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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000714
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [2] => [2]
=> 3
[[1],[2]]
=> [2] => [2]
=> 3
[[1,2,3]]
=> [3] => [3]
=> 4
[[1,3],[2]]
=> [2,1] => [2,1]
=> 2
[[1,2],[3]]
=> [3] => [3]
=> 4
[[1],[2],[3]]
=> [3] => [3]
=> 4
[[1,2,3,4]]
=> [4] => [4]
=> 5
[[1,3,4],[2]]
=> [2,2] => [2,2]
=> 1
[[1,2,4],[3]]
=> [3,1] => [3,1]
=> 3
[[1,2,3],[4]]
=> [4] => [4]
=> 5
[[1,3],[2,4]]
=> [2,2] => [2,2]
=> 1
[[1,2],[3,4]]
=> [3,1] => [3,1]
=> 3
[[1,4],[2],[3]]
=> [3,1] => [3,1]
=> 3
[[1,3],[2],[4]]
=> [2,2] => [2,2]
=> 1
[[1,2],[3],[4]]
=> [4] => [4]
=> 5
[[1],[2],[3],[4]]
=> [4] => [4]
=> 5
[[1,2,3,4,5]]
=> [5] => [5]
=> 6
[[1,3,4,5],[2]]
=> [2,3] => [3,2]
=> 2
[[1,2,4,5],[3]]
=> [3,2] => [3,2]
=> 2
[[1,2,3,5],[4]]
=> [4,1] => [4,1]
=> 4
[[1,2,3,4],[5]]
=> [5] => [5]
=> 6
[[1,3,5],[2,4]]
=> [2,2,1] => [2,2,1]
=> 0
[[1,2,5],[3,4]]
=> [3,2] => [3,2]
=> 2
[[1,3,4],[2,5]]
=> [2,3] => [3,2]
=> 2
[[1,2,4],[3,5]]
=> [3,2] => [3,2]
=> 2
[[1,2,3],[4,5]]
=> [4,1] => [4,1]
=> 4
[[1,4,5],[2],[3]]
=> [3,2] => [3,2]
=> 2
[[1,3,5],[2],[4]]
=> [2,2,1] => [2,2,1]
=> 0
[[1,2,5],[3],[4]]
=> [4,1] => [4,1]
=> 4
[[1,3,4],[2],[5]]
=> [2,3] => [3,2]
=> 2
[[1,2,4],[3],[5]]
=> [3,2] => [3,2]
=> 2
[[1,2,3],[4],[5]]
=> [5] => [5]
=> 6
[[1,4],[2,5],[3]]
=> [3,2] => [3,2]
=> 2
[[1,3],[2,5],[4]]
=> [2,2,1] => [2,2,1]
=> 0
[[1,2],[3,5],[4]]
=> [4,1] => [4,1]
=> 4
[[1,3],[2,4],[5]]
=> [2,3] => [3,2]
=> 2
[[1,2],[3,4],[5]]
=> [3,2] => [3,2]
=> 2
[[1,5],[2],[3],[4]]
=> [4,1] => [4,1]
=> 4
[[1,4],[2],[3],[5]]
=> [3,2] => [3,2]
=> 2
[[1,3],[2],[4],[5]]
=> [2,3] => [3,2]
=> 2
[[1,2],[3],[4],[5]]
=> [5] => [5]
=> 6
[[1],[2],[3],[4],[5]]
=> [5] => [5]
=> 6
[[1,2,3,4,5,6]]
=> [6] => [6]
=> 7
[[1,3,4,5,6],[2]]
=> [2,4] => [4,2]
=> 3
[[1,2,4,5,6],[3]]
=> [3,3] => [3,3]
=> 1
[[1,2,3,5,6],[4]]
=> [4,2] => [4,2]
=> 3
[[1,2,3,4,6],[5]]
=> [5,1] => [5,1]
=> 5
[[1,2,3,4,5],[6]]
=> [6] => [6]
=> 7
[[1,3,5,6],[2,4]]
=> [2,2,2] => [2,2,2]
=> 0
[[1,2,5,6],[3,4]]
=> [3,3] => [3,3]
=> 1
Description
The number of semistandard Young tableau of given shape, with entries at most 2.
This is also the dimension of the corresponding irreducible representation of $GL_2$.
Matching statistic: St001604
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 67%●distinct values known / distinct values provided: 20%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 67%●distinct values known / distinct values provided: 20%
Values
[[1,2]]
=> [2] => [1] => [1]
=> ? ∊ {3,3}
[[1],[2]]
=> [2] => [1] => [1]
=> ? ∊ {3,3}
[[1,2,3]]
=> [3] => [1] => [1]
=> ? ∊ {2,4,4,4}
[[1,3],[2]]
=> [2,1] => [1,1] => [1,1]
=> ? ∊ {2,4,4,4}
[[1,2],[3]]
=> [3] => [1] => [1]
=> ? ∊ {2,4,4,4}
[[1],[2],[3]]
=> [3] => [1] => [1]
=> ? ∊ {2,4,4,4}
[[1,2,3,4]]
=> [4] => [1] => [1]
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,3,4],[2]]
=> [2,2] => [2] => [2]
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,2,4],[3]]
=> [3,1] => [1,1] => [1,1]
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,2,3],[4]]
=> [4] => [1] => [1]
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,3],[2,4]]
=> [2,2] => [2] => [2]
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,2],[3,4]]
=> [3,1] => [1,1] => [1,1]
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,4],[2],[3]]
=> [3,1] => [1,1] => [1,1]
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,3],[2],[4]]
=> [2,2] => [2] => [2]
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,2],[3],[4]]
=> [4] => [1] => [1]
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1],[2],[3],[4]]
=> [4] => [1] => [1]
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,2,3,4,5]]
=> [5] => [1] => [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3,4,5],[2]]
=> [2,3] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,4,5],[3]]
=> [3,2] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,3,5],[4]]
=> [4,1] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,3,4],[5]]
=> [5] => [1] => [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3,5],[2,4]]
=> [2,2,1] => [2,1] => [2,1]
=> 0
[[1,2,5],[3,4]]
=> [3,2] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3,4],[2,5]]
=> [2,3] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,4],[3,5]]
=> [3,2] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,3],[4,5]]
=> [4,1] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,4,5],[2],[3]]
=> [3,2] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3,5],[2],[4]]
=> [2,2,1] => [2,1] => [2,1]
=> 0
[[1,2,5],[3],[4]]
=> [4,1] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3,4],[2],[5]]
=> [2,3] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,4],[3],[5]]
=> [3,2] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,3],[4],[5]]
=> [5] => [1] => [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,4],[2,5],[3]]
=> [3,2] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3],[2,5],[4]]
=> [2,2,1] => [2,1] => [2,1]
=> 0
[[1,2],[3,5],[4]]
=> [4,1] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3],[2,4],[5]]
=> [2,3] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2],[3,4],[5]]
=> [3,2] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,5],[2],[3],[4]]
=> [4,1] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,4],[2],[3],[5]]
=> [3,2] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3],[2],[4],[5]]
=> [2,3] => [1,1] => [1,1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2],[3],[4],[5]]
=> [5] => [1] => [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1],[2],[3],[4],[5]]
=> [5] => [1] => [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,3,4,5,6]]
=> [6] => [1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,3,4,5,6],[2]]
=> [2,4] => [1,1] => [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,2,4,5,6],[3]]
=> [3,3] => [2] => [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,2,3,5,6],[4]]
=> [4,2] => [1,1] => [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,2,3,4,6],[5]]
=> [5,1] => [1,1] => [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,2,3,4,5],[6]]
=> [6] => [1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,3,5,6],[2,4]]
=> [2,2,2] => [3] => [3]
=> 1
[[1,2,5,6],[3,4]]
=> [3,3] => [2] => [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,1,1] => [1,1,1]
=> 0
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 0
[[1,2,3,6],[4,5]]
=> [4,2] => [1,1] => [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,3,4,5],[2,6]]
=> [2,4] => [1,1] => [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,2,4,5],[3,6]]
=> [3,3] => [2] => [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,2,3,5],[4,6]]
=> [4,2] => [1,1] => [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [3] => [3]
=> 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,1,1] => [1,1,1]
=> 0
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 0
[[1,3,5],[2,4,6]]
=> [2,2,2] => [3] => [3]
=> 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,1,1] => [1,1,1]
=> 0
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 0
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 0
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [3] => [3]
=> 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,1,1] => [1,1,1]
=> 0
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 0
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [3] => [3]
=> 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,1] => [1,1,1]
=> 0
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 0
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [3] => [3]
=> 1
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 0
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,1] => [1,1,1]
=> 0
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [3] => [3]
=> 1
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [3] => [3]
=> 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,1] => [1,1,1]
=> 0
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 0
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 0
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,1] => [1,1,1]
=> 0
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [3] => [3]
=> 1
[[1,3,5,6,7],[2,4]]
=> [2,2,3] => [2,1] => [2,1]
=> 0
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [1,1,1] => [1,1,1]
=> 0
[[1,2,4,6,7],[3,5]]
=> [3,2,2] => [1,2] => [2,1]
=> 0
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [1,1,1] => [1,1,1]
=> 0
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => [2,1] => [2,1]
=> 0
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => [1,1,1]
=> 0
[[1,3,5,6,7],[2],[4]]
=> [2,2,3] => [2,1] => [2,1]
=> 0
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [1,1,1] => [1,1,1]
=> 0
[[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [1,2] => [2,1]
=> 0
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [1,1,1] => [1,1,1]
=> 0
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [2,1] => [2,1]
=> 0
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => [1,1,1]
=> 0
[[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [3,1] => [3,1]
=> 0
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => [2,1] => [2,1]
=> 0
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [1,1,1] => [1,1,1]
=> 0
[[1,2,4,7],[3,5,6]]
=> [3,2,2] => [1,2] => [2,1]
=> 0
[[1,3,5,6],[2,4,7]]
=> [2,2,3] => [2,1] => [2,1]
=> 0
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [1,1,1] => [1,1,1]
=> 0
[[1,2,4,6],[3,5,7]]
=> [3,2,2] => [1,2] => [2,1]
=> 0
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [1,1,1] => [1,1,1]
=> 0
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => [2,1] => [2,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001570
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 10%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 10%
Values
[[1,2]]
=> [2] => ([],2)
=> ([],1)
=> ? ∊ {3,3}
[[1],[2]]
=> [2] => ([],2)
=> ([],1)
=> ? ∊ {3,3}
[[1,2,3]]
=> [3] => ([],3)
=> ([],1)
=> ? ∊ {2,4,4,4}
[[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {2,4,4,4}
[[1,2],[3]]
=> [3] => ([],3)
=> ([],1)
=> ? ∊ {2,4,4,4}
[[1],[2],[3]]
=> [3] => ([],3)
=> ([],1)
=> ? ∊ {2,4,4,4}
[[1,2,3,4]]
=> [4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,2,3],[4]]
=> [4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,2],[3],[4]]
=> [4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1],[2],[3],[4]]
=> [4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,1,3,3,3,5,5,5,5}
[[1,2,3,4,5]]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3,4,5],[2]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,3,4],[5]]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,5],[3,4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3,4],[2,5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,4,5],[2],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3,4],[2],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,3],[4],[5]]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,4],[2,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3],[2,4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2],[3,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,4],[2],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,3],[2],[4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1],[2],[3],[4],[5]]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6}
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,3,4,5,6],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,3,5,6],[2,4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,5,6],[3,4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,3,4,6],[2,5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,3,6],[4,5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,3,4,5],[2,6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,7,7,7,7,7,7}
[[1,3,5,6],[2],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,4,6],[2],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,4],[2,5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,4],[3,5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,4,6],[2,5],[3]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,6],[2,5],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,6],[2,4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,6],[3,4],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,5],[2,6],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,4],[2,6],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2,4],[3,6],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,5],[2,4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3],[2,5],[4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3],[2,4],[5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2],[3,4],[5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St000454
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 60%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 60%
Values
[[1,2]]
=> [2] => [1,1] => ([(0,1)],2)
=> 1 = 3 - 2
[[1],[2]]
=> [2] => [1,1] => ([(0,1)],2)
=> 1 = 3 - 2
[[1,2,3]]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 4 - 2
[[1,3],[2]]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 4 - 2
[[1,2],[3]]
=> [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 2 - 2
[[1],[2],[3]]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 4 - 2
[[1,2,3,4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
[[1,3,4],[2]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
[[1,2,4],[3]]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,3,3,3} - 2
[[1,2,3],[4]]
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,3,3,3} - 2
[[1,3],[2,4]]
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,3,3,3} - 2
[[1,2],[3,4]]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,3,3,3} - 2
[[1,4],[2],[3]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
[[1,3],[2],[4]]
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,3,3,3} - 2
[[1,2],[3],[4]]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,3,3,3} - 2
[[1],[2],[3],[4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[[1,3,4,5],[2]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[[1,2,4,5],[3]]
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,2,3,5],[4]]
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,2,3,4],[5]]
=> [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,3,5],[2,4]]
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,2,5],[3,4]]
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,3,4],[2,5]]
=> [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,2,4],[3,5]]
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,2,3],[4,5]]
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,4,5],[2],[3]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[[1,3,5],[2],[4]]
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,2,5],[3],[4]]
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,3,4],[2],[5]]
=> [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,2,4],[3],[5]]
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,2,3],[4],[5]]
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,4],[2,5],[3]]
=> [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,3],[2,5],[4]]
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,2],[3,5],[4]]
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,3],[2,4],[5]]
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,2],[3,4],[5]]
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,5],[2],[3],[4]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[[1,4],[2],[3],[5]]
=> [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,3],[2],[4],[5]]
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1,2],[3],[4],[5]]
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4} - 2
[[1],[2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[[1,3,4,5,6],[2]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,2,3,4,6],[5]]
=> [4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,2,3,4,5],[6]]
=> [5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,3,5,6],[2,4]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,2,5,6],[3,4]]
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,3,4,6],[2,5]]
=> [4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,2,4,6],[3,5]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,2,3,6],[4,5]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,3,4,5],[2,6]]
=> [5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,2,4,5],[3,6]]
=> [2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,2,3,5],[4,6]]
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,2,3,4],[5,6]]
=> [4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,4,5,6],[2],[3]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[[1,3,5,6],[2],[4]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,2,5,6],[3],[4]]
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,3,4,6],[2],[5]]
=> [4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,2,4,6],[3],[5]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,2,3,6],[4],[5]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,3,4,5],[2],[6]]
=> [5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,2,4,5],[3],[6]]
=> [2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,2,3,5],[4],[6]]
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,2,3,4],[5],[6]]
=> [4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5} - 2
[[1,5,6],[2],[3],[4]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[[1,6],[2],[3],[4],[5]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[[1],[2],[3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[[1,2,3,4,5,6,7]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 8 - 2
[[1,3,4,5,6,7],[2]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 8 - 2
[[1,4,5,6,7],[2],[3]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 8 - 2
[[1,5,6,7],[2],[3],[4]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 8 - 2
[[1,6,7],[2],[3],[4],[5]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 8 - 2
[[1,7],[2],[3],[4],[5],[6]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 8 - 2
[[1],[2],[3],[4],[5],[6],[7]]
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 8 - 2
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.
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