- FindStat

Your data matches 52 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

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => ([],1)
 => [1]
 => 1
[1,0,1,0]
 => ([(0,1)],2)
 => [2]
 => 2
[1,1,0,0]
 => ([(0,1)],2)
 => [2]
 => 2
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 5
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 5
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => 5
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => 5
[1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 6
[1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 6
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 6
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 6
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 7
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 7
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => 6
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => 6
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 6
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 7
[1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 7

Description

The size of a partition. This statistic is the constant statistic of the level sets.

Matching statistic: St000479

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000479: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => ([],1)
 => ([],1)
 => 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([],2)
 => 2
[1,1,0,0]
 => ([(0,1)],2)
 => ([],2)
 => 2
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 4
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 5
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 5
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 5
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 5
[1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 6
[1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 6
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 6
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 6
[1,0,1,1,1,0,1,0,0,0]
 => ([(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
[1,0,1,1,1,1,0,0,0,0]
 => ([(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
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 6
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 6
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 6
[1,1,0,1,1,0,1,0,0,0]
 => ([(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
[1,1,0,1,1,1,0,0,0,0]
 => ([(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

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]

Matching statistic: St000548

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => ([],1)
 => [1]
 => [1]
 => 1
[1,0,1,0]
 => ([(0,1)],2)
 => [2]
 => [1,1]
 => 2
[1,1,0,0]
 => ([(0,1)],2)
 => [2]
 => [1,1]
 => 2
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 4
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [2,1,1,1]
 => 5
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
[1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [2,1,1,1]
 => 5
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [2,1,1,1]
 => 5
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [2,1,1,1]
 => 5
[1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 6
[1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 6
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 6
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 6
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [2,2,1,1,1]
 => 7
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [2,2,1,1,1]
 => 7
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 6
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 6
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 6
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [2,2,1,1,1]
 => 7
[1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [2,2,1,1,1]
 => 7

Description

The number of different non-empty partial sums of an integer partition.

Matching statistic: St001034
(load all 24 compositions to match this statistic)

Mapping

Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1,0]
 => [1,0]
 => 1
[1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 3
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 3
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 3
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 4
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 5
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 4
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 5
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 5
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 5
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 6
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 6
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 6
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 6
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 7
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 7
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 6
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 6
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 6
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 7
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 7

Description

The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].

Matching statistic: St001622
(load all 3 compositions to match this statistic)

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 90%

Values

[1,0]
 => ([],1)
 => ([(0,1)],2)
 => 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 2
[1,1,0,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 2
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 4
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,0,1,1,1,0,0,0]
 => ([(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)
 => 5
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,0,1,1,0,0,0]
 => ([(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)
 => 5
[1,1,1,0,0,0,1,0]
 => ([(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)
 => 5
[1,1,1,0,0,1,0,0]
 => ([(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)
 => 5
[1,1,1,0,1,0,0,0]
 => ([(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)
 => 6
[1,1,1,1,0,0,0,0]
 => ([(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)
 => 6
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => ([(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)
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => ([(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)
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => ([(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)
 => 6
[1,0,1,1,1,0,0,1,0,0]
 => ([(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)
 => 6
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(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)
 => 7
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(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)
 => 7
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => ([(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)
 => 6
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => ([(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)
 => 6
[1,1,0,1,1,0,0,0,1,0]
 => ([(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)
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => ([(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)
 => 6
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(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)
 => 7
[1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(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)
 => 7
[1,0,1,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ([(0,6),(1,10),(2,10),(4,9),(5,9),(6,7),(7,4),(7,5),(8,1),(8,2),(9,8),(10,3)],11)
 => ? = 8
[1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 8
[1,0,1,1,1,0,1,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 8
[1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 8
[1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 8
[1,1,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ([(0,6),(1,10),(2,10),(4,9),(5,9),(6,7),(7,4),(7,5),(8,1),(8,2),(9,8),(10,3)],11)
 => ? = 8
[1,1,0,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 8
[1,1,0,1,1,0,1,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 8
[1,1,0,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 8
[1,1,0,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
 => ? = 8
[1,1,1,0,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ([(0,8),(2,10),(3,10),(4,9),(5,9),(6,7),(7,2),(7,3),(8,4),(8,5),(9,6),(10,1)],11)
 => ? = 8
[1,1,1,0,0,1,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ([(0,8),(2,10),(3,10),(4,9),(5,9),(6,7),(7,2),(7,3),(8,4),(8,5),(9,6),(10,1)],11)
 => ? = 8
[1,1,1,0,0,1,1,0,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ([(0,8),(2,9),(3,9),(4,10),(5,10),(6,1),(7,4),(7,5),(8,2),(8,3),(9,7),(10,6)],11)
 => ? = 8
[1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ([(0,8),(2,9),(3,9),(4,10),(5,10),(6,1),(7,4),(7,5),(8,2),(8,3),(9,7),(10,6)],11)
 => ? = 8
[1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ([(0,9),(2,13),(3,12),(4,11),(5,11),(6,10),(7,6),(7,12),(8,3),(8,7),(9,4),(9,5),(10,13),(11,8),(12,2),(12,10),(13,1)],14)
 => ? = 9
[1,1,1,0,0,1,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ([(0,9),(2,13),(3,12),(4,11),(5,11),(6,10),(7,6),(7,12),(8,3),(8,7),(9,4),(9,5),(10,13),(11,8),(12,2),(12,10),(13,1)],14)
 => ? = 9
[1,1,1,0,1,0,0,0,1,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ([(0,8),(1,10),(2,11),(4,9),(5,3),(6,4),(6,11),(7,5),(8,2),(8,6),(9,10),(10,7),(11,1),(11,9)],12)
 => ? = 8
[1,1,1,0,1,0,0,0,1,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ([(0,8),(1,10),(2,11),(4,9),(5,3),(6,4),(6,11),(7,5),(8,2),(8,6),(9,10),(10,7),(11,1),(11,9)],12)
 => ? = 8
[1,1,1,0,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ([(0,8),(1,10),(2,11),(4,9),(5,3),(6,4),(6,11),(7,5),(8,2),(8,6),(9,10),(10,7),(11,1),(11,9)],12)
 => ? = 8
[1,1,1,0,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ([(0,8),(1,10),(2,11),(4,9),(5,3),(6,4),(6,11),(7,5),(8,2),(8,6),(9,10),(10,7),(11,1),(11,9)],12)
 => ? = 8
[1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ([(0,9),(1,11),(2,13),(4,12),(5,12),(6,10),(7,6),(7,13),(8,4),(8,5),(9,2),(9,7),(10,11),(11,8),(12,3),(13,1),(13,10)],14)
 => ? = 9
[1,1,1,1,0,0,0,0,1,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ([(0,8),(1,10),(2,11),(4,9),(5,3),(6,4),(6,11),(7,5),(8,2),(8,6),(9,10),(10,7),(11,1),(11,9)],12)
 => ? = 8
[1,1,1,1,0,0,0,0,1,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ([(0,8),(1,10),(2,11),(4,9),(5,3),(6,4),(6,11),(7,5),(8,2),(8,6),(9,10),(10,7),(11,1),(11,9)],12)
 => ? = 8
[1,1,1,1,0,0,0,1,0,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ([(0,8),(1,10),(2,11),(4,9),(5,3),(6,4),(6,11),(7,5),(8,2),(8,6),(9,10),(10,7),(11,1),(11,9)],12)
 => ? = 8
[1,1,1,1,0,0,0,1,0,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ([(0,8),(1,10),(2,11),(4,9),(5,3),(6,4),(6,11),(7,5),(8,2),(8,6),(9,10),(10,7),(11,1),(11,9)],12)
 => ? = 8
[1,1,1,1,0,0,0,1,1,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ([(0,9),(1,11),(2,13),(4,12),(5,12),(6,10),(7,6),(7,13),(8,4),(8,5),(9,2),(9,7),(10,11),(11,8),(12,3),(13,1),(13,10)],14)
 => ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ([(0,10),(2,3),(3,5),(4,2),(5,7),(6,4),(7,9),(8,6),(9,1),(10,8)],11)
 => ? = 10

Description

The number of join-irreducible elements of a lattice. An element

$j$ of a lattice

$L$ is '''join irreducible''' if it is not the least element and if

$j=x\vee y$ , then

$j\in\{x,y\}$ for all

$x,y\in L$ .

Matching statistic: St000395
(load all 21 compositions to match this statistic)

Mapping

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 90%

Values

[1,0]
 => [1,0]
 => 1
[1,0,1,0]
 => [1,1,0,0]
 => 2
[1,1,0,0]
 => [1,0,1,0]
 => 2
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 3
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 4
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 4
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 5
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 4
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 5
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 5
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 5
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 6
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 6
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 6
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 6
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 7
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 7
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 6
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 6
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 7
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 7
[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
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 8
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 8
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 8
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 8
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 8
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 8
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 8
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 8
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 8
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 8
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 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
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 9
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 9
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 9
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 9
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 9
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 9
[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
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10

Description

The sum of the heights of the peaks of a Dyck path.

Matching statistic: St000394
(load all 6 compositions to match this statistic)

Mapping

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1,0]
 => [1,1,0,0]
 => 1
[1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 4
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 4
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 5
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 5
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 5
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 5
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 6
[1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 6
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 6
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 6
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 7
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 7
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 6
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 6
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 6
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 7
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 7
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 8
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 8
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => ? = 8
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => ? = 8
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => ? = 8
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? = 8
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 8
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 8
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 8
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 8
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
 => ? = 8
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
 => ? = 8
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => ? = 8
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => ? = 8
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 8
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 8
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 8
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 8
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 8
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => ? = 8
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 8
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 8
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
 => ? = 8
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 8
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 8
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
 => ? = 8
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
 => ? = 8
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 8
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 8
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
 => ? = 8
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 8
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 8
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 8

Description

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

Matching statistic: St001330

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 80%

Values

[1,0]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2
[1,1,0,0]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,1,1,0,0,0]
 => ([(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)
 => ? = 5
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,1,1,0,0,0]
 => ([(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)
 => ? = 5
[1,1,1,0,0,0,1,0]
 => ([(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)
 => ? = 5
[1,1,1,0,0,1,0,0]
 => ([(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)
 => ? = 5
[1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[1,0,1,0,1,0,1,0,1,0]
 => ([(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)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => ([(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)
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => ([(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)
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => ([(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)
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[1,0,1,1,0,0,1,0,1,0]
 => ([(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)
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => ([(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)
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => ([(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)
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => ([(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)
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,1,0,0,1,0,1,0,1,0]
 => ([(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)
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => ([(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)
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => ([(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)
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => ([(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)
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[1,1,0,1,0,0,1,0,1,0]
 => ([(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)
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => ([(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)
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => ([(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)
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => ([(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)
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[1,1,1,0,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(3,6),(4,5)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,1,1,0,1,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,1,1,1,0,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,1,1,1,0,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,0,1,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,0,1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(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)
 => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[1,0,1,0,1,1,1,1,0,0,0,0]
 => ([(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)
 => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[1,0,1,1,0,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,1,0,0,1,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,0,1,1,0,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,0,1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,0,1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,0,1,1,0,1,1,0,1,0,0,0]
 => ([(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)
 => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[1,0,1,1,0,1,1,1,0,0,0,0]
 => ([(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)
 => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[1,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,0,1,1,1,0,0,0,1,1,0,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,0,1,1,1,0,0,1,0,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,0,1,1,1,0,0,1,0,1,0,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,0,1,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ([(4,7),(5,6)],8)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(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)
 => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[1,0,1,1,1,0,1,0,0,1,0,0]
 => ([(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)
 => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(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)
 => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(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)
 => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[1,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,0,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,0,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,0,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(5,6)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Matching statistic: St001318

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001318: Graphs ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 70%

Values

[1,0]
 => ([],1)
 => ([],1)
 => 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([],2)
 => 2
[1,1,0,0]
 => ([(0,1)],2)
 => ([],2)
 => 2
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 4
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 5
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 5
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 5
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 5
[1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 6
[1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 6
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 6
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 6
[1,0,1,1,1,0,1,0,0,0]
 => ([(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
[1,0,1,1,1,1,0,0,0,0]
 => ([(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
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 6
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 6
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 6
[1,1,0,1,1,0,1,0,0,0]
 => ([(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
[1,1,0,1,1,1,0,0,0,0]
 => ([(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
[1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(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
[1,0,1,0,1,1,1,1,0,0,0,0]
 => ([(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
[1,0,1,1,0,1,1,0,1,0,0,0]
 => ([(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
[1,0,1,1,0,1,1,1,0,0,0,0]
 => ([(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
[1,0,1,1,1,0,0,1,1,0,0,0]
 => ([(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
[1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(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
[1,0,1,1,1,0,1,0,0,1,0,0]
 => ([(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
[1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(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
[1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(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
[1,1,0,0,1,1,1,0,1,0,0,0]
 => ([(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
[1,1,0,0,1,1,1,1,0,0,0,0]
 => ([(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
[1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(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
[1,1,0,1,0,1,1,1,0,0,0,0]
 => ([(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
[1,1,0,1,1,0,0,1,1,0,0,0]
 => ([(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
[1,1,0,1,1,0,1,0,0,0,1,0]
 => ([(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
[1,1,0,1,1,0,1,0,0,1,0,0]
 => ([(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
[1,1,0,1,1,1,0,0,0,0,1,0]
 => ([(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
[1,1,0,1,1,1,0,0,0,1,0,0]
 => ([(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
[1,1,1,0,0,0,1,1,1,0,0,0]
 => ([(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
[1,1,1,0,0,1,0,1,1,0,0,0]
 => ([(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
[1,1,1,0,0,1,1,0,0,0,1,0]
 => ([(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
[1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(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
[1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(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
[1,1,1,0,0,1,1,1,0,0,0,0]
 => ([(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
[1,1,1,0,1,0,0,0,1,0,1,0]
 => ([(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
[1,1,1,0,1,0,0,0,1,1,0,0]
 => ([(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
[1,1,1,0,1,0,0,1,0,0,1,0]
 => ([(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
[1,1,1,0,1,0,0,1,0,1,0,0]
 => ([(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
[1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(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
[1,1,1,1,0,0,0,0,1,0,1,0]
 => ([(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
[1,1,1,1,0,0,0,0,1,1,0,0]
 => ([(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
[1,1,1,1,0,0,0,1,0,0,1,0]
 => ([(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
[1,1,1,1,0,0,0,1,0,1,0,0]
 => ([(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
[1,1,1,1,0,0,0,1,1,0,0,0]
 => ([(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
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8

Description

The number of vertices of the largest induced subforest with the same number of connected components of a graph.

Matching statistic: St001321

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001321: Graphs ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 70%

Values

[1,0]
 => ([],1)
 => ([],1)
 => 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([],2)
 => 2
[1,1,0,0]
 => ([(0,1)],2)
 => ([],2)
 => 2
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 4
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 5
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 5
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 5
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 5
[1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 6
[1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 6
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 6
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 6
[1,0,1,1,1,0,1,0,0,0]
 => ([(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
[1,0,1,1,1,1,0,0,0,0]
 => ([(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
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 6
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 6
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 6
[1,1,0,1,1,0,1,0,0,0]
 => ([(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
[1,1,0,1,1,1,0,0,0,0]
 => ([(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
[1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(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
[1,0,1,0,1,1,1,1,0,0,0,0]
 => ([(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
[1,0,1,1,0,1,1,0,1,0,0,0]
 => ([(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
[1,0,1,1,0,1,1,1,0,0,0,0]
 => ([(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
[1,0,1,1,1,0,0,1,1,0,0,0]
 => ([(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
[1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(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
[1,0,1,1,1,0,1,0,0,1,0,0]
 => ([(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
[1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(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
[1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(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
[1,1,0,0,1,1,1,0,1,0,0,0]
 => ([(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
[1,1,0,0,1,1,1,1,0,0,0,0]
 => ([(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
[1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(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
[1,1,0,1,0,1,1,1,0,0,0,0]
 => ([(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
[1,1,0,1,1,0,0,1,1,0,0,0]
 => ([(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
[1,1,0,1,1,0,1,0,0,0,1,0]
 => ([(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
[1,1,0,1,1,0,1,0,0,1,0,0]
 => ([(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
[1,1,0,1,1,1,0,0,0,0,1,0]
 => ([(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
[1,1,0,1,1,1,0,0,0,1,0,0]
 => ([(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
[1,1,1,0,0,0,1,1,1,0,0,0]
 => ([(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
[1,1,1,0,0,1,0,1,1,0,0,0]
 => ([(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
[1,1,1,0,0,1,1,0,0,0,1,0]
 => ([(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
[1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(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
[1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(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
[1,1,1,0,0,1,1,1,0,0,0,0]
 => ([(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
[1,1,1,0,1,0,0,0,1,0,1,0]
 => ([(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
[1,1,1,0,1,0,0,0,1,1,0,0]
 => ([(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
[1,1,1,0,1,0,0,1,0,0,1,0]
 => ([(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
[1,1,1,0,1,0,0,1,0,1,0,0]
 => ([(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
[1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(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
[1,1,1,1,0,0,0,0,1,0,1,0]
 => ([(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
[1,1,1,1,0,0,0,0,1,1,0,0]
 => ([(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
[1,1,1,1,0,0,0,1,0,0,1,0]
 => ([(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
[1,1,1,1,0,0,0,1,0,1,0,0]
 => ([(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
[1,1,1,1,0,0,0,1,1,0,0,0]
 => ([(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
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 8

Description

The number of vertices of the largest induced subforest of a graph.

The following 42 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001342The number of vertices in the center of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St000171The degree of the graph. St001120The length of a longest path in a graph. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000728The dimension of a set partition. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000874The position of the last double rise in a Dyck path. St000438The position of the last up step in a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000018The number of inversions of a permutation. 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. St000656The number of cuts of a poset. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000189The number of elements in the poset. St001725The harmonious chromatic number of a graph. St001875The number of simple modules with projective dimension at most 1. St000029The depth of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St000224The sorting index of a permutation. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001726The number of visible inversions of a permutation. St000030The sum of the descent differences of a permutations. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001869The maximum cut size of a graph. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000912The number of maximal antichains in a poset. St001894The depth of a signed permutation. St000680The Grundy value for Hackendot on posets. St000223The number of nestings in the permutation. St001727The number of invisible inversions of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.