Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000312
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000312: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => ([],1)
 => 0
[[1,2]]
 => [2] => ([],2)
 => 0
[[1],[2]]
 => [2] => ([],2)
 => 0
[[1,2,3]]
 => [3] => ([],3)
 => 0
[[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1,2],[3]]
 => [3] => ([],3)
 => 0
[[1],[2],[3]]
 => [3] => ([],3)
 => 0
[[1,2,3,4]]
 => [4] => ([],4)
 => 0
[[1,3,4],[2]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[1,2,3],[4]]
 => [4] => ([],4)
 => 0
[[1,3],[2,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[1,3],[2],[4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2],[3],[4]]
 => [4] => ([],4)
 => 0
[[1],[2],[3],[4]]
 => [4] => ([],4)
 => 0
[[1,2,3,4,5]]
 => [5] => ([],5)
 => 0
[[1,3,4,5],[2]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,4,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,2,3,4],[5]]
 => [5] => ([],5)
 => 0
[[1,3,5],[2,4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,2,5],[3,4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,3,4],[2,5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,4],[3,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,4,5],[2],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,3,5],[2],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,3,4],[2],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,4],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1,4],[2,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,3],[2,5],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,2],[3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,3],[2,4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2],[3,4],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,5],[2],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,4],[2],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,3],[2],[4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2],[3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => 0
[[1,3,4,5,6],[2]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 2
[[1,2,4,5,6],[3]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3,5,6],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[[1,2,3,4,5],[6]]
 => [6] => ([],6)
 => 0
[[1,3,5,6],[2,4]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
Description
The number of leaves in a graph. That is, the number of vertices of a graph that have degree 1.
Matching statistic: St001479
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001479: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => ([],1)
 => 0
[[1,2]]
 => [2] => ([],2)
 => 0
[[1],[2]]
 => [2] => ([],2)
 => 0
[[1,2,3]]
 => [3] => ([],3)
 => 0
[[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1,2],[3]]
 => [3] => ([],3)
 => 0
[[1],[2],[3]]
 => [3] => ([],3)
 => 0
[[1,2,3,4]]
 => [4] => ([],4)
 => 0
[[1,3,4],[2]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[1,2,3],[4]]
 => [4] => ([],4)
 => 0
[[1,3],[2,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[1,3],[2],[4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2],[3],[4]]
 => [4] => ([],4)
 => 0
[[1],[2],[3],[4]]
 => [4] => ([],4)
 => 0
[[1,2,3,4,5]]
 => [5] => ([],5)
 => 0
[[1,3,4,5],[2]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,4,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,2,3,4],[5]]
 => [5] => ([],5)
 => 0
[[1,3,5],[2,4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,2,5],[3,4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,3,4],[2,5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,4],[3,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,4,5],[2],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,3,5],[2],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,3,4],[2],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,4],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1,4],[2,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,3],[2,5],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,2],[3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,3],[2,4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2],[3,4],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,5],[2],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,4],[2],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,3],[2],[4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2],[3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => 0
[[1,3,4,5,6],[2]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 2
[[1,2,4,5,6],[3]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3,5,6],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[[1,2,3,4,5],[6]]
 => [6] => ([],6)
 => 0
[[1,3,5,6],[2,4]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
Description
The number of bridges of a graph. A bridge is an edge whose removal increases the number of connected components of the graph.
Matching statistic: St001826
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001826: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => ([],1)
 => 0
[[1,2]]
 => [2] => ([],2)
 => 0
[[1],[2]]
 => [2] => ([],2)
 => 0
[[1,2,3]]
 => [3] => ([],3)
 => 0
[[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1,2],[3]]
 => [3] => ([],3)
 => 0
[[1],[2],[3]]
 => [3] => ([],3)
 => 0
[[1,2,3,4]]
 => [4] => ([],4)
 => 0
[[1,3,4],[2]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[1,2,3],[4]]
 => [4] => ([],4)
 => 0
[[1,3],[2,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[1,3],[2],[4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2],[3],[4]]
 => [4] => ([],4)
 => 0
[[1],[2],[3],[4]]
 => [4] => ([],4)
 => 0
[[1,2,3,4,5]]
 => [5] => ([],5)
 => 0
[[1,3,4,5],[2]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,4,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,2,3,4],[5]]
 => [5] => ([],5)
 => 0
[[1,3,5],[2,4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,2,5],[3,4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,3,4],[2,5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,4],[3,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,4,5],[2],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,3,5],[2],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,3,4],[2],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,4],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1,4],[2,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,3],[2,5],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,2],[3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,3],[2,4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2],[3,4],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,5],[2],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,4],[2],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,3],[2],[4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2],[3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => 0
[[1,3,4,5,6],[2]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 2
[[1,2,4,5,6],[3]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3,5,6],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[[1,2,3,4,5],[6]]
 => [6] => ([],6)
 => 0
[[1,3,5,6],[2,4]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
Description
The maximal number of leaves on a vertex of a graph.
Matching statistic: St000422
(load all 5 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 43%
Values
[[1]]
 => [1] => [1] => ([],1)
 => 0
[[1,2]]
 => [2] => [2] => ([],2)
 => 0
[[1],[2]]
 => [2] => [2] => ([],2)
 => 0
[[1,2,3]]
 => [3] => [3] => ([],3)
 => 0
[[1,3],[2]]
 => [2,1] => [1,2] => ([(1,2)],3)
 => 2
[[1,2],[3]]
 => [3] => [3] => ([],3)
 => 0
[[1],[2],[3]]
 => [3] => [3] => ([],3)
 => 0
[[1,2,3,4]]
 => [4] => [4] => ([],4)
 => 0
[[1,3,4],[2]]
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3}
[[1,2,4],[3]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2
[[1,2,3],[4]]
 => [4] => [4] => ([],4)
 => 0
[[1,3],[2,4]]
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3}
[[1,2],[3,4]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2
[[1,4],[2],[3]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2
[[1,3],[2],[4]]
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3}
[[1,2],[3],[4]]
 => [4] => [4] => ([],4)
 => 0
[[1],[2],[3],[4]]
 => [4] => [4] => ([],4)
 => 0
[[1,2,3,4,5]]
 => [5] => [5] => ([],5)
 => 0
[[1,3,4,5],[2]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,4,5],[3]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3,5],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[[1,2,3,4],[5]]
 => [5] => [5] => ([],5)
 => 0
[[1,3,5],[2,4]]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,5],[3,4]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,4],[2,5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,4],[3,5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3],[4,5]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[[1,4,5],[2],[3]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,5],[2],[4]]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,5],[3],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[[1,3,4],[2],[5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,4],[3],[5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3],[4],[5]]
 => [5] => [5] => ([],5)
 => 0
[[1,4],[2,5],[3]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2,5],[4]]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3,5],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[[1,3],[2,4],[5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3,4],[5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,5],[2],[3],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[[1,4],[2],[3],[5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2],[4],[5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3],[4],[5]]
 => [5] => [5] => ([],5)
 => 0
[[1],[2],[3],[4],[5]]
 => [5] => [5] => ([],5)
 => 0
[[1,2,3,4,5,6]]
 => [6] => [6] => ([],6)
 => 0
[[1,3,4,5,6],[2]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,2,4,5,6],[3]]
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,5,6],[4]]
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,4,6],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[[1,2,3,4,5],[6]]
 => [6] => [6] => ([],6)
 => 0
[[1,3,5,6],[2,4]]
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,5,6],[3,4]]
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,4,6],[2,5]]
 => [2,3,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,4,6],[3,5]]
 => [3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,6],[4,5]]
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,4,5],[2,6]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,2,4,5],[3,6]]
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,5],[4,6]]
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,4],[5,6]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[[1,4,5,6],[2],[3]]
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,5,6],[2],[4]]
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,5,6],[3],[4]]
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,4,6],[2],[5]]
 => [2,3,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,4,6],[3],[5]]
 => [3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,6],[4],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[[1,3,4,5],[2],[6]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,2,4,5],[3],[6]]
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,5],[4],[6]]
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,4],[5],[6]]
 => [6] => [6] => ([],6)
 => 0
[[1,3,5],[2,4,6]]
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,5],[3,4,6]]
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,4],[2,5,6]]
 => [2,3,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,4],[3,5,6]]
 => [3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3],[4,5,6]]
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,4,6],[2,5],[3]]
 => [3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,6],[2,5],[4]]
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,6],[3,5],[4]]
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,6],[2,4],[5]]
 => [2,3,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,6],[3,4],[5]]
 => [3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,4,5],[2,6],[3]]
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,5],[2,6],[4]]
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,5],[3,6],[4]]
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,4],[2,6],[5]]
 => [2,3,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,4],[3,6],[5]]
 => [3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3],[4,6],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[[1,3,4],[2,5],[6]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,2,6],[3],[4],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[[1,3,4],[2],[5],[6]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,2,3],[4],[5],[6]]
 => [6] => [6] => ([],6)
 => 0
[[1,2],[3,6],[4],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[[1,3],[2,4],[5],[6]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,6],[2],[3],[4],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[[1,3],[2],[4],[5],[6]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,2],[3],[4],[5],[6]]
 => [6] => [6] => ([],6)
 => 0
[[1],[2],[3],[4],[5],[6]]
 => [6] => [6] => ([],6)
 => 0
[[1,2,3,4,5,6,7]]
 => [7] => [7] => ([],7)
 => 0
[[1,2,4,5,6,7],[3]]
 => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 4
[[1,2,3,4,5,7],[6]]
 => [6,1] => [1,6] => ([(5,6)],7)
 => 2
[[1,2,3,4,5,6],[7]]
 => [7] => [7] => ([],7)
 => 0
[[1,2,5,6,7],[3,4]]
 => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 4
[[1,2,4,5,6],[3,7]]
 => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 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: St000454
(load all 5 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 29%
Values
[[1]]
 => [1] => ([],1)
 => 0
[[1,2]]
 => [2] => ([],2)
 => 0
[[1],[2]]
 => [2] => ([],2)
 => 0
[[1,2,3]]
 => [3] => ([],3)
 => 0
[[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2
[[1,2],[3]]
 => [3] => ([],3)
 => 0
[[1],[2],[3]]
 => [3] => ([],3)
 => 0
[[1,2,3,4]]
 => [4] => ([],4)
 => 0
[[1,3,4],[2]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3,3}
[[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3,3}
[[1,2,3],[4]]
 => [4] => ([],4)
 => 0
[[1,3],[2,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3,3}
[[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3,3}
[[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3,3}
[[1,3],[2],[4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3,3}
[[1,2],[3],[4]]
 => [4] => ([],4)
 => 0
[[1],[2],[3],[4]]
 => [4] => ([],4)
 => 0
[[1,2,3,4,5]]
 => [5] => ([],5)
 => 0
[[1,3,4,5],[2]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,4,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,3,4],[5]]
 => [5] => ([],5)
 => 0
[[1,3,5],[2,4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,5],[3,4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,4],[2,5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,4],[3,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,4,5],[2],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,5],[2],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,3,4],[2],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,4],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1,4],[2,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2,5],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,3],[2,4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3,4],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,5],[2],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,4],[2],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2],[4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => 0
[[1,3,4,5,6],[2]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,4,5,6],[3]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,5,6],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,4,5],[6]]
 => [6] => ([],6)
 => 0
[[1,3,5,6],[2,4]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,5,6],[3,4]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,4,6],[2,5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,4,6],[3,5]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,6],[4,5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,3,4,5],[2,6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,4,5],[3,6]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,5],[4,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3,4],[5,6]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,4,5,6],[2],[3]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,5,6],[2],[4]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,5,6],[3],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,3,4,6],[2],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,4,6],[3],[5]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,6],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,4,5],[2],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,4,5],[3],[6]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,5],[4],[6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3,4],[5],[6]]
 => [6] => ([],6)
 => 0
[[1,3,5],[2,4,6]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,5],[3,4,6]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,4],[2,5,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,4],[3,5,6]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3],[4,5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,4,6],[2,5],[3]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,6],[2,5],[4]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,6],[3,5],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,3,6],[2,4],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,6],[3,4],[5]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,4,5],[2,6],[3]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,5],[2,6],[4]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,5],[3,6],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3],[4,5],[6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,5,6],[2],[3],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,5],[3],[4],[6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3],[4],[5],[6]]
 => [6] => ([],6)
 => 0
[[1,2],[3,5],[4,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,5],[2,6],[3],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,2],[3,5],[4],[6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,5],[2],[3],[4],[6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,2],[3],[4],[5],[6]]
 => [6] => ([],6)
 => 0
[[1],[2],[3],[4],[5],[6]]
 => [6] => ([],6)
 => 0
[[1,2,3,4,5,6,7]]
 => [7] => ([],7)
 => 0
[[1,2,3,5,6,7],[4]]
 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2
[[1,2,3,4,5,6],[7]]
 => [7] => ([],7)
 => 0
[[1,2,3,6,7],[4,5]]
 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2
[[1,2,3,5,6],[4,7]]
 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2
[[1,2,5,6,7],[3],[4]]
 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2
[[1,2,3,5,6],[4],[7]]
 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2
[[1,2,3,4,5],[6],[7]]
 => [7] => ([],7)
 => 0
[[1,2,3,7],[4,5,6]]
 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 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: St001629
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 14% ●values known / values provided: 20%●distinct values known / distinct values provided: 14%
Values
[[1]]
 => [1] => [1] => [1] => ? = 0
[[1,2]]
 => [2] => [1] => [1] => ? ∊ {0,0}
[[1],[2]]
 => [2] => [1] => [1] => ? ∊ {0,0}
[[1,2,3]]
 => [3] => [1] => [1] => ? ∊ {0,0,0,2}
[[1,3],[2]]
 => [2,1] => [1,1] => [2] => ? ∊ {0,0,0,2}
[[1,2],[3]]
 => [3] => [1] => [1] => ? ∊ {0,0,0,2}
[[1],[2],[3]]
 => [3] => [1] => [1] => ? ∊ {0,0,0,2}
[[1,2,3,4]]
 => [4] => [1] => [1] => ? ∊ {0,0,0,0,2,2,2,3,3,3}
[[1,3,4],[2]]
 => [2,2] => [2] => [1] => ? ∊ {0,0,0,0,2,2,2,3,3,3}
[[1,2,4],[3]]
 => [3,1] => [1,1] => [2] => ? ∊ {0,0,0,0,2,2,2,3,3,3}
[[1,2,3],[4]]
 => [4] => [1] => [1] => ? ∊ {0,0,0,0,2,2,2,3,3,3}
[[1,3],[2,4]]
 => [2,2] => [2] => [1] => ? ∊ {0,0,0,0,2,2,2,3,3,3}
[[1,2],[3,4]]
 => [3,1] => [1,1] => [2] => ? ∊ {0,0,0,0,2,2,2,3,3,3}
[[1,4],[2],[3]]
 => [3,1] => [1,1] => [2] => ? ∊ {0,0,0,0,2,2,2,3,3,3}
[[1,3],[2],[4]]
 => [2,2] => [2] => [1] => ? ∊ {0,0,0,0,2,2,2,3,3,3}
[[1,2],[3],[4]]
 => [4] => [1] => [1] => ? ∊ {0,0,0,0,2,2,2,3,3,3}
[[1],[2],[3],[4]]
 => [4] => [1] => [1] => ? ∊ {0,0,0,0,2,2,2,3,3,3}
[[1,2,3,4,5]]
 => [5] => [1] => [1] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,4,5],[2]]
 => [2,3] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,4,5],[3]]
 => [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3,5],[4]]
 => [4,1] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3,4],[5]]
 => [5] => [1] => [1] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,5],[2,4]]
 => [2,2,1] => [2,1] => [1,1] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,5],[3,4]]
 => [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,4],[2,5]]
 => [2,3] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,4],[3,5]]
 => [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3],[4,5]]
 => [4,1] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,4,5],[2],[3]]
 => [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,5],[2],[4]]
 => [2,2,1] => [2,1] => [1,1] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,5],[3],[4]]
 => [4,1] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,4],[2],[5]]
 => [2,3] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,4],[3],[5]]
 => [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3],[4],[5]]
 => [5] => [1] => [1] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,4],[2,5],[3]]
 => [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2,5],[4]]
 => [2,2,1] => [2,1] => [1,1] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3,5],[4]]
 => [4,1] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2,4],[5]]
 => [2,3] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3,4],[5]]
 => [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,4],[2],[3],[5]]
 => [3,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2],[4],[5]]
 => [2,3] => [1,1] => [2] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3],[4],[5]]
 => [5] => [1] => [1] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1],[2],[3],[4],[5]]
 => [5] => [1] => [1] => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3,4,5,6]]
 => [6] => [1] => [1] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,4,5,6],[2]]
 => [2,4] => [1,1] => [2] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,4,5,6],[3]]
 => [3,3] => [2] => [1] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,5,6],[4]]
 => [4,2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,4,6],[5]]
 => [5,1] => [1,1] => [2] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,4,5],[6]]
 => [6] => [1] => [1] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,5,6],[2,4]]
 => [2,2,2] => [3] => [1] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,4,6],[2,5]]
 => [2,3,1] => [1,1,1] => [3] => 1
[[1,2,4,6],[3,5]]
 => [3,2,1] => [1,1,1] => [3] => 1
[[1,3,4,6],[2],[5]]
 => [2,3,1] => [1,1,1] => [3] => 1
[[1,2,4,6],[3],[5]]
 => [3,2,1] => [1,1,1] => [3] => 1
[[1,3,4],[2,5,6]]
 => [2,3,1] => [1,1,1] => [3] => 1
[[1,2,4],[3,5,6]]
 => [3,2,1] => [1,1,1] => [3] => 1
[[1,4,6],[2,5],[3]]
 => [3,2,1] => [1,1,1] => [3] => 1
[[1,3,6],[2,4],[5]]
 => [2,3,1] => [1,1,1] => [3] => 1
[[1,2,6],[3,4],[5]]
 => [3,2,1] => [1,1,1] => [3] => 1
[[1,3,4],[2,6],[5]]
 => [2,3,1] => [1,1,1] => [3] => 1
[[1,2,4],[3,6],[5]]
 => [3,2,1] => [1,1,1] => [3] => 1
[[1,4,6],[2],[3],[5]]
 => [3,2,1] => [1,1,1] => [3] => 1
[[1,3,6],[2],[4],[5]]
 => [2,3,1] => [1,1,1] => [3] => 1
[[1,3],[2,4],[5,6]]
 => [2,3,1] => [1,1,1] => [3] => 1
[[1,2],[3,4],[5,6]]
 => [3,2,1] => [1,1,1] => [3] => 1
[[1,4],[2,6],[3],[5]]
 => [3,2,1] => [1,1,1] => [3] => 1
[[1,3],[2,6],[4],[5]]
 => [2,3,1] => [1,1,1] => [3] => 1
[[1,3,4,6,7],[2,5]]
 => [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4,5,7],[2,6]]
 => [2,4,1] => [1,1,1] => [3] => 1
[[1,2,3,5,7],[4,6]]
 => [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4,6,7],[2],[5]]
 => [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4,5,7],[2],[6]]
 => [2,4,1] => [1,1,1] => [3] => 1
[[1,2,3,5,7],[4],[6]]
 => [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4,7],[2,5,6]]
 => [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4,6],[2,5,7]]
 => [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4,5],[2,6,7]]
 => [2,4,1] => [1,1,1] => [3] => 1
[[1,2,3,5],[4,6,7]]
 => [4,2,1] => [1,1,1] => [3] => 1
[[1,3,6,7],[2,4],[5]]
 => [2,3,2] => [1,1,1] => [3] => 1
[[1,2,5,7],[3,6],[4]]
 => [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4,7],[2,6],[5]]
 => [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4,7],[2,5],[6]]
 => [2,4,1] => [1,1,1] => [3] => 1
[[1,2,3,7],[4,5],[6]]
 => [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4,6],[2,7],[5]]
 => [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4,5],[2,7],[6]]
 => [2,4,1] => [1,1,1] => [3] => 1
[[1,2,3,5],[4,7],[6]]
 => [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4,6],[2,5],[7]]
 => [2,3,2] => [1,1,1] => [3] => 1
[[1,3,6,7],[2],[4],[5]]
 => [2,3,2] => [1,1,1] => [3] => 1
[[1,2,5,7],[3],[4],[6]]
 => [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4,7],[2],[5],[6]]
 => [2,4,1] => [1,1,1] => [3] => 1
[[1,3,4,6],[2],[5],[7]]
 => [2,3,2] => [1,1,1] => [3] => 1
[[1,3,6],[2,4,7],[5]]
 => [2,3,2] => [1,1,1] => [3] => 1
[[1,2,5],[3,6,7],[4]]
 => [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4],[2,6,7],[5]]
 => [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4],[2,5,7],[6]]
 => [2,4,1] => [1,1,1] => [3] => 1
[[1,2,3],[4,5,7],[6]]
 => [4,2,1] => [1,1,1] => [3] => 1
[[1,3,4],[2,5,6],[7]]
 => [2,3,2] => [1,1,1] => [3] => 1
[[1,2,7],[3,5],[4,6]]
 => [4,2,1] => [1,1,1] => [3] => 1
[[1,3,7],[2,4],[5,6]]
 => [2,3,2] => [1,1,1] => [3] => 1
[[1,3,6],[2,4],[5,7]]
 => [2,3,2] => [1,1,1] => [3] => 1
[[1,3,4],[2,6],[5,7]]
 => [2,3,2] => [1,1,1] => [3] => 1
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St001060
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 43%
Values
[[1]]
 => [1] => [1] => ([],1)
 => ? = 0
[[1,2]]
 => [1,2] => [1,2] => ([],2)
 => ? ∊ {0,0}
[[1],[2]]
 => [2,1] => [1,2] => ([],2)
 => ? ∊ {0,0}
[[1,2,3]]
 => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,2}
[[1,3],[2]]
 => [2,1,3] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,2}
[[1,2],[3]]
 => [3,1,2] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,2}
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,2}
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,2,3,3,3}
[[1,3,4],[2]]
 => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2,4],[3]]
 => [3,1,2,4] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,2,3,3,3}
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,2,3,3,3}
[[1,3],[2,4]]
 => [2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,2,3,3,3}
[[1,2],[3,4]]
 => [3,4,1,2] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,2,3,3,3}
[[1,4],[2],[3]]
 => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,2,3,3,3}
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,2,3,3,3}
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,2,3,3,3}
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 2
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 2
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 2
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [1,2,3,4,6,5] => ([(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,2,3,4,5,6] => ([],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,4,6],[2,5]]
 => [2,5,1,3,4,6] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 2
[[1,2,4,6],[3,5]]
 => [3,5,1,2,4,6] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,6],[4,5]]
 => [4,5,1,2,3,6] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,4,5],[3,6]]
 => [3,6,1,2,4,5] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,5],[4,6]]
 => [4,6,1,2,3,5] => [1,2,3,5,4,6] => ([(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [1,2,3,4,5,6] => ([],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[[1,3,5,6],[2],[4]]
 => [4,2,1,3,5,6] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,3,4,6],[2],[5]]
 => [5,2,1,3,4,6] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 2
[[1,2,4,6],[3],[5]]
 => [5,3,1,2,4,6] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,2,3,6],[4],[5]]
 => [5,4,1,2,3,6] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[1,4,6],[2,5],[3]]
 => [3,2,5,1,4,6] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 2
[[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
[[1,3,6],[2,4],[5]]
 => [5,2,4,1,3,6] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 2
[[1,4,5],[2,6],[3]]
 => [3,2,6,1,4,5] => [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[[1,3,5],[2,6],[4]]
 => [4,2,6,1,3,5] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 2
[[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[[1,4,6],[2],[3],[5]]
 => [5,3,2,1,4,6] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[[1,3,6],[2],[4],[5]]
 => [5,4,2,1,3,6] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 2
[[1,5],[2,6],[3],[4]]
 => [4,3,2,6,1,5] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 2
[[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
Description
The distinguishing index of a graph. This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism. If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.