Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000734
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [[1,2]]
 => 2
([],2)
 => [2,2]
 => [[1,2],[3,4]]
 => 2
([(0,1)],2)
 => [3]
 => [[1,2,3]]
 => 3
([],3)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 2
([(1,2)],3)
 => [6]
 => [[1,2,3,4,5,6]]
 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => [[1,2,5],[3,4]]
 => 5
([(0,2),(2,1)],3)
 => [4]
 => [[1,2,3,4]]
 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => [[1,2,5],[3,4]]
 => 5
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 6
([(1,2),(2,3)],4)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 6
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 6
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => 8
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 7
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 7
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 7
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => 8
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => 8
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => 8
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => 8
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => 10
([(1,4),(3,2),(4,3)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 7
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => 8
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => 8
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => 10
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [[1,2,3,4,5,6]]
 => 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 7
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => 8
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [9]
 => [[1,2,3,4,5,6,7,8,9]]
 => 9
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => 8
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => 8
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => 8
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
 => [9]
 => [[1,2,3,4,5,6,7,8,9]]
 => 9
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 7
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [9]
 => [[1,2,3,4,5,6,7,8,9]]
 => 9
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => 8
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => [9]
 => [[1,2,3,4,5,6,7,8,9]]
 => 9
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => 10
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => 10
([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => 10
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => 8
([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => 10
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => 10
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000738
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 2
([],2)
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
([(0,1)],2)
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
([],3)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8]]
 => 2
([(1,2)],3)
 => [6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 5
([(0,2),(2,1)],3)
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 5
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 6
([(1,2),(2,3)],4)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [[1,5],[2,6],[3,7],[4,8]]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 6
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 6
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7],[8]]
 => 8
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7]]
 => 7
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 7
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8]]
 => 8
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8]]
 => 8
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8]]
 => 8
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 8
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => 10
([(1,4),(3,2),(4,3)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 7
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 8
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 8
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => 10
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 7
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8]]
 => 8
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [9]
 => [[1,2,3,4,5,6,7,8,9]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => 9
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8]]
 => 8
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8]]
 => 8
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8]]
 => 8
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
 => [9]
 => [[1,2,3,4,5,6,7,8,9]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => 9
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [9]
 => [[1,2,3,4,5,6,7,8,9]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => 9
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8]]
 => 8
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => [9]
 => [[1,2,3,4,5,6,7,8,9]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => 9
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => 10
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => 10
([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => 10
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 8
([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => 10
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => 10
Description
The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].
Matching statistic: St001879
(load all 2 compositions to match this statistic)
Mapping
St001879: Posets ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 78%
Values
([],1)
 => ? = 2 - 2
([],2)
 => ? = 2 - 2
([(0,1)],2)
 => ? = 3 - 2
([],3)
 => ? = 2 - 2
([(1,2)],3)
 => ? = 6 - 2
([(0,1),(0,2)],3)
 => ? = 5 - 2
([(0,2),(2,1)],3)
 => 2 = 4 - 2
([(0,2),(1,2)],3)
 => ? = 5 - 2
([(0,2),(0,3),(3,1)],4)
 => ? = 7 - 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => 4 = 6 - 2
([(1,2),(2,3)],4)
 => ? = 4 - 2
([(0,3),(3,1),(3,2)],4)
 => ? = 6 - 2
([(0,3),(1,3),(3,2)],4)
 => ? = 6 - 2
([(0,3),(1,2),(1,3)],4)
 => ? = 8 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 7 - 2
([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
([(0,3),(1,2),(2,3)],4)
 => ? = 7 - 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 5 = 7 - 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 8 - 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => ? = 8 - 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 7 - 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ? = 8 - 2
([(0,4),(1,2),(1,4),(4,3)],5)
 => ? = 10 - 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 6 = 8 - 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ? = 10 - 2
([(1,4),(3,2),(4,3)],5)
 => ? = 10 - 2
([(0,3),(3,4),(4,1),(4,2)],5)
 => ? = 7 - 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 8 - 2
([(0,4),(3,2),(4,1),(4,3)],5)
 => ? = 8 - 2
([(0,4),(1,2),(2,3),(2,4)],5)
 => ? = 10 - 2
([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5 = 7 - 2
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? = 8 - 2
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 7 = 9 - 2
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 6 = 8 - 2
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => ? = 8 - 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 6 = 8 - 2
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
 => ? = 9 - 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 9 - 2
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 6 = 8 - 2
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 7 = 9 - 2
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => 8 = 10 - 2
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => 8 = 10 - 2
([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
 => ? = 10 - 2
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => ? = 10 - 2
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
 => 8 = 10 - 2
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001330
(load all 3 compositions to match this statistic)
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 78%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 2
([],2)
 => ([(0,1)],2)
 => ([],2)
 => ([(0,2),(1,2)],3)
 => 2
([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 6
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5
([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 8
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 7
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(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
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 7
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 10
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(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)
 => ? = 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 10
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(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
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(3,5),(4,5)],6)
 => ([(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)
 => ([(0,3),(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)
 => ? = 9
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
 => ([(3,5),(4,5)],6)
 => ([(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)
 => ([(0,3),(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)
 => ? = 9
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],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)
 => ([(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
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => ([(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)
 => ([(0,3),(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)
 => ? = 9
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => ([(3,5),(4,5)],6)
 => ([(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)
 => ([(0,3),(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)
 => ? = 9
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => ([(4,6),(5,6)],7)
 => ([(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)
 => ([(0,3),(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)
 => ? = 10
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => ([(4,6),(5,6)],7)
 => ([(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)
 => ([(0,3),(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)
 => ? = 10
([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
 => ([(4,6),(5,6)],7)
 => ([(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)
 => ([(0,3),(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)
 => ? = 10
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],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)
 => ([(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
([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => ([(4,6),(5,6)],7)
 => ([(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)
 => ([(0,3),(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)
 => ? = 10
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
 => ([(4,6),(5,6)],7)
 => ([(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)
 => ([(0,3),(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)
 => ? = 10
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: St000454
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 67%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([],2)
 => ([(0,1)],2)
 => ([],2)
 => ([(0,2),(1,2)],3)
 => ? = 2 - 1
([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 2 - 1
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 6 - 1
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 - 1
([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 - 1
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 7 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 - 1
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 - 1
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 - 1
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 8 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 7 - 1
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(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)
 => 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 7 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 10 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 10 - 1
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(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)
 => ? = 10 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 10 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(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)
 => 5 = 6 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 7 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(3,5),(4,5)],6)
 => ([(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)
 => ([(0,3),(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)
 => ? = 9 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8 - 1
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
 => ([(3,5),(4,5)],6)
 => ([(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)
 => ([(0,3),(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)
 => ? = 9 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],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)
 => ([(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)
 => 6 = 7 - 1
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => ([(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)
 => ([(0,3),(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)
 => ? = 9 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8 - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => ([(3,5),(4,5)],6)
 => ([(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)
 => ([(0,3),(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)
 => ? = 9 - 1
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => ([(4,6),(5,6)],7)
 => ([(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)
 => ([(0,3),(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)
 => ? = 10 - 1
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => ([(4,6),(5,6)],7)
 => ([(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)
 => ([(0,3),(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)
 => ? = 10 - 1
([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
 => ([(4,6),(5,6)],7)
 => ([(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)
 => ([(0,3),(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)
 => ? = 10 - 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],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)
 => ([(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 - 1
([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => ([(4,6),(5,6)],7)
 => ([(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)
 => ([(0,3),(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)
 => ? = 10 - 1
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
 => ([(4,6),(5,6)],7)
 => ([(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)
 => ([(0,3),(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)
 => ? = 10 - 1
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.