- FindStat

Your data matches 17 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000356
(load all 2 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern

$13\!\!-\!\!2$ .

Matching statistic: St000223
(load all 4 compositions to match this statistic)

Mapping

Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 85%

Values

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

Description

The number of nestings in the permutation.

Matching statistic: St000358
(load all 6 compositions to match this statistic)

Mapping

Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 54%

Values

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

Description

The number of occurrences of the pattern 31-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern

$31\!\!-\!\!2$ .

Matching statistic: St000039
(load all 5 compositions to match this statistic)

Mapping

Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 54%

Values

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

Description

The number of crossings of a permutation. A crossing of a permutation

$\pi$ is given by a pair

$(i,j)$ such that either

$i < j \leq \pi(i) \leq \pi(j)$ or

$\pi(i) < \pi(j) < i < j$ . Pictorially, the diagram of a permutation is obtained by writing the numbers from

$1$ to

$n$ in this order on a line, and connecting

$i$ and

$\pi(i)$ with an arc above the line if

$i\leq\pi(i)$ and with an arc below the line if

$i > \pi(i)$ . Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.

Matching statistic: St000259

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%

Values

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

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

Matching statistic: St000260

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%

Values

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

Description

The radius of a connected graph. This is the minimum eccentricity of any vertex.

Matching statistic: St000302

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%

Values

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

Description

The determinant of the distance matrix of a connected graph.

Matching statistic: St000466

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000466: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%

Values

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

Description

The Gutman (or modified Schultz) index of a connected graph. This is

$\sum_{\{u,v\}\subseteq V} d(u)d(v)d(u,v)$ where

$d(u)$ is the degree of vertex

$u$ and

$d(u,v)$ is the distance between vertices

$u$ and

$v$ . For trees on

$n$ vertices, the modified Schultz index is related to the Wiener index via

$S^\ast(T)=4W(T)-(n-1)(2n-1)$ [1].

Matching statistic: St000467

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000467: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%

Values

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

Description

The hyper-Wiener index of a connected graph. This is

$\sum_{\{u,v\}\subseteq V} d(u,v)+d(u,v)^2.$

Matching statistic: St000771

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%

Values

[1] => [1] => ([],1)
 => ([],1)
 => 1 = 0 + 1
[1,2] => [1,2] => ([],2)
 => ([],1)
 => 1 = 0 + 1
[2,1] => [1,2] => ([],2)
 => ([],1)
 => 1 = 0 + 1
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[1,3,2] => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[2,1,3] => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[2,3,1] => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[3,1,2] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1 + 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,1,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2 + 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2 + 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2 + 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1

Description

The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums

$0$ , which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian

$\left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right).$ Its eigenvalues are

$0,4,4,6$ , so the statistic is

$2$ . The path on four vertices has eigenvalues

$0, 4.7\dots, 6, 9.2\dots$ and therefore statistic

$1$ .

The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001875The number of simple modules with projective dimension at most 1.