- FindStat

Your data matches 22 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001166
(load all 3 compositions to match this statistic)

Mapping

St001166: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra.

Matching statistic: St000259
(load all 21 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 71%

Values

[1,0]
 => [1] => [1] => ([],1)
 => 0 = 2 - 2
[1,0,1,0]
 => [2,1] => [1,2] => ([],2)
 => ? = 3 - 2
[1,1,0,0]
 => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 3 - 2
[1,0,1,0,1,0]
 => [3,2,1] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {3,3,3} - 2
[1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => ([],3)
 => ? ∊ {3,3,3} - 2
[1,1,0,0,1,0]
 => [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 4 - 2
[1,1,0,1,0,0]
 => [2,1,3] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {3,3,3} - 2
[1,1,1,0,0,0]
 => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,5} - 2
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,5} - 2
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,5} - 2
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,5} - 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,3,5} - 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 3 = 5 - 2
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {3,3,3,3,3,3,3,5} - 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,5} - 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 3 = 5 - 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,5} - 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3 = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 4 - 2
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 4 - 2
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3 = 5 - 2
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4 = 6 - 2
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 4 - 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3 = 5 - 2
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,6,6} - 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [5,4,3,1,2,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => [5,4,2,3,1,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => [5,4,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [5,4,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => [5,3,4,2,1,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [5,3,4,1,2,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,5,2,1] => [5,3,2,4,1,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,2,1] => [5,3,2,1,4,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,2,1] => [5,3,1,2,4,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,2,1] => [5,2,3,1,4,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,2,1] => [5,2,1,3,4,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => [4,5,3,2,1,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => [4,5,3,1,2,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => [6,4,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,1,2] => [6,4,3,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,1,2] => [6,4,2,3,1,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,1,2] => [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,1,2] => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,1,2] => [6,3,4,2,1,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,1,2] => [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [6,4,3,5,1,2] => [6,3,2,4,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,1,2] => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,1,2] => [6,2,3,4,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,1,2] => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,1,2] => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,1,2] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1,3] => [4,6,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,1,3] => [4,6,3,1,2,5] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,1,3] => [4,6,2,3,1,5] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [5,4,6,2,1,3] => [4,6,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 3 = 5 - 2
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,1,3] => [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1,4] => [4,3,6,2,1,5] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 5 - 2

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

Matching statistic: St001065

Mapping

Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001065: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 71%

Values

[1,0]
 => [[1],[]]
 => []
 => []
 => ? = 2
[1,0,1,0]
 => [[1,1],[]]
 => []
 => []
 => ? ∊ {3,3}
[1,1,0,0]
 => [[2],[]]
 => []
 => []
 => ? ∊ {3,3}
[1,0,1,0,1,0]
 => [[1,1,1],[]]
 => []
 => []
 => ? ∊ {3,3,4,4}
[1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => []
 => ? ∊ {3,3,4,4}
[1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,1,0,1,0,0]
 => [[3],[]]
 => []
 => []
 => ? ∊ {3,3,4,4}
[1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => []
 => ? ∊ {3,3,4,4}
[1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,4,4,5,5,5}
[1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,4,4,5,5,5}
[1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,4,4,5,5,5}
[1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,4,4,5,5,5}
[1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 4
[1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => []
 => ? ∊ {3,3,3,4,4,5,5,5}
[1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => []
 => ? ∊ {3,3,3,4,4,5,5,5}
[1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => []
 => ? ∊ {3,3,3,4,4,5,5,5}
[1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => []
 => ? ∊ {3,3,3,4,4,5,5,5}
[1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 5
[1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 4
[1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 4
[1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 4
[1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,1,1,1,0,1,0,0,0,0]
 => [[4,4],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,5,5,5,5,5,5,5,6,6,6}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [[4,1,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 5
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 5
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 4
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 5
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 4
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [[4,4,1],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 5
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [[5,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [[4,4,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 4
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 5
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => [1]
 => [1,0,1,0]
 => 3
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [[4,2,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [[4,3,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [[6],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [[5,2],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [[4,2,2],[]]
 => []
 => []
 => ? ∊ {3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7}

Description

Number of indecomposable reflexive modules in the corresponding Nakayama algebra.

Matching statistic: St000771
(load all 2 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 71%

Values

[1,0]
 => [1] => [] => ([],0)
 => ? = 2 - 2
[1,0,1,0]
 => [1,2] => [1] => ([],1)
 => 1 = 3 - 2
[1,1,0,0]
 => [2,1] => [1] => ([],1)
 => 1 = 3 - 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,2] => ([],2)
 => ? ∊ {4,4} - 2
[1,0,1,1,0,0]
 => [1,3,2] => [1,2] => ([],2)
 => ? ∊ {4,4} - 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1] => ([(0,1)],2)
 => 1 = 3 - 2
[1,1,0,1,0,0]
 => [2,3,1] => [2,1] => ([(0,1)],2)
 => 1 = 3 - 2
[1,1,1,0,0,0]
 => [3,2,1] => [2,1] => ([(0,1)],2)
 => 1 = 3 - 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => ? ∊ {3,3,3,4,5,5,5} - 2
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,3] => ([],3)
 => ? ∊ {3,3,3,4,5,5,5} - 2
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {3,3,3,4,5,5,5} - 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {3,3,3,4,5,5,5} - 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {3,3,3,4,5,5,5} - 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {3,3,3,4,5,5,5} - 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {3,3,3,4,5,5,5} - 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 4 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 5 - 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 5 - 2
[1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,6,4,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [2,3,4,5,1,6] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,3,5,4,1,6] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,4,6,1] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,6,4,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,5,4,1] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [2,4,3,5,1,6] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,3,5,6,1] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,3,6,5,1] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [2,5,3,4,1,6] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,5,3,4,6,1] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,6,3,4,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,6,3,5,4,1] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,5,4,3,1,6] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,4,3,6,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,6,4,3,5,1] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,6,4,5,3,1] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,5,4,3,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2

Description

The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.

Matching statistic: St000777
(load all 7 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => ([],1)
 => 1 = 2 - 1
[1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
[1,1,0,0]
 => [2] => ([],2)
 => ? = 3 - 1
[1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => ? ∊ {3,3,4} - 1
[1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 4 - 1
[1,1,0,1,0,0]
 => [3] => ([],3)
 => ? ∊ {3,3,4} - 1
[1,1,1,0,0,0]
 => [3] => ([],3)
 => ? ∊ {3,3,4} - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,5,5} - 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,5,5} - 1
[1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,5,5} - 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,5,5} - 1
[1,1,0,1,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,5,5} - 1
[1,1,0,1,1,0,0,0]
 => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,5,5} - 1
[1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,5,5} - 1
[1,1,1,0,1,0,0,0]
 => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,5,5} - 1
[1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,5,5} - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 6 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,0,0,1,1,0,0,0]
 => [5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,0,1,0,1,0,0,0]
 => [5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,0,1,1,0,0,0,0]
 => [5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,1,0,0,1,0,0,0]
 => [5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,1,0,1,0,0,0,0]
 => [5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,1,1,0,0,0,0,0]
 => [5] => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => ([(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)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7} - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7} - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7} - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => ([(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)
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7} - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7} - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7} - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7} - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7} - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7} - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,1,1,1] => ([(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)
 => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 7 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,1,1,1] => ([(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)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 6 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 6 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 6 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3 = 4 - 1

Description

The number of distinct eigenvalues of the distance Laplacian of a connected graph.

Matching statistic: St000260
(load all 12 compositions to match this statistic)

Mapping

Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 57%

Values

[1,0]
 => ([],1)
 => ([],1)
 => 0 = 2 - 2
[1,0,1,0]
 => ([(0,1)],2)
 => ([],2)
 => ? = 3 - 2
[1,1,0,0]
 => ([],2)
 => ([(0,1)],2)
 => 1 = 3 - 2
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {3,4,4} - 2
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {3,4,4} - 2
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => ? ∊ {3,4,4} - 2
[1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[1,1,1,0,0,0]
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 3 - 2
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? ∊ {3,3,3,4,4,4,5,5,5} - 2
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ? ∊ {3,3,3,4,4,4,5,5,5} - 2
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? ∊ {3,3,3,4,4,4,5,5,5} - 2
[1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,4,4,4,5,5,5} - 2
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,3,4,4,4,5,5,5} - 2
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ? ∊ {3,3,3,4,4,4,5,5,5} - 2
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ? ∊ {3,3,3,4,4,4,5,5,5} - 2
[1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,4,4,4,5,5,5} - 2
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 4 - 2
[1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,3,4,4,4,5,5,5} - 2
[1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,1,1,1,0,0,0,0]
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 4 - 2
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,0,0,1,1,0,0,0]
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,1,1,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,1,1,0,0,1,0,0,0]
 => ([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,1,1,0,1,0,0,0,0]
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,1,1,1,0,0,0,0,0]
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 3 = 5 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => 2 = 4 - 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
 => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(4,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 4 - 2
[1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,5),(2,3),(2,4),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 3 - 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => 2 = 4 - 2
[1,1,1,0,0,1,0,1,1,0,0,0]
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => 2 = 4 - 2
[1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 3 - 2
[1,1,1,0,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
 => ([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
 => ([(0,5),(1,2),(1,3),(1,4),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
 => ([(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 3 - 2

Description

The radius of a connected graph. This is the minimum eccentricity of any vertex.

Matching statistic: St001875
(load all 8 compositions to match this statistic)

Mapping

Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 71%

Values

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

Description

The number of simple modules with projective dimension at most 1.

Matching statistic: St001880
(load all 12 compositions to match this statistic)

Mapping

Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 71%

Values

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

Description

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

Matching statistic: St000741
(load all 5 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%

Values

[1,0]
 => [1] => [1] => ([],1)
 => 0 = 2 - 2
[1,0,1,0]
 => [1,2] => [1,2] => ([],2)
 => 1 = 3 - 2
[1,1,0,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1 = 3 - 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => ([],3)
 => 1 = 3 - 2
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {3,3} - 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {3,3} - 2
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 4 - 2
[1,1,1,0,0,0]
 => [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 1 = 3 - 2
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {3,4,5,5,5} - 2
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {3,4,5,5,5} - 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {3,4,5,5,5} - 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {3,4,5,5,5} - 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? ∊ {3,4,5,5,5} - 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 3 - 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 1 = 3 - 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 1 = 3 - 2
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 1 = 3 - 2
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => 1 = 3 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => 1 = 3 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,5,3] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
 => 1 = 3 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,6,4,5,3] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => [1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
 => 1 = 3 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,6,5,4] => [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
 => 1 = 3 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
 => 1 = 3 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,3,4,2,6,5] => [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
 => 1 = 3 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,2,6] => [1,5,2,3,4,6] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,3,4,6,5,2] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,5,4,2,6] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,4,6,2] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,3,6,4,5,2] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,3,6,5,4,2] => [1,5,4,6,2,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
 => 1 = 3 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,3,2,6,5] => [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
 => 1 = 3 - 2
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,3,5,6,2] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,4,3,6,5,2] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,5,3,4,2,6] => [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,5,3,4,6,2] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,6,3,4,5,2] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
 => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7} - 2

Description

The Colin de Verdière graph invariant.

Matching statistic: St000454
(load all 7 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 43%

Values

[1,0]
 => [1] => [.,.]
 => ([],1)
 => 0 = 2 - 2
[1,0,1,0]
 => [1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 3 - 2
[1,1,0,0]
 => [2,1] => [[.,.],.]
 => ([(0,1)],2)
 => 1 = 3 - 2
[1,0,1,0,1,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,4,4} - 2
[1,0,1,1,0,0]
 => [1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,4,4} - 2
[1,1,0,0,1,0]
 => [2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,4,4} - 2
[1,1,0,1,0,0]
 => [2,3,1] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,4,4} - 2
[1,1,1,0,0,0]
 => [3,2,1] => [[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,4,4} - 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5} - 2
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5} - 2
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5} - 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5} - 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5} - 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5} - 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5} - 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5} - 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5} - 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5} - 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5} - 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5} - 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5} - 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,3,4,4,4,4,5,5,5} - 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 4 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 4 - 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 4 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [6,2,4,3,5,1] => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 4 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [6,2,4,5,3,1] => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 4 - 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,2,5,4,3,7,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,2,5,4,6,3,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,5,4,6,7,3] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,5,4,7,6,3] => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,6,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,4,6,7,5] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,3,2,7,5,6,4] => [.,[[.,.],[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,3,4,2,6,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,3,4,2,6,7,5] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,3,4,6,5,2,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,3,4,6,5,7,2] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,3,7,5,4,6,2] => [.,[[.,.],[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,3,7,5,6,4,2] => [.,[[.,.],[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,7,4,3,5,6,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,7,4,3,6,5,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,7,4,5,3,6,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,7,4,5,6,3,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,7,4,6,5,3,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,5,4,3,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [2,1,5,4,6,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [2,1,5,4,6,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [2,1,5,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [2,5,3,4,1,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
 => [2,5,3,4,1,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => [2,5,3,4,6,1,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,5,3,4,6,7,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [2,5,3,4,7,6,1] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [2,5,4,3,1,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => [2,5,4,3,1,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [2,5,4,3,6,1,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [2,5,4,3,6,7,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [2,5,4,3,7,6,1] => [[.,.],[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 4 - 2
[1,1,1,0,1,0,0,0,1,1,0,0,1,0]
 => [4,2,3,1,6,5,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[1,1,1,0,1,0,0,0,1,1,0,1,0,0]
 => [4,2,3,1,6,7,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[1,1,1,0,1,0,0,1,1,0,0,0,1,0]
 => [4,2,3,6,5,1,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [4,2,3,6,5,7,1] => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [7,2,3,5,4,6,1] => [[[.,.],[.,[[.,.],[.,.]]]],.]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 4 - 2

Description

The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000455The second largest eigenvalue of a graph if it is integral. St000643The size of the largest orbit of antichains under Panyushev complementation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000264The girth of a graph, which is not a tree. St001644The dimension of a graph. St000907The number of maximal antichains of minimal length in a poset. St001330The hat guessing number of a graph. St001557The number of inversions of the second entry of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.