Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001541
Mapping
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001541: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[{1,2},{3}] => [2,1] => [[2,2],[1]]
 => [1]
 => 0
[{1,3},{2}] => [2,1] => [[2,2],[1]]
 => [1]
 => 0
[{2,3},{1}] => [2,1] => [[2,2],[1]]
 => [1]
 => 0
[{1},{2,3},{4}] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[{1},{2,4},{3}] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[{1},{3,4},{2}] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[{2},{1,3},{4}] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[{2},{1,4},{3}] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[{3},{1,2},{4}] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[{4},{1,2},{3}] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[{3},{1,4},{2}] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[{4},{1,3},{2}] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[{2},{3,4},{1}] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[{3},{2,4},{1}] => [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},{4}] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[{1,2},{4},{3}] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[{1,3},{2},{4}] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[{1,4},{2},{3}] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[{1,3},{4},{2}] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[{1,4},{3},{2}] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[{2,3},{1},{4}] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[{2,4},{1},{3}] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[{3,4},{1},{2}] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[{2,3},{4},{1}] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[{2,4},{3},{1}] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[{3,4},{2},{1}] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[{1,2},{3,4}] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[{1,3},{2,4}] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[{1,4},{2,3}] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[{2,3},{1,4}] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[{2,4},{1,3}] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[{3,4},{1,2}] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[{1,2,3},{4}] => [3,1] => [[3,3],[2]]
 => [2]
 => 1
[{1,2,4},{3}] => [3,1] => [[3,3],[2]]
 => [2]
 => 1
[{1,3,4},{2}] => [3,1] => [[3,3],[2]]
 => [2]
 => 1
[{2,3,4},{1}] => [3,1] => [[3,3],[2]]
 => [2]
 => 1
[{1},{2},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
[{1},{2},{3,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
[{1},{2},{4,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
[{1},{3},{2,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
[{1},{3},{2,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
[{1},{4},{2,3},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
[{1},{5},{2,3},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
[{1},{4},{2,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
[{1},{5},{2,4},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
[{1},{3},{4,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
[{1},{4},{3,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
[{1},{5},{3,4},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
[{2},{1},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
Description
The Gini index of an integer partition. As discussed in [1], this statistic is equal to [[St000567]] applied to the conjugate partition.
Matching statistic: St001330
Mapping
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 43%●distinct values known / distinct values provided: 10%
Values
[{1,2},{3}] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[{1,3},{2}] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[{2,3},{1}] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[{1},{2,3},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 0 + 2
[{1},{2,4},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 0 + 2
[{1},{3,4},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 0 + 2
[{2},{1,3},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 0 + 2
[{2},{1,4},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 0 + 2
[{3},{1,2},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 0 + 2
[{4},{1,2},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 0 + 2
[{3},{1,4},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 0 + 2
[{4},{1,3},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 0 + 2
[{2},{3,4},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 0 + 2
[{3},{2,4},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 0 + 2
[{4},{2,3},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 0 + 2
[{1,2},{3},{4}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[{1,2},{4},{3}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[{1,3},{2},{4}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[{1,4},{2},{3}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[{1,3},{4},{2}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[{1,4},{3},{2}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[{2,3},{1},{4}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[{2,4},{1},{3}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[{3,4},{1},{2}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[{2,3},{4},{1}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[{2,4},{3},{1}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[{3,4},{2},{1}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[{1,2},{3,4}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[{1,3},{2,4}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[{1,4},{2,3}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[{2,3},{1,4}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[{2,4},{1,3}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[{3,4},{1,2}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[{1,2,3},{4}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[{1,2,4},{3}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[{1,3,4},{2}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[{2,3,4},{1}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[{1},{2},{3,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{1},{2},{3,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{1},{2},{4,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{1},{3},{2,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{1},{3},{2,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{1},{4},{2,3},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{1},{5},{2,3},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{1},{4},{2,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{1},{5},{2,4},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{1},{3},{4,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{1},{4},{3,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{1},{5},{3,4},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{2},{1},{3,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{2},{1},{3,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{2},{1},{4,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{3},{1},{2,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{3},{1},{2,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{4},{1},{2,3},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{5},{1},{2,3},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{4},{1},{2,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{5},{1},{2,4},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{3},{1},{4,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{4},{1},{3,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
[{1},{2,3},{4,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{1},{2,4},{3,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{1},{2,5},{3,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{1},{3,4},{2,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{1},{3,5},{2,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{1},{4,5},{2,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{2},{1,3},{4,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{2},{1,4},{3,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{2},{1,5},{3,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{3},{1,2},{4,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{4},{1,2},{3,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{5},{1,2},{3,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{3},{1,4},{2,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{3},{1,5},{2,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{4},{1,3},{2,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{5},{1,3},{2,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{4},{1,5},{2,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{5},{1,4},{2,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{2},{3,4},{1,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{2},{3,5},{1,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{2},{4,5},{1,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{3},{2,4},{1,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{3},{2,5},{1,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{4},{2,3},{1,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{5},{2,3},{1,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{4},{2,5},{1,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{5},{2,4},{1,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{3},{4,5},{1,2}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{4},{3,5},{1,2}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{5},{3,4},{1,2}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[{1},{2,3,4},{5}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[{1},{2,3,5},{4}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[{1},{2,4,5},{3}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[{1},{3,4,5},{2}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[{2},{1,3,4},{5}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[{2},{1,3,5},{4}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[{2},{1,4,5},{3}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[{3},{1,2,4},{5}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[{3},{1,2,5},{4}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[{4},{1,2,3},{5}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 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: St001645
(load all 2 compositions to match this statistic)
Mapping
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 10%
Values
[{1,2},{3}] => [2,1] => [1,2] => ([(1,2)],3)
 => ? = 0 + 6
[{1,3},{2}] => [2,1] => [1,2] => ([(1,2)],3)
 => ? = 0 + 6
[{2,3},{1}] => [2,1] => [1,2] => ([(1,2)],3)
 => ? = 0 + 6
[{1},{2,3},{4}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{1},{2,4},{3}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{1},{3,4},{2}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{2},{1,3},{4}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{2},{1,4},{3}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{3},{1,2},{4}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{4},{1,2},{3}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{3},{1,4},{2}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{4},{1,3},{2}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{2},{3,4},{1}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{3},{2,4},{1}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{4},{2,3},{1}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{1,2},{3},{4}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{1,2},{4},{3}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{1,3},{2},{4}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{1,4},{2},{3}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{1,3},{4},{2}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{1,4},{3},{2}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{2,3},{1},{4}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{2,4},{1},{3}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{3,4},{1},{2}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{2,3},{4},{1}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{2,4},{3},{1}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{3,4},{2},{1}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 6
[{1,2},{3,4}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 6
[{1,3},{2,4}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 6
[{1,4},{2,3}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 6
[{2,3},{1,4}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 6
[{2,4},{1,3}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 6
[{3,4},{1,2}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 6
[{1,2,3},{4}] => [3,1] => [1,3] => ([(2,3)],4)
 => ? = 1 + 6
[{1,2,4},{3}] => [3,1] => [1,3] => ([(2,3)],4)
 => ? = 1 + 6
[{1,3,4},{2}] => [3,1] => [1,3] => ([(2,3)],4)
 => ? = 1 + 6
[{2,3,4},{1}] => [3,1] => [1,3] => ([(2,3)],4)
 => ? = 1 + 6
[{1},{2},{3,4},{5}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 6
[{1},{2},{3,5},{4}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 6
[{1},{2},{4,5},{3}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 6
[{1},{3},{2,4},{5}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 6
[{1},{3},{2,5},{4}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 6
[{1},{4},{2,3},{5}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 6
[{1},{5},{2,3},{4}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 6
[{1},{4},{2,5},{3}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 6
[{1},{5},{2,4},{3}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 6
[{1},{3},{4,5},{2}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 6
[{1},{4},{3,5},{2}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 6
[{1},{5},{3,4},{2}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 6
[{2},{1},{3,4},{5}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 6
[{1},{2},{3},{4,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{2},{3},{4,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{2},{3},{5,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{2},{4},{3,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{2},{4},{3,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{2},{5},{3,4},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{2},{6},{3,4},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{2},{5},{3,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{2},{6},{3,5},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{2},{4},{5,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{2},{5},{4,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{2},{6},{4,5},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{3},{2},{4,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{3},{2},{4,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{3},{2},{5,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{4},{2},{3,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{4},{2},{3,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{5},{2},{3,4},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{6},{2},{3,4},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{5},{2},{3,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{6},{2},{3,5},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{4},{2},{5,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{5},{2},{4,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{6},{2},{4,5},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{3},{4},{2,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{3},{4},{2,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{3},{5},{2,4},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{3},{6},{2,4},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{3},{5},{2,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{3},{6},{2,5},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{4},{3},{2,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{4},{3},{2,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{5},{3},{2,4},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{6},{3},{2,4},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{5},{3},{2,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{6},{3},{2,5},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{4},{5},{2,3},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{4},{6},{2,3},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{5},{4},{2,3},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{6},{4},{2,3},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{5},{6},{2,3},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{6},{5},{2,3},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{4},{5},{2,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{4},{6},{2,5},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{5},{4},{2,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{6},{4},{2,5},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{5},{6},{2,4},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{6},{5},{2,4},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{3},{4},{5,6},{2}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
[{1},{3},{5},{4,6},{2}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 0 + 6
Description
The pebbling number of a connected graph.
Matching statistic: St000454
Mapping
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[{1,2},{3}] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 0 - 1
[{1,3},{2}] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 0 - 1
[{2,3},{1}] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 0 - 1
[{1},{2,3},{4}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{1},{2,4},{3}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{1},{3,4},{2}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{2},{1,3},{4}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{2},{1,4},{3}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{3},{1,2},{4}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{4},{1,2},{3}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{3},{1,4},{2}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{4},{1,3},{2}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{2},{3,4},{1}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{3},{2,4},{1}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{4},{2,3},{1}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{1,2},{3},{4}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{1,2},{4},{3}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{1,3},{2},{4}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{1,4},{2},{3}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{1,3},{4},{2}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{1,4},{3},{2}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{2,3},{1},{4}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{2,4},{1},{3}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{3,4},{1},{2}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{2,3},{4},{1}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{2,4},{3},{1}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{3,4},{2},{1}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[{1,2},{3,4}] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 - 1
[{1,3},{2,4}] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 - 1
[{1,4},{2,3}] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 - 1
[{2,3},{1,4}] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 - 1
[{2,4},{1,3}] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 - 1
[{3,4},{1,2}] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 - 1
[{1,2,3},{4}] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 - 1
[{1,2,4},{3}] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 - 1
[{1,3,4},{2}] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 - 1
[{2,3,4},{1}] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 - 1
[{1},{2},{3,4},{5}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[{1},{2},{3,5},{4}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[{1},{2},{4,5},{3}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[{1},{3},{2,4},{5}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[{1},{3},{2,5},{4}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[{1},{4},{2,3},{5}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[{1},{5},{2,3},{4}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[{1},{4},{2,5},{3}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[{1},{5},{2,4},{3}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[{1},{3},{4,5},{2}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[{1},{4},{3,5},{2}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[{1},{5},{3,4},{2}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[{2},{1},{3,4},{5}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[{1,2,3},{4},{5}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{1,2,3},{5},{4}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{1,2,4},{3},{5}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{1,2,5},{3},{4}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{1,2,4},{5},{3}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{1,2,5},{4},{3}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{1,3,4},{2},{5}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{1,3,5},{2},{4}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{1,4,5},{2},{3}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{1,3,4},{5},{2}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{1,3,5},{4},{2}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{1,4,5},{3},{2}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{2,3,4},{1},{5}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{2,3,5},{1},{4}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{2,4,5},{1},{3}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{3,4,5},{1},{2}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{2,3,4},{5},{1}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{2,3,5},{4},{1}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{2,4,5},{3},{1}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{3,4,5},{2},{1}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[{1,2,3,4},{5}] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[{1,2,3,5},{4}] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[{1,2,4,5},{3}] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[{1,3,4,5},{2}] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[{2,3,4,5},{1}] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[{1,2,3},{4},{5,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,2,3},{5},{4,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,2,3},{6},{4,5}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,2,4},{3},{5,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,2,5},{3},{4,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,2,6},{3},{4,5}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,2,4},{5},{3,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,2,4},{6},{3,5}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 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)
 => 3 = 4 - 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)
 => 3 = 4 - 1
[{1,2,5},{6},{3,4}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 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)
 => 3 = 4 - 1
[{1,3,4},{2},{5,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,3,5},{2},{4,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,3,6},{2},{4,5}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,4,5},{2},{3,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,4,6},{2},{3,5}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,5,6},{2},{3,4}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,3,4},{5},{2,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,3,4},{6},{2,5}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 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)
 => 3 = 4 - 1
[{1,3,6},{4},{2,5}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,3,5},{6},{2,4}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,3,6},{5},{2,4}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[{1,4,5},{3},{2,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
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.