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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000566
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,4,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[3,4,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[4,1,3,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[4,2,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,2,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,5,2,4,3] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[2,3,1,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,3,5,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 3
[2,4,1,5,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,4,3,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,4,3,5,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,4,5,3,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 3
[2,5,1,4,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 3
[2,5,3,1,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,5,3,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 3
[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[3,1,2,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[3,2,1,5,4] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[3,2,5,4,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[3,4,1,5,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[3,4,2,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[3,4,2,5,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[3,4,5,2,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 3
[3,5,1,4,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 3
[3,5,2,1,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[3,5,2,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 3
[3,5,4,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[4,1,2,5,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[4,1,3,2,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[4,1,3,5,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[4,1,5,3,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[4,2,1,5,3] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[4,2,3,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[4,2,3,5,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[4,2,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[4,3,5,2,1] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[4,5,1,3,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 3
[4,5,2,1,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[4,5,2,3,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 3
[4,5,3,1,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[4,5,3,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[5,1,2,4,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 3
[5,1,3,2,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St000479
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St000479: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 6%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St000479: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 6%
Values
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 18 = 1 + 17
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 18 = 1 + 17
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 18 = 1 + 17
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 18 = 1 + 17
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 17
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 18 = 1 + 17
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 17
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 17
[2,5,3,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 18 = 1 + 17
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 17
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 17
[3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[3,4,2,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 18 = 1 + 17
[3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 17
[3,5,1,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 17
[3,5,2,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 18 = 1 + 17
[3,5,2,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 17
[3,5,4,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[4,1,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 18 = 1 + 17
[4,1,3,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[4,1,5,3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[4,2,1,5,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 17
[4,2,3,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 18 = 1 + 17
[4,2,3,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[4,2,5,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[4,3,5,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[4,5,1,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 17
[4,5,2,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 18 = 1 + 17
[4,5,2,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 17
[4,5,3,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 18 = 1 + 17
[4,5,3,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 17
[5,1,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 17
[5,1,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 18 = 1 + 17
[5,1,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 17
[5,1,4,3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[5,2,1,4,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[5,2,3,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 18 = 1 + 17
[5,2,3,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 17
[5,2,4,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[5,3,1,4,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[5,3,2,4,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[5,3,4,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[5,4,1,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[5,4,2,3,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 17
[5,4,3,2,1] => [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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 17
[1,2,3,4,6,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? = 3 + 17
[1,2,3,5,4,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 17
[2,4,3,1,5,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[2,5,3,1,4,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[2,6,3,1,4,5] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[3,4,2,1,5,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[3,5,2,1,4,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[3,6,2,1,4,5] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[4,1,3,2,5,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[4,2,3,1,5,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[4,5,2,1,3,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[4,5,3,1,2,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[4,6,2,1,3,5] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[4,6,3,1,2,5] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[5,1,3,2,4,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[5,2,3,1,4,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[5,6,2,1,3,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[5,6,3,1,2,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[6,1,3,2,4,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[6,2,3,1,4,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 18 = 1 + 17
[2,4,3,1,5,6,7] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[2,5,3,1,4,6,7] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[2,6,3,1,4,5,7] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[2,7,3,1,4,5,6] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[3,4,2,1,5,6,7] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[3,5,2,1,4,6,7] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[3,6,2,1,4,5,7] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[3,7,2,1,4,5,6] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[4,1,3,2,5,6,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[4,2,3,1,5,6,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[4,5,2,1,3,6,7] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[4,5,3,1,2,6,7] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[4,6,2,1,3,5,7] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[4,6,3,1,2,5,7] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[4,7,2,1,3,5,6] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[4,7,3,1,2,5,6] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[5,1,3,2,4,6,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
[5,2,3,1,4,6,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 18 = 1 + 17
Description
The Ramsey number of a graph.
This is the smallest integer $n$ such that every two-colouring of the edges of the complete graph $K_n$ contains a (not necessarily induced) monochromatic copy of the given graph. [1]
Thus, the Ramsey number of the complete graph $K_n$ is the ordinary Ramsey number $R(n,n)$. Very few of these numbers are known, in particular, it is only known that $43\leq R(5,5)\leq 48$. [2,3,4,5]
Matching statistic: St001645
Mp00069: Permutations —complement⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 6%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 6%
Values
[2,4,3,1] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? = 1 + 6
[3,4,2,1] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> ? = 1 + 6
[4,1,3,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 6
[4,2,3,1] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 6
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[1,4,5,3,2] => [5,2,1,3,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[1,5,2,4,3] => [5,1,4,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[1,5,3,4,2] => [5,1,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[2,1,3,5,4] => [4,5,3,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[2,1,5,4,3] => [4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 6
[2,3,1,5,4] => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[2,3,5,4,1] => [4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3 + 6
[2,4,1,5,3] => [4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[2,4,3,1,5] => [4,2,3,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[2,4,3,5,1] => [4,2,3,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[2,4,5,3,1] => [4,2,1,3,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3 + 6
[2,5,1,4,3] => [4,1,5,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 6
[2,5,3,1,4] => [4,1,3,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[2,5,3,4,1] => [4,1,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 6
[2,5,4,3,1] => [4,1,2,3,5] => [1,4] => ([(3,4)],5)
=> ? = 1 + 6
[3,1,2,5,4] => [3,5,4,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[3,1,5,4,2] => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 6
[3,2,1,5,4] => [3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 6
[3,2,5,4,1] => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 6
[3,4,1,5,2] => [3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[3,4,2,1,5] => [3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[3,4,2,5,1] => [3,2,4,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[3,4,5,2,1] => [3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3 + 6
[3,5,1,4,2] => [3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 6
[3,5,2,1,4] => [3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[3,5,2,4,1] => [3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 6
[3,5,4,2,1] => [3,1,2,4,5] => [1,4] => ([(3,4)],5)
=> ? = 1 + 6
[4,1,2,5,3] => [2,5,4,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[4,1,3,2,5] => [2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[4,1,3,5,2] => [2,5,3,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[4,1,5,3,2] => [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 6
[4,2,1,5,3] => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 6
[4,2,3,1,5] => [2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[4,2,3,5,1] => [2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[4,2,5,3,1] => [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 6
[4,3,5,2,1] => [2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 6
[4,5,1,3,2] => [2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 6
[4,5,2,1,3] => [2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[4,5,2,3,1] => [2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 6
[4,5,3,1,2] => [2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[4,5,3,2,1] => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> ? = 2 + 6
[5,1,2,4,3] => [1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 6
[5,1,3,2,4] => [1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
[2,4,3,1,5,6,7] => [6,4,5,7,3,2,1] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[2,5,3,1,4,6,7] => [6,3,5,7,4,2,1] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[2,6,3,1,4,5,7] => [6,2,5,7,4,3,1] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[2,7,3,1,4,5,6] => [6,1,5,7,4,3,2] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[3,4,2,1,5,6,7] => [5,4,6,7,3,2,1] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[3,5,2,1,4,6,7] => [5,3,6,7,4,2,1] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[3,6,2,1,4,5,7] => [5,2,6,7,4,3,1] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[3,7,2,1,4,5,6] => [5,1,6,7,4,3,2] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[4,1,3,2,5,6,7] => [4,7,5,6,3,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[4,2,3,1,5,6,7] => [4,6,5,7,3,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[4,5,2,1,3,6,7] => [4,3,6,7,5,2,1] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[4,5,3,1,2,6,7] => [4,3,5,7,6,2,1] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[4,6,2,1,3,5,7] => [4,2,6,7,5,3,1] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[4,6,3,1,2,5,7] => [4,2,5,7,6,3,1] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[4,7,2,1,3,5,6] => [4,1,6,7,5,3,2] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[4,7,3,1,2,5,6] => [4,1,5,7,6,3,2] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[5,1,3,2,4,6,7] => [3,7,5,6,4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[5,2,3,1,4,6,7] => [3,6,5,7,4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[5,6,2,1,3,4,7] => [3,2,6,7,5,4,1] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[5,6,3,1,2,4,7] => [3,2,5,7,6,4,1] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[5,7,2,1,3,4,6] => [3,1,6,7,5,4,2] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[5,7,3,1,2,4,6] => [3,1,5,7,6,4,2] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[6,1,3,2,4,5,7] => [2,7,5,6,4,3,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[6,2,3,1,4,5,7] => [2,6,5,7,4,3,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[6,7,2,1,3,4,5] => [2,1,6,7,5,4,3] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[6,7,3,1,2,4,5] => [2,1,5,7,6,4,3] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[7,1,3,2,4,5,6] => [1,7,5,6,4,3,2] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
[7,2,3,1,4,5,6] => [1,6,5,7,4,3,2] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 7 = 1 + 6
Description
The pebbling number of a connected graph.
Matching statistic: St000422
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 12%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 12%
Values
[2,4,3,1] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 4
[3,4,2,1] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 4
[4,1,3,2] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 4
[4,2,3,1] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 4
[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)
=> ? = 1 + 4
[1,2,5,4,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[1,3,5,4,2] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[1,4,5,3,2] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[1,5,2,4,3] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[1,5,3,4,2] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[2,1,3,5,4] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[2,1,5,4,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[2,3,1,5,4] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[2,3,5,4,1] => [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)
=> ? = 3 + 4
[2,4,1,5,3] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[2,4,3,1,5] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[2,4,3,5,1] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[2,4,5,3,1] => [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)
=> ? = 3 + 4
[2,5,1,4,3] => [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)
=> ? = 3 + 4
[2,5,3,1,4] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[2,5,3,4,1] => [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)
=> ? = 3 + 4
[2,5,4,3,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[3,1,2,5,4] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[3,1,5,4,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[3,2,1,5,4] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 4
[3,2,5,4,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[3,4,1,5,2] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[3,4,2,1,5] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[3,4,2,5,1] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[3,4,5,2,1] => [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)
=> ? = 3 + 4
[3,5,1,4,2] => [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)
=> ? = 3 + 4
[3,5,2,1,4] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[3,5,2,4,1] => [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)
=> ? = 3 + 4
[3,5,4,2,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[4,1,2,5,3] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[4,1,3,2,5] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[4,1,3,5,2] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[4,1,5,3,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[4,2,1,5,3] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 4
[4,2,3,1,5] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[4,2,3,5,1] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[4,2,5,3,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[4,3,5,2,1] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[4,5,1,3,2] => [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)
=> ? = 3 + 4
[4,5,2,1,3] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[4,5,2,3,1] => [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)
=> ? = 3 + 4
[4,5,3,1,2] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[4,5,3,2,1] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6 = 2 + 4
[5,1,2,4,3] => [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)
=> ? = 3 + 4
[5,1,3,2,4] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[5,1,3,4,2] => [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)
=> ? = 3 + 4
[1,5,6,4,3,2] => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 2 + 4
[5,3,6,4,2,1] => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 2 + 4
[5,4,6,3,2,1] => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 2 + 4
[6,3,5,4,2,1] => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 2 + 4
[6,4,5,3,2,1] => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 2 + 4
[6,5,1,4,3,2] => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 2 + 4
[6,5,2,4,3,1] => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 2 + 4
[6,5,3,4,2,1] => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 2 + 4
[1,6,4,7,5,3,2] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[1,6,5,7,4,3,2] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[1,7,4,6,5,3,2] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[1,7,5,6,4,3,2] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[1,7,6,2,5,4,3] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[1,7,6,3,5,4,2] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[1,7,6,4,5,3,2] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[6,5,4,7,3,2,1] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[7,5,4,6,3,2,1] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[7,6,4,1,5,3,2] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[7,6,4,2,5,3,1] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[7,6,4,3,5,2,1] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[7,6,4,5,3,2,1] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[7,6,5,1,4,3,2] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[7,6,5,2,4,3,1] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[7,6,5,3,4,2,1] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 2 + 4
[7,6,5,4,3,2,1] => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 0 + 4
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
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