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Your data matches 96 different statistics following compositions of up to 3 maps.
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Matching statistic: St000002
St000002: 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] => 1
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 4
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 2
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 1
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 1
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 10
[1,2,3,5,4] => 7
[1,2,4,3,5] => 7
[1,2,4,5,3] => 5
[1,2,5,3,4] => 5
[1,2,5,4,3] => 3
[1,3,2,4,5] => 7
[1,3,2,5,4] => 4
[1,3,4,2,5] => 5
[1,3,4,5,2] => 4
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 5
[1,4,2,5,3] => 3
[1,4,3,2,5] => 3
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
Description
The number of occurrences of the pattern 123 in a permutation.
Matching statistic: St000119
St000119: 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] => 0
[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] => 0
[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] => 0
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 2
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 4
[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] => 0
[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] => 0
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 0
Description
The number of occurrences of the pattern 321 in a permutation.
Matching statistic: St000095
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St000095: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000095: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,2] => ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> 0
[1,2,3] => ([],3)
=> 0
[1,3,2] => ([(1,2)],3)
=> 0
[2,1,3] => ([(1,2)],3)
=> 0
[2,3,1] => ([(0,2),(1,2)],3)
=> 0
[3,1,2] => ([(0,2),(1,2)],3)
=> 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> 0
[1,3,2,4] => ([(2,3)],4)
=> 0
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => ([(2,3)],4)
=> 0
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 0
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> 0
[1,2,4,3,5] => ([(3,4)],5)
=> 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
Description
The number of triangles of a graph.
A triangle $T$ of a graph $G$ is a collection of three vertices $\{u,v,w\} \in G$ such that they form $K_3$, the complete graph on three vertices.
Matching statistic: St001328
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St001328: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001328: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,2] => ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> 0
[1,2,3] => ([],3)
=> 0
[1,3,2] => ([(1,2)],3)
=> 0
[2,1,3] => ([(1,2)],3)
=> 0
[2,3,1] => ([(0,2),(1,2)],3)
=> 0
[3,1,2] => ([(0,2),(1,2)],3)
=> 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> 0
[1,3,2,4] => ([(2,3)],4)
=> 0
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => ([(2,3)],4)
=> 0
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 0
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> 0
[1,2,4,3,5] => ([(3,4)],5)
=> 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
Description
The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph.
A graph is bipartite if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001396
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
St001396: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001396: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> 0
[2,1] => ([],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,3,2] => ([(0,1),(0,2)],3)
=> 0
[2,1,3] => ([(0,2),(1,2)],3)
=> 0
[2,3,1] => ([(1,2)],3)
=> 0
[3,1,2] => ([(1,2)],3)
=> 0
[3,2,1] => ([],3)
=> 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 0
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 0
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 0
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 0
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 0
[2,3,4,1] => ([(1,2),(2,3)],4)
=> 0
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 0
[2,4,3,1] => ([(1,2),(1,3)],4)
=> 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 0
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => ([(0,3),(1,2)],4)
=> 0
[3,4,2,1] => ([(2,3)],4)
=> 2
[4,1,2,3] => ([(1,2),(2,3)],4)
=> 0
[4,1,3,2] => ([(1,2),(1,3)],4)
=> 1
[4,2,1,3] => ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => ([(2,3)],4)
=> 2
[4,3,1,2] => ([(2,3)],4)
=> 2
[4,3,2,1] => ([],4)
=> 4
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 0
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 0
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 0
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 0
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 0
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 0
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 0
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 0
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 0
Description
Number of triples of incomparable elements in a finite poset.
For a finite poset this is the number of 3-element sets $S \in \binom{P}{3}$ that are pairwise incomparable.
Matching statistic: St000940
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00194: Signed permutations —Foata-Han inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 40%●distinct values known / distinct values provided: 38%
Mp00194: Signed permutations —Foata-Han inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 40%●distinct values known / distinct values provided: 38%
Values
[1] => [1] => [1] => []
=> ? = 0
[1,2] => [1,2] => [1,2] => []
=> ? = 0
[2,1] => [2,1] => [-2,1] => [2]
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => []
=> ? ∊ {0,0}
[1,3,2] => [1,3,2] => [-3,1,2] => [3]
=> 0
[2,1,3] => [2,1,3] => [-2,1,3] => [2]
=> 0
[2,3,1] => [2,3,1] => [-3,-2,1] => [2,1]
=> 1
[3,1,2] => [3,1,2] => [3,1,2] => []
=> ? ∊ {0,0}
[3,2,1] => [3,2,1] => [2,-3,1] => [3]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => []
=> ? ∊ {0,0,0,0,0,0,0,2,4}
[1,2,4,3] => [1,2,4,3] => [-4,1,2,3] => [4]
=> 1
[1,3,2,4] => [1,3,2,4] => [-3,1,2,4] => [3]
=> 0
[1,3,4,2] => [1,3,4,2] => [-4,-3,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,2,4}
[1,4,2,3] => [1,4,2,3] => [4,1,2,3] => []
=> ? ∊ {0,0,0,0,0,0,0,2,4}
[1,4,3,2] => [1,4,3,2] => [3,-4,1,2] => [2]
=> 0
[2,1,3,4] => [2,1,3,4] => [-2,1,3,4] => [2]
=> 0
[2,1,4,3] => [2,1,4,3] => [-4,-2,1,3] => [3,1]
=> 2
[2,3,1,4] => [2,3,1,4] => [-3,-2,1,4] => [2,1]
=> 1
[2,3,4,1] => [2,3,4,1] => [-4,-3,-2,1] => [2]
=> 0
[2,4,1,3] => [2,4,1,3] => [2,-4,1,3] => [4]
=> 1
[2,4,3,1] => [2,4,3,1] => [3,-4,-2,1] => []
=> ? ∊ {0,0,0,0,0,0,0,2,4}
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => []
=> ? ∊ {0,0,0,0,0,0,0,2,4}
[3,1,4,2] => [3,1,4,2] => [4,-3,1,2] => [4]
=> 1
[3,2,1,4] => [3,2,1,4] => [2,-3,1,4] => [3]
=> 0
[3,2,4,1] => [3,2,4,1] => [2,-4,-3,1] => [3,1]
=> 2
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,2,4}
[3,4,2,1] => [3,4,2,1] => [2,4,-3,1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,2,4}
[4,1,2,3] => [4,1,2,3] => [1,-4,2,3] => [3]
=> 0
[4,1,3,2] => [4,1,3,2] => [-3,-4,1,2] => [2,2]
=> 0
[4,2,1,3] => [4,2,1,3] => [-2,-4,1,3] => []
=> ? ∊ {0,0,0,0,0,0,0,2,4}
[4,2,3,1] => [4,2,3,1] => [2,3,-4,1] => [4]
=> 1
[4,3,1,2] => [4,3,1,2] => [-4,3,1,2] => [4]
=> 1
[4,3,2,1] => [4,3,2,1] => [-3,2,-4,1] => []
=> ? ∊ {0,0,0,0,0,0,0,2,4}
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,2,3,5,4] => [1,2,3,5,4] => [-5,1,2,3,4] => [5]
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => [-4,1,2,3,5] => [4]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [-5,-4,1,2,3] => [3,2]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [5,1,2,3,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,2,5,4,3] => [1,2,5,4,3] => [4,-5,1,2,3] => [5]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [-3,1,2,4,5] => [3]
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [-5,-3,1,2,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,3,4,2,5] => [1,3,4,2,5] => [-4,-3,1,2,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,3,4,5,2] => [1,3,4,5,2] => [-5,-4,-3,1,2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,3,5,2,4] => [1,3,5,2,4] => [3,-5,1,2,4] => [3]
=> 0
[1,3,5,4,2] => [1,3,5,4,2] => [4,-5,-3,1,2] => [2,1]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [4,1,2,3,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,4,2,5,3] => [1,4,2,5,3] => [5,-4,1,2,3] => [2]
=> 0
[1,4,3,2,5] => [1,4,3,2,5] => [3,-4,1,2,5] => [2]
=> 0
[1,4,3,5,2] => [1,4,3,5,2] => [3,-5,-4,1,2] => [3,2]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,4,5,3,2] => [1,4,5,3,2] => [3,5,-4,1,2] => [3]
=> 0
[1,5,2,3,4] => [1,5,2,3,4] => [1,-5,2,3,4] => [4]
=> 1
[1,5,2,4,3] => [1,5,2,4,3] => [-4,-5,1,2,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,5,3,2,4] => [1,5,3,2,4] => [-3,-5,1,2,4] => [3,2]
=> 1
[1,5,3,4,2] => [1,5,3,4,2] => [3,4,-5,1,2] => [5]
=> 2
[1,5,4,2,3] => [1,5,4,2,3] => [-5,4,1,2,3] => [3]
=> 0
[1,5,4,3,2] => [1,5,4,3,2] => [-4,3,-5,1,2] => [3,2]
=> 1
[2,1,3,4,5] => [2,1,3,4,5] => [-2,1,3,4,5] => [2]
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => [-5,-2,1,3,4] => [4,1]
=> 3
[2,1,4,3,5] => [2,1,4,3,5] => [-4,-2,1,3,5] => [3,1]
=> 2
[2,1,4,5,3] => [2,1,4,5,3] => [-5,-4,-2,1,3] => [5]
=> 2
[2,1,5,3,4] => [2,1,5,3,4] => [2,-5,1,3,4] => [5]
=> 2
[2,1,5,4,3] => [2,1,5,4,3] => [4,-5,-2,1,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,3,1,4,5] => [2,3,1,4,5] => [-3,-2,1,4,5] => [2,1]
=> 1
[2,3,1,5,4] => [2,3,1,5,4] => [-5,-3,-2,1,4] => [3]
=> 0
[2,3,4,1,5] => [2,3,4,1,5] => [-4,-3,-2,1,5] => [2]
=> 0
[2,3,4,5,1] => [2,3,4,5,1] => [-5,-4,-3,-2,1] => [2,1]
=> 1
[2,3,5,1,4] => [2,3,5,1,4] => [3,-5,-2,1,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,3,5,4,1] => [2,3,5,4,1] => [4,-5,-3,-2,1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,4,1,3,5] => [2,4,1,3,5] => [2,-4,1,3,5] => [4]
=> 1
[2,4,1,5,3] => [2,4,1,5,3] => [2,-5,-4,1,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,4,3,1,5] => [2,4,3,1,5] => [3,-4,-2,1,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,4,3,5,1] => [2,4,3,5,1] => [3,-5,-4,-2,1] => [5]
=> 2
[2,4,5,1,3] => [2,4,5,1,3] => [2,5,-4,1,3] => [5]
=> 2
[2,4,5,3,1] => [2,4,5,3,1] => [3,5,-4,-2,1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,5,1,3,4] => [2,5,1,3,4] => [-2,-5,1,3,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,5,1,4,3] => [2,5,1,4,3] => [2,4,-5,1,3] => [2]
=> 0
[2,5,3,1,4] => [2,5,3,1,4] => [2,3,-5,1,4] => [5]
=> 2
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,1,5,2,4] => [3,1,5,2,4] => [3,5,1,2,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,2,5,1,4] => [3,2,5,1,4] => [2,5,-3,1,4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,2,5,4,1] => [3,2,5,4,1] => [2,4,-5,-3,1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,4,1,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,4,1,5,2] => [3,4,1,5,2] => [4,5,-3,1,2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,4,2,1,5] => [3,4,2,1,5] => [2,4,-3,1,5] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,4,2,5,1] => [3,4,2,5,1] => [2,5,-4,-3,1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,4,5,1,2] => [3,4,5,1,2] => [3,4,5,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,5,2,1,4] => [3,5,2,1,4] => [-3,2,-5,1,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[4,1,2,5,3] => [4,1,2,5,3] => [1,-5,-4,2,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[4,1,5,3,2] => [4,1,5,3,2] => [-5,3,-4,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[4,2,1,3,5] => [4,2,1,3,5] => [-2,-4,1,3,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[4,2,5,1,3] => [4,2,5,1,3] => [-5,2,-4,1,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[4,2,5,3,1] => [4,2,5,3,1] => [2,3,5,-4,1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[4,3,1,5,2] => [4,3,1,5,2] => [-3,5,-4,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[4,3,2,1,5] => [4,3,2,1,5] => [-3,2,-4,1,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[4,3,2,5,1] => [4,3,2,5,1] => [-3,2,-5,-4,1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[4,5,1,2,3] => [4,5,1,2,3] => [-5,1,-4,2,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[4,5,3,1,2] => [4,5,3,1,2] => [-5,-4,3,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[5,1,2,3,4] => [5,1,2,3,4] => [1,5,2,3,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[5,1,3,4,2] => [5,1,3,4,2] => [-3,4,-5,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
Description
The number of characters of the symmetric group whose value on the partition is zero.
The maximal value for any given size is recorded in [2].
Matching statistic: St000771
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 39%●distinct values known / distinct values provided: 31%
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 39%●distinct values known / distinct values provided: 31%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => [1,2] => ([],2)
=> ? = 0 + 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,3,1] => [2,3,1] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1} + 1
[3,1,2] => [3,1,2] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,1} + 1
[3,2,1] => [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1} + 1
[1,2,3,4] => [1,4,3,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[1,2,4,3] => [1,4,3,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[1,3,2,4] => [1,4,3,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[1,3,4,2] => [1,4,3,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[1,4,2,3] => [1,4,3,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[1,4,3,2] => [1,4,3,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[2,1,3,4] => [2,1,4,3] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,3,1,4] => [2,4,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,1] => [2,4,3,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[2,4,1,3] => [2,4,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,3,1] => [2,4,3,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[3,1,2,4] => [3,1,4,2] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[3,1,4,2] => [3,1,4,2] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[3,2,1,4] => [3,2,1,4] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[3,2,4,1] => [3,2,4,1] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[3,4,1,2] => [3,4,1,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[3,4,2,1] => [3,4,2,1] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[4,1,2,3] => [4,1,3,2] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,1,3,2] => [4,1,3,2] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,2,1,3] => [4,2,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[4,2,3,1] => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[4,3,1,2] => [4,3,1,2] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[4,3,2,1] => [4,3,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,4} + 1
[1,2,3,4,5] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,2,3,5,4] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,2,4,3,5] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,2,4,5,3] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,2,5,3,4] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,2,5,4,3] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,3,2,4,5] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,3,2,5,4] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,3,4,2,5] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,3,4,5,2] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,3,5,2,4] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,3,5,4,2] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,4,2,3,5] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,4,2,5,3] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,4,3,2,5] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,4,3,5,2] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,4,5,2,3] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,4,5,3,2] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,5,2,3,4] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,5,2,4,3] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,5,3,2,4] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,5,3,4,2] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,5,4,2,3] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[1,5,4,3,2] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,1,3,4,5] => [2,1,5,4,3] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,3,5,4] => [2,1,5,4,3] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,4,3,5] => [2,1,5,4,3] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,3,1,4,5] => [2,5,1,4,3] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,1,5,4] => [2,5,1,4,3] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,4,1,5] => [2,5,4,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,3,4,5,1] => [2,5,4,3,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,3,5,1,4] => [2,5,4,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,3,5,4,1] => [2,5,4,3,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,4,1,3,5] => [2,5,1,4,3] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,1,5,3] => [2,5,1,4,3] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,3,1,5] => [2,5,4,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,4,3,5,1] => [2,5,4,3,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,4,5,1,3] => [2,5,4,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,4,5,3,1] => [2,5,4,3,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,5,1,3,4] => [2,5,1,4,3] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,1,4,3] => [2,5,1,4,3] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,3,1,4] => [2,5,4,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,5,3,4,1] => [2,5,4,3,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,5,4,1,3] => [2,5,4,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[3,2,4,1,5] => [3,2,5,1,4] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,5,1,4] => [3,2,5,1,4] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,4,1,2,5] => [3,5,1,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,4,1,5,2] => [3,5,1,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,4,2,5,1] => [3,5,2,4,1] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[3,5,1,2,4] => [3,5,1,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,5,1,4,2] => [3,5,1,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,5,2,4,1] => [3,5,2,4,1] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[4,2,3,1,5] => [4,2,5,1,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,2,3,5,1] => [4,2,5,3,1] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,2,5,1,3] => [4,2,5,1,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,2,5,3,1] => [4,2,5,3,1] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,5,1,2,3] => [4,5,1,3,2] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,5,1,3,2] => [4,5,1,3,2] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,5,2,1,3] => [4,5,2,1,3] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[4,5,2,3,1] => [4,5,2,3,1] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,2,3,1,4] => [5,2,4,1,3] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[5,2,3,4,1] => [5,2,4,3,1] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,2,4,1,3] => [5,2,4,1,3] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[5,2,4,3,1] => [5,2,4,3,1] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 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 $1$.
Matching statistic: St000772
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 37%●distinct values known / distinct values provided: 31%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 37%●distinct values known / distinct values provided: 31%
Values
[1] => [1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,1,0,0]
=> [1,2] => ([],2)
=> ? = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0} + 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0} + 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0} + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,4} + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,4} + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,4} + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,4} + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,4} + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,4} + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,4} + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,4} + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,4} + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,4} + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,4} + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,4} + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,4} + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 0 + 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10} + 1
Description
The multiplicity of the largest 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 $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St001605
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 35%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 35%●distinct values known / distinct values provided: 25%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,1}
[1,3,2] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1}
[2,1,3] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1}
[3,2,1] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[1,3,4,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[1,4,2,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[2,1,4,3] => [2,2]
=> [2]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[2,3,1,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[3,1,2,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[3,4,1,2] => [2,2]
=> [2]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[4,3,2,1] => [2,2]
=> [2]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,4}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,3,4,5,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,3,5,2,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,4,2,5,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[1,4,5,3,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,5,2,3,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[4,2,5,1,3] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[4,3,2,1,5] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[4,5,3,1,2] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[5,2,3,4,1] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[5,2,4,3,1] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[5,3,2,4,1] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[5,4,3,2,1] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [4,1]
=> 1
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,2,3,6,4,5] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,2,3,6,5,4] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,2,4,3,6,5] => [2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> 3
[1,2,4,5,3,6] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,2,4,6,5,3] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,2,5,3,4,6] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,2,5,4,3,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,2,5,4,6,3] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,2,5,6,3,4] => [2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> 3
[1,2,6,3,5,4] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,2,6,4,3,5] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,2,6,4,5,3] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,2,6,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> 3
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,3,2,4,6,5] => [2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> 3
[1,3,2,5,4,6] => [2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> 3
[1,3,2,5,6,4] => [3,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[1,3,2,6,4,5] => [3,2,1]
=> [2,1]
=> [1,1,1]
=> 2
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000454
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 33%●distinct values known / distinct values provided: 25%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 33%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [2] => [2] => ([],2)
=> 0
[2,1] => [2] => [2] => ([],2)
=> 0
[1,2,3] => [3] => [3] => ([],3)
=> 0
[1,3,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0
[2,1,3] => [3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => [3] => ([],3)
=> 0
[3,1,2] => [3] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => [1,2] => ([(1,2)],3)
=> 1
[1,2,3,4] => [4] => [4] => ([],4)
=> 0
[1,2,4,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,2,4}
[1,3,2,4] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,2,4}
[1,3,4,2] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,2,4}
[1,4,2,3] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,2,4}
[1,4,3,2] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [4] => [4] => ([],4)
=> 0
[2,1,4,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,2,4}
[2,3,1,4] => [4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[3,1,2,4] => [4] => [4] => ([],4)
=> 0
[3,1,4,2] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,2,4}
[3,2,1,4] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,2,4}
[3,2,4,1] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,2,4}
[3,4,1,2] => [4] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,1,2,3] => [4] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,2,1,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,2,4}
[4,2,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,3,1,2] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,2,4}
[4,3,2,1] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,2,4,3,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,2,4,5,3] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,2,5,3,4] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,2,5,4,3] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,3,2,4,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,3,4,2,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,4,3,2,5] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,4,3,5,2] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,4,5,2,3] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,3,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,5,2,3,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,5,3,2,4] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[1,5,3,4,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,5,4,2,3] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,4,3,2] => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,3,4,5] => [5] => [5] => ([],5)
=> 0
[2,1,3,5,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,1,4,3,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,1,4,5,3] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,1,5,3,4] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,1,5,4,3] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,3,1,4,5] => [5] => [5] => ([],5)
=> 0
[2,3,1,5,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,3,4,1,5] => [5] => [5] => ([],5)
=> 0
[2,3,4,5,1] => [5] => [5] => ([],5)
=> 0
[2,3,5,1,4] => [5] => [5] => ([],5)
=> 0
[2,3,5,4,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,4,1,3,5] => [5] => [5] => ([],5)
=> 0
[2,4,1,5,3] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,4,3,1,5] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,4,3,5,1] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,4,5,1,3] => [5] => [5] => ([],5)
=> 0
[2,4,5,3,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,5,1,3,4] => [5] => [5] => ([],5)
=> 0
[2,5,1,4,3] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,5,3,1,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,5,3,4,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,5,4,1,3] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[2,5,4,3,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,1,2,4,5] => [5] => [5] => ([],5)
=> 0
[3,1,2,5,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,1,4,2,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,1,4,5,2] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,1,5,2,4] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,1,5,4,2] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,2,1,4,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,2,1,5,4] => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,2,4,1,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,2,4,5,1] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,2,5,1,4] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,2,5,4,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,4,1,2,5] => [5] => [5] => ([],5)
=> 0
[3,4,1,5,2] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,4,2,1,5] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,4,2,5,1] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,5,2,1,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,5,4,1,2] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
[3,5,4,2,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,7,7,7,7,10}
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 86 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000567The sum of the products of all pairs of parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001964The interval resolution global dimension of a poset. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001651The Frankl number of a lattice. St000284The Plancherel distribution on integer partitions. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000259The diameter of a connected graph. St000456The monochromatic index of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000260The radius of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000422The energy of a graph, if it is integral. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000264The girth of a graph, which is not a tree. St001845The number of join irreducibles minus the rank of a lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001846The number of elements which do not have a complement in the lattice. St000909The number of maximal chains of maximal size in a poset. St001875The number of simple modules with projective dimension at most 1. St001435The number of missing boxes in the first row. St000527The width of the poset. 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001438The number of missing boxes of a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001902The number of potential covers of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001487The number of inner corners of a skew partition.
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