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Your data matches 88 different statistics following compositions of up to 3 maps.
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Matching statistic: St001727
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St001727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 2
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
Description
The number of invisible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
Matching statistic: St000356
Mp00223: Permutations —runsort⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000356: 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] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [1,3,2] => 1
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => 1
[1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,2,3] => 2
[2,1,3,4] => [1,3,4,2] => 1
[2,1,4,3] => [1,4,2,3] => 2
[2,3,1,4] => [1,4,2,3] => 2
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => 1
[2,4,3,1] => [1,2,4,3] => 1
[3,1,2,4] => [1,2,4,3] => 1
[3,1,4,2] => [1,4,2,3] => 2
[3,2,1,4] => [1,4,2,3] => 2
[3,2,4,1] => [1,2,4,3] => 1
[3,4,1,2] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => 1
[4,2,1,3] => [1,3,2,4] => 1
[4,2,3,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => 2
[1,2,5,4,3] => [1,2,5,3,4] => 2
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,3,4,2,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => 1
[1,3,5,2,4] => [1,3,5,2,4] => 2
[1,3,5,4,2] => [1,3,5,2,4] => 2
[1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,2,5,3] => [1,4,2,5,3] => 3
[1,4,3,2,5] => [1,4,2,5,3] => 3
[1,4,3,5,2] => [1,4,2,3,5] => 2
[1,4,5,2,3] => [1,4,5,2,3] => 2
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: St000358
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 1
[2,1,3] => [1,3,2] => [3,1,2] => 1
[2,3,1] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [1,2,3] => [3,2,1] => 0
[3,2,1] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 1
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 2
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 2
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => 1
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => 2
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => 2
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 1
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => 1
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => 2
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => 2
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 0
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 0
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => 2
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => 3
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => 3
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => 2
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: St001726
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001726: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001726: Permutations ⟶ 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
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 2
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 2
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => 1
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => 2
[4,2,1,3] => [1,4,3,2] => [1,4,2,3] => 1
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 2
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 2
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
Description
The number of visible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$.
Matching statistic: St000223
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000223: Permutations ⟶ 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] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [3,2,1] => 1
[2,1,3] => [1,3,2] => [2,3,1] => [3,2,1] => 1
[2,3,1] => [1,2,3] => [3,2,1] => [2,3,1] => 0
[3,1,2] => [1,2,3] => [3,2,1] => [2,3,1] => 0
[3,2,1] => [1,2,3] => [3,2,1] => [2,3,1] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [2,4,3,1] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => [3,2,4,1] => 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => [3,2,4,1] => 1
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [2,4,3,1] => 1
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => [2,4,3,1] => 1
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => [2,4,3,1] => 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [2,3,5,4,1] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [2,4,5,3,1] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [2,4,3,5,1] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [2,5,4,3,1] => 2
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [2,5,4,3,1] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [3,4,2,5,1] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [3,5,2,4,1] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [3,4,5,2,1] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => [3,2,4,5,1] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [3,5,4,2,1] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => [3,5,4,2,1] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [4,3,5,2,1] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => [4,5,3,2,1] => 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => [4,5,3,2,1] => 3
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => [4,3,5,2,1] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => [4,3,2,5,1] => 2
Description
The number of nestings in the permutation.
Matching statistic: St000497
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000497: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000497: Set partitions ⟶ 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] => {{1,2}}
=> 0
[1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => {{1},{2,3}}
=> 0
[2,1,3] => {{1,2},{3}}
=> 0
[2,3,1] => {{1,2,3}}
=> 0
[3,1,2] => {{1,3},{2}}
=> 1
[3,2,1] => {{1,3},{2}}
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => {{1},{2},{3,4}}
=> 0
[1,3,2,4] => {{1},{2,3},{4}}
=> 0
[1,3,4,2] => {{1},{2,3,4}}
=> 0
[1,4,2,3] => {{1},{2,4},{3}}
=> 1
[1,4,3,2] => {{1},{2,4},{3}}
=> 1
[2,1,3,4] => {{1,2},{3},{4}}
=> 0
[2,1,4,3] => {{1,2},{3,4}}
=> 0
[2,3,1,4] => {{1,2,3},{4}}
=> 0
[2,3,4,1] => {{1,2,3,4}}
=> 0
[2,4,1,3] => {{1,2,4},{3}}
=> 1
[2,4,3,1] => {{1,2,4},{3}}
=> 1
[3,1,2,4] => {{1,3},{2},{4}}
=> 1
[3,1,4,2] => {{1,3,4},{2}}
=> 1
[3,2,1,4] => {{1,3},{2},{4}}
=> 1
[3,2,4,1] => {{1,3,4},{2}}
=> 1
[3,4,1,2] => {{1,3},{2,4}}
=> 1
[3,4,2,1] => {{1,3},{2,4}}
=> 1
[4,1,2,3] => {{1,4},{2},{3}}
=> 2
[4,1,3,2] => {{1,4},{2},{3}}
=> 2
[4,2,1,3] => {{1,4},{2},{3}}
=> 2
[4,2,3,1] => {{1,4},{2},{3}}
=> 2
[4,3,1,2] => {{1,4},{2,3}}
=> 2
[4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 0
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 0
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 0
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 0
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 0
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 1
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> 1
Description
The lcb statistic of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1, Definition 3], a '''lcb''' (left-closer-bigger) of $S$ is given by a pair $i < j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a > b$.
Matching statistic: St000491
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00215: Set partitions —Wachs-White⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00215: Set partitions —Wachs-White⟶ Set partitions
St000491: Set partitions ⟶ 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
[2,1] => {{1,2}}
=> {{1,2}}
=> 0
[1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[1,3,2] => {{1},{2,3}}
=> {{1,2},{3}}
=> 0
[2,1,3] => {{1,2},{3}}
=> {{1},{2,3}}
=> 0
[2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 0
[3,1,2] => {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[3,2,1] => {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 0
[1,3,2,4] => {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
[1,3,4,2] => {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 0
[1,4,2,3] => {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1
[1,4,3,2] => {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1
[2,1,3,4] => {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 0
[2,1,4,3] => {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
[2,3,1,4] => {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 0
[2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[2,4,1,3] => {{1,2,4},{3}}
=> {{1,3},{2,4}}
=> 1
[2,4,3,1] => {{1,2,4},{3}}
=> {{1,3},{2,4}}
=> 1
[3,1,2,4] => {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
[3,1,4,2] => {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 1
[3,2,1,4] => {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
[3,2,4,1] => {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 1
[3,4,1,2] => {{1,3},{2,4}}
=> {{1,3,4},{2}}
=> 2
[3,4,2,1] => {{1,3},{2,4}}
=> {{1,3,4},{2}}
=> 2
[4,1,2,3] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
[4,1,3,2] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
[4,2,1,3] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
[4,2,3,1] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
[4,3,1,2] => {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1
[4,3,2,1] => {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> 0
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> 0
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> 0
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> 0
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> 0
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 0
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> 0
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> {{1,3},{2,4},{5}}
=> 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> {{1,3},{2,4},{5}}
=> 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> {{1,3,4},{2},{5}}
=> 2
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> {{1,3,4},{2},{5}}
=> 2
Description
The number of inversions of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1], see also [2,3], an inversion of $S$ is given by a pair $i > j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$.
This statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller".
This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.
Matching statistic: St000609
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000609: Set partitions ⟶ 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
[2,1] => {{1,2}}
=> {{1,2}}
=> 0
[1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> 0
[2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 0
[2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 0
[3,1,2] => {{1,3},{2}}
=> {{1},{2,3}}
=> 1
[3,2,1] => {{1,3},{2}}
=> {{1},{2,3}}
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 0
[1,3,2,4] => {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 0
[1,3,4,2] => {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 0
[1,4,2,3] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1
[1,4,3,2] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1
[2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 0
[2,1,4,3] => {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 0
[2,3,1,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 0
[2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[2,4,1,3] => {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 1
[2,4,3,1] => {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 1
[3,1,2,4] => {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 1
[3,1,4,2] => {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> 1
[3,2,1,4] => {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 1
[3,2,4,1] => {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> 1
[3,4,1,2] => {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> 2
[3,4,2,1] => {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> 2
[4,1,2,3] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 2
[4,1,3,2] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 2
[4,2,1,3] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 2
[4,2,3,1] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 2
[4,3,1,2] => {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 1
[4,3,2,1] => {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> 0
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> 0
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 0
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 0
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> 0
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1,3,4},{2},{5}}
=> 0
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1,3,4,5},{2}}
=> 0
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> {{1,3},{2,5},{4}}
=> 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> {{1,3},{2,5},{4}}
=> 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> {{1,5},{2,4},{3}}
=> 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> {{1,5},{2,4},{3}}
=> 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> {{1},{2,4,5},{3}}
=> 2
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> {{1},{2,4,5},{3}}
=> 2
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
Matching statistic: St000809
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000809: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000809: Permutations ⟶ 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
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 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] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 2
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 1
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 2
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => 2
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 2
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 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] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 2
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,4,2,3] => 3
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,4,2,3] => 3
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => 2
[1,4,5,3,2] => [1,4,5,2,3] => [1,5,2,4,3] => 2
Description
The reduced reflection length of the permutation.
Let $T$ be the set of reflections in a Coxeter group and let $\ell(w)$ be the usual length function. Then the reduced reflection length of $w$ is
$$\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.$$
In the case of the symmetric group, this is twice the depth [[St000029]] minus the usual length [[St000018]].
Matching statistic: St000454
(load all 43 compositions to match this statistic)
(load all 43 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 71%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [1,2] => ([],2)
=> 0
[2,1] => [1,2] => [1,2] => ([],2)
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 1
[2,1,3] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[3,2,1] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1}
[1,4,2,3] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1}
[2,1,4,3] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[3,1,4,2] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[1,3,5,4,2] => [1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[1,4,5,3,2] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[1,5,2,3,4] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,2,4,3] => [1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[1,5,3,2,4] => [1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[1,5,3,4,2] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,4,2,3] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,4,3,2] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,3,4,5] => [1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[2,1,3,5,4] => [1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[2,1,4,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 2
[2,1,4,5,3] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[2,1,5,3,4] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,5,4,3] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3,1,4,5] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[2,3,1,5,4] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3,4,1,5] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[2,4,1,3,5] => [1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[2,4,1,5,3] => [1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[2,4,3,1,5] => [1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[2,4,5,3,1] => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[2,5,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[2,5,1,4,3] => [1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[2,5,3,1,4] => [1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[3,1,2,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[3,1,4,2,5] => [1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[3,1,4,5,2] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[3,1,5,2,4] => [1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[3,2,1,4,5] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[3,2,4,1,5] => [1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[3,2,4,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[3,2,5,1,4] => [1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[4,1,3,5,2] => [1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[4,2,1,3,5] => [1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[5,1,3,4,2] => [1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[5,2,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4}
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [1,2,6,5,3,4] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [1,2,6,5,3,4] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,3,4,2,5,6] => [1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
The following 78 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001060The distinguishing index of a graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St000264The girth of a graph, which is not a tree. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000260The radius of a connected graph. St000668The least common multiple of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St000937The number of positive values of the symmetric group character corresponding to the partition. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000259The diameter of a connected graph. St001438The number of missing boxes of a skew partition. St000516The number of stretching pairs of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000567The sum of the products of all pairs of parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001568The smallest positive integer that does not appear twice in the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St001881The number of factors of a lattice as a Cartesian product of lattices. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001893The flag descent of a signed permutation. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001645The pebbling number of a connected graph. St001875The number of simple modules with projective dimension at most 1. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001867The number of alignments of type EN of a signed permutation. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001209The pmaj statistic of a parking function. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001570The minimal number of edges to add to make a graph Hamiltonian.
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