searching the database
Your data matches 196 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001694
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 1
([(0,1)],2)
=> 1
([],3)
=> 1
([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> 1
([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> 3
([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 6
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([],5)
=> 1
([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> 3
([(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> 1
([(1,4),(2,3),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> 3
([(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> 6
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 10
Description
The number of maximal dissociation sets in a graph.
Matching statistic: St001527
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001527: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 52%●distinct values known / distinct values provided: 27%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001527: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 52%●distinct values known / distinct values provided: 27%
Values
([],1)
=> []
=> ?
=> ? = 1
([],2)
=> []
=> ?
=> ? ∊ {1,1}
([(0,1)],2)
=> [1]
=> []
=> ? ∊ {1,1}
([],3)
=> []
=> ?
=> ? ∊ {1,3,3}
([(1,2)],3)
=> [1]
=> []
=> ? ∊ {1,3,3}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,3,3}
([],4)
=> []
=> ?
=> ? ∊ {2,3,3,6,6,6}
([(2,3)],4)
=> [1]
=> []
=> ? ∊ {2,3,3,6,6,6}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> ? ∊ {2,3,3,6,6,6}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {2,3,3,6,6,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> ? ∊ {2,3,3,6,6,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> ? ∊ {2,3,3,6,6,6}
([],5)
=> []
=> ?
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(3,4)],5)
=> [1]
=> []
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> ? ∊ {1,2,2,3,3,3,4,4,5,5,6,6,6,10,10,10}
([],6)
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(4,5)],6)
=> [1]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [8]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [8]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [9]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
Description
The cyclic permutation representation number of an integer partition.
This is the size of the largest cyclic group $C$ such that the fake degree is the character of a permutation representation of $C$.
Matching statistic: St000456
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1
([],2)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1
([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,2)],3)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> ? ∊ {1,3}
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,3}
([],4)
=> ([],4)
=> ([(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)
=> 6
([(2,3)],4)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ? ∊ {1,1,2,3,6,6}
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {1,1,2,3,6,6}
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> ? ∊ {1,1,2,3,6,6}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {1,1,2,3,6,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> ? ∊ {1,1,2,3,6,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,2,3,6,6}
([],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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
([(3,4)],5)
=> ([],4)
=> ([(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)
=> 6
([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(1,4),(2,3)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,4,5,6,6,10,10}
([],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),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 15
([(4,5)],6)
=> ([],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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
([(3,5),(4,5)],6)
=> ([(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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
([(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(2,5),(3,4)],6)
=> ([],4)
=> ([(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)
=> 6
([(2,5),(3,4),(4,5)],6)
=> ([(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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
([(1,2),(3,5),(4,5)],6)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([(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)
=> 6
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 8
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,9,9,9,10,10,15,15,15}
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001570
Values
([],1)
=> ([],1)
=> ([],1)
=> ? = 1
([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1}
([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> ? ∊ {1,1}
([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,3,3}
([(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,1,3,3}
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],1)
=> ? ∊ {1,1,3,3}
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {1,1,3,3}
([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,3,3,6,6,6}
([(2,3)],4)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,1,1,2,3,3,6,6,6}
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,1,1,2,3,3,6,6,6}
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> ? ∊ {1,1,1,2,3,3,6,6,6}
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ? ∊ {1,1,1,2,3,3,6,6,6}
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {1,1,1,2,3,3,6,6,6}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> ? ∊ {1,1,1,2,3,3,6,6,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {1,1,1,2,3,3,6,6,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,1,1,2,3,3,6,6,6}
([],5)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(3,4)],5)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 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)
=> ([],2)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(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)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(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)
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([],6)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(4,5)],6)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 3
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 3
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 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)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St001118
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1
([],2)
=> ([],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1}
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1}
([],3)
=> ([],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,3,3}
([(1,2)],3)
=> ([],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,3,3}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,3,3}
([],4)
=> ([],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,2,3,6,6,6}
([(2,3)],4)
=> ([],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,2,3,6,6,6}
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,2,3,6,6,6}
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,2,3,6,6,6}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,2,3,6,6,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,2,3,6,6,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,2,3,6,6,6}
([],5)
=> ([],5)
=> ([],5)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(3,4)],5)
=> ([],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(0,4),(1,3),(2,3),(2,4),(3,4)],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),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(0,3),(1,2),(1,4),(2,4),(3,4)],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),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4,5,6,6,6,10,10,10}
([],6)
=> ([],6)
=> ([],6)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(4,5)],6)
=> ([],5)
=> ([],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(0,2),(0,3),(0,4),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St000454
Values
([],1)
=> ([],1)
=> ([(0,1)],2)
=> 1
([],2)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? = 1
([(0,1)],2)
=> ([],1)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,3,3}
([(1,2)],3)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,3,3}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,3,3}
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([(0,1)],2)
=> 1
([],4)
=> ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(2,3)],4)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,3,6,6,6}
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,3,6,6,6}
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,3,6,6,6}
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,6,6,6}
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,3,6,6,6}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,3,6,6,6}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 1
([],5)
=> ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(3,4)],5)
=> ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,4),(2,3)],5)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1
([],6)
=> ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(4,5)],6)
=> ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,5),(3,4)],6)
=> ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(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)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(0,6),(0,8),(1,7),(1,8),(2,7),(2,8),(3,4),(3,5),(3,8),(4,6),(4,8),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 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.
Matching statistic: St000207
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 45%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 45%
Values
([],1)
=> [1]
=> []
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> 1
([(0,1)],2)
=> [2]
=> []
=> ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {3,3}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {3,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,3,3,6,6,6}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,3,3,6,6,6}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,3,3,6,6,6}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {1,3,3,6,6,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,3,3,6,6,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,3,3,6,6,6}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 4
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 3
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 5
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St000667
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 31%●distinct values known / distinct values provided: 27%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 31%●distinct values known / distinct values provided: 27%
Values
([],1)
=> [1]
=> []
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> 1
([(0,1)],2)
=> [2]
=> []
=> ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {3,3}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {3,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,3,3,6,6,6}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,3,3,6,6,6}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,3,3,6,6,6}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {1,3,3,6,6,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,3,3,6,6,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,3,3,6,6,6}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000781
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 31%●distinct values known / distinct values provided: 9%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 31%●distinct values known / distinct values provided: 9%
Values
([],1)
=> [1]
=> []
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> 1
([(0,1)],2)
=> [2]
=> []
=> ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {3,3}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {3,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {2,3,3,6,6,6}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {2,3,3,6,6,6}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {2,3,3,6,6,6}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {2,3,3,6,6,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {2,3,3,6,6,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {2,3,3,6,6,6}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001432
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 31%●distinct values known / distinct values provided: 18%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 31%●distinct values known / distinct values provided: 18%
Values
([],1)
=> [1]
=> []
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> 1
([(0,1)],2)
=> [2]
=> []
=> ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {3,3}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {3,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {2,3,3,6,6,6}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {2,3,3,6,6,6}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {2,3,3,6,6,6}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {2,3,3,6,6,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {2,3,3,6,6,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {2,3,3,6,6,6}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3,4,4,5,5,6,6,6,10,10,10}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,15,15,15,15}
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
The following 186 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001571The Cartan determinant of the integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000003The number of standard Young tableaux of the partition. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000026The position of the first return of a Dyck path. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000053The number of valleys of the Dyck path. St000075The orbit size of a standard tableau under promotion. St000120The number of left tunnels of a Dyck path. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000331The number of upper interactions of a Dyck path. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000347The inversion sum of a binary word. St000390The number of runs of ones in a binary word. St000391The sum of the positions of the ones in a binary word. St000517The Kreweras number of an integer partition. St000529The number of permutations whose descent word is the given binary word. St000531The leading coefficient of the rook polynomial of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000543The size of the conjugacy class of a binary word. St000548The number of different non-empty partial sums of an integer partition. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000691The number of changes of a binary word. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000734The last entry in the first row of a standard tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000847The number of standard Young tableaux whose descent set is the binary word. St000869The sum of the hook lengths of an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000947The major index east count of a Dyck path. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000983The length of the longest alternating subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001129The product of the squares of the parts of a partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001389The number of partitions of the same length below the given integer partition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001564The value of the forgotten symmetric functions when all variables set to 1. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001721The degree of a binary word. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001838The number of nonempty primitive factors of a binary word. St001884The number of borders of a binary word. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001930The weak major index of a binary word. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000264The girth of a graph, which is not a tree. St000422The energy of a graph, if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001875The number of simple modules with projective dimension at most 1. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001060The distinguishing index of a graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001877Number of indecomposable injective modules with projective dimension 2. St000618The number of self-evacuating tableaux of given shape. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!