- FindStat

Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000651
(load all 4 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal size of a rise in a permutation. This is

$\max_i \sigma_{i+1}-\sigma_i$ , except for the permutations without rises, where it is

$0$ .

Matching statistic: St000141
(load all 2 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 92%

Values

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

Description

The maximum drop size of a permutation. The maximum drop size of a permutation

$\pi$ of

$[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of

$i-\pi(i)$ .

Matching statistic: St000209
(load all 4 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%

Values

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

Description

Maximum difference of elements in cycles. Given a cycle

$C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is

$\max \{ a_i-a_j | a_i, a_j \in C \}$ . The statistic is then the maximum of this value over all cycles in the permutation.

Matching statistic: St000956
(load all 4 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%

Values

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

Description

The maximal displacement of a permutation. This is

$\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation

$\pi$ of

$\{1,\ldots,n\}$ . This statistic without the absolute value is the maximal drop size [[St000141]].