Processing math: 32%

Your data matches 18 different statistics following compositions of up to 3 maps.
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Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> 2
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 8
([(1,2)],3)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> 16
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 12
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 9
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 8
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 9
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 9
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 7
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
([(3,4)],5)
=> [6,6,6,6]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> 24
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> 17
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 13
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 10
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 12
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 10
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 9
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 8
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 16
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 12
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 10
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 12
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 8
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 10
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> 17
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 12
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 11
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> 14
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 13
([(1,4),(2,3)],5)
=> [6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> 18
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 9
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 9
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 8
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Mp00198: Posets incomparability graphGraphs
Mp00111: Graphs complementGraphs
St000300: Graphs ⟶ ℤResult quality: 93% values known / values provided: 93%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> ([],1)
=> 2
([],2)
=> ([(0,1)],2)
=> ([],2)
=> 4
([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 3
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 8
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 6
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 5
([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 5
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 16
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 12
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 10
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 9
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 8
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 10
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 9
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 9
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 7
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 7
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> 24
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 17
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 13
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 12
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 10
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 9
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 8
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 16
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 12
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 12
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 8
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 17
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 12
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 11
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 14
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 13
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 18
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 9
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 9
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 8
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 10
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(5,4),(6,3)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 16
([(0,6),(1,6),(2,3),(3,5),(6,4)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 24
([(0,5),(0,6),(1,4),(2,6),(3,5),(3,6),(4,2),(4,3)],7)
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 16
([(0,5),(1,3),(1,4),(3,6),(4,6),(5,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 24
([(0,5),(0,6),(1,3),(1,5),(1,6),(5,4),(6,2),(6,4)],7)
=> ([(0,3),(1,5),(1,6),(2,4),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 20
([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,6),(4,6),(5,3),(5,6)],7)
=> ([(0,4),(1,3),(2,5),(2,6),(3,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 16
([(0,2),(0,3),(0,5),(1,5),(1,6),(2,4),(3,6),(6,4)],7)
=> ([(0,5),(1,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 20
([(0,6),(1,5),(5,4),(6,2),(6,3)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 24
([(0,3),(1,2),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 16
([(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,5),(1,6),(2,4),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 20
([(0,5),(0,6),(1,3),(1,4),(2,6),(3,6),(4,2),(4,5)],7)
=> ([(0,5),(1,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 20
([(0,5),(0,6),(1,3),(1,5),(1,6),(3,4),(4,2),(6,4)],7)
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 16
([(0,5),(0,6),(1,4),(1,6),(2,5),(3,2),(4,3)],7)
=> ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 16
([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 18
([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 15
([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 16
([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 16
([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 17
([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 17
Description
The number of independent sets of vertices of a graph. An independent set of vertices of a graph G is a subset UV(G) such that no two vertices in U are adjacent. This is also the number of vertex covers of G as the map UV(G)U is a bijection between independent sets of vertices and vertex covers. The size of the largest independent set, also called independence number of G, is [[St000093]]
Matching statistic: St001279
Mp00306: Posets rowmotion cycle typeInteger partitions
St001279: Integer partitions ⟶ ℤResult quality: 76% values known / values provided: 92%distinct values known / distinct values provided: 76%
Values
([],1)
=> [2]
=> 2
([],2)
=> [2,2]
=> 4
([(0,1)],2)
=> [3]
=> 3
([],3)
=> [2,2,2,2]
=> 8
([(1,2)],3)
=> [6]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> 5
([],4)
=> [2,2,2,2,2,2,2,2]
=> 16
([(2,3)],4)
=> [6,6]
=> 12
([(1,2),(1,3)],4)
=> [6,2,2]
=> 10
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 9
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 6
([(1,2),(2,3)],4)
=> [4,4]
=> 8
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 6
([(1,3),(2,3)],4)
=> [6,2,2]
=> 10
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 6
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 9
([(0,3),(1,2)],4)
=> [3,3,3]
=> 9
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 7
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 7
([(3,4)],5)
=> [6,6,6,6]
=> ? = 24
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> 17
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> 13
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> 10
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> 12
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 7
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> 10
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> 9
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> 8
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> 16
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> 12
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> 10
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> 12
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> 8
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> 10
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> 17
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> 12
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> 11
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 7
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> 14
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> 13
([(1,4),(2,3)],5)
=> [6,6,6]
=> ? = 18
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> 9
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> 9
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> 8
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> 10
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> 14
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> 9
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> ? = 19
([(2,3),(3,5),(5,4)],6)
=> [10,10]
=> ? = 20
([(1,5),(2,5),(3,4)],6)
=> [6,6,6,6,6]
=> ? = 30
([(1,5),(2,3),(2,5),(5,4)],6)
=> [10,10]
=> ? = 20
([(1,5),(2,3),(2,4)],6)
=> [6,6,6,6,6]
=> ? = 30
([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> [10,10]
=> ? = 20
([(1,5),(2,3),(3,4),(3,5)],6)
=> [10,10]
=> ? = 20
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [7,6,6]
=> ? = 19
([(0,6),(1,6),(2,3),(3,5),(6,4)],7)
=> [4,4,4,4,4,4]
=> ? = 24
([(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
=> [10,10]
=> ? = 20
([(1,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> [10,10]
=> ? = 20
([(0,5),(1,3),(1,4),(3,6),(4,6),(5,2)],7)
=> [4,4,4,4,4,4]
=> ? = 24
([(0,5),(0,6),(1,3),(1,5),(1,6),(5,4),(6,2),(6,4)],7)
=> [10,10]
=> ? = 20
([(0,2),(0,3),(0,5),(1,5),(1,6),(2,4),(3,6),(6,4)],7)
=> [10,10]
=> ? = 20
([(1,5),(3,6),(4,2),(4,6),(5,3),(5,4)],7)
=> [10,10]
=> ? = 20
([(1,6),(2,3),(3,5),(3,6),(6,4)],7)
=> [6,6,6,6]
=> ? = 24
([(2,6),(4,5),(5,3),(6,4)],7)
=> [6,6,6,6]
=> ? = 24
([(0,6),(1,5),(5,4),(6,2),(6,3)],7)
=> [4,4,4,4,4,4]
=> ? = 24
([(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [10,10]
=> ? = 20
([(0,5),(0,6),(1,3),(1,4),(2,6),(3,6),(4,2),(4,5)],7)
=> [10,10]
=> ? = 20
([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> [6,6,6]
=> ? = 18
Description
The sum of the parts of an integer partition that are at least two.
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000293: Binary words ⟶ ℤResult quality: 81% values known / values provided: 83%distinct values known / distinct values provided: 81%
Values
([],1)
=> [2]
=> [1,1]
=> 110 => 2
([],2)
=> [2,2]
=> [2,2]
=> 1100 => 4
([(0,1)],2)
=> [3]
=> [1,1,1]
=> 1110 => 3
([],3)
=> [2,2,2,2]
=> [4,4]
=> 110000 => 8
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> 1111110 => 6
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> 11010 => 5
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 11110 => 4
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 11010 => 5
([],4)
=> [2,2,2,2,2,2,2,2]
=> [8,8]
=> 1100000000 => ? = 16
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> 11111100 => 12
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 110011110 => 10
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 1100010 => 9
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 11111110 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 6
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> 111100 => 8
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 6
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 110011110 => 10
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 6
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 1100010 => 9
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3,3]
=> 111000 => 9
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> 1110110 => 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [3,3,1]
=> 110010 => 7
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 111110 => 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 11111110 => 7
([(3,4)],5)
=> [6,6,6,6]
=> [4,4,4,4,4,4]
=> 1111110000 => ? = 24
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> 11000000010 => ? = 17
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> 111111010 => 13
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 11000110 => 10
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => 12
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 7
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 1110010 => 10
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 9
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 1100110 => 8
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4,4]
=> 11110000 => 16
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => 12
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 11000110 => 10
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => 12
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 1100110 => 8
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 11000110 => 10
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> 11000000010 => ? = 17
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> 110010110 => 12
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> 11000010 => 11
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 7
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> 111011010 => 14
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> 111111010 => 13
([(1,4),(2,3)],5)
=> [6,6,6]
=> [3,3,3,3,3,3]
=> 111111000 => 18
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 9
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> 110111110 => 9
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 1100110 => 8
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 10
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> 111011010 => 14
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 9
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 111111110 => 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> 110010110 => 12
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [3,3,3,3,3,3,1]
=> 1111110010 => ? = 19
([(2,3),(3,5),(5,4)],6)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> 111111111100 => ? = 20
([(1,5),(2,5),(3,4)],6)
=> [6,6,6,6,6]
=> [5,5,5,5,5,5]
=> 11111100000 => ? = 30
([(1,5),(2,3),(2,5),(5,4)],6)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> 111111111100 => ? = 20
([(1,5),(2,3),(2,4)],6)
=> [6,6,6,6,6]
=> [5,5,5,5,5,5]
=> 11111100000 => ? = 30
([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? = 16
([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> 111111111100 => ? = 20
([(0,3),(0,4),(1,2),(1,4),(2,5),(3,5)],6)
=> [8,7]
=> [2,2,2,2,2,2,2,1]
=> 1111111010 => ? = 15
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [8,7]
=> [2,2,2,2,2,2,2,1]
=> 1111111010 => ? = 15
([(1,5),(2,3),(3,5),(5,4)],6)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? = 16
([(1,5),(4,3),(5,2),(5,4)],6)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? = 16
([(1,5),(2,3),(3,4),(3,5)],6)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> 111111111100 => ? = 20
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 15
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [7,6,6]
=> [3,3,3,3,3,3,1]
=> 1111110010 => ? = 19
([(0,3),(0,4),(3,5),(3,6),(4,2),(4,5),(4,6),(6,1)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> 11111111010 => ? = 17
([(0,3),(0,4),(1,5),(2,5),(3,2),(3,6),(4,1),(4,6)],7)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? = 16
([(0,6),(1,6),(2,3),(3,5),(6,4)],7)
=> [4,4,4,4,4,4]
=> [6,6,6,6]
=> 1111000000 => ? = 24
([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> 11111111010 => ? = 17
([(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> 111111111100 => ? = 20
([(0,3),(0,4),(0,5),(1,6),(4,6),(5,1),(6,2)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> 11111111010 => ? = 17
([(1,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> 111111111100 => ? = 20
([(0,5),(1,3),(1,4),(3,6),(4,6),(5,2)],7)
=> [4,4,4,4,4,4]
=> [6,6,6,6]
=> 1111000000 => ? = 24
([(0,5),(0,6),(1,3),(1,5),(1,6),(5,4),(6,2),(6,4)],7)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> 111111111100 => ? = 20
([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(5,4),(6,4)],7)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? = 16
([(0,2),(0,3),(0,5),(1,5),(1,6),(2,4),(3,6),(6,4)],7)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> 111111111100 => ? = 20
([(0,2),(0,3),(0,4),(1,5),(1,6),(2,6),(3,5),(3,6),(4,1)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> 11111111010 => ? = 17
([(0,2),(0,3),(1,5),(1,6),(2,6),(3,5),(5,4),(6,4)],7)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? = 16
([(1,5),(3,6),(4,2),(4,6),(5,3),(5,4)],7)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> 111111111100 => ? = 20
([(1,6),(2,3),(3,5),(3,6),(6,4)],7)
=> [6,6,6,6]
=> [4,4,4,4,4,4]
=> 1111110000 => ? = 24
([(2,6),(4,5),(5,3),(6,4)],7)
=> [6,6,6,6]
=> [4,4,4,4,4,4]
=> 1111110000 => ? = 24
([(0,6),(1,5),(5,4),(6,2),(6,3)],7)
=> [4,4,4,4,4,4]
=> [6,6,6,6]
=> 1111000000 => ? = 24
([(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> 111111111100 => ? = 20
([(0,5),(0,6),(1,3),(1,4),(2,6),(3,6),(4,2),(4,5)],7)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> 111111111100 => ? = 20
([(0,6),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> 11111111010 => ? = 17
([(0,3),(0,5),(4,2),(5,6),(6,1),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> 11111111010 => ? = 17
([(0,3),(0,4),(1,6),(2,5),(3,5),(3,6),(4,1),(4,2)],7)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? = 16
([(0,6),(1,3),(1,4),(2,5),(3,5),(4,2),(5,6)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> 11111111010 => ? = 17
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,2)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> 11111111010 => ? = 17
([(0,3),(0,5),(1,6),(2,6),(4,2),(5,1),(5,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> 11111111010 => ? = 17
([(0,6),(1,5),(2,6),(3,6),(4,3),(5,2),(5,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> 11111111010 => ? = 17
([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [8,7]
=> [2,2,2,2,2,2,2,1]
=> 1111111010 => ? = 15
([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? = 16
([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? = 16
([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> 11111111010 => ? = 17
([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> 11111111010 => ? = 17
Description
The number of inversions of a binary word.
Matching statistic: St000290
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00095: Integer partitions to binary wordBinary words
Mp00316: Binary words inverse Foata bijectionBinary words
St000290: Binary words ⟶ ℤResult quality: 81% values known / values provided: 82%distinct values known / distinct values provided: 81%
Values
([],1)
=> [2]
=> 100 => 010 => 2
([],2)
=> [2,2]
=> 1100 => 1010 => 4
([(0,1)],2)
=> [3]
=> 1000 => 0010 => 3
([],3)
=> [2,2,2,2]
=> 111100 => 111010 => 8
([(1,2)],3)
=> [6]
=> 1000000 => 0000010 => 6
([(0,1),(0,2)],3)
=> [3,2]
=> 10100 => 10010 => 5
([(0,2),(2,1)],3)
=> [4]
=> 10000 => 00010 => 4
([(0,2),(1,2)],3)
=> [3,2]
=> 10100 => 10010 => 5
([],4)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? => ? = 16
([(2,3)],4)
=> [6,6]
=> 11000000 => 00001010 => 12
([(1,2),(1,3)],4)
=> [6,2,2]
=> 100001100 => 110000010 => 10
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 1011100 => 1110010 => 9
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 10000000 => 00000010 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 100100 => 100010 => 6
([(1,2),(2,3)],4)
=> [4,4]
=> 110000 => 001010 => 8
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 100100 => 100010 => 6
([(1,3),(2,3)],4)
=> [6,2,2]
=> 100001100 => 110000010 => 10
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 100100 => 100010 => 6
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 1011100 => 1110010 => 9
([(0,3),(1,2)],4)
=> [3,3,3]
=> 111000 => 101010 => 9
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 1001000 => 0100010 => 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 101100 => 110010 => 7
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 100000 => 000010 => 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 10000000 => 00000010 => 7
([(3,4)],5)
=> [6,6,6,6]
=> 1111000000 => ? => ? = 24
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> 10111111100 => ? => ? = 17
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> 101000000 => 000010010 => 13
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> 10011100 => 11100010 => 10
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> 11001100 => 00111010 => 12
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 1000100 => 1000010 => 7
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> 1011000 => 1010010 => 10
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> 1010000 => 0010010 => 9
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> 1001100 => 1100010 => 8
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> 11110000 => 10101010 => 16
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> 11001100 => 00111010 => 12
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> 10011100 => 11100010 => 10
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> 11001100 => 00111010 => 12
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> 1001100 => 1100010 => 8
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> 10011100 => 11100010 => 10
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> 10111111100 => ? => ? = 17
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> 100101100 => 011100010 => 12
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> 10111100 => 11110010 => 11
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 1000100 => 1000010 => 7
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> 101001000 => 100010010 => 14
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> 101000000 => 000010010 => 13
([(1,4),(2,3)],5)
=> [6,6,6]
=> 111000000 => 000101010 => 18
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> 1010000 => 0010010 => 9
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> 100000100 => 100000010 => 9
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> 1001100 => 1100010 => 8
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> 10000000000 => 00000000010 => 10
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> 101001000 => 100010010 => 14
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> 1010000 => 0010010 => 9
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> 100000000 => 000000010 => 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> 100101100 => 011100010 => 12
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> 1011000000 => ? => ? = 19
([(2,3),(3,5),(5,4)],6)
=> [10,10]
=> 110000000000 => ? => ? = 20
([(1,5),(2,5),(3,4)],6)
=> [6,6,6,6,6]
=> 11111000000 => ? => ? = 30
([(1,5),(2,3),(2,5),(5,4)],6)
=> [10,10]
=> 110000000000 => ? => ? = 20
([(1,5),(2,3),(2,4)],6)
=> [6,6,6,6,6]
=> 11111000000 => ? => ? = 30
([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [8,8]
=> 1100000000 => ? => ? = 16
([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> [10,10]
=> 110000000000 => ? => ? = 20
([(0,3),(0,4),(1,2),(1,4),(2,5),(3,5)],6)
=> [8,7]
=> 1010000000 => ? => ? = 15
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [8,7]
=> 1010000000 => ? => ? = 15
([(1,5),(2,3),(3,5),(5,4)],6)
=> [8,8]
=> 1100000000 => ? => ? = 16
([(1,5),(4,3),(5,2),(5,4)],6)
=> [8,8]
=> 1100000000 => ? => ? = 16
([(1,5),(2,3),(3,4),(3,5)],6)
=> [10,10]
=> 110000000000 => ? => ? = 20
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [3,2,2,2,2,2,2]
=> 1011111100 => ? => ? = 15
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [7,6,6]
=> 1011000000 => ? => ? = 19
([(0,3),(0,4),(3,5),(3,6),(4,2),(4,5),(4,6),(6,1)],7)
=> [9,8]
=> 10100000000 => ? => ? = 17
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(5,4),(6,3)],7)
=> [4,3,3,2,2,2]
=> 1011011100 => ? => ? = 16
([(0,3),(0,4),(1,5),(2,5),(3,2),(3,6),(4,1),(4,6)],7)
=> [8,8]
=> 1100000000 => ? => ? = 16
([(0,6),(1,6),(2,3),(3,5),(6,4)],7)
=> [4,4,4,4,4,4]
=> 1111110000 => ? => ? = 24
([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> 10100000000 => ? => ? = 17
([(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
=> [10,10]
=> 110000000000 => ? => ? = 20
([(0,3),(0,4),(0,5),(1,6),(4,6),(5,1),(6,2)],7)
=> [9,8]
=> 10100000000 => ? => ? = 17
([(1,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> [10,10]
=> 110000000000 => ? => ? = 20
([(0,5),(1,3),(1,4),(3,6),(4,6),(5,2)],7)
=> [4,4,4,4,4,4]
=> 1111110000 => ? => ? = 24
([(0,5),(0,6),(1,3),(1,5),(1,6),(5,4),(6,2),(6,4)],7)
=> [10,10]
=> 110000000000 => ? => ? = 20
([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(5,4),(6,4)],7)
=> [8,8]
=> 1100000000 => ? => ? = 16
([(0,2),(0,3),(0,5),(1,5),(1,6),(2,4),(3,6),(6,4)],7)
=> [10,10]
=> 110000000000 => ? => ? = 20
([(0,2),(0,3),(0,4),(1,5),(1,6),(2,6),(3,5),(3,6),(4,1)],7)
=> [9,8]
=> 10100000000 => ? => ? = 17
([(0,2),(0,3),(1,5),(1,6),(2,6),(3,5),(5,4),(6,4)],7)
=> [8,8]
=> 1100000000 => ? => ? = 16
([(1,5),(3,6),(4,2),(4,6),(5,3),(5,4)],7)
=> [10,10]
=> 110000000000 => ? => ? = 20
([(1,6),(2,3),(3,5),(3,6),(6,4)],7)
=> [6,6,6,6]
=> 1111000000 => ? => ? = 24
([(2,6),(4,5),(5,3),(6,4)],7)
=> [6,6,6,6]
=> 1111000000 => ? => ? = 24
([(0,6),(1,5),(5,4),(6,2),(6,3)],7)
=> [4,4,4,4,4,4]
=> 1111110000 => ? => ? = 24
([(0,3),(1,2),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [4,3,3,2,2,2]
=> 1011011100 => ? => ? = 16
([(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [10,10]
=> 110000000000 => ? => ? = 20
([(0,5),(0,6),(1,3),(1,4),(2,6),(3,6),(4,2),(4,5)],7)
=> [10,10]
=> 110000000000 => ? => ? = 20
([(0,6),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> [9,8]
=> 10100000000 => ? => ? = 17
([(0,3),(0,5),(4,2),(5,6),(6,1),(6,4)],7)
=> [9,8]
=> 10100000000 => ? => ? = 17
([(0,3),(0,4),(1,6),(2,5),(3,5),(3,6),(4,1),(4,2)],7)
=> [8,8]
=> 1100000000 => ? => ? = 16
([(0,6),(1,3),(1,4),(2,5),(3,5),(4,2),(5,6)],7)
=> [9,8]
=> 10100000000 => ? => ? = 17
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,2)],7)
=> [9,8]
=> 10100000000 => ? => ? = 17
([(0,3),(0,5),(1,6),(2,6),(4,2),(5,1),(5,4)],7)
=> [9,8]
=> 10100000000 => ? => ? = 17
([(0,6),(1,5),(2,6),(3,6),(4,3),(5,2),(5,4)],7)
=> [9,8]
=> 10100000000 => ? => ? = 17
([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [8,7]
=> 1010000000 => ? => ? = 15
([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [8,8]
=> 1100000000 => ? => ? = 16
([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> [8,8]
=> 1100000000 => ? => ? = 16
([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> [9,8]
=> 10100000000 => ? => ? = 17
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length n with a zeros, the generating function for the major index is the q-binomial coefficient \binom{n}{a}_q.
St000070: Posets ⟶ ℤResult quality: 63% values known / values provided: 63%distinct values known / distinct values provided: 100%
Values
([],1)
=> 2
([],2)
=> 4
([(0,1)],2)
=> 3
([],3)
=> 8
([(1,2)],3)
=> 6
([(0,1),(0,2)],3)
=> 5
([(0,2),(2,1)],3)
=> 4
([(0,2),(1,2)],3)
=> 5
([],4)
=> 16
([(2,3)],4)
=> 12
([(1,2),(1,3)],4)
=> 10
([(0,1),(0,2),(0,3)],4)
=> 9
([(0,2),(0,3),(3,1)],4)
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> 6
([(1,2),(2,3)],4)
=> 8
([(0,3),(3,1),(3,2)],4)
=> 6
([(1,3),(2,3)],4)
=> 10
([(0,3),(1,3),(3,2)],4)
=> 6
([(0,3),(1,3),(2,3)],4)
=> 9
([(0,3),(1,2)],4)
=> 9
([(0,3),(1,2),(1,3)],4)
=> 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> 7
([(0,3),(2,1),(3,2)],4)
=> 5
([(0,3),(1,2),(2,3)],4)
=> 7
([(3,4)],5)
=> 24
([(0,1),(0,2),(0,3),(0,4)],5)
=> 17
([(0,2),(0,3),(0,4),(4,1)],5)
=> 13
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 10
([(1,2),(1,3),(2,4),(3,4)],5)
=> 12
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 7
([(0,3),(0,4),(3,2),(4,1)],5)
=> 10
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 8
([(2,3),(3,4)],5)
=> 16
([(1,4),(4,2),(4,3)],5)
=> 12
([(0,4),(4,1),(4,2),(4,3)],5)
=> 10
([(1,4),(2,4),(4,3)],5)
=> 12
([(0,4),(1,4),(4,2),(4,3)],5)
=> 8
([(0,4),(1,4),(2,4),(4,3)],5)
=> 10
([(0,4),(1,4),(2,4),(3,4)],5)
=> 17
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 12
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 11
([(0,4),(1,4),(2,3),(4,2)],5)
=> 7
([(0,4),(1,4),(2,3),(2,4)],5)
=> 14
([(0,4),(1,4),(2,3),(3,4)],5)
=> 13
([(1,4),(2,3)],5)
=> 18
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 9
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 9
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 8
([(0,4),(1,2),(1,4),(4,3)],5)
=> 10
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 13
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> ? = 15
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> ? = 15
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> ? = 13
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> ? = 15
([(1,5),(2,3),(2,4)],6)
=> ? = 30
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ? = 15
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ? = 13
([(0,3),(0,4),(1,2),(1,4),(2,5),(3,5)],6)
=> ? = 15
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ? = 15
([(1,5),(2,3),(3,4),(3,5)],6)
=> ? = 20
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> ? = 10
([(0,3),(0,4),(3,5),(3,6),(4,5),(4,6),(5,2),(6,1)],7)
=> ? = 13
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
=> ? = 12
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> ? = 11
([(0,3),(0,4),(3,5),(3,6),(4,2),(4,5),(4,6),(6,1)],7)
=> ? = 17
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(5,4),(6,3)],7)
=> ? = 16
([(0,6),(1,4),(1,6),(2,4),(2,6),(4,5),(5,3),(6,5)],7)
=> ? = 14
([(0,6),(1,6),(2,3),(2,6),(3,5),(5,4),(6,5)],7)
=> ? = 16
([(0,3),(0,4),(1,5),(2,5),(3,2),(3,6),(4,1),(4,6)],7)
=> ? = 16
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
=> ? = 10
([(0,6),(1,6),(2,3),(3,5),(6,4)],7)
=> ? = 24
([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7)
=> ? = 12
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
=> ? = 10
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
=> ? = 10
([(0,6),(1,6),(4,3),(5,2),(6,4),(6,5)],7)
=> ? = 13
([(0,6),(1,6),(3,5),(4,2),(4,5),(6,3),(6,4)],7)
=> ? = 12
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7)
=> ? = 11
([(0,5),(0,6),(1,5),(1,6),(2,4),(3,4),(5,3),(6,2)],7)
=> ? = 13
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,3),(5,4),(6,2),(6,4)],7)
=> ? = 12
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> ? = 11
([(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
=> ? = 20
([(0,5),(1,4),(1,5),(4,6),(5,6),(6,2),(6,3)],7)
=> ? = 12
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7)
=> ? = 11
([(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1)],7)
=> ? = 13
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7)
=> ? = 10
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(5,2),(6,4)],7)
=> ? = 12
([(0,5),(0,6),(1,4),(2,6),(3,5),(3,6),(4,2),(4,3)],7)
=> ? = 16
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,1),(4,2),(4,3)],7)
=> ? = 14
([(0,5),(1,2),(1,3),(1,5),(2,6),(3,6),(5,6),(6,4)],7)
=> ? = 16
([(0,3),(0,4),(0,5),(1,6),(4,6),(5,1),(6,2)],7)
=> ? = 17
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
=> ? = 12
([(1,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 20
([(0,3),(0,5),(3,6),(4,1),(4,6),(5,4),(6,2)],7)
=> ? = 13
([(0,3),(0,4),(2,6),(3,5),(3,6),(4,2),(4,5),(6,1)],7)
=> ? = 13
([(0,3),(0,4),(2,5),(3,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ? = 14
([(0,4),(0,5),(2,6),(4,2),(5,1),(5,6),(6,3)],7)
=> ? = 14
([(0,3),(1,2),(1,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 14
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> ? = 12
([(0,2),(0,5),(2,6),(3,4),(4,1),(4,6),(5,3)],7)
=> ? = 13
Description
The number of antichains in a poset. An antichain in a poset P is a subset of elements of P which are pairwise incomparable. An order ideal is a subset I of P such that a\in I and b \leq_P a implies b \in I. Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.
Mp00306: Posets rowmotion cycle typeInteger partitions
St000228: Integer partitions ⟶ ℤResult quality: 34% values known / values provided: 34%distinct values known / distinct values provided: 43%
Values
([],1)
=> [2]
=> 2
([],2)
=> [2,2]
=> 4
([(0,1)],2)
=> [3]
=> 3
([],3)
=> [2,2,2,2]
=> 8
([(1,2)],3)
=> [6]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> 5
([],4)
=> [2,2,2,2,2,2,2,2]
=> ? = 16
([(2,3)],4)
=> [6,6]
=> ? = 12
([(1,2),(1,3)],4)
=> [6,2,2]
=> 10
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 9
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 6
([(1,2),(2,3)],4)
=> [4,4]
=> 8
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 6
([(1,3),(2,3)],4)
=> [6,2,2]
=> 10
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 6
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 9
([(0,3),(1,2)],4)
=> [3,3,3]
=> 9
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 7
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 7
([(3,4)],5)
=> [6,6,6,6]
=> ? = 24
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> ? = 17
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> ? = 13
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> 10
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> ? = 12
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 7
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> 10
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> 9
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> 8
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> ? = 16
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> ? = 12
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> 10
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> ? = 12
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> 8
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> 10
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> ? = 17
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> ? = 12
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> ? = 11
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 7
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> ? = 14
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> ? = 13
([(1,4),(2,3)],5)
=> [6,6,6]
=> ? = 18
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> 9
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> 9
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> 8
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> 10
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> ? = 14
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> 9
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> ? = 12
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> ? = 11
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> 9
([(1,4),(3,2),(4,3)],5)
=> [10]
=> 10
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> 7
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 8
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> 8
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> 10
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 6
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 10
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 9
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> 7
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> ? = 11
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> ? = 13
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> ? = 12
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> ? = 15
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> ? = 13
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> ? = 19
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> 9
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> 10
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> ? = 15
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> ? = 13
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> ? = 12
([(2,3),(3,5),(5,4)],6)
=> [10,10]
=> ? = 20
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> ? = 11
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> ? = 12
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> ? = 12
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> ? = 11
([(1,5),(2,5),(3,4)],6)
=> [6,6,6,6,6]
=> ? = 30
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> ? = 13
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> ? = 12
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [6,5,4]
=> ? = 15
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [5,4,2,2]
=> ? = 13
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [5,3,3]
=> ? = 11
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> ? = 17
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,4,2,2]
=> ? = 13
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [5,2,2,2]
=> ? = 11
([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 13
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> [6,4,3]
=> ? = 13
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> [5,4,2]
=> ? = 11
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> [4,3,3,2]
=> ? = 12
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> [5,4,2]
=> ? = 11
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [5,4,2]
=> ? = 11
([(1,5),(2,3),(2,5),(5,4)],6)
=> [10,10]
=> ? = 20
([(1,5),(2,3),(2,4)],6)
=> [6,6,6,6,6]
=> ? = 30
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [6,5,4]
=> ? = 15
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000395
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00227: Dyck paths Delest-Viennot-inverseDyck paths
St000395: Dyck paths ⟶ ℤResult quality: 30% values known / values provided: 30%distinct values known / distinct values provided: 62%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 4
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 8
([(1,2)],3)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 16
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 12
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 10
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 9
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 6
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 8
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 6
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 10
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 6
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 9
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 9
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 7
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7
([(3,4)],5)
=> [6,6,6,6]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 24
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 17
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 13
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 10
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 12
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 7
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 10
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 9
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 8
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 16
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 12
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 10
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 12
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 8
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 10
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 17
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 12
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 11
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 7
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 14
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 13
([(1,4),(2,3)],5)
=> [6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 18
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 9
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 9
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 8
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 10
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 14
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 9
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 12
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 11
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 9
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 10
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 7
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 10
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 10
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 9
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 7
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 11
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 13
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 12
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 15
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 13
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> ? = 19
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 9
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 10
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 9
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 15
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 13
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 12
([(2,3),(3,5),(5,4)],6)
=> [10,10]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 20
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 11
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 12
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 12
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 11
([(1,5),(2,5),(3,4)],6)
=> [6,6,6,6,6]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 30
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 13
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 12
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 10
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 15
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 9
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 13
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> 11
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 8
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 17
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 13
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 11
([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 13
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 9
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> [6,4,3]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 13
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 10
([(1,5),(2,3),(2,5),(5,4)],6)
=> [10,10]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 20
Description
The sum of the heights of the peaks of a Dyck path.
Mp00195: Posets order idealsLattices
St001616: Lattices ⟶ ℤResult quality: 20% values known / values provided: 20%distinct values known / distinct values provided: 38%
Values
([],1)
=> ([(0,1)],2)
=> 2
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3
([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 8
([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 16
([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 12
([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 10
([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 9
([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 8
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 10
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 9
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 9
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 7
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 7
([(3,4)],5)
=> ?
=> ? = 24
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 17
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 13
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 10
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> ? = 12
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 10
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> 9
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> 8
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? = 16
([(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> ? = 12
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 10
([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ? = 12
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> 8
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 10
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 17
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> ? = 12
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> ? = 11
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,1),(12,13),(13,8)],14)
=> ? = 14
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 13
([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ? = 18
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> 9
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> 9
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> 8
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 10
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1),(0,2),(1,11),(2,4),(2,5),(2,11),(3,6),(3,7),(4,8),(4,10),(5,8),(5,9),(6,13),(7,13),(8,12),(9,6),(9,12),(10,7),(10,12),(11,3),(11,9),(11,10),(12,13)],14)
=> ? = 14
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> 9
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> ? = 12
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> ? = 11
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> 9
([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 10
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 8
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> 8
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 10
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 10
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 9
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,5)],11)
=> ? = 11
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,10),(2,9),(3,8),(3,11),(4,7),(4,11),(5,7),(5,8),(6,3),(6,4),(6,5),(7,12),(8,12),(9,10),(11,2),(11,12),(12,1),(12,9)],13)
=> ? = 13
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,7),(2,7),(3,9),(3,10),(4,8),(4,10),(5,8),(5,9),(6,3),(6,4),(6,5),(8,11),(9,11),(10,11),(11,1),(11,2)],12)
=> ? = 12
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> ([(0,6),(1,11),(1,12),(2,10),(3,7),(3,9),(4,7),(4,8),(5,1),(5,8),(5,9),(6,3),(6,4),(6,5),(7,14),(8,11),(8,14),(9,12),(9,14),(11,13),(12,13),(13,10),(14,2),(14,13)],15)
=> ? = 15
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(1,11),(2,7),(2,8),(3,8),(3,9),(4,7),(4,9),(5,1),(5,10),(6,2),(6,3),(6,4),(7,12),(8,12),(9,5),(9,12),(10,11),(12,10)],13)
=> ? = 13
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,1),(1,2),(1,3),(1,4),(2,12),(2,13),(3,6),(3,13),(3,14),(4,5),(4,12),(4,14),(5,8),(5,10),(6,9),(6,11),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18)],19)
=> ? = 19
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=> 9
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
=> ? = 10
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> 9
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> ([(0,6),(1,12),(2,10),(2,11),(3,7),(3,9),(4,7),(4,8),(5,3),(5,4),(5,12),(6,1),(6,5),(7,14),(8,10),(8,14),(9,11),(9,14),(10,13),(11,13),(12,2),(12,8),(12,9),(14,13)],15)
=> ? = 15
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> ([(0,6),(1,11),(2,8),(2,10),(3,8),(3,9),(4,7),(5,4),(5,11),(6,1),(6,5),(7,9),(7,10),(8,12),(9,12),(10,12),(11,2),(11,3),(11,7)],13)
=> ? = 13
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,6),(1,10),(2,10),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(6,1),(6,2),(7,11),(8,11),(9,11),(10,3),(10,4),(10,5)],12)
=> ? = 12
([(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,12),(1,13),(2,8),(2,9),(3,2),(3,14),(3,15),(4,7),(4,11),(5,7),(5,10),(6,1),(6,10),(6,11),(7,16),(8,17),(9,17),(10,12),(10,16),(11,13),(11,16),(12,14),(12,18),(13,15),(13,18),(14,8),(14,19),(15,9),(15,19),(16,18),(18,19),(19,17)],20)
=> ? = 20
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> ([(0,5),(1,8),(1,9),(2,7),(2,9),(3,7),(3,8),(4,6),(5,4),(6,1),(6,2),(6,3),(7,10),(8,10),(9,10)],11)
=> ? = 11
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,8),(3,9),(4,7),(5,7),(6,1),(6,2),(6,3),(7,6),(8,11),(9,11),(10,11)],12)
=> ? = 12
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> ([(0,3),(0,4),(0,5),(1,7),(2,7),(3,9),(3,10),(4,8),(4,10),(5,8),(5,9),(6,1),(6,2),(8,11),(9,11),(10,11),(11,6)],12)
=> ? = 12
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ? = 11
([(1,5),(2,5),(3,4)],6)
=> ?
=> ? = 30
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,9),(4,8),(4,10),(5,7),(5,10),(6,7),(6,8),(7,12),(8,12),(9,11),(10,3),(10,12),(11,1),(12,2),(12,9)],13)
=> ? = 13
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(0,6),(2,10),(3,10),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(7,11),(8,11),(9,11),(10,1),(11,2),(11,3)],12)
=> ? = 12
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(0,3),(0,4),(1,9),(2,8),(3,7),(4,7),(5,1),(5,8),(6,2),(6,5),(7,6),(8,9)],10)
=> ? = 10
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,12),(3,10),(3,11),(4,7),(4,9),(5,7),(5,8),(6,3),(6,8),(6,9),(7,14),(8,10),(8,14),(9,11),(9,14),(10,13),(11,13),(12,2),(13,12),(14,1),(14,13)],15)
=> ? = 15
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> 9
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> ([(0,6),(1,10),(1,11),(2,9),(2,11),(3,7),(4,8),(5,1),(5,2),(5,7),(6,3),(6,5),(7,9),(7,10),(9,12),(10,12),(11,4),(11,12),(12,8)],13)
=> ? = 13
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> 8
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
=> 9
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
=> 9
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> 9
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 8
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> 8
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 8
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
=> 9
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 9
Description
The number of neutral elements in a lattice. An element e of the lattice L is neutral if the sublattice generated by e, x and y is distributive for all x, y \in L.
Matching statistic: St000479
Mp00195: Posets order idealsLattices
Mp00193: Lattices to posetPosets
Mp00198: Posets incomparability graphGraphs
St000479: Graphs ⟶ ℤResult quality: 10% values known / values provided: 10%distinct values known / distinct values provided: 38%
Values
([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 8
([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 6
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 5
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 5
([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ([(2,5),(2,7),(2,8),(2,9),(2,11),(2,14),(2,15),(3,4),(3,6),(3,7),(3,9),(3,11),(3,13),(3,15),(4,6),(4,7),(4,8),(4,11),(4,12),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(6,7),(6,10),(6,14),(6,15),(7,10),(7,12),(7,13),(8,9),(8,10),(8,13),(8,15),(9,10),(9,12),(9,14),(10,11),(10,12),(10,13),(10,14),(10,15),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(14,15)],16)
=> ? = 16
([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ([(2,8),(2,9),(2,11),(3,6),(3,7),(3,10),(4,5),(4,7),(4,9),(4,10),(4,11),(5,6),(5,8),(5,10),(5,11),(6,7),(6,9),(6,11),(7,8),(7,11),(8,9),(8,10),(9,10),(10,11)],12)
=> ? = 12
([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ([(2,9),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,9),(7,9),(8,9)],10)
=> ? = 10
([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ([(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7)],9)
=> ? = 9
([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 6
([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 8
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 6
([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ([(2,9),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,9),(7,9),(8,9)],10)
=> ? = 10
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 6
([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7)],9)
=> ? = 9
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 9
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> 7
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 7
([(3,4)],5)
=> ?
=> ?
=> ?
=> ? = 24
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ([(3,6),(3,8),(3,9),(3,10),(3,12),(3,15),(3,16),(4,5),(4,7),(4,8),(4,10),(4,12),(4,14),(4,16),(5,7),(5,8),(5,9),(5,12),(5,13),(5,15),(6,7),(6,9),(6,10),(6,12),(6,13),(6,14),(7,8),(7,11),(7,15),(7,16),(8,11),(8,13),(8,14),(9,10),(9,11),(9,14),(9,16),(10,11),(10,13),(10,15),(11,12),(11,13),(11,14),(11,15),(11,16),(12,13),(12,14),(12,15),(12,16),(13,14),(13,15),(13,16),(14,15),(14,16),(15,16)],17)
=> ? = 17
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ([(3,9),(3,10),(3,12),(4,7),(4,8),(4,11),(5,6),(5,8),(5,10),(5,11),(5,12),(6,7),(6,9),(6,11),(6,12),(7,8),(7,10),(7,12),(8,9),(8,12),(9,10),(9,11),(10,11),(11,12)],13)
=> ? = 13
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ([(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8)],10)
=> ? = 10
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> ([(2,11),(3,10),(4,5),(4,7),(4,9),(4,10),(5,7),(5,8),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,9),(8,11),(9,11),(10,11)],12)
=> ? = 12
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 7
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 10
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 9
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(4,7),(5,6)],8)
=> 8
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ([(2,12),(2,13),(2,15),(3,10),(3,11),(3,14),(4,5),(4,6),(4,7),(4,10),(4,11),(4,14),(5,8),(5,9),(5,12),(5,13),(5,15),(6,7),(6,9),(6,11),(6,13),(6,14),(6,15),(7,8),(7,10),(7,12),(7,14),(7,15),(8,9),(8,11),(8,13),(8,14),(8,15),(9,10),(9,12),(9,14),(9,15),(10,11),(10,13),(10,15),(11,12),(11,15),(12,13),(12,14),(13,14),(14,15)],16)
=> ? = 16
([(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> ([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,6),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(7,10),(8,9),(8,11),(9,11),(10,11)],12)
=> ? = 12
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ([(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8)],10)
=> ? = 10
([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,6),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(7,10),(8,9),(8,11),(9,11),(10,11)],12)
=> ? = 12
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(4,7),(5,6)],8)
=> 8
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ([(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8)],10)
=> ? = 10
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ([(3,6),(3,8),(3,9),(3,10),(3,12),(3,15),(3,16),(4,5),(4,7),(4,8),(4,10),(4,12),(4,14),(4,16),(5,7),(5,8),(5,9),(5,12),(5,13),(5,15),(6,7),(6,9),(6,10),(6,12),(6,13),(6,14),(7,8),(7,11),(7,15),(7,16),(8,11),(8,13),(8,14),(9,10),(9,11),(9,14),(9,16),(10,11),(10,13),(10,15),(11,12),(11,13),(11,14),(11,15),(11,16),(12,13),(12,14),(12,15),(12,16),(13,14),(13,15),(13,16),(14,15),(14,16),(15,16)],17)
=> ? = 17
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> ([(2,11),(3,4),(4,11),(5,6),(5,8),(5,10),(6,8),(6,9),(7,8),(7,9),(7,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 12
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> ([(3,4),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,9)],11)
=> ? = 11
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 7
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,1),(12,13),(13,8)],14)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,1),(12,13),(13,8)],14)
=> ([(2,5),(3,8),(3,9),(3,10),(4,11),(4,12),(4,13),(5,11),(5,12),(5,13),(6,7),(6,9),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,9),(8,12),(8,13),(9,11),(9,13),(10,11),(10,12),(10,13),(11,12)],14)
=> ? = 14
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ([(3,9),(3,10),(3,12),(4,7),(4,8),(4,11),(5,6),(5,8),(5,10),(5,11),(5,12),(6,7),(6,9),(6,11),(6,12),(7,8),(7,10),(7,12),(8,9),(8,12),(9,10),(9,11),(10,11),(11,12)],13)
=> ? = 13
([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ([(2,3),(2,8),(2,11),(2,15),(2,17),(3,8),(3,10),(3,14),(3,16),(4,5),(4,9),(4,13),(4,14),(4,16),(5,9),(5,12),(5,15),(5,17),(6,9),(6,12),(6,13),(6,14),(6,15),(6,16),(6,17),(7,8),(7,10),(7,11),(7,14),(7,15),(7,16),(7,17),(8,9),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,16),(9,17),(10,11),(10,12),(10,13),(10,14),(10,15),(10,17),(11,12),(11,13),(11,14),(11,15),(11,16),(12,13),(12,14),(12,16),(12,17),(13,15),(13,16),(13,17),(14,15),(14,17),(15,16),(16,17)],18)
=> ? = 18
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 9
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(3,4),(5,8),(6,7),(7,8)],9)
=> 9
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(4,7),(5,6)],8)
=> 8
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(2,9),(3,8),(4,5),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ? = 10
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1),(0,2),(1,11),(2,4),(2,5),(2,11),(3,6),(3,7),(4,8),(4,10),(5,8),(5,9),(6,13),(7,13),(8,12),(9,6),(9,12),(10,7),(10,12),(11,3),(11,9),(11,10),(12,13)],14)
=> ([(0,1),(0,2),(1,11),(2,4),(2,5),(2,11),(3,6),(3,7),(4,8),(4,10),(5,8),(5,9),(6,13),(7,13),(8,12),(9,6),(9,12),(10,7),(10,12),(11,3),(11,9),(11,10),(12,13)],14)
=> ([(2,5),(3,8),(3,9),(3,10),(4,11),(4,12),(4,13),(5,11),(5,12),(5,13),(6,7),(6,9),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,9),(8,12),(8,13),(9,11),(9,13),(10,11),(10,12),(10,13),(11,12)],14)
=> ? = 14
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 9
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> ([(2,11),(3,4),(4,11),(5,6),(5,8),(5,10),(6,8),(6,9),(7,8),(7,9),(7,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 12
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> ([(3,4),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,9)],11)
=> ? = 11
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(2,9),(3,8),(4,6),(5,7),(6,8),(7,9),(8,9)],10)
=> ? = 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(3,4),(5,8),(6,7),(7,8)],9)
=> 9
([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 10
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 7
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> 8
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> 8
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(2,9),(3,8),(4,5),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ? = 10
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 10
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 9
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 7
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,5)],11)
=> ([(0,6),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,5)],11)
=> ?
=> ? = 11
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,10),(2,9),(3,8),(3,11),(4,7),(4,11),(5,7),(5,8),(6,3),(6,4),(6,5),(7,12),(8,12),(9,10),(11,2),(11,12),(12,1),(12,9)],13)
=> ?
=> ?
=> ? = 13
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,7),(2,7),(3,9),(3,10),(4,8),(4,10),(5,8),(5,9),(6,3),(6,4),(6,5),(8,11),(9,11),(10,11),(11,1),(11,2)],12)
=> ?
=> ?
=> ? = 12
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> ([(0,6),(1,11),(1,12),(2,10),(3,7),(3,9),(4,7),(4,8),(5,1),(5,8),(5,9),(6,3),(6,4),(6,5),(7,14),(8,11),(8,14),(9,12),(9,14),(11,13),(12,13),(13,10),(14,2),(14,13)],15)
=> ?
=> ?
=> ? = 15
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(1,11),(2,7),(2,8),(3,8),(3,9),(4,7),(4,9),(5,1),(5,10),(6,2),(6,3),(6,4),(7,12),(8,12),(9,5),(9,12),(10,11),(12,10)],13)
=> ?
=> ?
=> ? = 13
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,1),(1,2),(1,3),(1,4),(2,12),(2,13),(3,6),(3,13),(3,14),(4,5),(4,12),(4,14),(5,8),(5,10),(6,9),(6,11),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18)],19)
=> ([(0,1),(1,2),(1,3),(1,4),(2,12),(2,13),(3,6),(3,13),(3,14),(4,5),(4,12),(4,14),(5,8),(5,10),(6,9),(6,11),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18)],19)
=> ?
=> ? = 19
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=> ?
=> ? = 9
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
=> ?
=> ?
=> ? = 10
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ?
=> ? = 9
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> ([(0,6),(1,12),(2,10),(2,11),(3,7),(3,9),(4,7),(4,8),(5,3),(5,4),(5,12),(6,1),(6,5),(7,14),(8,10),(8,14),(9,11),(9,14),(10,13),(11,13),(12,2),(12,8),(12,9),(14,13)],15)
=> ?
=> ?
=> ? = 15
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> ([(0,6),(1,11),(2,8),(2,10),(3,8),(3,9),(4,7),(5,4),(5,11),(6,1),(6,5),(7,9),(7,10),(8,12),(9,12),(10,12),(11,2),(11,3),(11,7)],13)
=> ?
=> ?
=> ? = 13
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> 8
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 7
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> 8
Description
The Ramsey number of a graph. This is the smallest integer n such that every two-colouring of the edges of the complete graph K_n contains a (not necessarily induced) monochromatic copy of the given graph. [1] Thus, the Ramsey number of the complete graph K_n is the ordinary Ramsey number R(n,n). Very few of these numbers are known, in particular, it is only known that 43\leq R(5,5)\leq 48. [2,3,4,5]
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001622The number of join-irreducible elements of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001614The cyclic permutation representation number of a skew partition. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph.