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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000454
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => ([],1)
=> ([],1)
=> 0
[2,1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2,3] => [1,2] => ([],2)
=> ([],2)
=> 0
[1,3,2] => [1,2] => ([],2)
=> ([],2)
=> 0
[2,1,3] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,3,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,2] => [1,2] => ([],2)
=> ([],2)
=> 0
[3,2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3,4] => [1,2,3] => ([],3)
=> ([],3)
=> 0
[1,2,4,3] => [1,2,3] => ([],3)
=> ([],3)
=> 0
[1,3,2,4] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[1,3,4,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[1,4,2,3] => [1,2,3] => ([],3)
=> ([],3)
=> 0
[1,4,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,1,4,3] => [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,4,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,1,2,3] => [1,2,3] => ([],3)
=> ([],3)
=> 0
[4,1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[4,2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4,5] => [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[1,2,3,5,4] => [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,2,4,5,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,2,5,3,4] => [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,2,4,5] => [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,2,5,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,5,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,5,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,5,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,5,2,3,4] => [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[1,5,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,5,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,3,5,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,4,3,5] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[2,1,4,5,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[2,1,5,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[2,5,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,5,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,5,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001645
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[2,1] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2,3] => [1,2] => ([],2)
=> ([],2)
=> ? = 0 + 1
[1,3,2] => [1,2] => ([],2)
=> ([],2)
=> ? = 0 + 1
[2,1,3] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2] => [1,2] => ([],2)
=> ([],2)
=> ? = 0 + 1
[3,2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3] => ([],3)
=> ([],3)
=> ? = 0 + 1
[1,2,4,3] => [1,2,3] => ([],3)
=> ([],3)
=> ? = 0 + 1
[1,3,2,4] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,3,4,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,4,2,3] => [1,2,3] => ([],3)
=> ([],3)
=> ? = 0 + 1
[1,4,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,1,4,3] => [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,4,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,2,3] => [1,2,3] => ([],3)
=> ([],3)
=> ? = 0 + 1
[4,1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[4,2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4] => ([],4)
=> ([],4)
=> ? = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => ([],4)
=> ([],4)
=> ? = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[1,2,4,5,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[1,2,5,3,4] => [1,2,3,4] => ([],4)
=> ([],4)
=> ? = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[1,3,2,4,5] => [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[1,3,2,5,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[1,3,5,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,4,3,5,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,4,5,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,5,2,3,4] => [1,2,3,4] => ([],4)
=> ([],4)
=> ? = 0 + 1
[1,5,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[1,5,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[2,1,3,5,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[2,1,4,3,5] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[2,1,4,5,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[2,1,5,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[2,5,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[2,5,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,2,1,5,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,2,5,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,4,1,5,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,4,2,1,5] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,4,2,5,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,4,5,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,4,5,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,5,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,5,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,5,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,2,3,1,5] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,2,3,5,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,2,5,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3,1,2,5] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3,1,5,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3,2,1,5] => [4,3,2,1] => ([(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)
=> 4 = 3 + 1
[4,3,2,5,1] => [4,3,2,1] => ([(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)
=> 4 = 3 + 1
[4,3,5,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3,5,2,1] => [4,3,2,1] => ([(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)
=> 4 = 3 + 1
[4,5,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,5,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,5,3,2,1] => [4,3,2,1] => ([(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)
=> 4 = 3 + 1
[5,1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],4)
=> ? = 0 + 1
[5,1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[5,1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[5,1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[5,2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[5,2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[5,3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[5,3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[5,3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[5,4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[5,4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[5,4,3,2,1] => [4,3,2,1] => ([(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)
=> 4 = 3 + 1
[3,5,1,4,2,6] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,1,4,6,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,1,6,4,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,2,4,1,6] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,2,4,6,1] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,2,6,4,1] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,4,1,2,6] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,4,1,6,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,4,2,1,6] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,4,2,6,1] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,4,6,1,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,4,6,2,1] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,6,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,6,2,4,1] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,6,4,1,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,6,4,2,1] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St001207
(load all 55 compositions to match this statistic)
(load all 55 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 57%
St001207: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 57%
Values
[1,2] => [1] => ? = 0
[2,1] => [1] => ? = 0
[1,2,3] => [1,2] => 0
[1,3,2] => [1,2] => 0
[2,1,3] => [2,1] => 1
[2,3,1] => [2,1] => 1
[3,1,2] => [1,2] => 0
[3,2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => 1
[2,4,1,3] => [2,1,3] => 1
[3,2,1,4] => [3,2,1] => 2
[3,2,4,1] => [3,2,1] => 2
[3,4,2,1] => [3,2,1] => 2
[4,1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => 1
[4,3,2,1] => [3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => 1
[1,2,4,5,3] => [1,2,4,3] => 1
[1,2,5,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => 1
[1,3,2,4,5] => [1,3,2,4] => 1
[1,3,2,5,4] => [1,3,2,4] => 1
[1,3,5,2,4] => [1,3,2,4] => 1
[1,4,3,2,5] => [1,4,3,2] => 2
[1,4,3,5,2] => [1,4,3,2] => 2
[1,4,5,3,2] => [1,4,3,2] => 2
[1,5,2,3,4] => [1,2,3,4] => 0
[1,5,2,4,3] => [1,2,4,3] => 1
[1,5,3,2,4] => [1,3,2,4] => 1
[1,5,4,3,2] => [1,4,3,2] => 2
[2,1,3,4,5] => [2,1,3,4] => 1
[2,1,3,5,4] => [2,1,3,4] => 1
[2,1,4,3,5] => [2,1,4,3] => 1
[2,1,4,5,3] => [2,1,4,3] => 1
[2,1,5,3,4] => [2,1,3,4] => 1
[2,1,5,4,3] => [2,1,4,3] => 1
[2,5,1,3,4] => [2,1,3,4] => 1
[2,5,1,4,3] => [2,1,4,3] => 1
[3,2,1,4,5] => [3,2,1,4] => 2
[3,2,1,5,4] => [3,2,1,4] => 2
[3,2,5,1,4] => [3,2,1,4] => 2
[3,4,1,2,5] => [3,4,1,2] => 3
[1,2,3,4,5,6] => [1,2,3,4,5] => ? = 0
[1,2,3,4,6,5] => [1,2,3,4,5] => ? = 0
[1,2,3,5,4,6] => [1,2,3,5,4] => ? = 1
[1,2,3,5,6,4] => [1,2,3,5,4] => ? = 1
[1,2,3,6,4,5] => [1,2,3,4,5] => ? = 0
[1,2,3,6,5,4] => [1,2,3,5,4] => ? = 1
[1,2,4,3,5,6] => [1,2,4,3,5] => ? = 1
[1,2,4,3,6,5] => [1,2,4,3,5] => ? = 1
[1,2,4,6,3,5] => [1,2,4,3,5] => ? = 1
[1,2,5,4,3,6] => [1,2,5,4,3] => ? = 2
[1,2,5,4,6,3] => [1,2,5,4,3] => ? = 2
[1,2,5,6,4,3] => [1,2,5,4,3] => ? = 2
[1,2,6,3,4,5] => [1,2,3,4,5] => ? = 0
[1,2,6,3,5,4] => [1,2,3,5,4] => ? = 1
[1,2,6,4,3,5] => [1,2,4,3,5] => ? = 1
[1,2,6,5,4,3] => [1,2,5,4,3] => ? = 2
[1,3,2,4,5,6] => [1,3,2,4,5] => ? = 1
[1,3,2,4,6,5] => [1,3,2,4,5] => ? = 1
[1,3,2,5,4,6] => [1,3,2,5,4] => ? = 1
[1,3,2,5,6,4] => [1,3,2,5,4] => ? = 1
[1,3,2,6,4,5] => [1,3,2,4,5] => ? = 1
[1,3,2,6,5,4] => [1,3,2,5,4] => ? = 1
[1,3,6,2,4,5] => [1,3,2,4,5] => ? = 1
[1,3,6,2,5,4] => [1,3,2,5,4] => ? = 1
[1,4,3,2,5,6] => [1,4,3,2,5] => ? = 2
[1,4,3,2,6,5] => [1,4,3,2,5] => ? = 2
[1,4,3,6,2,5] => [1,4,3,2,5] => ? = 2
[1,4,5,2,3,6] => [1,4,5,2,3] => ? = 3
[1,4,5,2,6,3] => [1,4,5,2,3] => ? = 3
[1,4,5,3,2,6] => [1,4,5,3,2] => ? = 3
[1,4,5,3,6,2] => [1,4,5,3,2] => ? = 3
[1,4,5,6,2,3] => [1,4,5,2,3] => ? = 3
[1,4,5,6,3,2] => [1,4,5,3,2] => ? = 3
[1,4,6,3,2,5] => [1,4,3,2,5] => ? = 2
[1,4,6,5,2,3] => [1,4,5,2,3] => ? = 3
[1,4,6,5,3,2] => [1,4,5,3,2] => ? = 3
[1,5,3,4,2,6] => [1,5,3,4,2] => ? = 3
[1,5,3,4,6,2] => [1,5,3,4,2] => ? = 3
[1,5,3,6,4,2] => [1,5,3,4,2] => ? = 3
[1,5,4,2,3,6] => [1,5,4,2,3] => ? = 3
[1,5,4,2,6,3] => [1,5,4,2,3] => ? = 3
[1,5,4,3,2,6] => [1,5,4,3,2] => ? = 3
[1,5,4,3,6,2] => [1,5,4,3,2] => ? = 3
[1,5,4,6,2,3] => [1,5,4,2,3] => ? = 3
[1,5,4,6,3,2] => [1,5,4,3,2] => ? = 3
[1,5,6,3,4,2] => [1,5,3,4,2] => ? = 3
[1,5,6,4,2,3] => [1,5,4,2,3] => ? = 3
[1,5,6,4,3,2] => [1,5,4,3,2] => ? = 3
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001624
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Values
[1,2] => [1] => ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1] => ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1] => [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2] => [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,2,1] => [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[1,3,4,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[2,1,4,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[2,4,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,2,1,4] => [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 1
[3,2,4,1] => [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 1
[3,4,2,1] => [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 1
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[4,3,2,1] => [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 2 + 1
[1,4,3,5,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 2 + 1
[1,4,5,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 2 + 1
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,5,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,5,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,5,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 2 + 1
[2,1,3,4,5] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[2,1,3,5,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[2,1,4,3,5] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2 = 1 + 1
[2,1,4,5,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2 = 1 + 1
[2,1,5,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[2,1,5,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2 = 1 + 1
[2,5,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[2,5,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2 = 1 + 1
[3,2,1,4,5] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 2 + 1
[3,2,1,5,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 2 + 1
[3,2,5,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 2 + 1
[3,4,1,2,5] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 + 1
[3,4,1,5,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 + 1
[3,4,2,1,5] => [3,4,2,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3 + 1
[3,4,2,5,1] => [3,4,2,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3 + 1
[3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 + 1
[3,4,5,2,1] => [3,4,2,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3 + 1
[3,5,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 2 + 1
[3,5,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 + 1
[3,5,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3 + 1
[4,2,3,1,5] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3 + 1
[4,2,3,5,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3 + 1
[4,2,5,3,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3 + 1
[4,3,1,2,5] => [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3 + 1
[4,3,1,5,2] => [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3 + 1
[4,3,2,1,5] => [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3 + 1
[4,3,2,5,1] => [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3 + 1
[4,3,5,1,2] => [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3 + 1
[4,3,5,2,1] => [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3 + 1
[4,5,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3 + 1
[4,5,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3 + 1
[4,5,3,2,1] => [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3 + 1
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[5,1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[5,1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 2 + 1
[5,2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[5,2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2 = 1 + 1
[5,3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 2 + 1
[5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 + 1
[5,3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3 + 1
[5,4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3 + 1
[5,4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3 + 1
[5,4,3,2,1] => [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3 + 1
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 2 = 1 + 1
[1,2,3,5,6,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 2 = 1 + 1
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,5,4,3,6] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 2 + 1
[1,2,5,4,6,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 2 + 1
[1,2,5,6,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 2 + 1
[1,2,6,5,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 2 + 1
[1,3,2,5,4,6] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 1 + 1
[1,3,2,5,6,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 1 + 1
[1,3,2,6,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 1 + 1
[1,3,6,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 1 + 1
[1,4,3,2,5,6] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[1,4,3,2,6,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[1,4,3,6,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
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