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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000259
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,1] => [2] => [1] => ([],1)
=> 0
[2] => [1] => [1] => ([],1)
=> 0
[1,1,1] => [3] => [1] => ([],1)
=> 0
[3] => [1] => [1] => ([],1)
=> 0
[1,1,1,1] => [4] => [1] => ([],1)
=> 0
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[2,2] => [2] => [1] => ([],1)
=> 0
[4] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1] => [5] => [1] => ([],1)
=> 0
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[5] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1] => [6] => [1] => ([],1)
=> 0
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,2,2] => [3] => [1] => ([],1)
=> 0
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[3,3] => [2] => [1] => ([],1)
=> 0
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[6] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
=> 0
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001093
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ 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
[1,1] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[2] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1] => [3] => [1] => ([],1)
=> 1 = 0 + 1
[3] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1] => [4] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,2] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[4] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1] => [5] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[5] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1] => [6] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,2,2] => [3] => [1] => ([],1)
=> 1 = 0 + 1
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,3] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[6] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 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: St001418
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 97%●distinct values known / distinct values provided: 67%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 97%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1] => [2] => [1] => [1,0]
=> ? = 0 - 1
[2] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1] => [3] => [1] => [1,0]
=> ? = 0 - 1
[3] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1] => [4] => [1] => [1,0]
=> ? = 0 - 1
[1,1,2] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,1,1] => [1,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,2] => [2] => [1] => [1,0]
=> ? = 0 - 1
[4] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1] => [5] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,2] => [3,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,3] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,2] => [1,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,1,1,1] => [1,3] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,2,1] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[3,1,1] => [1,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[5] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1,1] => [6] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,2] => [4,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,3] => [3,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,4] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,1,1,1] => [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,2,1] => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1,1,1] => [1,4] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,1,1,2] => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,2,2] => [3] => [1] => [1,0]
=> ? = 0 - 1
[3,1,1,1] => [1,3] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[3,3] => [2] => [1] => [1,0]
=> ? = 0 - 1
[4,1,1] => [1,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[6] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1,1,1] => [7] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1,2] => [5,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,3] => [4,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,2,2] => [3,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,4] => [3,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,2,2,1] => [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,3,1,1] => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,5] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,2,2] => [1,3] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,3,1,1,1] => [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,3] => [1,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,4,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1,1,1,1] => [1,5] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,1,1,1,2] => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,1,3] => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,2,1,1] => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,1,2,2] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,2,1,1,1] => [2,3] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,2,2,1] => [3,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,2,3] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,3,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,1,1,1] => [1,4] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[3,1,1,2] => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,2,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,2] => [1,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[3,3,1] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[4,1,1,1] => [1,3] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[7] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1,1,1,1] => [8] => [1] => [1,0]
=> ? = 0 - 1
[2,2,2,2] => [4] => [1] => [1,0]
=> ? = 0 - 1
[4,4] => [2] => [1] => [1,0]
=> ? = 0 - 1
[8] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1,1,1,1,1] => [9] => [1] => [1,0]
=> ? = 0 - 1
[3,3,3] => [3] => [1] => [1,0]
=> ? = 0 - 1
[9] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1,1,1,1,1,1] => [10] => [1] => [1,0]
=> ? = 0 - 1
[2,2,2,2,2] => [5] => [1] => [1,0]
=> ? = 0 - 1
[5,5] => [2] => [1] => [1,0]
=> ? = 0 - 1
[10] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1,1,1,1,1,1,1,1] => [12] => [1] => [1,0]
=> ? = 0 - 1
[2,2,2,2,2,2] => [6] => [1] => [1,0]
=> ? = 0 - 1
[3,3,3,3] => [4] => [1] => [1,0]
=> ? = 0 - 1
[6,6] => [2] => [1] => [1,0]
=> ? = 0 - 1
[4,4,4] => [3] => [1] => [1,0]
=> ? = 0 - 1
[12] => [1] => [1] => [1,0]
=> ? = 0 - 1
[11] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1,1,1,1,1,1,1] => [11] => [1] => [1,0]
=> ? = 0 - 1
Description
Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.
Matching statistic: St001431
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 91%●distinct values known / distinct values provided: 67%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 91%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1] => [2] => [1] => [1,0]
=> ? = 0 - 1
[2] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1] => [3] => [1] => [1,0]
=> ? = 0 - 1
[3] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1] => [4] => [1] => [1,0]
=> ? = 0 - 1
[1,1,2] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,1,1] => [1,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,2] => [2] => [1] => [1,0]
=> ? = 0 - 1
[4] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1] => [5] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,2] => [3,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,3] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,2] => [1,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,1,1,1] => [1,3] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,2,1] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[3,1,1] => [1,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[5] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1,1] => [6] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,2] => [4,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,3] => [3,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,4] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,1,1,1] => [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,2,1] => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1,1,1] => [1,4] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,1,1,2] => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,2,2] => [3] => [1] => [1,0]
=> ? = 0 - 1
[3,1,1,1] => [1,3] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[3,3] => [2] => [1] => [1,0]
=> ? = 0 - 1
[4,1,1] => [1,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[6] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1,1,1] => [7] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1,2] => [5,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,3] => [4,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,2,2] => [3,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,4] => [3,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,2,2,1] => [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,3,1,1] => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,5] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,2,2] => [1,3] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,3,1,1,1] => [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,3] => [1,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,4,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1,1,1,1] => [1,5] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,1,1,1,2] => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,1,3] => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,2,1,1] => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,1,2,2] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,2,1,1,1] => [2,3] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,2,2,1] => [3,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,2,3] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,3,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,1,1,1] => [1,4] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[3,1,1,2] => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,2,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,2] => [1,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[3,3,1] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[4,1,1,1] => [1,3] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[7] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1,1,1,1] => [8] => [1] => [1,0]
=> ? = 0 - 1
[2,2,2,2] => [4] => [1] => [1,0]
=> ? = 0 - 1
[4,4] => [2] => [1] => [1,0]
=> ? = 0 - 1
[8] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1,1,1,1,1] => [9] => [1] => [1,0]
=> ? = 0 - 1
[3,3,3] => [3] => [1] => [1,0]
=> ? = 0 - 1
[9] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1,1,1,1,1,1] => [10] => [1] => [1,0]
=> ? = 0 - 1
[1,2,1,2,1,1,2] => [1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[2,1,2,1,2,1,1] => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[2,2,2,2,2] => [5] => [1] => [1,0]
=> ? = 0 - 1
[5,5] => [2] => [1] => [1,0]
=> ? = 0 - 1
[10] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,1,1,1,1,1,1,1,1,1,1,1] => [12] => [1] => [1,0]
=> ? = 0 - 1
[1,1,2,1,2,3,1,1] => [2,1,1,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,2,1,2,1,1,3] => [2,1,1,1,2,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,1,3,2,1,2,1,1] => [2,1,1,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[2,2,2,2,2,2] => [6] => [1] => [1,0]
=> ? = 0 - 1
[2,1,2,2,1,2,1,1] => [1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
[2,1,2,1,1,3,1,1] => [1,1,1,2,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[3,3,3,3] => [4] => [1] => [1,0]
=> ? = 0 - 1
[3,1,1,2,1,2,1,1] => [1,2,1,1,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2 - 1
[6,6] => [2] => [1] => [1,0]
=> ? = 0 - 1
[4,4,4] => [3] => [1] => [1,0]
=> ? = 0 - 1
[1,2,1,3,1,1,4] => [1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,2,1,2,3,2,2] => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,1,2,1,3,3] => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,1,2,1,1,5] => [1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[12] => [1] => [1] => [1,0]
=> ? = 0 - 1
[1,3,1,2,1,1,3] => [1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[3,1,2,1,2,2,1] => [1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,3,2,1,2,2,1] => [1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,2,4,1,2,1,1] => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,4,2,1,2,1,1] => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
Description
Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 84%●distinct values known / distinct values provided: 67%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 84%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0 - 2
[1,1] => [2] => [1] => ([],1)
=> ? = 0 - 2
[2] => [1] => [1] => ([],1)
=> ? = 0 - 2
[1,1,1] => [3] => [1] => ([],1)
=> ? = 0 - 2
[3] => [1] => [1] => ([],1)
=> ? = 0 - 2
[1,1,1,1] => [4] => [1] => ([],1)
=> ? = 0 - 2
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[2,2] => [2] => [1] => ([],1)
=> ? = 0 - 2
[4] => [1] => [1] => ([],1)
=> ? = 0 - 2
[1,1,1,1,1] => [5] => [1] => ([],1)
=> ? = 0 - 2
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[5] => [1] => [1] => ([],1)
=> ? = 0 - 2
[1,1,1,1,1,1] => [6] => [1] => ([],1)
=> ? = 0 - 2
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[2,2,2] => [3] => [1] => ([],1)
=> ? = 0 - 2
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[3,3] => [2] => [1] => ([],1)
=> ? = 0 - 2
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[6] => [1] => [1] => ([],1)
=> ? = 0 - 2
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
=> ? = 0 - 2
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,1,1,1,1,1] => [1,5] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[2,1,1,1,2] => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[2,1,1,3] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[2,1,2,1,1] => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,1,2,2] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,2,1,1,1] => [2,3] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[2,2,2,1] => [3,1] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[2,2,3] => [2,1] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[2,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[3,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[3,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[3,3,1] => [2,1] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[4,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[7] => [1] => [1] => ([],1)
=> ? = 0 - 2
[1,1,1,1,1,1,1,1] => [8] => [1] => ([],1)
=> ? = 0 - 2
[2,2,2,2] => [4] => [1] => ([],1)
=> ? = 0 - 2
[4,4] => [2] => [1] => ([],1)
=> ? = 0 - 2
[8] => [1] => [1] => ([],1)
=> ? = 0 - 2
[1,1,1,1,1,1,1,1,1] => [9] => [1] => ([],1)
=> ? = 0 - 2
[1,1,2,1,2,1,1] => [2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,1,2,1,2,2] => [2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[1,1,2,3,1,1] => [2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[1,1,3,2,1,1] => [2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[1,2,2,1,1,2] => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[2,1,1,2,2,1] => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[2,2,1,2,1,1] => [2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[3,3,3] => [3] => [1] => ([],1)
=> ? = 0 - 2
[9] => [1] => [1] => ([],1)
=> ? = 0 - 2
[1,1,1,1,1,1,1,1,1,1] => [10] => [1] => ([],1)
=> ? = 0 - 2
[1,1,1,2,1,2,1,1] => [3,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,1,1,2,1,2,2] => [3,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[1,1,1,2,3,1,1] => [3,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[1,1,1,3,2,1,1] => [3,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[1,1,2,1,2,1,1,1] => [2,1,1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,1,2,1,2,2,1] => [2,1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,1,2,1,3,1,1] => [2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,1,2,3,1,1,1] => [2,1,1,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[1,1,2,4,1,1] => [2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[1,1,3,1,2,1,1] => [2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,1,3,1,2,2] => [2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[1,1,3,2,1,1,1] => [2,1,1,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[1,1,4,2,1,1] => [2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[1,2,1,1,2,2,1] => [1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,2,2,1,1,3] => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[1,2,2,1,2,1,1] => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,2,1,2,1,1,1] => [2,1,1,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[2,2,1,3,1,1] => [2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[2,2,2,2,2] => [5] => [1] => ([],1)
=> ? = 0 - 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: St001207
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => ? = 0
[1,1] => [2] => [1,1,0,0]
=> [1,2] => 0
[2] => [1] => [1,0]
=> [1] => ? = 0
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[3] => [1] => [1,0]
=> [1] => ? = 0
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[2,2] => [2] => [1,1,0,0]
=> [1,2] => 0
[4] => [1] => [1,0]
=> [1] => ? = 0
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ? = 0
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[5] => [1] => [1,0]
=> [1] => ? = 0
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ? = 0
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ? = 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ? = 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ? = 2
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ? = 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1
[2,2,2] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[3,3] => [2] => [1,1,0,0]
=> [1,2] => 0
[4,1,1] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[6] => [1] => [1,0]
=> [1] => ? = 0
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => ? = 0
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ? = 1
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ? = 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => ? = 1
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ? = 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => ? = 1
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ? = 2
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ? = 1
[1,1,5] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => ? = 2
[1,2,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ? = 2
[1,2,2,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,3,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ? = 2
[1,3,3] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[1,4,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[2,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ? = 1
[2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ? = 1
[2,1,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1
[2,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ? = 2
[2,1,2,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ? = 1
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[2,2,3] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[2,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[3,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ? = 1
[3,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1
[3,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[3,2,2] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[3,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[4,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[5,1,1] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[7] => [1] => [1,0]
=> [1] => ? = 0
[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => ? = 1
[1,1,1,1,1,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ? = 1
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,5,6,4,7] => ? = 1
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => ? = 1
[1,1,1,1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ? = 1
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,5,3,6,7] => ? = 1
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5] => ? = 1
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => ? = 1
[1,1,1,5] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => ? = 1
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => ? = 1
[1,1,2,2,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ? = 1
[1,1,3,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => ? = 1
[1,1,4,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ? = 1
[1,1,6] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[1,2,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => ? = 2
[1,2,1,1,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => ? = 2
[1,2,1,1,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ? = 2
[1,2,1,2,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => ? = 2
[1,2,1,2,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ? = 2
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => ? = 1
[1,2,2,2,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ? = 1
[1,2,2,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1
[1,3,2,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[1,3,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1
[1,5,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[2,1,1,4] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1
[2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[2,2,4] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[2,3,3] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[2,4,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[3,1,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1
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: St001330
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 67%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1] => [2] => [2] => ([],2)
=> 1 = 0 + 1
[2] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1] => [3] => [3] => ([],3)
=> 1 = 0 + 1
[3] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1] => [4] => [4] => ([],4)
=> 1 = 0 + 1
[1,1,2] => [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,2] => [2] => [2] => ([],2)
=> 1 = 0 + 1
[4] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1] => [5] => [5] => ([],5)
=> 1 = 0 + 1
[1,1,1,2] => [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,1,3] => [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,2,1,1] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,1,1] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2,1] => [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[5] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1] => [6] => [6] => ([],6)
=> 1 = 0 + 1
[1,1,1,1,2] => [4,1] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,1,1,3] => [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,1,2,1,1] => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,4] => [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,2,1,1,1] => [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,2,1] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,3,1,1] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,1,1,1,1] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,1,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[2,2,2] => [3] => [3] => ([],3)
=> 1 = 0 + 1
[3,1,1,1] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,3] => [2] => [2] => ([],2)
=> 1 = 0 + 1
[4,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[6] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1,1] => [7] => [7] => ([],7)
=> ? = 0 + 1
[1,1,1,1,1,2] => [5,1] => [1,5] => ([(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,3] => [4,1] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,1,1,2,1,1] => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,1,2,2] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,4] => [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,1,2,1,1,1] => [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,2,2,1] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,3,1,1] => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,5] => [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,2,1,1,1,1] => [1,1,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,1,1,2] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,2,2] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,1,1,1] => [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,3] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,4,1,1] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,1,1,1,1,1] => [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[2,1,1,1,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[2,1,1,3] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[2,1,2,1,1] => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,2,2] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,2,1,1,1] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,2,2,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[2,2,3] => [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1,1] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,1,1,1,1] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,1,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[3,2,1,1] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,3,1] => [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[4,1,1,1] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[7] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1,1,1] => [8] => [8] => ([],8)
=> ? = 0 + 1
[1,1,1,1,1,1,2] => [6,1] => [1,6] => ([(5,6)],7)
=> ? = 1 + 1
[1,1,1,1,1,3] => [5,1] => [1,5] => ([(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,2,1,1] => [4,1,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,1,1,2,2] => [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,4] => [4,1] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,1,1,2,1,1,1] => [3,1,3] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,1,2,2,1] => [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,1,3,1,1] => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,1,5] => [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,1,2,1,1,1,1] => [2,1,4] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,2,1,1,2] => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,2,2,2] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,3,1,1,1] => [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,4,1,1] => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,2,1,1,1,1,1] => [1,1,5] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,2,1,1,1,2] => [1,1,3,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,1,1,3] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,1,2,1,1] => [1,1,1,1,2] => [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)
=> ? = 2 + 1
[1,2,1,2,2] => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,2,1,1,1] => [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,2,2,2,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,2,2,3] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,2,3,1,1] => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,1,1,1,1] => [1,1,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,1,1,2] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,2,1,1] => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,2,2] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,3,3,1] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,4,1,1,1] => [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,5,1,1] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,1,1,1,1,2] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[2,1,1,1,3] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001645
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1] => [2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
[2] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1,1] => [3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[3] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1,1,1] => [4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,2] => [2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
[4] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1,1,1,1] => [5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,2,2] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,1,1,1] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,1] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[5] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1] => [6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,1,1,1,1] => [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[2,2,2] => [3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[3,1,1,1] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[3,3] => [2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
[4,1,1] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[6] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1,1] => [7] => ([],7)
=> ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,2,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,1,1,2,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,4] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,2,1,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,1,2,2,1] => [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 + 1
[1,1,3,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,1,5] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,2,1,1,2] => [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 + 1
[1,2,2,2] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,3,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,3,3] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,4,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,1,1,1,1,1] => [1,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[2,1,1,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[2,1,2,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,2,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,2,1,1,1] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[2,2,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,2,3] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,1,1,1,1] => [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[3,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[3,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,2,2] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[3,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[4,1,1,1] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[5,1,1] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[7] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1,1,1] => [8] => ([],8)
=> ?
=> ? = 0 + 1
[1,1,1,1,1,1,2] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,1,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,2,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,1,1,1,2,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,2,1,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,1,1,2,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,1,1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,1,1,5] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,2,1,1,1,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,1,2,1,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,2,2,2] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,3,1,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,1,4,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,1,6] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1,1,1,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,2,1,1,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,2,2,2] => [4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[2,2,4] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[4,4] => [2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
[8] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1,3] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,1,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,5] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,6] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,7] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,2,2,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,2,2,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,2,5] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,3,3] => [3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[4,4,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
Description
The pebbling number of a connected graph.
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