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Matching statistic: St000341
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St000341: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 1
[2,1] => 0
[1,2,3] => 4
[1,3,2] => 3
[2,1,3] => 3
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 10
[1,2,4,3] => 9
[1,3,2,4] => 9
[1,3,4,2] => 7
[1,4,2,3] => 7
[1,4,3,2] => 6
[2,1,3,4] => 9
[2,1,4,3] => 8
[2,3,1,4] => 7
[2,3,4,1] => 4
[2,4,1,3] => 5
[2,4,3,1] => 3
[3,1,2,4] => 7
[3,1,4,2] => 5
[3,2,1,4] => 6
[3,2,4,1] => 3
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 4
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 0
[5,1,2,3,4] => 10
[5,1,2,4,3] => 9
[5,1,3,2,4] => 9
[5,1,3,4,2] => 7
[5,1,4,2,3] => 7
[5,1,4,3,2] => 6
[5,2,1,3,4] => 9
[5,2,1,4,3] => 8
[5,2,3,1,4] => 7
[5,2,3,4,1] => 4
[5,2,4,1,3] => 5
[5,2,4,3,1] => 3
[5,3,1,2,4] => 7
[5,3,1,4,2] => 5
[5,3,2,1,4] => 6
[5,3,2,4,1] => 3
[5,3,4,1,2] => 2
Description
The non-inversion sum of a permutation.
A pair a<b is an noninversion of a permutation π if π(a)<π(b). The non-inversion sum is given by ∑(b−a) over all non-inversions of π.
Matching statistic: St000055
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Mp00069: Permutations —complement⟶ Permutations
St000055: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000055: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [2,1] => 1
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 4
[1,3,2] => [3,1,2] => 3
[2,1,3] => [2,3,1] => 3
[2,3,1] => [2,1,3] => 1
[3,1,2] => [1,3,2] => 1
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 10
[1,2,4,3] => [4,3,1,2] => 9
[1,3,2,4] => [4,2,3,1] => 9
[1,3,4,2] => [4,2,1,3] => 7
[1,4,2,3] => [4,1,3,2] => 7
[1,4,3,2] => [4,1,2,3] => 6
[2,1,3,4] => [3,4,2,1] => 9
[2,1,4,3] => [3,4,1,2] => 8
[2,3,1,4] => [3,2,4,1] => 7
[2,3,4,1] => [3,2,1,4] => 4
[2,4,1,3] => [3,1,4,2] => 5
[2,4,3,1] => [3,1,2,4] => 3
[3,1,2,4] => [2,4,3,1] => 7
[3,1,4,2] => [2,4,1,3] => 5
[3,2,1,4] => [2,3,4,1] => 6
[3,2,4,1] => [2,3,1,4] => 3
[3,4,1,2] => [2,1,4,3] => 2
[3,4,2,1] => [2,1,3,4] => 1
[4,1,2,3] => [1,4,3,2] => 4
[4,1,3,2] => [1,4,2,3] => 3
[4,2,1,3] => [1,3,4,2] => 3
[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
[5,1,2,3,4] => [1,5,4,3,2] => 10
[5,1,2,4,3] => [1,5,4,2,3] => 9
[5,1,3,2,4] => [1,5,3,4,2] => 9
[5,1,3,4,2] => [1,5,3,2,4] => 7
[5,1,4,2,3] => [1,5,2,4,3] => 7
[5,1,4,3,2] => [1,5,2,3,4] => 6
[5,2,1,3,4] => [1,4,5,3,2] => 9
[5,2,1,4,3] => [1,4,5,2,3] => 8
[5,2,3,1,4] => [1,4,3,5,2] => 7
[5,2,3,4,1] => [1,4,3,2,5] => 4
[5,2,4,1,3] => [1,4,2,5,3] => 5
[5,2,4,3,1] => [1,4,2,3,5] => 3
[5,3,1,2,4] => [1,3,5,4,2] => 7
[5,3,1,4,2] => [1,3,5,2,4] => 5
[5,3,2,1,4] => [1,3,4,5,2] => 6
[5,3,2,4,1] => [1,3,4,2,5] => 3
[5,3,4,1,2] => [1,3,2,5,4] => 2
Description
The inversion sum of a permutation.
A pair a<b is an inversion of a permutation π if π(a)>π(b). The inversion sum is given by ∑(b−a) over all inversions of π.
This is also half of the metric associated with Spearmans coefficient of association ρ, ∑i(πi−i)2, see [5].
This is also equal to the total number of occurrences of the classical permutation patterns [2,1],[2,3,1],[3,1,2], and [3,2,1], see [2].
This is also equal to the rank of the permutation inside the alternating sign matrix lattice, see references [2] and [3].
This lattice is the MacNeille completion of the strong Bruhat order on the symmetric group [1], which means it is the smallest lattice containing the Bruhat order as a subposet. This is a distributive lattice, so the rank of each element is given by the cardinality of the associated order ideal. The rank is calculated by summing the entries of the corresponding ''monotone triangle'' and subtracting \binom{n+2}{3}, which is the sum of the entries of the monotone triangle corresponding to the identity permutation of n.
This is also the number of bigrassmannian permutations (that is, permutations with exactly one left descent and one right descent) below a given permutation \pi in Bruhat order, see Theorem 1 of [6].
Matching statistic: St000076
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Mp00069: Permutations —complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000076: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000076: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> 0
[1,2] => [2,1] => [[0,1],[1,0]]
=> 1
[2,1] => [1,2] => [[1,0],[0,1]]
=> 0
[1,2,3] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 4
[1,3,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 3
[2,1,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 3
[2,3,1] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[3,1,2] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[3,2,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[1,2,3,4] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 10
[1,2,4,3] => [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 9
[1,3,2,4] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 9
[1,3,4,2] => [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 7
[1,4,2,3] => [4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 7
[1,4,3,2] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 6
[2,1,3,4] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 9
[2,1,4,3] => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 8
[2,3,1,4] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 7
[2,3,4,1] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 4
[2,4,1,3] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 5
[2,4,3,1] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 3
[3,1,2,4] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 7
[3,1,4,2] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 5
[3,2,1,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 6
[3,2,4,1] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 3
[3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[3,4,2,1] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[4,1,2,3] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 4
[4,1,3,2] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 3
[4,2,1,3] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 3
[4,2,3,1] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[4,3,1,2] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[4,3,2,1] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[5,1,2,3,4] => [1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 10
[5,1,2,4,3] => [1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> 9
[5,1,3,2,4] => [1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 9
[5,1,3,4,2] => [1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 7
[5,1,4,2,3] => [1,5,2,4,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> 7
[5,1,4,3,2] => [1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 6
[5,2,1,3,4] => [1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 9
[5,2,1,4,3] => [1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> 8
[5,2,3,1,4] => [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 7
[5,2,3,4,1] => [1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 4
[5,2,4,1,3] => [1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> 5
[5,2,4,3,1] => [1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3
[5,3,1,2,4] => [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 7
[5,3,1,4,2] => [1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 5
[5,3,2,1,4] => [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 6
[5,3,2,4,1] => [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 3
[5,3,4,1,2] => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2
Description
The rank of the alternating sign matrix in the alternating sign matrix poset.
This rank is the sum of the entries of the monotone triangle minus \binom{n+2}{3}, which is the smallest sum of the entries in the set of all monotone triangles with bottom row 1\dots n.
Alternatively, rank(A)=\frac{1}{2} \sum_{i,j=1}^n (i-j)^2 a_{ij}, see [3, thm.5.1].
Matching statistic: St001848
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Mp00069: Permutations —complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001848: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001848: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => 4
[1,3,2] => [3,1,2] => [3,1,2] => 3
[2,1,3] => [2,3,1] => [2,3,1] => 3
[2,3,1] => [2,1,3] => [2,1,3] => 1
[3,1,2] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 10
[1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 9
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 9
[1,3,4,2] => [4,2,1,3] => [4,2,1,3] => 7
[1,4,2,3] => [4,1,3,2] => [4,1,3,2] => 7
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 6
[2,1,3,4] => [3,4,2,1] => [3,4,2,1] => 9
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 8
[2,3,1,4] => [3,2,4,1] => [3,2,4,1] => 7
[2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 4
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 5
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 3
[3,1,2,4] => [2,4,3,1] => [2,4,3,1] => 7
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 5
[3,2,1,4] => [2,3,4,1] => [2,3,4,1] => 6
[3,2,4,1] => [2,3,1,4] => [2,3,1,4] => 3
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 4
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 3
[4,2,1,3] => [1,3,4,2] => [1,3,4,2] => 3
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 10
[5,1,2,4,3] => [1,5,4,2,3] => [1,5,4,2,3] => 9
[5,1,3,2,4] => [1,5,3,4,2] => [1,5,3,4,2] => 9
[5,1,3,4,2] => [1,5,3,2,4] => [1,5,3,2,4] => 7
[5,1,4,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => 7
[5,1,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 6
[5,2,1,3,4] => [1,4,5,3,2] => [1,4,5,3,2] => 9
[5,2,1,4,3] => [1,4,5,2,3] => [1,4,5,2,3] => 8
[5,2,3,1,4] => [1,4,3,5,2] => [1,4,3,5,2] => 7
[5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2,5] => 4
[5,2,4,1,3] => [1,4,2,5,3] => [1,4,2,5,3] => 5
[5,2,4,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => 3
[5,3,1,2,4] => [1,3,5,4,2] => [1,3,5,4,2] => 7
[5,3,1,4,2] => [1,3,5,2,4] => [1,3,5,2,4] => 5
[5,3,2,1,4] => [1,3,4,5,2] => [1,3,4,5,2] => 6
[5,3,2,4,1] => [1,3,4,2,5] => [1,3,4,2,5] => 3
[5,3,4,1,2] => [1,3,2,5,4] => [1,3,2,5,4] => 2
Description
The atomic length of a signed permutation.
The atomic length of an element w of a Weyl group is the sum of the heights of the inversions of w.
Matching statistic: St001171
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(load all 6 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
St001171: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
St001171: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [2,1] => 1
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 4
[1,3,2] => [3,1,2] => 3
[2,1,3] => [2,3,1] => 3
[2,3,1] => [2,1,3] => 1
[3,1,2] => [1,3,2] => 1
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 10
[1,2,4,3] => [4,3,1,2] => 9
[1,3,2,4] => [4,2,3,1] => 9
[1,3,4,2] => [4,2,1,3] => 7
[1,4,2,3] => [4,1,3,2] => 7
[1,4,3,2] => [4,1,2,3] => 6
[2,1,3,4] => [3,4,2,1] => 9
[2,1,4,3] => [3,4,1,2] => 8
[2,3,1,4] => [3,2,4,1] => 7
[2,3,4,1] => [3,2,1,4] => 4
[2,4,1,3] => [3,1,4,2] => 5
[2,4,3,1] => [3,1,2,4] => 3
[3,1,2,4] => [2,4,3,1] => 7
[3,1,4,2] => [2,4,1,3] => 5
[3,2,1,4] => [2,3,4,1] => 6
[3,2,4,1] => [2,3,1,4] => 3
[3,4,1,2] => [2,1,4,3] => 2
[3,4,2,1] => [2,1,3,4] => 1
[4,1,2,3] => [1,4,3,2] => 4
[4,1,3,2] => [1,4,2,3] => 3
[4,2,1,3] => [1,3,4,2] => 3
[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
[5,1,2,3,4] => [1,5,4,3,2] => ? = 10
[5,1,2,4,3] => [1,5,4,2,3] => ? = 9
[5,1,3,2,4] => [1,5,3,4,2] => ? = 9
[5,1,3,4,2] => [1,5,3,2,4] => ? = 7
[5,1,4,2,3] => [1,5,2,4,3] => ? = 7
[5,1,4,3,2] => [1,5,2,3,4] => ? = 6
[5,2,1,3,4] => [1,4,5,3,2] => ? = 9
[5,2,1,4,3] => [1,4,5,2,3] => ? = 8
[5,2,3,1,4] => [1,4,3,5,2] => ? = 7
[5,2,3,4,1] => [1,4,3,2,5] => ? = 4
[5,2,4,1,3] => [1,4,2,5,3] => ? = 5
[5,2,4,3,1] => [1,4,2,3,5] => ? = 3
[5,3,1,2,4] => [1,3,5,4,2] => ? = 7
[5,3,1,4,2] => [1,3,5,2,4] => ? = 5
[5,3,2,1,4] => [1,3,4,5,2] => ? = 6
[5,3,2,4,1] => [1,3,4,2,5] => ? = 3
[5,3,4,1,2] => [1,3,2,5,4] => ? = 2
[5,3,4,2,1] => [1,3,2,4,5] => ? = 1
[5,4,1,2,3] => [1,2,5,4,3] => ? = 4
[5,4,1,3,2] => [1,2,5,3,4] => ? = 3
[5,4,2,1,3] => [1,2,4,5,3] => ? = 3
[5,4,2,3,1] => [1,2,4,3,5] => ? = 1
[5,4,3,1,2] => [1,2,3,5,4] => ? = 1
[5,4,3,2,1] => [1,2,3,4,5] => ? = 0
Description
The vector space dimension of Ext_A^1(I_o,A) when I_o is the tilting module corresponding to the permutation o in the Auslander algebra A of K[x]/(x^n).
Matching statistic: St001330
Values
[1] => ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,1] => ([],2)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4 + 2
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 + 2
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 + 2
[2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[3,2,1] => ([],3)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 10 + 2
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 9 + 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 9 + 2
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 7 + 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 7 + 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 + 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 9 + 2
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 8 + 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 7 + 2
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 + 2
[2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 7 + 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 + 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 + 2
[3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 2
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 2
[4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 2
[4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 2
[4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,3,2,1] => ([],4)
=> ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 10 + 2
[5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 9 + 2
[5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 9 + 2
[5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 7 + 2
[5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 7 + 2
[5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 + 2
[5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 9 + 2
[5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 8 + 2
[5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 7 + 2
[5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 2
[5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 2
[5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
[5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 7 + 2
[5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 2
[5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 + 2
[5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
[5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 2
[5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 2
[5,4,1,3,2] => ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
[5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
[5,4,2,3,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[5,4,3,2,1] => ([],5)
=> ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number HG(G) of a graph G is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001060
Values
[1] => ([],1)
=> ([],0)
=> ([],0)
=> ? = 0 - 5
[1,2] => ([],2)
=> ([],0)
=> ([],0)
=> ? = 1 - 5
[2,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> ? = 0 - 5
[1,2,3] => ([],3)
=> ([],0)
=> ([],0)
=> ? = 4 - 5
[1,3,2] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> ? = 3 - 5
[2,1,3] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> ? = 3 - 5
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 5
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 5
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 0 - 5
[1,2,3,4] => ([],4)
=> ([],0)
=> ([],0)
=> ? = 10 - 5
[1,2,4,3] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 9 - 5
[1,3,2,4] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 9 - 5
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? = 7 - 5
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? = 7 - 5
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 6 - 5
[2,1,3,4] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 9 - 5
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> ? = 8 - 5
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? = 7 - 5
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 4 - 5
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 5 - 5
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? = 3 - 5
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? = 7 - 5
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 5 - 5
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 6 - 5
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? = 3 - 5
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 5
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1 - 5
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 4 - 5
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? = 3 - 5
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? = 3 - 5
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1 - 5
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1 - 5
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ? = 0 - 5
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 10 - 5
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 9 - 5
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 9 - 5
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 7 - 5
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 7 - 5
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 6 - 5
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 9 - 5
[5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3 = 8 - 5
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 7 - 5
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ? = 4 - 5
[5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 5 - 5
[5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3 - 5
[5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 7 - 5
[5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 5 - 5
[5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 6 - 5
[5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3 - 5
[5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ? = 2 - 5
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? = 1 - 5
[5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ? = 4 - 5
[5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3 - 5
[5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3 - 5
[5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? = 1 - 5
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
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