Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000935
Mapping
St000935: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 2
[1,1]
 => 1
[3]
 => 3
[2,1]
 => 2
[1,1,1]
 => 1
[4]
 => 5
[3,1]
 => 3
[2,2]
 => 3
[2,1,1]
 => 2
[1,1,1,1]
 => 1
[5]
 => 7
[4,1]
 => 5
[3,2]
 => 4
[3,1,1]
 => 3
[2,2,1]
 => 3
[2,1,1,1]
 => 2
[1,1,1,1,1]
 => 1
[6]
 => 11
[5,1]
 => 7
[4,2]
 => 7
[4,1,1]
 => 5
[3,3]
 => 5
[3,2,1]
 => 4
[3,1,1,1]
 => 3
[2,2,2]
 => 4
[2,2,1,1]
 => 3
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 1
[7]
 => 15
[6,1]
 => 11
[5,2]
 => 9
[5,1,1]
 => 7
[4,3]
 => 8
[4,2,1]
 => 7
[4,1,1,1]
 => 5
[3,3,1]
 => 5
[3,2,2]
 => 5
[3,2,1,1]
 => 4
[3,1,1,1,1]
 => 3
[2,2,2,1]
 => 4
[2,2,1,1,1]
 => 3
[2,1,1,1,1,1]
 => 2
[1,1,1,1,1,1,1]
 => 1
[8]
 => 22
[7,1]
 => 15
[6,2]
 => 15
[6,1,1]
 => 11
[5,3]
 => 10
[5,2,1]
 => 9
Description
The number of ordered refinements of an integer partition. This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.
Matching statistic: St001232
(load all 47 compositions to match this statistic)
Mapping
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 29%
Values
[1]
 => [1]
 => [1,0,1,0]
 => 1
[2]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,1]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[3]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? ∊ {2,3}
[2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? ∊ {2,3}
[1,1,1]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[4]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {3,3,5}
[3,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,2]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {3,3,5}
[2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {3,3,5}
[1,1,1,1]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[5]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? ∊ {2,3,4,5,7}
[4,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,3,4,5,7}
[3,2]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => ? ∊ {2,3,4,5,7}
[3,1,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,3,4,5,7}
[2,2,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[2,1,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? ∊ {2,3,4,5,7}
[1,1,1,1,1]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[6]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ? ∊ {3,4,4,7,7,11}
[5,1]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? ∊ {3,4,4,7,7,11}
[4,2]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {3,4,4,7,7,11}
[4,1,1]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[3,3]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[3,2,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {3,4,4,7,7,11}
[3,1,1,1]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? ∊ {3,4,4,7,7,11}
[2,2,2]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[2,2,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 5
[2,1,1,1,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ? ∊ {3,4,4,7,7,11}
[1,1,1,1,1,1]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[7]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? ∊ {1,2,3,3,4,5,5,5,7,8,9,11,15}
[6,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ? ∊ {1,2,3,3,4,5,5,5,7,8,9,11,15}
[5,2]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? ∊ {1,2,3,3,4,5,5,5,7,8,9,11,15}
[5,1,1]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {1,2,3,3,4,5,5,5,7,8,9,11,15}
[4,3]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? ∊ {1,2,3,3,4,5,5,5,7,8,9,11,15}
[4,2,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => ? ∊ {1,2,3,3,4,5,5,5,7,8,9,11,15}
[4,1,1,1]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {1,2,3,3,4,5,5,5,7,8,9,11,15}
[3,3,1]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? ∊ {1,2,3,3,4,5,5,5,7,8,9,11,15}
[3,2,2]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? ∊ {1,2,3,3,4,5,5,5,7,8,9,11,15}
[3,2,1,1]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? ∊ {1,2,3,3,4,5,5,5,7,8,9,11,15}
[3,1,1,1,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ? ∊ {1,2,3,3,4,5,5,5,7,8,9,11,15}
[2,2,2,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[2,2,1,1,1]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 7
[2,1,1,1,1,1]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? ∊ {1,2,3,3,4,5,5,5,7,8,9,11,15}
[1,1,1,1,1,1,1]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {1,2,3,3,4,5,5,5,7,8,9,11,15}
[8]
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? ∊ {1,4,5,5,5,5,7,8,9,10,11,11,15,15,22}
[7,1]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? ∊ {1,4,5,5,5,5,7,8,9,10,11,11,15,15,22}
[6,2]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? ∊ {1,4,5,5,5,5,7,8,9,10,11,11,15,15,22}
[6,1,1]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => ? ∊ {1,4,5,5,5,5,7,8,9,10,11,11,15,15,22}
[5,3]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[5,2,1]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {1,4,5,5,5,5,7,8,9,10,11,11,15,15,22}
[5,1,1,1]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
[4,4]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,4,5,5,5,5,7,8,9,10,11,11,15,15,22}
[4,3,1]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,4,5,5,5,5,7,8,9,10,11,11,15,15,22}
[4,2,2]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[4,2,1,1]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {1,4,5,5,5,5,7,8,9,10,11,11,15,15,22}
[4,1,1,1,1]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => ? ∊ {1,4,5,5,5,5,7,8,9,10,11,11,15,15,22}
[3,3,2]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? ∊ {1,4,5,5,5,5,7,8,9,10,11,11,15,15,22}
[3,3,1,1]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[3,2,2,1]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? ∊ {1,4,5,5,5,5,7,8,9,10,11,11,15,15,22}
[3,2,1,1,1]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? ∊ {1,4,5,5,5,5,7,8,9,10,11,11,15,15,22}
[3,1,1,1,1,1]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? ∊ {1,4,5,5,5,5,7,8,9,10,11,11,15,15,22}
[2,2,2,2]
 => [4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 6
[2,2,2,1,1]
 => [5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 7
[2,2,1,1,1,1]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 9
[2,1,1,1,1,1,1]
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? ∊ {1,4,5,5,5,5,7,8,9,10,11,11,15,15,22}
[1,1,1,1,1,1,1,1]
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {1,4,5,5,5,5,7,8,9,10,11,11,15,15,22}
[9]
 => [8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {1,2,3,4,4,5,6,6,7,7,7,8,9,9,9,11,11,11,13,15,15,17,19,22,30}
[8,1]
 => [7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => ? ∊ {1,2,3,4,4,5,6,6,7,7,7,8,9,9,9,11,11,11,13,15,15,17,19,22,30}
[7,2]
 => [6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => ? ∊ {1,2,3,4,4,5,6,6,7,7,7,8,9,9,9,11,11,11,13,15,15,17,19,22,30}
[7,1,1]
 => [6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => ? ∊ {1,2,3,4,4,5,6,6,7,7,7,8,9,9,9,11,11,11,13,15,15,17,19,22,30}
[6,3]
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 5
[6,2,1]
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => ? ∊ {1,2,3,4,4,5,6,6,7,7,7,8,9,9,9,11,11,11,13,15,15,17,19,22,30}
[6,1,1,1]
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? ∊ {1,2,3,4,4,5,6,6,7,7,7,8,9,9,9,11,11,11,13,15,15,17,19,22,30}
[4,4,1]
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3
[3,3,1,1,1]
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 5
[2,2,2,2,1]
 => [5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
[2,2,2,1,1,1]
 => [6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => 10
[7,3]
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => 7
[6,1,1,1,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 2
[5,2,2,1]
 => [4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 6
[3,3,3,1]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 5
[3,3,1,1,1,1]
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => 7
[2,2,2,2,2]
 => [5,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => 7
[2,2,2,2,1,1]
 => [6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => 9
[6,4,1]
 => [5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
[5,2,2,2]
 => [4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5
[4,4,1,1,1]
 => [5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
[3,3,3,1,1]
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 4
[2,2,2,2,2,1]
 => [6,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => 6
[7,4,1]
 => [6,3,3]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => 5
[6,2,2,1,1]
 => [5,5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
 => 8
[5,5,1,1]
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => 3
[5,3,3,1]
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4
[4,4,4]
 => [3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => 4
[4,4,1,1,1,1]
 => [6,3,3]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => 5
[3,3,3,3]
 => [4,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[3,3,3,1,1,1]
 => [6,2,2,2]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => 7
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 17%
Values
[1]
 => [1,0]
 => [1] => ([],1)
 => 0 = 1 - 1
[2]
 => [1,0,1,0]
 => [1,2] => ([],2)
 => 0 = 1 - 1
[1,1]
 => [1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => 1 = 2 - 1
[3]
 => [1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => 0 = 1 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => 1 = 2 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ? = 3 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => 0 = 1 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => 1 = 2 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,5} - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {3,5} - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => 0 = 1 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => 1 = 2 - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {4,5,7} - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {4,5,7} - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {4,5,7} - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ([],6)
 => 0 = 1 - 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => ([(4,5)],6)
 => 1 = 2 - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ? ∊ {4,5,5,7,7,11} - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {4,5,5,7,7,11} - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,5,5,7,7,11} - 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,5,5,7,7,11} - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,5,5,7,7,11} - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 3 - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,5,5,7,7,11} - 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => ([],7)
 => 0 = 1 - 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => ([(5,6)],7)
 => 1 = 2 - 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
 => ? ∊ {4,5,5,5,7,7,8,9,11,15} - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,5,5,5,7,7,8,9,11,15} - 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,5,5,5,7,7,8,9,11,15} - 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => ([(3,6),(4,6),(5,6)],7)
 => ? ∊ {4,5,5,5,7,7,8,9,11,15} - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,5,5,5,7,7,8,9,11,15} - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,3,5,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,5,5,5,7,7,8,9,11,15} - 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 3 - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {4,5,5,5,7,7,8,9,11,15} - 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,5,5,5,7,7,8,9,11,15} - 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {4,5,5,5,7,7,8,9,11,15} - 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {4,5,5,5,7,7,8,9,11,15} - 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8] => ([],8)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
 => 2 = 3 - 1
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,6,5,7,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,5] => ([(4,7),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,5,3,4,6,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,5,4,6,7,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5,4,3,6,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,4,3,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,2,5,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} - 1
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8,9] => ([],9)
 => ? ∊ {1,2,3,3,5,5,5,6,6,7,7,7,8,9,9,9,10,11,11,11,13,15,15,17,19,22,30} - 1
[8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
 => ? ∊ {1,2,3,3,5,5,5,6,6,7,7,7,8,9,9,9,10,11,11,11,13,15,15,17,19,22,30} - 1
[7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,3,5,5,5,6,6,7,7,7,8,9,9,9,10,11,11,11,13,15,15,17,19,22,30} - 1
[7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,6,8,9,7] => ([(6,8),(7,8)],9)
 => ? ∊ {1,2,3,3,5,5,5,6,6,7,7,7,8,9,9,9,10,11,11,11,13,15,15,17,19,22,30} - 1
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,7,5,6,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,3,5,5,5,6,6,7,7,7,8,9,9,9,10,11,11,11,13,15,15,17,19,22,30} - 1
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,4,7,6,8,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,3,5,5,5,6,6,7,7,7,8,9,9,9,10,11,11,11,13,15,15,17,19,22,30} - 1
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,5,7,8,9,6] => ([(5,8),(6,8),(7,8)],9)
 => ? ∊ {1,2,3,3,5,5,5,6,6,7,7,7,8,9,9,9,10,11,11,11,13,15,15,17,19,22,30} - 1
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,6,4,5,7,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,3,5,5,5,6,6,7,7,7,8,9,9,9,10,11,11,11,13,15,15,17,19,22,30} - 1
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,7,6,5,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 4 - 1
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,3,6,5,7,8,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,3,5,5,5,6,6,7,7,7,8,9,9,9,10,11,11,11,13,15,15,17,19,22,30} - 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,2,7,4,5,6,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 4 - 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
[5,3,3]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,7,6,5,4,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4 = 5 - 1
[3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000422
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 11%
Values
[1]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 2 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 4 = 3 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,3,5} + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {2,3,5} + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,3,5} + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {2,3,4,7} + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6 = 5 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,3,4,7} + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,3,4,7} + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,3,4,7} + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,5] => ([(4,5)],6)
 => 2 = 1 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {2,3,3,4,4,7,7,11} + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,3,3,4,4,7,7,11} + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6 = 5 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,4,4,7,7,11} + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,4,4,7,7,11} + 1
[3,2,1]
 => [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)
 => 6 = 5 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,4,4,7,7,11} + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {2,3,3,4,4,7,7,11} + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,4,4,7,7,11} + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,3,3,4,4,7,7,11} + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => 2 = 1 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,4,4,5,5,7,7,8,9,11,15} + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,4,4,5,5,7,7,8,9,11,15} + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,3,4,4,5,5,7,7,8,9,11,15} + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,3,4,4,5,5,7,7,8,9,11,15} + 1
[4,3]
 => [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)
 => 6 = 5 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,3,4,4,5,5,7,7,8,9,11,15} + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,3,4,4,5,5,7,7,8,9,11,15} + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,3,4,4,5,5,7,7,8,9,11,15} + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,3,4,4,5,5,7,7,8,9,11,15} + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,3,4,4,5,5,7,7,8,9,11,15} + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,3,4,4,5,5,7,7,8,9,11,15} + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,3,4,4,5,5,7,7,8,9,11,15} + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,4,4,5,5,7,7,8,9,11,15} + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,7] => ([(6,7)],8)
 => ? ∊ {1,2,3,4,4,5,5,7,7,8,9,11,15} + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [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)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4 = 3 + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[4,2,2]
 => [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)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [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)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,4,4,5,5,5,5,6,7,7,8,9,9,10,11,11,15,15,22} + 1
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[3,3,2,1]
 => [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)
 => 6 = 5 + 1
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[4,3,2,1]
 => [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)
 => 8 = 7 + 1
[4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 4 = 3 + 1
[4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 4 = 3 + 1
[3,3,3,2,1]
 => [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)
 => 6 = 5 + 1
Description
The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.