searching the database
Your data matches 271 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001037
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001195
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001276
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
Description
The number of 2-regular indecomposable modules in the corresponding Nakayama algebra.
Generalising the notion of k-regular modules from simple to arbitrary indecomposable modules, we call an indecomposable module $M$ over an algebra $A$ k-regular in case it has projective dimension k and $Ext_A^i(M,A)=0$ for $i \neq k$ and $Ext_A^k(M,A)$ is 1-dimensional.
The number of Dyck paths where the statistic returns 0 might be given by [[OEIS:A035929]] .
Matching statistic: St000264
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 2
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 2
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ? = 0 + 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 3 = 1 + 2
([(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1 + 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1 + 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ? = 0 + 2
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ? = 0 + 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1 + 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 2
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001570
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ? = 0 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(3,6),(4,5),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 1
([(2,3),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 - 1
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
=> ([],3)
=> ([],1)
=> ? = 0 - 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 - 1
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 1
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ?
=> ? = 1 - 1
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St001060
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 + 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 1 + 2
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 + 2
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ? = 0 + 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 1 + 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(3,6),(4,5),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 2
([(2,3),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 + 2
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,6),(2,5),(3,4)],7)
=> ([],3)
=> ([],1)
=> ? = 0 + 2
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 2
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 2
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ?
=> ? = 1 + 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000260
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00157: Graphs —connected complement⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(2,3),(4,6),(5,6)],7)
=> ([(2,3),(4,6),(5,6)],7)
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 0
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> [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
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [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
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,6),(5,6)],7)
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> [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
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [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
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [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
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [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
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [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
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 0
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [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
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(4,6),(5,6)],7)
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 0
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(4,5),(4,6),(5,6)],7)
=> [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
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001704
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 0 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 0 + 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ? = 0 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 1 + 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 1 + 1
([(2,3),(4,6),(5,6)],7)
=> ([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 0 + 1
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 1 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ? = 0 + 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 0 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ?
=> ? = 1 + 1
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? = 1 + 1
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ?
=> ? = 1 + 1
Description
The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph.
The deck of a graph is the multiset of induced subgraphs obtained by deleting a single vertex.
The graph reconstruction conjecture states that the deck of a graph with at least three vertices determines the graph.
This statistic is only defined for graphs with at least two vertices, because there is only a single graph of the given size otherwise.
Matching statistic: St000455
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Values
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> -1 = 1 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 1 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 1 - 2
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 1 - 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 0 - 2
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> -1 = 1 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> -1 = 1 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ? = 1 - 2
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 1 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 - 2
([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 0 - 2
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 1 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> -1 = 1 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ? = 1 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ? = 0 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 1 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> -1 = 1 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ? = 1 - 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> -1 = 1 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 1 - 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],2)
=> ([],1)
=> ? = 1 - 2
([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 1 - 2
([(2,3),(4,6),(5,6)],7)
=> ([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 0 - 2
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],2)
=> ([],1)
=> ? = 1 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 1 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([],2)
=> ([],1)
=> ? = 1 - 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ? = 0 - 2
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 0 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> -1 = 1 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ?
=> ? = 1 - 2
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> -1 = 1 - 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([],2)
=> ([],1)
=> ? = 1 - 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? = 1 - 2
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ?
=> ? = 1 - 2
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000781
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 52%●distinct values known / distinct values provided: 50%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 52%●distinct values known / distinct values provided: 50%
Values
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [3]
=> []
=> ? = 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [3]
=> []
=> ? = 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? = 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? = 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [2,2,2]
=> [2,2]
=> 1
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(3,6),(4,5),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 1
([(2,3),(4,6),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 0
([(4,5),(4,6),(5,6)],7)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 0
([(2,6),(3,5),(4,5),(4,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 0
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3]
=> [2,2,2]
=> [2,2]
=> 1
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [3,3,1]
=> [3,2,2]
=> [2,2]
=> 1
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> [4,3]
=> [2,2,2,1]
=> [2,2,1]
=> 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
The following 261 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001593This is the number of standard Young tableaux of the given shifted shape. St001118The acyclic chromatic index of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000454The largest eigenvalue of a graph if it is integral. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001651The Frankl number of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St000322The skewness of a graph. St000379The number of Hamiltonian cycles in a graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000997The even-odd crank of an integer partition. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001307The number of induced stars on four vertices in a graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000741The Colin de Verdière graph invariant. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001626The number of maximal proper sublattices of a lattice. St000785The number of distinct colouring schemes of a graph. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000449The number of pairs of vertices of a graph with distance 4. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000053The number of valleys of the Dyck path. St000079The number of alternating sign matrices for a given Dyck path. St000287The number of connected components of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000553The number of blocks of a graph. St000675The number of centered multitunnels of a Dyck path. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000916The packing number of a graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001352The number of internal nodes in the modular decomposition of a graph. St001463The number of distinct columns in the nullspace of a graph. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000095The number of triangles of a graph. St000323The minimal crossing number of a graph. St000370The genus of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000552The number of cut vertices of a graph. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. 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. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001305The number of induced cycles on four vertices in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001471The magnitude of a Dyck path. St001500The global dimension of magnitude 1 Nakayama algebras. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001573The minimal number of edges to remove to make a graph triangle-free. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001584The area statistic between a Dyck path and its bounce path. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001689The number of celebrities in a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001692The number of vertices with higher degree than the average degree in a graph. St001708The number of pairs of vertices of different degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001871The number of triconnected components of a graph. St001910The height of the middle non-run of a Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001644The dimension of a graph. St000069The number of maximal elements of a poset. St000456The monochromatic index of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001393The induced matching number of a graph. St001432The order dimension of the partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000315The number of isolated vertices of a graph. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001248Sum of the even parts of a partition. St001367The smallest number which does not occur as degree of a vertex in a graph. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001256Number of simple reflexive modules that are 2-stable reflexive. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St000464The Schultz index of a connected graph. St000879The number of long braid edges in the graph of braid moves of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001545The second Elser number of a connected graph. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St000331The number of upper interactions of a Dyck path. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001722The number of minimal chains with small intervals between a binary word and the top element. St001820The size of the image of the pop stack sorting operator. St000015The number of peaks of a Dyck path. St000117The number of centered tunnels of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001530The depth of a Dyck path. St001846The number of elements which do not have a complement in the lattice. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!