Your data matches 25 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000228
(load all 7 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
([(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
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => 10
([(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
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => 10
([(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),(2,3),(4,2)],5)
 => [5,2]
 => 7
([(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
([(0,2),(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),(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
([(0,3),(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
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St001279
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
([(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
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => 10
([(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
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => 10
([(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),(2,3),(4,2)],5)
 => [5,2]
 => 7
([(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
([(0,2),(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),(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
([(0,3),(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
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St000293
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => 100 => 2
([],2)
 => [2,2]
 => 1100 => 4
([(0,1)],2)
 => [3]
 => 1000 => 3
([],3)
 => [2,2,2,2]
 => 111100 => 8
([(1,2)],3)
 => [6]
 => 1000000 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => 10100 => 5
([(0,2),(2,1)],3)
 => [4]
 => 10000 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => 10100 => 5
([(1,2),(1,3)],4)
 => [6,2,2]
 => 100001100 => 10
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => 1011100 => 9
([(0,2),(0,3),(3,1)],4)
 => [7]
 => 10000000 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => 100100 => 6
([(1,2),(2,3)],4)
 => [4,4]
 => 110000 => 8
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => 100100 => 6
([(1,3),(2,3)],4)
 => [6,2,2]
 => 100001100 => 10
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 100100 => 6
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => 1011100 => 9
([(0,3),(1,2)],4)
 => [3,3,3]
 => 111000 => 9
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => 1001000 => 8
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => 101100 => 7
([(0,3),(2,1),(3,2)],4)
 => [5]
 => 100000 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => 10000000 => 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => 10011100 => 10
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => 1000100 => 7
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => 1011000 => 10
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => 1010000 => 9
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => 1001100 => 8
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => 10011100 => 10
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => 1001100 => 8
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => 10011100 => 10
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => 1000100 => 7
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => 1010000 => 9
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => 100000100 => 9
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => 1001100 => 8
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => 10000000000 => 10
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => 1010000 => 9
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => 100000000 => 8
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [10]
 => 10000000000 => 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [7,2]
 => 100000100 => 9
([(1,4),(3,2),(4,3)],5)
 => [10]
 => 10000000000 => 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => 1000100 => 7
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => 100000000 => 8
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => 100000000 => 8
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => 10000000000 => 10
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1000000 => 6
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => 1011000 => 10
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => 1010000 => 9
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => 1000100 => 7
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => 10001100 => 9
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [8,2]
 => 1000000100 => 10
Description
The number of inversions of a binary word.
Matching statistic: St000300
(load all 2 compositions to match this statistic)
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000300: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
([(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
([(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
([(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
([(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
([(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),(2,3),(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
([(0,3),(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
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 9
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 9
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 10
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 10
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 9
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 9
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 10
Description
The number of independent sets of vertices of a graph. An independent set of vertices of a graph $G$ is a subset $U \subset V(G)$ such that no two vertices in $U$ are adjacent. This is also the number of vertex covers of $G$ as the map $U \mapsto V(G)\setminus U$ 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: St001034
(load all 3 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck 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
([(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
([(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
([(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
([(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
([(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),(2,3),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 7
([(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
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 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]
 => 8
([(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]
 => 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]
 => 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]
 => 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 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]
 => 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]
 => 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]
 => 10
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 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]
 => 10
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 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]
 => 7
([(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]
 => 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]
 => 10
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Matching statistic: St000290
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
([(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
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => 10011100 => 11100010 => 10
([(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
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => 10011100 => 11100010 => 10
([(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),(2,3),(4,2)],5)
 => [5,2]
 => 1000100 => 1000010 => 7
([(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
([(0,2),(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),(2,4)],5)
 => [10]
 => 10000000000 => 00000000010 => 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [7,2]
 => 100000100 => 100000010 => 9
([(1,4),(3,2),(4,3)],5)
 => [10]
 => 10000000000 => 00000000010 => 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => 1000100 => 1000010 => 7
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => 100000000 => 000000010 => 8
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => 100000000 => 000000010 => 8
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => 10000000000 => 00000000010 => 10
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1000000 => 0000010 => 6
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => 1011000 => 1010010 => 10
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => 1010000 => 0010010 => 9
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => 1000100 => 1000010 => 7
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => 10001100 => 11000010 => 9
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [8,2]
 => 1000000100 => 1000000010 => 10
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 $\binom{n}{a}_q$.
Matching statistic: St000070
(load all 2 compositions to match this statistic)
Mapping
St000070: Posets ⟶ ℤResult quality: 91% ●values known / values provided: 91%●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
([(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
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 10
([(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
([(0,4),(4,1),(4,2),(4,3)],5)
 => 10
([(0,4),(1,4),(4,2),(4,3)],5)
 => 8
([(0,4),(1,4),(2,4),(4,3)],5)
 => 10
([(0,4),(1,4),(2,3),(4,2)],5)
 => 7
([(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
([(0,2),(0,4),(3,1),(4,3)],5)
 => 9
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 8
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => 9
([(1,4),(3,2),(4,3)],5)
 => 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => 7
([(0,4),(1,2),(2,4),(4,3)],5)
 => 8
([(0,4),(3,2),(4,1),(4,3)],5)
 => 8
([(0,4),(1,2),(2,3),(2,4)],5)
 => 10
([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
([(0,3),(1,2),(2,4),(3,4)],5)
 => 10
([(0,4),(1,2),(2,3),(3,4)],5)
 => 9
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 7
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => 9
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => 10
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
 => ? = 10
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
 => ? = 10
([(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
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7)
 => ? = 10
([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
 => ? = 10
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => ? = 10
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => ? = 10
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => ? = 10
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.
Matching statistic: St001616
(load all 2 compositions to match this statistic)
Mapping
Mp00195: Posets —order ideals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 89%
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
([(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
([(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
([(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
([(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
([(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),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 7
([(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
([(0,2),(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),(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
([(0,3),(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
([(0,5),(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),(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
([(0,5),(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),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,4),(0,6),(1,8),(3,7),(4,9),(5,2),(6,3),(6,9),(7,8),(8,5),(9,1),(9,7)],10)
 => ? = 10
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(8,1),(8,2),(9,3),(9,4)],10)
 => ? = 10
([(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
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,4),(0,5),(1,8),(3,7),(4,9),(5,9),(6,1),(6,7),(7,8),(8,2),(9,3),(9,6)],10)
 => ? = 10
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(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)
 => ? = 10
([(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),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
 => ? = 10
([(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
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,6),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,4),(6,5),(8,9),(9,2),(9,3)],10)
 => ? = 10
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,5),(0,6),(1,8),(2,8),(4,9),(5,7),(6,4),(6,7),(7,9),(8,3),(9,1),(9,2)],10)
 => ? = 10
([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,4),(0,5),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,2),(6,3),(8,9),(9,6)],10)
 => ? = 10
([(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
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => ([(0,5),(1,7),(2,8),(3,4),(3,7),(4,2),(4,9),(5,6),(6,1),(6,3),(7,9),(9,8)],10)
 => ? = 10
([(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
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 10
([(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
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(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)
 => ? = 10
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
 => ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],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
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
 => ([(0,7),(1,9),(2,9),(4,8),(5,8),(6,3),(7,1),(7,2),(8,6),(9,4),(9,5)],10)
 => ? = 10
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
 => ([(0,4),(0,5),(2,9),(3,9),(4,8),(5,8),(6,7),(7,2),(7,3),(8,6),(9,1)],10)
 => ? = 10
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
 => ? = 10
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
 => ([(0,3),(0,4),(1,8),(2,8),(3,9),(4,9),(5,7),(6,5),(7,1),(7,2),(9,6)],10)
 => ? = 10
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
 => ([(0,4),(0,5),(2,9),(3,9),(4,8),(5,8),(6,1),(7,2),(7,3),(8,7),(9,6)],10)
 => ? = 10
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7)
 => ([(0,4),(0,5),(2,8),(3,8),(4,9),(5,9),(6,1),(7,6),(8,7),(9,2),(9,3)],10)
 => ? = 10
([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
 => ([(0,7),(1,9),(2,9),(3,8),(4,8),(5,6),(6,3),(6,4),(7,1),(7,2),(9,5)],10)
 => ? = 10
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => ([(0,6),(1,9),(2,9),(3,8),(4,8),(6,7),(7,1),(7,2),(8,5),(9,3),(9,4)],10)
 => ? = 10
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => ([(0,5),(1,8),(2,8),(3,9),(4,9),(5,6),(6,3),(6,4),(7,1),(7,2),(9,7)],10)
 => ? = 10
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => ([(0,6),(2,9),(3,8),(4,2),(4,8),(5,1),(6,7),(7,3),(7,4),(8,9),(9,5)],10)
 => ? = 10
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => ([(0,7),(1,9),(3,8),(4,2),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,6)],10)
 => ? = 10
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => ([(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,5),(7,1),(7,2),(9,3),(9,4)],10)
 => ? = 10
([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
 => ([(0,6),(1,8),(2,8),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],9)
 => ? = 9
([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
 => ([(0,6),(1,9),(2,8),(3,4),(4,7),(5,1),(5,8),(6,3),(7,2),(7,5),(8,9)],10)
 => ? = 10
([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
 => ? = 10
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
 => ([(0,6),(1,9),(3,8),(4,7),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,2)],10)
 => ? = 10
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
 => ? = 9
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],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: St000395
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
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
([(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
([(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
([(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
([(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
([(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),(2,3),(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
([(0,3),(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
([(0,2),(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),(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
([(0,3),(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
([(0,5),(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),(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
([(0,5),(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
([(0,4),(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),(2,5),(4,5),(5,3)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 10
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => [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),(0,5),(1,4),(1,5),(3,2),(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
([(0,4),(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
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 10
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 9
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 10
([(0,2),(0,3),(2,5),(3,5),(4,1),(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
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],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
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(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
([(0,5),(1,2),(2,5),(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
([(0,4),(3,5),(4,3),(5,1),(5,2)],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),(3,4),(4,2),(5,1),(5,3)],6)
 => [6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 10
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 9
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 9
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 9
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
 => [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
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
 => [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),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => [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
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
 => [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
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => [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
([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
 => [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
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
 => [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
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7)
 => [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
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => [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),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
 => [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),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => [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
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => [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
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => [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),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => [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),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => [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
([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
 => [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
([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
 => [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
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [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
([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => [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),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
 => [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
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => [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
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => [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
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St000479
Mapping
Mp00195: Posets —order ideals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000479: Graphs ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 89%
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
([(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
([(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
([(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
([(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
([(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),(2,3),(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
([(0,3),(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
([(0,2),(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),(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
([(0,3),(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
([(0,5),(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),(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)
 => ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
 => ?
 => ? = 9
([(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)
 => ([(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)
 => ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
 => ?
 => ? = 9
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,4),(0,6),(1,8),(3,7),(4,9),(5,2),(6,3),(6,9),(7,8),(8,5),(9,1),(9,7)],10)
 => ?
 => ?
 => ? = 10
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(8,1),(8,2),(9,3),(9,4)],10)
 => ?
 => ?
 => ? = 10
([(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)
 => ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
 => ?
 => ? = 9
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,4),(0,5),(1,8),(3,7),(4,9),(5,9),(6,1),(6,7),(7,8),(8,2),(9,3),(9,6)],10)
 => ?
 => ?
 => ? = 10
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(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)
 => ([(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)
 => ? = 10
([(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)
 => ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
 => ?
 => ? = 9
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
 => ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
 => ?
 => ? = 10
([(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)
 => ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
 => ?
 => ? = 8
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,6),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,4),(6,5),(8,9),(9,2),(9,3)],10)
 => ?
 => ?
 => ? = 10
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,5),(0,6),(1,8),(2,8),(4,9),(5,7),(6,4),(6,7),(7,9),(8,3),(9,1),(9,2)],10)
 => ?
 => ?
 => ? = 10
([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,4),(0,5),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,2),(6,3),(8,9),(9,6)],10)
 => ?
 => ?
 => ? = 10
([(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)
 => ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
 => ?
 => ? = 8
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => ([(0,5),(1,7),(2,8),(3,4),(3,7),(4,2),(4,9),(5,6),(6,1),(6,3),(7,9),(9,8)],10)
 => ([(0,5),(1,7),(2,8),(3,4),(3,7),(4,2),(4,9),(5,6),(6,1),(6,3),(7,9),(9,8)],10)
 => ?
 => ? = 10
([(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),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ?
 => ? = 10
([(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)
 => ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
 => ?
 => ? = 9
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(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)
 => ?
 => ?
 => ? = 10
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
 => ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
 => ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],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)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],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)
 => ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
 => ?
 => ? = 9
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => ?
 => ? = 8
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
 => ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
 => ?
 => ? = 9
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
 => ([(0,7),(1,9),(2,9),(4,8),(5,8),(6,3),(7,1),(7,2),(8,6),(9,4),(9,5)],10)
 => ?
 => ?
 => ? = 10
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
 => ([(0,4),(0,5),(2,9),(3,9),(4,8),(5,8),(6,7),(7,2),(7,3),(8,6),(9,1)],10)
 => ?
 => ?
 => ? = 10
([(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 15 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000656The number of cuts of a poset. St001717The largest size of an interval in a poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000189The number of elements in the poset.