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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St001232
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => [1,0]
=> 0
([],2)
=> [2] => [2] => [1,1,0,0]
=> 0
([(0,1)],2)
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
([],3)
=> [3] => [3] => [1,1,1,0,0,0]
=> 0
([(1,2)],3)
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
([],4)
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
([(2,3)],4)
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
([],5)
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
([(3,4)],5)
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
([],6)
=> [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(4,5)],6)
=> [1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1,4] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
([(2,5),(3,4)],6)
=> [2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,3,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,2] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,2] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,2] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001227
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 86%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 86%
Values
([],1)
=> [1] => [1] => [1,0]
=> 0
([],2)
=> [2] => [2] => [1,1,0,0]
=> 0
([(0,1)],2)
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
([],3)
=> [3] => [3] => [1,1,1,0,0,0]
=> 0
([(1,2)],3)
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
([],4)
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
([(2,3)],4)
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
([],5)
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
([(3,4)],5)
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
([],6)
=> [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(4,5)],6)
=> [1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1,4] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
([(2,5),(3,4)],6)
=> [2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,3,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,2] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,2] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,2] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
([],7)
=> [7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
([(5,6)],7)
=> [1,6] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(4,6),(5,6)],7)
=> [1,1,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6
([(3,6),(4,6),(5,6)],7)
=> [1,2,4] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
([(2,6),(3,6),(4,6),(5,6)],7)
=> [1,3,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,4,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
([(3,6),(4,5)],7)
=> [2,5] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(4,5),(4,6),(5,6)],7)
=> [2,5] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> [1,1,2,3] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,3,2] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 6
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,2,3] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,3,2] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 6
([(3,5),(3,6),(4,5),(4,6)],7)
=> [1,2,4] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2,3] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [1,1,3,2] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 6
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,1,4] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 6
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,2,3] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,3] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,3,2] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 6
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,3,2] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
([(1,6),(2,5),(3,4)],7)
=> [3,4] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
([(2,3),(4,5),(4,6),(5,6)],7)
=> [2,1,4] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 6
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [1,2,2,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,3] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [2,1,2,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,3,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,3] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [1,2,2,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,3] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [1,2,2,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,4] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,2,2,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,3,2] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 6
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,3,2] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,2,2,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,1,2,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [2,2,3] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> [1,2,2,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,3] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [2,1,2,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [1,4,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
([(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,3,2] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 6
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [2,1,2,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,2] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [2,2,3] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,1,2,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [4,3] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,3] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,1,2,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [1,2,2,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001645
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([],1)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],3)
=> ([],1)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([],4)
=> ([],1)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 3 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 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)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 4 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
([(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)
=> ? = 3 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
([],6)
=> ([],1)
=> 1 = 0 + 1
([(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 5 + 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 3 + 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ? = 5 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 4 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ? = 4 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 5 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 5 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4 + 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ? = 3 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ? = 4 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ? = 3 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 5 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4 + 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ? = 4 + 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
([],7)
=> ([],1)
=> 1 = 0 + 1
([(5,6)],7)
=> ([(1,2)],3)
=> ? = 1 + 1
([(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ? = 6 + 1
([(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ? = 5 + 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ? = 4 + 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ? = 3 + 1
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> ? = 6 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> ? = 6 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 + 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St001330
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Values
([],1)
=> ([(0,1)],2)
=> 2 = 0 + 2
([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 2
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 4 + 2
([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 2
([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 2
([(2,5),(3,4)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
([(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 5 + 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 5 + 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 5 + 2
([],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 0 + 2
([(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
([(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 + 2
([(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 2
([(3,6),(4,5)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 2
([(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,2),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 + 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 + 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 + 2
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(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)
=> 8 = 6 + 2
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.
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