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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St001261
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 1
{{1,2}}
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 2
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([],3)
=> 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 3
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 3
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 3
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
Description
The Castelnuovo-Mumford regularity of a graph.
Matching statistic: St001393
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001393: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001393: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 0 = 1 - 1
{{1,2}}
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 1 = 2 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 2 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 3 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1 = 2 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1 = 2 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 2 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 2 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 2 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 2 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
Description
The induced matching number of a graph.
An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.
Matching statistic: St000455
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
{{1}}
=> [1] => ([],1)
=> ([(0,1)],2)
=> -1 = 1 - 2
{{1,2}}
=> [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
{{1},{2}}
=> [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
{{1,2,3}}
=> [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
{{1,2},{3}}
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
{{1,3},{2}}
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
{{1},{2,3}}
=> [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
{{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
{{1,2,3,4}}
=> [4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
{{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
{{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
{{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
{{1,2},{3},{4}}
=> [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),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
{{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
{{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
{{1,3},{2},{4}}
=> [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),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
{{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
{{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
{{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
{{1,4},{2},{3}}
=> [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),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
{{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
{{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 1 - 2
{{1,2,3,4,5}}
=> [5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
{{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
{{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
{{1,2,3},{4,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
{{1,2,3},{4},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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 = 2 - 2
{{1,2,4,5},{3}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
{{1,2,4},{3,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1,2,4},{3},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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 = 2 - 2
{{1,2,5},{3,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1,2},{3,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
{{1,2},{3,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
{{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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 = 2 - 2
{{1,2},{3,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
{{1,2},{3},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
{{1,3,4,5},{2}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
{{1,3,4},{2,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1,3,4},{2},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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 = 2 - 2
{{1,3,5},{2,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1,3},{2,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
{{1,3},{2,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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 = 2 - 2
{{1,3},{2,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1,3},{2},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
{{1,4,5},{2,3}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
{{1,4},{2,3,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1,4},{2,3},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1,5},{2,3,4}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1,5},{2,3},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1},{2,3},{4,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
{{1},{2,3},{4},{5}}
=> [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),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1,4,5},{2},{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),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
{{1,4},{2,5},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1,4},{2},{3,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
{{1,5},{2,4},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1},{2,4},{3,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1},{2,4},{3},{5}}
=> [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),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1,5},{2},{3,4}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1},{2,5},{3,4}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
{{1},{2,5},{3},{4}}
=> [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),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 1 - 2
{{1,2,3,4,5,6}}
=> [6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,2,3,4,5},{6}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,2,3,4,6},{5}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,2,3,4},{5,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 2
{{1,2,3,4},{5},{6}}
=> [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),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,2,3,5,6},{4}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,2,3,5},{4,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
{{1,2,3,5},{4},{6}}
=> [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),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,2,3,6},{4,5}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
{{1,2,3},{4,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 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)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 2
{{1,2,3,6},{4},{5}}
=> [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),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,2,3},{4,6},{5}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 2
{{1,2,3},{4},{5},{6}}
=> [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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,2,4,5,6},{3}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,2,4,5},{3},{6}}
=> [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),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,2,4,6},{3},{5}}
=> [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),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,2,4},{3},{5},{6}}
=> [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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,2,5,6},{3},{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),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,2,5},{3},{4},{6}}
=> [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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,2,6},{3},{4},{5}}
=> [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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,3,4,5,6},{2}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,3,4,5},{2},{6}}
=> [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),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,3,4,6},{2},{5}}
=> [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),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
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: St001582
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 75%
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> [1] => [1] => [1] => ? = 1 - 1
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 1 = 2 - 1
{{1},{2}}
=> [1,2] => [2,1] => [2,1] => 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [1,3,2] => [1,3,2] => 1 = 2 - 1
{{1,2},{3}}
=> [2,1,3] => [3,1,2] => [3,1,2] => 1 = 2 - 1
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => [1,3,2] => 1 = 2 - 1
{{1},{2,3}}
=> [1,3,2] => [2,3,1] => [2,3,1] => 1 = 2 - 1
{{1},{2},{3}}
=> [1,2,3] => [3,2,1] => [3,2,1] => 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 1 = 2 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 2 = 3 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 1 = 2 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,4,2,3] => [1,4,3,2] => 1 = 2 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,1,2,3] => [4,1,3,2] => 1 = 2 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 1 = 2 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [2,4,3,1] => [2,4,3,1] => 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1 = 2 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 1 = 2 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,3,4,1] => [2,4,3,1] => 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 2 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,5,4,3,2] => ? = 2 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 3 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => ? = 2 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,5,4,3,2] => ? = 2 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 2 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [5,1,3,4,2] => [5,1,4,3,2] => ? = 2 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,5,4,3,2] => ? = 2 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 3 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [5,3,4,1,2] => [5,3,4,1,2] => ? = 3 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,4,3,5,2] => [1,5,4,3,2] => ? = 2 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [3,4,5,1,2] => [3,5,4,1,2] => ? = 3 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 3 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 2 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,3,2] => ? = 2 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,1,4,5,3] => [2,1,5,4,3] => ? = 2 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [5,1,4,2,3] => [5,1,4,3,2] => ? = 2 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,5,4,3,2] => ? = 2 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 3 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 2 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,4,5,2,3] => [1,5,4,3,2] => ? = 2 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [2,4,1,5,3] => [2,5,1,4,3] => ? = 2 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [4,5,1,2,3] => [4,5,1,3,2] => ? = 3 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,3,2] => ? = 2 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,5,4,3,2] => ? = 3 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [2,1,5,3,4] => [2,1,5,4,3] => ? = 2 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [5,1,2,3,4] => [5,1,4,3,2] => ? = 2 - 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,2,4,3,5] => [1,5,4,3,2] => ? = 2 - 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 2 - 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 2 - 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,4,2,3,5] => [1,5,4,3,2] => ? = 2 - 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [2,4,5,3,1] => [2,5,4,3,1] => ? = 2 - 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 3 - 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 2 - 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,5,3,2,4] => [1,5,4,3,2] => ? = 2 - 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [2,1,3,5,4] => [2,1,5,4,3] => ? = 2 - 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [3,1,5,2,4] => [3,1,5,4,2] => ? = 3 - 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [5,1,3,2,4] => [5,1,4,3,2] => ? = 2 - 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,5,4,3,2] => ? = 2 - 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [2,5,3,4,1] => [2,5,4,3,1] => ? = 2 - 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 2 - 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => ? = 2 - 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,3,4,2,5] => [1,5,4,3,2] => ? = 2 - 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [2,3,4,5,1] => [2,5,4,3,1] => ? = 2 - 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 2 - 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 2 - 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,4,3,2,5] => [1,5,4,3,2] => ? = 2 - 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [2,4,3,5,1] => [2,5,4,3,1] => ? = 2 - 1
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Matching statistic: St001860
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001860: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 75%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001860: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> [1] => [1] => [1] => ? = 1 - 1
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [3,2,1] => [3,2,1] => 1 = 2 - 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [3,2,1] => 1 = 2 - 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 1 = 2 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,3,2,1] => [4,3,2,1] => 1 = 2 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,3,1] => [4,2,3,1] => 1 = 2 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1 = 2 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1 = 2 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 1 = 2 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 2 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 2 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 2 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 3 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 2 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 2 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 2 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 3 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 3 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 3 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 3 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 2 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 2 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 3 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 2 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 2 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 3 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 2 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 3 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 2 - 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => ? = 2 - 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => ? = 2 - 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 2 - 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 3 - 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 2 - 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2 - 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 - 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 2 - 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => ? = 2 - 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 2 - 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => ? = 2 - 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 2 - 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 2 - 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 2 - 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 2 - 1
Description
The number of factors of the Stanley symmetric function associated with a signed permutation.
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