Your data matches 17 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001473
(load all 2 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001473: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 4
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 4
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 5
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 4
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 5
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 4
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 5
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 5
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 5
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 6
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 7
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 7
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 7
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 7
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 5
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 5
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 7
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 5
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 6
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 4
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 4
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 5
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 6
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 5
Description
The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra.
Matching statistic: St001872
(load all 3 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001872: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 4
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 4
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 5
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 4
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 4
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 5
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 5
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 5
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 5
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 6
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 7
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 7
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 7
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 7
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 7
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 4
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 5
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 5
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 5
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 5
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 5
Description
The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St000967
Mapping
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000967: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Values
[1,0]
 => [[1],[]]
 => []
 => []
 => ? = 1
[1,0,1,0]
 => [[1,1],[]]
 => []
 => []
 => ? ∊ {1,4}
[1,1,0,0]
 => [[2],[]]
 => []
 => []
 => ? ∊ {1,4}
[1,0,1,0,1,0]
 => [[1,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,5}
[1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,5}
[1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,0,1,0,0]
 => [[3],[]]
 => []
 => []
 => ? ∊ {1,4,4,5}
[1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,5}
[1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[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]
 => 6
[1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[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]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[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]
 => 6
[1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 6
[1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[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]
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,1,0,1,0,0,0,0]
 => [[4,4],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
Description
The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra.
Matching statistic: St001213
Mapping
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001213: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Values
[1,0]
 => [[1],[]]
 => []
 => []
 => ? = 1
[1,0,1,0]
 => [[1,1],[]]
 => []
 => []
 => ? ∊ {1,4}
[1,1,0,0]
 => [[2],[]]
 => []
 => []
 => ? ∊ {1,4}
[1,0,1,0,1,0]
 => [[1,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,5}
[1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,5}
[1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,0,1,0,0]
 => [[3],[]]
 => []
 => []
 => ? ∊ {1,4,4,5}
[1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,5}
[1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
[1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
[1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
[1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
[1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
[1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
[1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
[1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
[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]
 => 6
[1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 6
[1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[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]
 => 6
[1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 6
[1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
[1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
[1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
[1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
[1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
[1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
[1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,1,0,1,0,0,0,0]
 => [[4,4],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
[1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,7,7,7,7,7,7}
Description
The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module.
Matching statistic: St001218
Mapping
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001218: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Values
[1,0]
 => [[1],[]]
 => []
 => []
 => ? = 1
[1,0,1,0]
 => [[1,1],[]]
 => []
 => []
 => ? ∊ {1,4}
[1,1,0,0]
 => [[2],[]]
 => []
 => []
 => ? ∊ {1,4}
[1,0,1,0,1,0]
 => [[1,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,5}
[1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,5}
[1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,0,1,0,0]
 => [[3],[]]
 => []
 => []
 => ? ∊ {1,4,4,5}
[1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,5}
[1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,5,5,6}
[1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[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]
 => 6
[1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[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]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[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]
 => 6
[1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 6
[1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[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]
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 5
[1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 5
[1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,1,0,1,0,0,0,0]
 => [[4,4],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
[1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => []
 => []
 => ? ∊ {1,4,4,4,4,4,4,5,6,6,7,7,7,7,7,7}
Description
Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. It returns zero in case there is no such k.
Matching statistic: St001239
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 80%
Values
[1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 4
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 4
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 4
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 5
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 4
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 5
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 5
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 4
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 5
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
Matching statistic: St001192
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 80%
Values
[1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 5 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4 = 5 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 6 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Matching statistic: St001194
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001194: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 80%
Values
[1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 5 = 6 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7} - 1
Description
The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module
Matching statistic: St000264
(load all 7 compositions to match this statistic)
Mapping
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 40%
Values
[1,0]
 => ([],1)
 => ([],1)
 => ? = 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,4}
[1,1,0,0]
 => ([],2)
 => ([],2)
 => ? ∊ {1,4}
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {1,4,4,4,5}
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {1,4,4,4,5}
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {1,4,4,4,5}
[1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {1,4,4,4,5}
[1,1,1,0,0,0]
 => ([],3)
 => ([],3)
 => ? ∊ {1,4,4,4,5}
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,0,0,0,0]
 => ([],4)
 => ([],4)
 => ? ∊ {1,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[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)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,1,1,0,0,1,1,0,0,0]
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,0,1,1,0,0,0,0]
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,1,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,1,0,0,1,0,0,0]
 => ([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,1,0,1,0,0,0,0]
 => ([(3,4)],5)
 => ([(3,4)],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
[1,1,1,1,1,0,0,0,0,0]
 => ([],5)
 => ([],5)
 => ? ∊ {1,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7}
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Matching statistic: St000422
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 60%
Values
[1,0]
 => [1,1,0,0]
 => [1,2] => ([],2)
 => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ? = 4 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4 = 5 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4} - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4} - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {4,4,4} - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {4,4,4,4,4,4,4,4,5,6} - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {4,4,4,4,4,4,4,4,5,6} - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,4,4,4,4,5,6} - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,4,4,4,4,5,6} - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,4,4,4,4,5,6} - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,4,4,4,4,5,6} - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {4,4,4,4,4,4,4,4,5,6} - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,4,4,4,4,5,6} - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,4,4,4,4,5,6} - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4,4,4,4,4,5,6} - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([],5)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [4,1,5,2,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => 6 = 7 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [4,1,2,3,5,6] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => 4 = 5 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,5,2,3,4,6] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [5,1,2,6,3,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4 = 5 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4 = 5 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7} - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([],6)
 => 0 = 1 - 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.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000635The number of strictly order preserving maps of a poset into itself. St000495The number of inversions of distance at most 2 of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001645The pebbling number of a connected graph. St000034The maximum defect over any reduced expression for a permutation and any subexpression.