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Your data matches 119 different statistics following compositions of up to 3 maps.
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Matching statistic: St000225
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000225: 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
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[2,3,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,4,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,3,2,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[1,3,4,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,4,2,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,1,4,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,3,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[2,3,4,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[2,4,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[2,4,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[3,1,4,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[3,2,4,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[3,4,1,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,4,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[4,1,3,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[4,2,3,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[1,2,3,5,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,2,4,3,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,2,4,5,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,2,5,3,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,2,5,4,3] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,3,2,4,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,3,2,5,4] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,3,4,2,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,3,4,5,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,3,5,2,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,3,5,4,2] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,4,2,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,4,2,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,4,3,2,5] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,4,3,5,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,4,5,2,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,4,5,3,2] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,5,2,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,5,2,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,5,3,2,4] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,5,3,4,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,5,4,2,3] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,5,4,3,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[2,1,3,5,4] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[2,1,4,3,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[2,1,4,5,3] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[2,1,5,3,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[2,1,5,4,3] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[2,3,1,4,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,3,1,5,4] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[2,3,4,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,3,4,5,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000455
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 25%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 25%
Values
[1,3,2] => [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 0
[2,3,1] => [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 0
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,3,4,2] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,4,2,3] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[2,3,1,4] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0
[2,3,4,1] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[2,4,1,3] => [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0
[2,4,3,1] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,1,4,2] => [[1,3],[2,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[3,2,4,1] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,4,2,1] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0
[4,1,3,2] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,4,5,3,2] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,5,2,4,3] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,5,3,2,4] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,5,3,4,2] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,5,4,2,3] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,1,3,5,4] => [[1,3,4],[2,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,1,4,3,5] => [[1,3,5],[2,4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,1,4,5,3] => [[1,3,4],[2,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,1,5,3,4] => [[1,3,5],[2,4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,1,5,4,3] => [[1,3],[2,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,3,1,5,4] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,3,4,1,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,3,4,5,1] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,3,5,1,4] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,3,5,4,1] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,4,1,5,3] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,4,3,1,5] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,4,3,5,1] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,4,5,1,3] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,4,5,3,1] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,5,1,4,3] => [[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,5,3,4,1] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,4,1,5,2] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,4,2,5,1] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,5,1,4,2] => [[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,5,2,4,1] => [[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,5,1,3,2] => [[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,5,2,3,1] => [[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,2,4,3,6,5] => [[1,2,3,5],[4,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,2,5,3,6,4] => [[1,2,3,5],[4,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,2,5,4,6,3] => [[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,2,6,3,5,4] => [[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,2,6,4,5,3] => [[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,3,2,4,6,5] => [[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,3,2,5,4,6] => [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,3,2,5,6,4] => [[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,3,2,6,4,5] => [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,3,2,6,5,4] => [[1,2,4],[3,5],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,3,4,2,6,5] => [[1,2,3,5],[4,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,3,5,2,6,4] => [[1,2,3,5],[4,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,3,5,4,6,2] => [[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,3,6,2,5,4] => [[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)
=> ? = 1
[1,3,6,4,5,2] => [[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,4,2,3,6,5] => [[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,4,2,5,3,6] => [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,4,2,5,6,3] => [[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,4,2,6,3,5] => [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,4,2,6,5,3] => [[1,2,4],[3,5],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,4,3,2,6,5] => [[1,2,5],[3,6],[4]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,4,3,5,2,6] => [[1,2,4,6],[3],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,4,3,5,6,2] => [[1,2,4,5],[3],[6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,4,3,6,2,5] => [[1,2,4],[3,6],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,4,3,6,5,2] => [[1,2,4],[3,5],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,4,5,2,6,3] => [[1,2,3,5],[4,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,4,5,3,6,2] => [[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,4,6,2,5,3] => [[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)
=> ? = 1
[1,4,6,3,5,2] => [[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)
=> ? = 1
[1,5,2,3,6,4] => [[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,5,2,4,3,6] => [[1,2,4,6],[3],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,5,2,4,6,3] => [[1,2,4,5],[3],[6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,5,2,6,3,4] => [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,5,2,6,4,3] => [[1,2,4],[3,5],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001577
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001577: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001577: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 0
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 0
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 0
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0
[1,5,4,3,2] => [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),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 0
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 0
[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),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0
[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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0
[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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 0
[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),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0
[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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[4,2,1,5,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0
[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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[4,3,1,5,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0
[4,3,2,5,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0
[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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[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),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0
[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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[5,2,1,4,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0
[5,2,4,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[5,3,1,4,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0
[5,3,2,4,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0
[5,3,4,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[5,4,1,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0
[5,4,2,3,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,2,3,6,5,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 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),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,4,6,5,3] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 0
[1,2,5,3,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,5,4,3,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 0
[1,2,5,4,6,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,5,6,4,3] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 0
[1,2,6,3,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,6,4,3,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 0
[1,2,6,4,5,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,6,5,3,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 0
[1,2,6,5,4,3] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 0
[1,3,2,4,6,5] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,3,2,5,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,3,2,6,5,4] => [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)
=> ?
=> ? = 1
[1,3,4,2,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,3,4,6,5,2] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 0
[1,3,5,2,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,3,5,4,2,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 0
[1,3,5,4,6,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,3,5,6,4,2] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 0
[1,3,6,2,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
Description
The minimal number of edges to add or remove to make a graph a cograph.
A cograph is a graph that can be obtained from the one vertex graph by complementation and disjoint union.
Matching statistic: St000454
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ 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: 25%
Mp00294: Standard tableaux —peak composition⟶ 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: 25%
Values
[1,3,2] => [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 2
[2,3,1] => [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 2
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[1,3,4,2] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,4,2,3] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[2,3,1,4] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[2,3,4,1] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[2,4,1,3] => [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[2,4,3,1] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[3,1,4,2] => [[1,3],[2,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[3,2,4,1] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[3,4,2,1] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[4,1,3,2] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,4,5,3,2] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[1,5,2,4,3] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,3,2,4] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[1,5,3,4,2] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,4,2,3] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[2,1,3,5,4] => [[1,3,4],[2,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,4,3,5] => [[1,3,5],[2,4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[2,1,4,5,3] => [[1,3,4],[2,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,3,4] => [[1,3,5],[2,4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[2,1,5,4,3] => [[1,3],[2,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[2,3,1,5,4] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,3,4,1,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[2,3,4,5,1] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,5,1,4] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[2,3,5,4,1] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[2,4,1,5,3] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,4,3,1,5] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[2,4,3,5,1] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,1,2,5,4] => [[1,3,4],[2,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,1,4,5,2] => [[1,3,4],[2,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,2,1,5,4] => [[1,4],[2,5],[3]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,2,4,5,1] => [[1,3,4],[2],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,1,2,5,3] => [[1,3,4],[2,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,1,3,5,2] => [[1,3,4],[2],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,2,1,5,3] => [[1,4],[2,5],[3]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,2,3,5,1] => [[1,3,4],[2],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,3,1,5,2] => [[1,4],[2,5],[3]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,1,2,4,3] => [[1,3,4],[2],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,1,3,4,2] => [[1,3,4],[2],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,2,1,4,3] => [[1,4],[2,5],[3]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,2,3,4,1] => [[1,3,4],[2],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,3,1,4,2] => [[1,4],[2,5],[3]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,4,1,3,2] => [[1,4],[2],[3],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,3,5,4,6] => [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,2,3,6,4,5] => [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,2,3,6,5,4] => [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,2,4,5,3,6] => [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,2,4,6,3,5] => [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,2,4,6,5,3] => [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,2,5,6,3,4] => [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,2,5,6,4,3] => [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,3,4,5,2,6] => [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,3,4,6,2,5] => [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,3,4,6,5,2] => [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,3,5,6,2,4] => [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,3,5,6,4,2] => [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,4,5,6,2,3] => [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,4,5,6,3,2] => [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,1,3,5,4,6] => [[1,3,4,6],[2,5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,1,3,6,4,5] => [[1,3,4,6],[2,5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,1,3,6,5,4] => [[1,3,4],[2,5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,1,4,5,3,6] => [[1,3,4,6],[2,5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,1,4,6,3,5] => [[1,3,4],[2,5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,1,4,6,5,3] => [[1,3,4],[2,5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,1,5,6,3,4] => [[1,3,4],[2,5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,1,5,6,4,3] => [[1,3,4],[2,5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,3,4,5,1,6] => [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,3,4,6,1,5] => [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,3,4,6,5,1] => [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 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.
Matching statistic: St000422
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[1,3,2] => [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 4
[2,3,1] => [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 4
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[1,3,4,2] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[1,4,2,3] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[2,3,1,4] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[2,3,4,1] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[2,4,1,3] => [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[2,4,3,1] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[3,1,4,2] => [[1,3],[2,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[3,2,4,1] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[3,4,2,1] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[4,1,3,2] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,4,5,3,2] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[1,5,2,4,3] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[1,5,3,2,4] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[1,5,3,4,2] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[1,5,4,2,3] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[2,1,3,5,4] => [[1,3,4],[2,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[2,1,4,3,5] => [[1,3,5],[2,4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[2,1,4,5,3] => [[1,3,4],[2,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[2,1,5,3,4] => [[1,3,5],[2,4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[2,1,5,4,3] => [[1,3],[2,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[2,3,1,5,4] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[2,3,4,1,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[2,3,4,5,1] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[2,3,5,1,4] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[2,3,5,4,1] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[2,4,1,5,3] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[2,4,3,1,5] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[2,4,3,5,1] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[3,1,2,5,4] => [[1,3,4],[2,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[3,1,4,5,2] => [[1,3,4],[2,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[3,2,1,5,4] => [[1,4],[2,5],[3]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[3,2,4,5,1] => [[1,3,4],[2],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[4,1,2,5,3] => [[1,3,4],[2,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[4,1,3,5,2] => [[1,3,4],[2],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[4,2,1,5,3] => [[1,4],[2,5],[3]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[4,2,3,5,1] => [[1,3,4],[2],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[4,3,1,5,2] => [[1,4],[2,5],[3]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[5,1,2,4,3] => [[1,3,4],[2],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[5,1,3,4,2] => [[1,3,4],[2],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[5,2,1,4,3] => [[1,4],[2,5],[3]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[5,2,3,4,1] => [[1,3,4],[2],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[5,3,1,4,2] => [[1,4],[2,5],[3]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[5,4,1,3,2] => [[1,4],[2],[3],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[1,2,3,5,4,6] => [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,2,3,6,4,5] => [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,2,3,6,5,4] => [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,2,4,5,3,6] => [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,2,4,6,3,5] => [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,2,4,6,5,3] => [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,2,5,6,3,4] => [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,2,5,6,4,3] => [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,3,4,5,2,6] => [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,3,4,6,2,5] => [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,3,4,6,5,2] => [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,3,5,6,2,4] => [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,3,5,6,4,2] => [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,4,5,6,2,3] => [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,4,5,6,3,2] => [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,1,3,5,4,6] => [[1,3,4,6],[2,5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,1,3,6,4,5] => [[1,3,4,6],[2,5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,1,3,6,5,4] => [[1,3,4],[2,5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,1,4,5,3,6] => [[1,3,4,6],[2,5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,1,4,6,3,5] => [[1,3,4],[2,5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,1,4,6,5,3] => [[1,3,4],[2,5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,1,5,6,3,4] => [[1,3,4],[2,5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,1,5,6,4,3] => [[1,3,4],[2,5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,3,4,5,1,6] => [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,3,4,6,1,5] => [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,3,4,6,5,1] => [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 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.
Matching statistic: St001330
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[1,3,2] => [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[2,3,1] => [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,3,4,2] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,4,2,3] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,3,1,4] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,3,4,1] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,4,1,3] => [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,4,3,1] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1,4,2] => [[1,3],[2,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,4,1] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,4,2,1] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,1,3,2] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,5,3,2] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,2,4,3] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,3,2,4] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,3,4,2] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,4,2,3] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,3,5,4] => [[1,3,4],[2,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,4,3,5] => [[1,3,5],[2,4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,4,5,3] => [[1,3,4],[2,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,3,4] => [[1,3,5],[2,4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,4,3] => [[1,3],[2,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,1,5,4] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,3,4,1,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,4,5,1] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,5,1,4] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,5,4,1] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,4,1,5,3] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,4,3,1,5] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,4,3,5,1] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,4,5,1,3] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,4,5,3,1] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,5,1,4,3] => [[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,5,3,4,1] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,4,1,5,2] => [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,4,2,5,1] => [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,5,1,4,2] => [[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,5,2,4,1] => [[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,5,1,3,2] => [[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,5,2,3,1] => [[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,2,4,3,6,5] => [[1,2,3,5],[4,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,2,5,3,6,4] => [[1,2,3,5],[4,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,2,5,4,6,3] => [[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,2,6,3,5,4] => [[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,2,6,4,5,3] => [[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,3,2,4,6,5] => [[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,3,2,5,4,6] => [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,3,2,5,6,4] => [[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,3,2,6,4,5] => [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,3,2,6,5,4] => [[1,2,4],[3,5],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,3,4,2,6,5] => [[1,2,3,5],[4,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,3,5,2,6,4] => [[1,2,3,5],[4,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,3,5,4,6,2] => [[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,3,6,2,5,4] => [[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)
=> ? = 1 + 2
[1,3,6,4,5,2] => [[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,4,2,3,6,5] => [[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,4,2,5,3,6] => [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,4,2,5,6,3] => [[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,4,2,6,3,5] => [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,4,2,6,5,3] => [[1,2,4],[3,5],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,4,3,2,6,5] => [[1,2,5],[3,6],[4]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,4,3,5,2,6] => [[1,2,4,6],[3],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,4,3,5,6,2] => [[1,2,4,5],[3],[6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,4,3,6,2,5] => [[1,2,4],[3,6],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,4,3,6,5,2] => [[1,2,4],[3,5],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,4,5,2,6,3] => [[1,2,3,5],[4,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,4,5,3,6,2] => [[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,4,6,2,5,3] => [[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)
=> ? = 1 + 2
[1,4,6,3,5,2] => [[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)
=> ? = 1 + 2
[1,5,2,3,6,4] => [[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,5,2,4,3,6] => [[1,2,4,6],[3],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,5,2,4,6,3] => [[1,2,4,5],[3],[6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,5,2,6,3,4] => [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,5,2,6,4,3] => [[1,2,4],[3,5],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000834
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 50%
Values
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 0 + 2
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 0 + 2
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 0 + 2
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 0 + 2
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 0 + 2
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 0 + 2
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2 = 0 + 2
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 0 + 2
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2 = 0 + 2
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2 = 0 + 2
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 3 = 1 + 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2 = 0 + 2
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 0 + 2
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2 = 0 + 2
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2 = 0 + 2
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 3 = 1 + 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2 = 0 + 2
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 3 = 1 + 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2 = 0 + 2
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2 = 0 + 2
[1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 3 = 1 + 2
[1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2 = 0 + 2
[1,5,3,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 3 = 1 + 2
[1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2 = 0 + 2
[1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 0 + 2
[2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2 = 0 + 2
[2,1,4,3,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 2 = 0 + 2
[2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2 = 0 + 2
[2,1,5,3,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 2 = 0 + 2
[2,1,5,4,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 2 = 0 + 2
[2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2 = 0 + 2
[2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 3 = 1 + 2
[2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2 = 0 + 2
[2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 0 + 2
[1,2,4,3,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 0 + 2
[1,2,4,3,6,5] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1 + 2
[1,2,5,3,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 0 + 2
[1,2,5,3,6,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1 + 2
[1,2,5,4,3,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,2,5,4,6,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1 + 2
[1,2,6,3,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 0 + 2
[1,2,6,3,5,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1 + 2
[1,2,6,4,3,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,2,6,4,5,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1 + 2
[1,2,6,5,3,4] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,2,6,5,4,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,3,2,4,5,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 0 + 2
[1,3,2,4,6,5] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2 + 2
[1,3,2,5,4,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1 + 2
[1,3,2,5,6,4] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2 + 2
[1,3,2,6,4,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1 + 2
[1,3,2,6,5,4] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 1 + 2
[1,3,4,2,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 0 + 2
[1,3,4,2,6,5] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1 + 2
[1,3,5,2,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 0 + 2
[1,3,5,2,6,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1 + 2
[1,3,5,4,2,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,3,5,4,6,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1 + 2
[1,3,6,2,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 0 + 2
[1,3,6,2,5,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1 + 2
[1,3,6,4,2,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,3,6,4,5,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1 + 2
[1,3,6,5,2,4] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,3,6,5,4,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,4,2,3,5,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 0 + 2
[1,4,2,3,6,5] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2 + 2
[1,4,2,5,3,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1 + 2
[1,4,2,5,6,3] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2 + 2
[1,4,2,6,3,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1 + 2
[1,4,2,6,5,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 1 + 2
[1,4,3,2,5,6] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 0 + 2
[1,4,3,2,6,5] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 1 + 2
[1,4,3,5,2,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1 + 2
[1,4,3,5,6,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2 + 2
[1,4,3,6,2,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1 + 2
[1,4,3,6,5,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 1 + 2
[1,4,5,2,3,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 0 + 2
[1,4,5,2,6,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1 + 2
[1,4,5,3,2,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,4,5,3,6,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1 + 2
[1,4,6,2,3,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 0 + 2
[1,4,6,2,5,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1 + 2
[1,4,6,3,2,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,4,6,3,5,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1 + 2
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000256
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 50%
Values
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1 = 0 + 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1 = 0 + 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 0 + 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1 = 0 + 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 0 + 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1 = 0 + 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 0 + 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1 = 0 + 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1 = 0 + 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 0 + 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1 = 0 + 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 0 + 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1 = 0 + 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1 = 0 + 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1 = 0 + 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 0 + 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1 = 0 + 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1 = 0 + 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1 = 0 + 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1 = 0 + 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1 = 0 + 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1 = 0 + 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1 = 0 + 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2 = 1 + 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1 = 0 + 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1 = 0 + 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1 = 0 + 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1 = 0 + 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1 = 0 + 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2 = 1 + 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1 = 0 + 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2 = 1 + 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1 = 0 + 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1 = 0 + 1
[1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1 = 0 + 1
[1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2 = 1 + 1
[1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1 = 0 + 1
[1,5,3,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2 = 1 + 1
[1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1 = 0 + 1
[1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1 = 0 + 1
[2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1 = 0 + 1
[2,1,4,3,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1 = 0 + 1
[2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1 = 0 + 1
[2,1,5,3,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1 = 0 + 1
[2,1,5,4,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1 = 0 + 1
[2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1 = 0 + 1
[2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2 = 1 + 1
[2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1 = 0 + 1
[2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1 = 0 + 1
[1,2,4,3,6,5] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 1 + 1
[1,2,5,3,6,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 1 + 1
[1,2,5,4,3,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 0 + 1
[1,2,5,4,6,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 1 + 1
[1,2,6,3,5,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 1 + 1
[1,2,6,4,3,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 0 + 1
[1,2,6,4,5,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 1 + 1
[1,2,6,5,3,4] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 0 + 1
[1,2,6,5,4,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 0 + 1
[1,3,2,4,6,5] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 2 + 1
[1,3,2,5,4,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 1 + 1
[1,3,2,5,6,4] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 2 + 1
[1,3,2,6,4,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 1 + 1
[1,3,2,6,5,4] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 1 + 1
[1,3,4,2,6,5] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 1 + 1
[1,3,5,2,6,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 1 + 1
[1,3,5,4,2,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 0 + 1
[1,3,5,4,6,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 1 + 1
[1,3,6,2,5,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 1 + 1
[1,3,6,4,2,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 0 + 1
[1,3,6,4,5,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 1 + 1
[1,3,6,5,2,4] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 0 + 1
[1,3,6,5,4,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 0 + 1
[1,4,2,3,6,5] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 2 + 1
[1,4,2,5,3,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 1 + 1
[1,4,2,5,6,3] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 2 + 1
[1,4,2,6,3,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 1 + 1
[1,4,2,6,5,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 1 + 1
[1,4,3,2,5,6] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? = 0 + 1
[1,4,3,2,6,5] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 1 + 1
[1,4,3,5,2,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 1 + 1
[1,4,3,5,6,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 2 + 1
[1,4,3,6,2,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 1 + 1
[1,4,3,6,5,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 1 + 1
[1,4,5,2,6,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 1 + 1
[1,4,5,3,2,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 0 + 1
[1,4,5,3,6,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 1 + 1
[1,4,6,2,5,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 1 + 1
[1,4,6,3,2,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 0 + 1
[1,4,6,3,5,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 1 + 1
[1,4,6,5,2,3] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 0 + 1
[1,4,6,5,3,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 0 + 1
[1,5,2,3,6,4] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 2 + 1
[1,5,2,4,3,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 1 + 1
[1,5,2,4,6,3] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 2 + 1
[1,5,2,6,3,4] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 1 + 1
[1,5,2,6,4,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 1 + 1
[1,5,3,2,4,6] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? = 0 + 1
[1,5,3,2,6,4] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 1 + 1
[1,5,3,4,2,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 1 + 1
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St001651
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001651: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 25%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001651: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 25%
Values
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 0
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 0
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 0
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[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),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 0
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[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),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[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),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 0
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 0
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[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),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 0
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[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),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 1
[1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 0
[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),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 1
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 0
[1,5,4,3,2] => [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),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
=> ? = 0
[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),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0
[2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0
[2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[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),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 0
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[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),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 1
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[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),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 0
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[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),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 1
[2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 0
[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),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 1
[2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[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),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 0
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[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),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 1
[2,5,3,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 0
[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),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 1
[2,5,4,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 0
[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),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
=> ? = 0
[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),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0
[3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0
[3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[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),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 0
[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),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 0
[3,2,4,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0
[3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[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),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 0
[3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[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),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 1
[3,4,2,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 0
[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),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 1
[3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[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),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 0
[3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[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),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0
[1,2,3,5,4,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
[1,2,3,5,6,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0
[1,2,3,6,4,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
[1,2,4,3,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[1,2,4,5,3,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
[1,2,4,5,6,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0
[1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
[1,2,5,3,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
[1,2,6,3,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[1,3,2,4,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,3,4,2,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[1,3,4,5,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
[1,3,4,5,6,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0
[1,3,4,6,2,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
[1,3,5,2,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
Description
The Frankl number of a lattice.
For a lattice $L$ on at least two elements, this is
$$
\max_x(|L|-2|[x, 1]|),
$$
where we maximize over all join irreducible elements and $[x, 1]$ denotes the interval from $x$ to the top element. Frankl's conjecture asserts that this number is non-negative, and zero if and only if $L$ is a Boolean lattice.
Matching statistic: St000068
(load all 89 compositions to match this statistic)
(load all 89 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000068: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 25%
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000068: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 25%
Values
[1,3,2] => 01 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,3,1] => 01 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,2,4,3] => 001 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,3,2,4] => 010 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,3,4,2] => 001 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,4,2,3] => 010 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,4,3,2] => 011 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[2,1,4,3] => 101 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
[2,3,1,4] => 010 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[2,3,4,1] => 001 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[2,4,1,3] => 010 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[2,4,3,1] => 011 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,1,4,2] => 101 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
[3,2,4,1] => 101 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
[3,4,1,2] => 010 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,4,2,1] => 011 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[4,1,3,2] => 101 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
[4,2,3,1] => 101 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,4] => 0001 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[1,2,4,3,5] => 0010 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[1,2,4,5,3] => 0001 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[1,2,5,3,4] => 0010 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[1,2,5,4,3] => 0011 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[1,3,2,4,5] => 0100 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[1,3,2,5,4] => 0101 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[1,3,4,2,5] => 0010 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[1,3,4,5,2] => 0001 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[1,3,5,2,4] => 0010 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[1,3,5,4,2] => 0011 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[1,4,2,3,5] => 0100 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[1,4,2,5,3] => 0101 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[1,4,3,2,5] => 0110 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[1,4,3,5,2] => 0101 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[1,4,5,2,3] => 0010 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[1,4,5,3,2] => 0011 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[1,5,2,3,4] => 0100 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[1,5,2,4,3] => 0101 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[1,5,3,2,4] => 0110 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[1,5,3,4,2] => 0101 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[1,5,4,2,3] => 0110 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[1,5,4,3,2] => 0111 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[2,1,3,5,4] => 1001 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 0 + 1
[2,1,4,3,5] => 1010 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0 + 1
[2,1,4,5,3] => 1001 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 0 + 1
[2,1,5,3,4] => 1010 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0 + 1
[2,1,5,4,3] => 1011 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[2,3,1,4,5] => 0100 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[2,3,1,5,4] => 0101 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[2,3,4,1,5] => 0010 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[2,3,4,5,1] => 0001 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[2,3,5,1,4] => 0010 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[2,3,5,4,1] => 0011 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[2,4,1,3,5] => 0100 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[2,4,1,5,3] => 0101 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[2,4,3,1,5] => 0110 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[2,4,3,5,1] => 0101 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[2,4,5,1,3] => 0010 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[2,4,5,3,1] => 0011 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[2,5,1,3,4] => 0100 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[2,5,1,4,3] => 0101 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[2,5,3,1,4] => 0110 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[2,5,3,4,1] => 0101 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[2,5,4,1,3] => 0110 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[2,5,4,3,1] => 0111 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[3,1,2,5,4] => 1001 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 0 + 1
[3,1,4,2,5] => 1010 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0 + 1
[3,1,4,5,2] => 1001 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 0 + 1
[3,1,5,2,4] => 1010 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0 + 1
[3,1,5,4,2] => 1011 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[3,2,1,5,4] => 1101 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[3,2,4,1,5] => 1010 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0 + 1
[3,2,4,5,1] => 1001 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 0 + 1
[3,2,5,1,4] => 1010 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0 + 1
[3,2,5,4,1] => 1011 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[3,4,1,2,5] => 0100 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[3,4,1,5,2] => 0101 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[3,4,2,1,5] => 0110 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[3,4,2,5,1] => 0101 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[3,4,5,1,2] => 0010 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[3,4,5,2,1] => 0011 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[3,5,1,2,4] => 0100 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[3,5,1,4,2] => 0101 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[3,5,2,1,4] => 0110 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[3,5,2,4,1] => 0101 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[3,5,4,1,2] => 0110 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[3,5,4,2,1] => 0111 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[4,1,2,5,3] => 1001 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 0 + 1
[4,1,3,2,5] => 1010 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0 + 1
[4,1,3,5,2] => 1001 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 0 + 1
[4,1,5,2,3] => 1010 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0 + 1
[4,1,5,3,2] => 1011 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[4,2,1,5,3] => 1101 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[4,2,3,1,5] => 1010 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0 + 1
[4,2,3,5,1] => 1001 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 0 + 1
[4,2,5,1,3] => 1010 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0 + 1
[4,2,5,3,1] => 1011 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[4,3,1,5,2] => 1101 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[4,3,2,5,1] => 1101 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[4,3,5,1,2] => 1010 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0 + 1
[4,5,2,1,3] => 0110 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
Description
The number of minimal elements in a poset.
The following 109 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001487The number of inner corners of a skew partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001960The number of descents of a permutation minus one if its first entry is not one. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000354The number of recoils of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001618The cardinality of the Frattini sublattice of a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001568The smallest positive integer that does not appear twice in the partition. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001625The Möbius invariant of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001754The number of tolerances of a finite lattice. St000312The number of leaves in a graph. St000627The exponent of a binary word. St000878The number of ones minus the number of zeros of a binary word. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001851The number of Hecke atoms of a signed permutation. St001862The number of crossings of a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001769The reflection length of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000098The chromatic number of a graph. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001071The beta invariant of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001743The discrepancy of a graph. St000364The exponent of the automorphism group of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001654The monophonic hull number of a graph. St001716The 1-improper chromatic number of a graph. St001396Number of triples of incomparable elements in a finite poset. St001335The cardinality of a minimal cycle-isolating set of a graph. St001472The permanent of the Coxeter matrix of the poset. St001532The leading coefficient of the Poincare polynomial of the poset cone.
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