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Your data matches 24 different statistics following compositions of up to 3 maps.
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Matching statistic: St000259
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(load all 3 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => [3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001093
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(load all 3 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => [2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
[2,1] => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => [3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => [4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4,5] => [5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
Description
The detour number of a graph.
This is the number of vertices in a longest induced path in a graph.
Note that [1] defines the detour number as the number of edges in a longest induced path, which is unsuitable for the empty graph.
Matching statistic: St000640
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00243: Graphs —weak duplicate order⟶ Posets
St000640: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00243: Graphs —weak duplicate order⟶ Posets
St000640: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 0 - 1
[1,2] => [2] => ([],2)
=> ([],1)
=> ? = 0 - 1
[2,1] => [1,1] => ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[1,2,3] => [3] => ([],3)
=> ([],1)
=> ? = 0 - 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([],2)
=> 0 = 1 - 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([],2)
=> 0 = 1 - 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
[1,2,3,4] => [4] => ([],4)
=> ([],1)
=> ? = 0 - 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> 0 = 1 - 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> 0 = 1 - 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> 0 = 1 - 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> 0 = 1 - 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> 0 = 1 - 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> 0 = 1 - 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
[1,2,3,4,5] => [5] => ([],5)
=> ([],1)
=> ? = 0 - 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 0 = 1 - 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 0 = 1 - 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)
=> 0 = 1 - 1
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 0 = 1 - 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)
=> 0 = 1 - 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 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)
=> 0 = 1 - 1
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],4)
=> 0 = 1 - 1
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 0 = 1 - 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)
=> 0 = 1 - 1
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 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)
=> 0 = 1 - 1
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],4)
=> 0 = 1 - 1
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 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)
=> 0 = 1 - 1
[3,5,1,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,2,3,4,5,6] => [6] => ([],6)
=> ([],1)
=> ? = 0 - 1
[1,2,3,4,5,6,7] => [7] => ([],7)
=> ([],1)
=> ? = 0 - 1
[5,6,7,8,4,3,2,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[4,6,7,8,5,3,2,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[4,5,7,8,6,3,2,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[4,5,6,8,7,3,2,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[3,6,7,8,5,4,2,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[3,5,7,8,6,4,2,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[3,5,6,8,7,4,2,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[3,4,7,8,6,5,2,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[3,4,6,8,7,5,2,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[3,4,5,8,7,6,2,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[2,6,7,8,5,4,3,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[2,5,7,8,6,4,3,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[2,5,6,8,7,4,3,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[2,4,7,8,6,5,3,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[2,4,6,8,7,5,3,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[2,3,7,8,6,5,4,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[2,3,6,8,7,5,4,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[2,3,4,8,7,6,5,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,6,7,8,5,4,3,2] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,5,7,8,6,4,3,2] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,5,6,8,7,4,3,2] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,4,7,8,6,5,3,2] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,4,6,8,7,5,3,2] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,4,5,8,7,6,3,2] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,3,7,8,6,5,4,2] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,3,6,8,7,5,4,2] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,3,5,8,7,6,4,2] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,3,4,8,7,6,5,2] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,2,7,8,6,5,4,3] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,2,6,8,7,5,4,3] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,2,5,8,7,6,4,3] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,2,4,8,7,6,5,3] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,2,3,8,7,6,5,4] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
Description
The rank of the largest boolean interval in a poset.
Matching statistic: St000264
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 95%●distinct values known / distinct values provided: 33%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 95%●distinct values known / distinct values provided: 33%
Values
[1] => => [] => ?
=> ? = 0 + 1
[1,2] => 0 => [1] => ([],1)
=> ? = 0 + 1
[2,1] => 1 => [1] => ([],1)
=> ? = 1 + 1
[1,2,3] => 00 => [2] => ([],2)
=> ? = 0 + 1
[1,3,2] => 01 => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
[2,3,1] => 01 => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
[3,2,1] => 11 => [2] => ([],2)
=> ? = 1 + 1
[1,2,3,4] => 000 => [3] => ([],3)
=> ? = 0 + 1
[1,2,4,3] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 1
[1,3,4,2] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 1
[1,4,3,2] => 011 => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[2,1,4,3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,4,1] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 1
[2,4,3,1] => 011 => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[3,1,4,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,4,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,2,1] => 011 => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[4,1,3,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2,3,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,3,2,1] => 111 => [3] => ([],3)
=> ? = 1 + 1
[1,2,3,4,5] => 0000 => [4] => ([],4)
=> ? = 0 + 1
[1,2,3,5,4] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,2,4,5,3] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,2,5,4,3] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,3,2,5,4] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,4,5,2] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,3,5,4,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,4,2,5,3] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,5,2] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,5,3,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,5,2,4,3] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,5,3,4,2] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,5,4,3,2] => 0111 => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
[2,1,3,5,4] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,4,5,3] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,5,4,3] => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,1,5,4] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,5,1] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 1
[2,3,5,4,1] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[2,4,1,5,3] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,3,5,1] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,5,3,1] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[2,5,1,4,3] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,5,3,4,1] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,5,4,3,1] => 0111 => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
[3,1,2,5,4] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,5,2] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,5,4,2] => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,1,5,4] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,4,5,1] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,5,4,1] => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,1,5,2] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,5,1] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,5,2,1] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[3,5,1,4,2] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,5,2,4,1] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,5,4,2,1] => 0111 => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
[4,1,2,5,3] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,3,5,2] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,5,3,2] => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,1,5,3] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,3,5,1] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,5,3,1] => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,1,5,2] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,2,5,1] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,5,2,1] => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,5,1,3,2] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,5,2,3,1] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,5,3,2,1] => 0111 => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
[5,1,2,4,3] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,1,3,4,2] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,1,4,3,2] => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,2,1,4,3] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,2,3,4,1] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,2,4,3,1] => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,3,1,4,2] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,3,2,4,1] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,3,4,2,1] => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,4,1,3,2] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,4,2,3,1] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,4,3,2,1] => 1111 => [4] => ([],4)
=> ? = 1 + 1
[1,2,3,4,5,6] => 00000 => [5] => ([],5)
=> ? = 0 + 1
[1,2,3,4,6,5] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,2,3,5,6,4] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,2,3,6,5,4] => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,2,4,5,6,3] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,2,4,6,5,3] => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,2,5,6,4,3] => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,2,6,5,4,3] => 00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[1,3,4,5,6,2] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,3,4,6,5,2] => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,3,5,6,4,2] => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,3,6,5,4,2] => 00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[1,4,5,6,3,2] => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,4,6,5,3,2] => 00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[1,5,6,4,3,2] => 00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[1,6,5,4,3,2] => 01111 => [1,4] => ([(3,4)],5)
=> ? = 1 + 1
[2,3,4,5,6,1] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[2,3,4,6,5,1] => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[2,3,5,6,4,1] => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000455
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 55%●distinct values known / distinct values provided: 33%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 55%●distinct values known / distinct values provided: 33%
Values
[1] => [.,.]
=> ([],1)
=> ([],1)
=> ? = 0 - 2
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 - 2
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 2
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 - 2
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 2
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 2
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 - 2
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0 = 2 - 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0 = 2 - 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 - 2
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[3,5,2,4,1] => [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,5,4,2,1] => [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[4,1,2,5,3] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[4,1,3,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[4,1,5,3,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[4,2,1,5,3] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[4,2,5,3,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[4,3,1,5,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[4,5,1,3,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[5,1,2,4,3] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[5,1,3,4,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[5,1,4,3,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[5,2,1,4,3] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[5,3,1,4,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[5,4,1,3,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 0 - 2
[1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,2,3,5,6,4] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,2,4,5,6,3] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,2,4,6,5,3] => [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,2,5,6,4,3] => [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,3,4,5,6,2] => [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,3,4,6,5,2] => [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,3,5,6,4,2] => [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,3,6,5,4,2] => [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,4,5,6,3,2] => [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,4,6,5,3,2] => [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,5,6,4,3,2] => [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[2,1,4,3,6,5] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[2,1,5,3,6,4] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[2,1,5,4,6,3] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 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: St001668
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00243: Graphs —weak duplicate order⟶ Posets
St001668: Posets ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00243: Graphs —weak duplicate order⟶ Posets
St001668: Posets ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 0 - 1
[1,2] => [2] => ([],2)
=> ([],1)
=> ? = 0 - 1
[2,1] => [1,1] => ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[1,2,3] => [3] => ([],3)
=> ([],1)
=> ? = 0 - 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([],2)
=> 0 = 1 - 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([],2)
=> 0 = 1 - 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
[1,2,3,4] => [4] => ([],4)
=> ([],1)
=> ? = 0 - 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> 0 = 1 - 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> 0 = 1 - 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> 0 = 1 - 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> 0 = 1 - 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> 0 = 1 - 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> 0 = 1 - 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
[1,2,3,4,5] => [5] => ([],5)
=> ([],1)
=> ? = 0 - 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 0 = 1 - 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 0 = 1 - 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)
=> 0 = 1 - 1
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 0 = 1 - 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)
=> 0 = 1 - 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 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)
=> 0 = 1 - 1
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],4)
=> 0 = 1 - 1
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 0 = 1 - 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)
=> 0 = 1 - 1
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 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)
=> 0 = 1 - 1
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],4)
=> 0 = 1 - 1
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 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)
=> 0 = 1 - 1
[3,5,1,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,2,3,4,5,6] => [6] => ([],6)
=> ([],1)
=> ? = 0 - 1
[2,1,4,3,6,5] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[2,1,5,3,6,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[2,1,5,4,6,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[2,1,6,3,5,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[2,1,6,4,5,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[2,1,6,5,4,3] => [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)
=> ([(3,5),(4,5)],6)
=> ? = 2 - 1
[3,1,4,2,6,5] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[3,1,5,2,6,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[3,1,5,4,6,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[3,1,6,2,5,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[3,1,6,4,5,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[3,1,6,5,4,2] => [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)
=> ([(3,5),(4,5)],6)
=> ? = 2 - 1
[3,2,1,6,5,4] => [1,1,2,1,1] => ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[3,2,4,1,6,5] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[3,2,5,1,6,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[3,2,5,4,6,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[3,2,6,1,5,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[3,2,6,4,5,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[3,2,6,5,4,1] => [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)
=> ([(3,5),(4,5)],6)
=> ? = 2 - 1
[4,1,3,2,6,5] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[4,1,5,2,6,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[4,1,5,3,6,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[4,1,6,2,5,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[4,1,6,3,5,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[4,1,6,5,3,2] => [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)
=> ([(3,5),(4,5)],6)
=> ? = 2 - 1
[4,2,1,6,5,3] => [1,1,2,1,1] => ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[4,2,3,1,6,5] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[4,2,5,1,6,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[4,2,5,3,6,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[4,2,6,1,5,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[4,2,6,3,5,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[4,2,6,5,3,1] => [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)
=> ([(3,5),(4,5)],6)
=> ? = 2 - 1
[4,3,1,6,5,2] => [1,1,2,1,1] => ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[4,3,2,1,6,5] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[4,3,2,6,5,1] => [1,1,2,1,1] => ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[4,3,5,1,6,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[4,3,5,2,6,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[4,3,6,1,5,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[4,3,6,2,5,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[4,3,6,5,2,1] => [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)
=> ([(3,5),(4,5)],6)
=> ? = 2 - 1
[5,1,3,2,6,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[5,1,4,2,6,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[5,1,4,3,6,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
[5,1,6,2,4,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St001603
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1]
=> []
=> ? = 0 - 1
[1,2] => [1,2] => [1,1]
=> [1]
=> ? = 0 - 1
[2,1] => [2,1] => [2]
=> []
=> ? = 1 - 1
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? = 0 - 1
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? = 1 - 1
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? = 1 - 1
[3,2,1] => [3,2,1] => [3]
=> []
=> ? = 1 - 1
[1,2,3,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? = 0 - 1
[1,2,4,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? = 1 - 1
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? = 1 - 1
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? = 1 - 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> ? = 2 - 1
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? = 1 - 1
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? = 1 - 1
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> ? = 2 - 1
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? = 2 - 1
[3,4,2,1] => [3,4,2,1] => [3,1]
=> [1]
=> ? = 1 - 1
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? = 2 - 1
[4,2,3,1] => [4,2,3,1] => [3,1]
=> [1]
=> ? = 2 - 1
[4,3,2,1] => [4,3,2,1] => [4]
=> []
=> ? = 1 - 1
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 0 - 1
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 1 - 1
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 1 - 1
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 1 - 1
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 2 - 1
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 1 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 1 - 1
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 2 - 1
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 2 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 1 - 1
[1,5,2,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 2 - 1
[1,5,3,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 2 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 1 - 1
[2,1,3,5,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> ? = 2 - 1
[2,1,4,5,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> ? = 2 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> ? = 2 - 1
[2,3,1,5,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> ? = 2 - 1
[2,3,4,5,1] => [2,5,4,3,1] => [4,1]
=> [1]
=> ? = 1 - 1
[2,3,5,4,1] => [2,5,4,3,1] => [4,1]
=> [1]
=> ? = 1 - 1
[2,4,1,5,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> ? = 2 - 1
[2,4,3,5,1] => [2,5,4,3,1] => [4,1]
=> [1]
=> ? = 2 - 1
[2,4,5,3,1] => [2,5,4,3,1] => [4,1]
=> [1]
=> ? = 1 - 1
[2,5,1,4,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> ? = 2 - 1
[2,5,3,4,1] => [2,5,4,3,1] => [4,1]
=> [1]
=> ? = 2 - 1
[2,5,4,3,1] => [2,5,4,3,1] => [4,1]
=> [1]
=> ? = 1 - 1
[3,1,2,5,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> ? = 2 - 1
[3,1,4,5,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> ? = 2 - 1
[3,1,5,4,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> ? = 2 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [3,2]
=> [2]
=> ? = 2 - 1
[3,2,4,5,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> ? = 2 - 1
[3,2,1,4,6,5] => [3,2,1,6,5,4] => [3,3]
=> [3]
=> 1 = 2 - 1
[3,2,1,5,6,4] => [3,2,1,6,5,4] => [3,3]
=> [3]
=> 1 = 2 - 1
[3,2,1,6,5,4] => [3,2,1,6,5,4] => [3,3]
=> [3]
=> 1 = 2 - 1
[3,2,4,1,6,5] => [3,2,6,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,2,5,1,6,4] => [3,2,6,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,2,6,1,5,4] => [3,2,6,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,4,2,1,6,5] => [3,6,2,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,5,2,1,6,4] => [3,6,2,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,6,2,1,5,4] => [3,6,2,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,2,1,3,6,5] => [4,2,1,6,5,3] => [3,3]
=> [3]
=> 1 = 2 - 1
[4,2,1,5,6,3] => [4,2,1,6,5,3] => [3,3]
=> [3]
=> 1 = 2 - 1
[4,2,1,6,5,3] => [4,2,1,6,5,3] => [3,3]
=> [3]
=> 1 = 2 - 1
[4,2,3,1,6,5] => [4,2,6,1,5,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,2,5,1,6,3] => [4,2,6,1,5,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,2,6,1,5,3] => [4,2,6,1,5,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,3,1,2,6,5] => [4,3,1,6,5,2] => [3,3]
=> [3]
=> 1 = 2 - 1
[4,3,1,5,6,2] => [4,3,1,6,5,2] => [3,3]
=> [3]
=> 1 = 2 - 1
[4,3,1,6,5,2] => [4,3,1,6,5,2] => [3,3]
=> [3]
=> 1 = 2 - 1
[4,3,5,1,6,2] => [4,3,6,1,5,2] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,3,6,1,5,2] => [4,3,6,1,5,2] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,5,2,1,6,3] => [4,6,2,1,5,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,5,3,1,6,2] => [4,6,3,1,5,2] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,6,2,1,5,3] => [4,6,2,1,5,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,6,3,1,5,2] => [4,6,3,1,5,2] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[5,2,1,3,6,4] => [5,2,1,6,4,3] => [3,3]
=> [3]
=> 1 = 2 - 1
[5,2,1,4,6,3] => [5,2,1,6,4,3] => [3,3]
=> [3]
=> 1 = 2 - 1
[5,2,1,6,4,3] => [5,2,1,6,4,3] => [3,3]
=> [3]
=> 1 = 2 - 1
[5,2,3,1,6,4] => [5,2,6,1,4,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[5,2,4,1,6,3] => [5,2,6,1,4,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[5,2,6,1,4,3] => [5,2,6,1,4,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[5,3,1,2,6,4] => [5,3,1,6,4,2] => [3,3]
=> [3]
=> 1 = 2 - 1
[5,3,1,4,6,2] => [5,3,1,6,4,2] => [3,3]
=> [3]
=> 1 = 2 - 1
[5,3,1,6,4,2] => [5,3,1,6,4,2] => [3,3]
=> [3]
=> 1 = 2 - 1
[5,3,4,1,6,2] => [5,3,6,1,4,2] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[5,3,6,1,4,2] => [5,3,6,1,4,2] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[5,6,2,1,4,3] => [5,6,2,1,4,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[5,6,3,1,4,2] => [5,6,3,1,4,2] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,2,1,4,5,7,6] => [3,2,1,7,6,5,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[3,2,1,4,6,7,5] => [3,2,1,7,6,5,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[3,2,1,4,7,6,5] => [3,2,1,7,6,5,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[3,2,1,5,4,7,6] => [3,2,1,7,6,5,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[3,2,1,5,6,7,4] => [3,2,1,7,6,5,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[3,2,1,5,7,6,4] => [3,2,1,7,6,5,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[3,2,1,6,4,7,5] => [3,2,1,7,6,5,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[3,2,1,6,5,7,4] => [3,2,1,7,6,5,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[3,2,1,6,7,5,4] => [3,2,1,7,6,5,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[3,2,1,7,4,6,5] => [3,2,1,7,6,5,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[3,2,4,1,5,7,6] => [3,2,7,1,6,5,4] => [4,2,1]
=> [2,1]
=> 1 = 2 - 1
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001200
Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Values
[1] => => [] => ?
=> ? = 0 + 1
[1,2] => 0 => [1] => [1,0]
=> ? = 0 + 1
[2,1] => 1 => [1] => [1,0]
=> ? = 1 + 1
[1,2,3] => 00 => [2] => [1,1,0,0]
=> ? = 0 + 1
[1,3,2] => 01 => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[2,3,1] => 01 => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[3,2,1] => 11 => [2] => [1,1,0,0]
=> ? = 1 + 1
[1,2,3,4] => 000 => [3] => [1,1,1,0,0,0]
=> ? = 0 + 1
[1,2,4,3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,4,3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,3,4,1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,4,3,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,4,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,4,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,4,2,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,3,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[4,2,3,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[4,3,2,1] => 111 => [3] => [1,1,1,0,0,0]
=> ? = 1 + 1
[1,2,3,4,5] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,2,3,5,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,2,5,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,4,5,2] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,3,5,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,4,5,3,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,5,2,4,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,5,3,4,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,5,4,3,2] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,5,4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,4,5,3] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,5,4,3] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,1,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,3,4,5,1] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,3,5,4,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,1,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,4,3,5,1] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,4,5,3,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,5,1,4,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,5,3,4,1] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,5,4,3,1] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,2,5,4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[3,1,4,5,2] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[3,1,5,4,2] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,1,5,4] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,4,5,1] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,5,4,1] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[3,4,1,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[3,4,2,5,1] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[3,4,5,2,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,5,1,4,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[3,5,2,4,1] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[3,5,4,2,1] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,1,2,5,3] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[5,4,3,2,1] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,5,6] => 00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[6,5,4,3,2,1] => 11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,5,6,7] => 000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,4,5,7,6] => 000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,2,3,4,6,7,5] => 000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,2,3,4,7,6,5] => 000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[1,2,3,5,4,7,6] => 000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,2,3,5,6,7,4] => 000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,2,3,5,7,6,4] => 000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[1,2,3,6,4,7,5] => 000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,2,3,6,5,7,4] => 000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,2,3,6,7,5,4] => 000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[1,2,3,7,4,6,5] => 000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,2,3,7,5,6,4] => 000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,2,3,7,6,5,4] => 000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[1,2,4,3,5,7,6] => 001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,2,4,3,6,7,5] => 001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,2,4,3,7,6,5] => 001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[1,2,4,5,3,7,6] => 000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,2,4,5,6,7,3] => 000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,2,4,5,7,6,3] => 000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[1,2,4,6,3,7,5] => 000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,2,4,6,5,7,3] => 000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,2,4,6,7,5,3] => 000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[1,2,4,7,3,6,5] => 000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,2,4,7,5,6,3] => 000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,2,4,7,6,5,3] => 000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[1,2,5,3,4,7,6] => 001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,2,5,3,6,7,4] => 001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,2,5,3,7,6,4] => 001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[1,2,5,4,3,7,6] => 001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 + 1
[1,2,5,4,6,7,3] => 001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,2,5,4,7,6,3] => 001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[1,2,5,6,3,7,4] => 000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,2,5,6,4,7,3] => 000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,2,5,6,7,4,3] => 000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[1,2,5,7,3,6,4] => 000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,2,5,7,4,6,3] => 000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,2,5,7,6,4,3] => 000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[1,2,6,3,4,7,5] => 001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,2,6,3,5,7,4] => 001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 + 1
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001060
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 33%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 0 + 1
[1,2] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1] => [1,2] => ([],2)
=> ([],1)
=> ? = 1 + 1
[1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,3,2] => [2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,2,1] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 1 + 1
[1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,4,3,2] => [2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,4,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,4,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,4,3,1] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,4,1] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,2,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[4,3,2,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 1 + 1
[1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,5,4,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,4,5,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,3,5,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,4,2,5,3] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,3,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,5,3,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,5,2,4,3] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,5,3,4,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,3,5,4] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,1,4,5,3] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,1,5,4,3] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,1,5,4] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,4,5,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,3,5,4,1] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,4,1,5,3] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,3,5,1] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,5,3,1] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,5,1,4,3] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[2,5,3,4,1] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[2,5,4,3,1] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,1,2,5,4] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,4,5,2] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,5,4,2] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,1,5,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,4,5,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,5,4,1] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,1,5,2] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,2,5,1] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,5,2,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,5,1,4,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,5,2,4,1] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[3,5,4,2,1] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[4,1,2,5,3] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,3,5,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,5,3,2] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2,1,5,3] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,3,5,1] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2,5,3,1] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,3,1,5,2] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,2,5,1] => [5,4,3,1,2] => ([(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),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,5,2,1] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,5,1,3,2] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,5,2,3,1] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,5,3,2,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[5,1,2,4,3] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[5,1,3,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[5,1,4,3,2] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[5,2,1,4,3] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,2,3,4,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[5,2,4,3,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[5,3,1,4,2] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,3,2,4,1] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,3,4,2,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[5,4,1,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[5,4,2,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[5,4,3,2,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 1 + 1
[1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,2,3,4,6,5] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,3,5,6,4] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,3,6,5,4] => [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,4,3,6,5] => [2,3,5,4,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,5,3,6,4] => [2,3,6,4,1,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,5,4,6,3] => [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,2,4,6,5] => [2,4,3,5,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,2,5,6,4] => [2,4,3,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,2,6,5,4] => [2,4,3,1,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,4,2,6,5] => [2,4,5,3,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,5,2,6,4] => [2,4,6,3,1,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,5,4,6,2] => [2,4,6,5,1,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,2,3,6,5] => [2,5,3,4,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,2,5,6,3] => [2,5,3,6,1,4] => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,2,6,5,3] => [2,5,3,1,4,6] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,3,2,6,5] => [2,5,4,3,1,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,5,6,2] => [2,5,4,6,1,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,3,6,5,2] => [2,5,4,1,3,6] => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
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.
Matching statistic: St001570
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 33%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 0 - 2
[1,2] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 - 2
[2,1] => [1,2] => ([],2)
=> ([],1)
=> ? = 1 - 2
[1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 - 2
[1,3,2] => [2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 2
[3,2,1] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 1 - 2
[1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 - 2
[1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,4,3,2] => [2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,1,4,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,3,4,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,4,3,1] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 2
[3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,2,4,1] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,4,2,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 2
[4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[4,3,2,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 1 - 2
[1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 - 2
[1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,2,5,4,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,3,4,5,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,3,5,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,4,2,5,3] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,4,3,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,4,5,3,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,5,2,4,3] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,5,3,4,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,1,3,5,4] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,1,4,5,3] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,1,5,4,3] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,3,1,5,4] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,3,4,5,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,3,5,4,1] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,4,1,5,3] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,3,5,1] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,5,3,1] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,5,1,4,3] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,5,3,4,1] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,5,4,3,1] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[3,1,2,5,4] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,4,5,2] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,5,4,2] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,2,1,5,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,4,5,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,2,5,4,1] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,4,1,5,2] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,4,2,5,1] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,4,5,2,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[3,5,1,4,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,5,2,4,1] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[3,5,4,2,1] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[4,1,2,5,3] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,1,3,5,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,1,5,3,2] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,2,1,5,3] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,2,3,5,1] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,2,5,3,1] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,3,1,5,2] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,3,2,5,1] => [5,4,3,1,2] => ([(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),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,3,5,2,1] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,5,1,3,2] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,5,2,3,1] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,5,3,2,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[5,1,2,4,3] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[5,1,3,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[5,1,4,3,2] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[5,2,1,4,3] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[5,2,3,4,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[5,2,4,3,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[5,3,1,4,2] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[5,3,2,4,1] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[5,3,4,2,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[5,4,1,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[5,4,2,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[5,4,3,2,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 1 - 2
[1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 - 2
[1,2,3,4,6,5] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,2,3,5,6,4] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,2,3,6,5,4] => [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,2,4,3,6,5] => [2,3,5,4,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,2,5,3,6,4] => [2,3,6,4,1,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,2,5,4,6,3] => [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,3,2,4,6,5] => [2,4,3,5,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,3,2,5,6,4] => [2,4,3,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,3,2,6,5,4] => [2,4,3,1,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,3,4,2,6,5] => [2,4,5,3,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,3,5,2,6,4] => [2,4,6,3,1,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,3,5,4,6,2] => [2,4,6,5,1,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,4,2,3,6,5] => [2,5,3,4,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,4,2,5,6,3] => [2,5,3,6,1,4] => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,4,2,6,5,3] => [2,5,3,1,4,6] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,4,3,2,6,5] => [2,5,4,3,1,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,4,3,5,6,2] => [2,5,4,6,1,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,4,3,6,5,2] => [2,5,4,1,3,6] => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000897The number of different multiplicities of parts of an integer partition. St000353The number of inner valleys of a permutation. St000711The number of big exceedences of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000092The number of outer peaks of a permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St000699The toughness times the least common multiple of 1,. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function.
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