6,3,4,5,7] => [7,5,1,2,3,4,6] => [1,2,3,5,4,7,6] => ? = 10
[1,2,6,7,3,4,5] => [7,5,6,1,2,3,4] => [1,2,5,3,7,6,4] => ? = 14
[1,2,7,3,4,5,6] => [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => ? = 11
[1,3,2,4,5,6,7] => [2,7,1,3,4,5,6] => [2,1,3,4,7,5,6] => ? = 6
[1,3,4,5,2,6,7] => [2,3,4,7,1,5,6] => [2,7,3,4,1,5,6] => ? = 6
[1,3,4,5,6,2,7] => [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => ? = 6
[1,3,4,5,6,7,2] => [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => ? = 6
[1,3,4,5,7,2,6] => [2,3,4,6,7,1,5] => [6,7,2,3,4,1,5] => ? = 7
[1,3,4,6,7,2,5] => [2,3,5,6,7,1,4] => [5,6,7,2,3,1,4] => ? = 8
[1,3,5,4,6,7,2] => [2,4,1,3,5,6,7] => [4,2,1,3,5,6,7] => ? = 3
[1,3,6,2,4,5,7] => [2,5,7,1,3,4,6] => [2,1,7,5,3,4,6] => ? = 8
[1,3,6,4,5,7,2] => [2,5,1,3,4,6,7] => [2,1,5,3,4,6,7] => ? = 4
[1,3,7,2,4,5,6] => [2,6,7,1,3,4,5] => [2,1,6,3,7,4,5] => ? = 9
[1,3,7,4,5,6,2] => [2,6,1,3,4,5,7] => [2,1,3,6,4,5,7] => ? = 5
[1,4,3,5,6,2,7] => [3,1,2,4,5,7,6] => [1,3,7,2,4,5,6] => ? = 3
[1,4,3,5,6,7,2] => [3,1,2,4,5,6,7] => [1,3,2,4,5,6,7] => ? = 2
[1,4,5,3,6,7,2] => [3,4,1,2,5,6,7] => [3,1,4,2,5,6,7] => ? = 4
[1,4,5,6,7,2,3] => [3,4,5,6,7,1,2] => [3,4,5,6,1,7,2] => ? = 10
[1,4,6,7,3,5,2] => [3,5,6,1,2,4,7] => [5,1,6,3,2,4,7] => ? = 8
[1,4,7,3,5,6,2] => [3,6,1,2,4,5,7] => [1,3,2,6,4,5,7] => ? = 6
[1,5,3,4,6,2,7] => [4,1,2,3,5,7,6] => [1,2,4,7,3,5,6] => ? = 4
[1,5,3,4,6,7,2] => [4,1,2,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 3
[1,5,6,7,3,4,2] => [4,5,6,1,2,3,7] => [4,1,5,2,6,3,7] => ? = 9
[1,5,7,3,4,2,6] => [4,6,1,2,3,7,5] => [1,2,6,4,7,3,5] => ? = 8
[1,5,7,3,4,6,2] => [4,6,1,2,3,5,7] => [1,2,6,4,3,5,7] => ? = 7
[1,6,3,4,2,5,7] => [5,1,2,3,7,4,6] => [5,1,2,3,7,4,6] => ? = 6
[1,6,3,4,5,2,7] => [5,1,2,3,4,7,6] => [1,2,3,5,7,4,6] => ? = 5
[1,6,3,4,5,7,2] => [5,1,2,3,4,6,7] => [1,2,3,5,4,6,7] => ? = 4
[1,6,3,4,7,2,5] => [5,1,2,3,6,7,4] => [1,2,3,5,6,7,4] => ? = 6
[1,6,3,4,7,5,2] => [5,1,2,3,6,4,7] => [1,2,3,5,6,4,7] => ? = 5
[1,6,3,7,4,5,2] => [5,1,2,6,3,4,7] => [5,1,2,3,6,4,7] => ? = 6
[1,6,7,2,3,4,5] => [5,6,7,1,2,3,4] => [1,5,2,6,3,7,4] => ? = 12
[1,6,7,3,4,2,5] => [5,6,1,2,3,7,4] => [1,2,5,3,6,7,4] => ? = 9
[1,6,7,3,4,5,2] => [5,6,1,2,3,4,7] => [1,2,5,3,6,4,7] => ? = 8
[1,7,2,3,4,5,6] => [6,7,1,2,3,4,5] => [1,2,3,6,4,7,5] => ? = 10
[1,7,3,2,4,5,6] => [6,1,2,7,3,4,5] => [1,6,2,3,4,7,5] => ? = 8
[1,7,3,4,2,5,6] => [6,1,2,3,7,4,5] => [6,1,2,3,4,7,5] => ? = 7
[1,7,3,4,5,2,6] => [6,1,2,3,4,7,5] => [1,2,3,4,6,7,5] => ? = 6
[1,7,3,4,5,6,2] => [6,1,2,3,4,5,7] => [1,2,3,4,6,5,7] => ? = 5
[2,1,3,4,5,6,7] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => ? = 6
[2,3,1,4,5,6,7] => [1,7,2,3,4,5,6] => [1,2,3,4,7,5,6] => ? = 5
[2,3,4,1,5,6,7] => [7,2,1,3,4,5,6] => [2,1,3,4,5,7,6] => ? = 7
[2,3,4,5,1,6,7] => [7,2,3,1,4,5,6] => [2,3,1,4,5,7,6] => ? = 8
[2,3,4,5,6,1,7] => [7,2,3,4,1,5,6] => [2,3,4,1,5,7,6] => ? = 9
[2,3,4,5,6,7,1] => [7,2,3,4,5,1,6] => [2,3,4,5,1,7,6] => ? = 10
[2,4,1,6,7,5,3] => [1,7,3,5,6,4,2] => [3,7,5,6,4,1,2] => ? = 11

Description

The major index of a permutation. This is the sum of the positions of its descents,

$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$ Its generating function is

$[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for

$[k]_q = 1 + q + q^2 + \dots q^{k-1}$ . A statistic equidistributed with the major index is called '''Mahonian statistic'''.

The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000305The inverse major index of a permutation. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St001428The number of B-inversions of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001433The flag major index of a signed permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001877Number of indecomposable injective modules with projective dimension 2.