searching the database
Your data matches 362 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: St001029
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 1
([(0,1)],2)
=> 2
([],3)
=> 1
([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> 1
([(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> 1
([(3,4)],5)
=> 2
([(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
Description
The size of the core of a graph.
The core of the graph $G$ is the smallest graph $C$ such that there is a graph homomorphism from $G$ to $C$ and a graph homomorphism from $C$ to $G$.
Matching statistic: St000741
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([],1)
=> 0 = 1 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([],3)
=> ([],1)
=> 0 = 1 - 1
([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 - 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([],4)
=> ([],1)
=> 0 = 1 - 1
([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 - 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(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 = 4 - 1
([],5)
=> ([],1)
=> 0 = 1 - 1
([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 - 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ? = 5 - 1
([(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)
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(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)
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 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),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(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)
=> 4 = 5 - 1
([],6)
=> ([],1)
=> 0 = 1 - 1
([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2 - 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2 - 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2 - 1
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 2 - 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ? = 2 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ? = 5 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 2 - 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 5 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ? = 5 - 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ? = 2 - 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)
=> ? = 3 - 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 6 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ? = 3 - 1
([(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)
=> ? = 3 - 1
([(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)
=> ? = 3 - 1
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
([(5,6)],7)
=> ([(1,2)],3)
=> ? = 2 - 1
([(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ? = 2 - 1
([(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ? = 2 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ? = 2 - 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ? = 2 - 1
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> ? = 2 - 1
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> ? = 2 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> ? = 2 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> ? = 2 - 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St001291
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 71%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 71%
Values
([],1)
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
([],2)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
([],3)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
([],4)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
([],5)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 4 = 3 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 4 = 3 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 3 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5 = 4 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5 = 4 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 4 = 3 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 5 = 4 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 5 + 1
([],6)
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
([(3,5),(4,5)],6)
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 3 = 2 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 5 + 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 5 + 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 3 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 3 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 3 + 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 1
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 3 + 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 + 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 3 + 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 3 + 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 3 + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St001227
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 71%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 71%
Values
([],1)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
([],2)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
([],3)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
([],4)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,1),(0,2),(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]
=> 4
([],5)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
([(0,1),(0,2),(0,3),(0,4),(1,2),(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]
=> 5
([],6)
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
([(3,5),(4,5)],6)
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 5
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 5
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> ? = 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> ? = 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 5
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> ? = 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> ? = 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 3
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> ? = 3
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> ? = 3
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> ? = 3
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 3
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 3
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 3
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001480
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 71%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 71%
Values
([],1)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
([],2)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
([],3)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
([],4)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([],5)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
([],6)
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2
([(3,5),(4,5)],6)
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 5
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 3
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 3
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 3
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3
Description
The number of simple summands of the module J^2/J^3. Here J is the Jacobson radical of the Nakayama algebra algebra corresponding to the Dyck path.
Matching statistic: St000722
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(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)
=> 4
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,2),(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(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)
=> 4
([(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)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(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
([],7)
=> ?
=> ? = 1
([(5,6)],7)
=> ?
=> ? = 2
([(4,6),(5,6)],7)
=> ?
=> ? = 2
([(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(3,6),(4,5)],7)
=> ?
=> ? = 2
([(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(2,3),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? = 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? = 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
Description
The number of different neighbourhoods in a graph.
Matching statistic: St001110
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(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)
=> 4
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,2),(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(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)
=> 4
([(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)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(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
([],7)
=> ?
=> ? = 1
([(5,6)],7)
=> ?
=> ? = 2
([(4,6),(5,6)],7)
=> ?
=> ? = 2
([(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(3,6),(4,5)],7)
=> ?
=> ? = 2
([(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(2,3),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? = 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? = 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
Description
The 3-dynamic chromatic number of a graph.
A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring.
This statistic records the $3$-dynamic chromatic number of a graph.
Matching statistic: St001302
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(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)
=> 4
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,2),(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(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)
=> 4
([(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)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(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
([],7)
=> ?
=> ? = 1
([(5,6)],7)
=> ?
=> ? = 2
([(4,6),(5,6)],7)
=> ?
=> ? = 2
([(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(3,6),(4,5)],7)
=> ?
=> ? = 2
([(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(2,3),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? = 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? = 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
Description
The number of minimally dominating sets of vertices of a graph.
A subset of vertices is '''dominating''' if every vertex is either in this subset or adjacent to an element therein [1]. If a set of vertices is dominating, then so is every superset of this set. This statistic counts the minimally dominating sets.
Matching statistic: St001304
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(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)
=> 4
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,2),(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(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)
=> 4
([(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)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(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
([],7)
=> ?
=> ? = 1
([(5,6)],7)
=> ?
=> ? = 2
([(4,6),(5,6)],7)
=> ?
=> ? = 2
([(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(3,6),(4,5)],7)
=> ?
=> ? = 2
([(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(2,3),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? = 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? = 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
Description
The number of maximally independent sets of vertices of a graph.
An '''independent set''' of vertices of a graph is a set of vertices no two of which are adjacent. If a set of vertices is independent then so is every subset. This statistic counts the number of maximally independent sets of vertices.
Matching statistic: St001645
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(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)
=> 4
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,2),(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(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)
=> 4
([(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)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(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
([],7)
=> ?
=> ? = 1
([(5,6)],7)
=> ?
=> ? = 2
([(4,6),(5,6)],7)
=> ?
=> ? = 2
([(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(3,6),(4,5)],7)
=> ?
=> ? = 2
([(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(2,3),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? = 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? = 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
Description
The pebbling number of a connected graph.
The following 352 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000087The number of induced subgraphs. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000171The degree of the graph. St000172The Grundy number of a graph. St000184The size of the centralizer of any permutation of given cycle type. St000189The number of elements in the poset. St000228The size of a partition. St000286The number of connected components of the complement of a graph. St000309The number of vertices with even degree. St000363The number of minimal vertex covers of a graph. St000384The maximal part of the shifted composition of an integer partition. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000527The width of the poset. St000531The leading coefficient of the rook polynomial of an integer partition. St000636The hull number of a graph. St000667The greatest common divisor of the parts of the partition. St000784The maximum of the length and the largest part of the integer partition. St000822The Hadwiger number of the graph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000908The length of the shortest maximal antichain in a poset. St000909The number of maximal chains of maximal size in a poset. St000926The clique-coclique number of a graph. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001268The size of the largest ordinal summand in the poset. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001360The number of covering relations in Young's lattice below a partition. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. St001399The distinguishing number of a poset. St001494The Alon-Tarsi number of a graph. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001670The connected partition number of a graph. St001779The order of promotion on the set of linear extensions of a poset. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000145The Dyson rank of a partition. St000180The number of chains of a poset. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000362The size of a minimal vertex cover of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000532The total number of rook placements on a Ferrers board. St000536The pathwidth of a graph. St000632The jump number of the poset. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000778The metric dimension of a graph. St001119The length of a shortest maximal path in a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001391The disjunction number of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001644The dimension of a graph. St001664The number of non-isomorphic subposets of a poset. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001814The number of partitions interlacing the given partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000144The pyramid weight of the Dyck path. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000258The burning number of a graph. St000271The chromatic index of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000293The number of inversions of a binary word. St000315The number of isolated vertices of a graph. St000395The sum of the heights of the peaks of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000444The length of the maximal rise of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000475The number of parts equal to 1 in a partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000507The number of ascents of a standard tableau. St000519The largest length of a factor maximising the subword complexity. St000544The cop number of a graph. St000548The number of different non-empty partial sums of an integer partition. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000734The last entry in the first row of a standard tableau. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000867The sum of the hook lengths in the first row of an integer partition. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000922The minimal number such that all substrings of this length are unique. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000947The major index east count of a Dyck path. St000982The length of the longest constant subword. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001118The acyclic chromatic index of a graph. St001127The sum of the squares of the parts of a partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. 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:
St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001363The Euler characteristic of a graph according to Knill. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001441The number of non-empty connected induced subgraphs of a graph. St001462The number of factors of a standard tableaux under concatenation. St001463The number of distinct columns in the nullspace of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001523The degree of symmetry of a Dyck path. St001530The depth of a Dyck path. St001614The cyclic permutation representation number of a skew partition. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001672The restrained domination number of a graph. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001691The number of kings in a graph. St001733The number of weak left to right maxima of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001933The largest multiplicity of a part in an integer partition. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000377The dinv defect of an integer partition. St000393The number of strictly increasing runs in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000420The number of Dyck paths that are weakly above a Dyck path. St000439The position of the first down step of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000921The number of internal inversions of a binary word. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001117The game chromatic index of a graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule 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. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001267The length of the Lyndon factorization of the binary word. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St001437The flex of a binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001808The box weight or horizontal decoration of a Dyck path. St001955The number of natural descents for set-valued two row standard Young tableaux. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St000181The number of connected components of the Hasse diagram for the poset. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001812The biclique partition number of a graph. St001706The number of closed sets in a graph. St000005The bounce statistic of a Dyck path. St000120The number of left tunnels of a Dyck path. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001959The product of the heights of the peaks of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001834The number of non-isomorphic minors of a graph. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000477The weight of a partition according to Alladi. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000656The number of cuts of a poset. St000674The number of hills of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000993The multiplicity of the largest part of an integer partition. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St000806The semiperimeter of the associated bargraph. St000932The number of occurrences of the pattern UDU in a Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000299The number of nonisomorphic vertex-induced subtrees. St001093The detour number of a graph. St001512The minimum rank of a graph. St001651The Frankl number of a lattice. St001570The minimal number of edges to add to make a graph Hamiltonian. St001592The maximal number of simple paths between any two different vertices of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000474Dyson's crank of a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001890The maximum magnitude of the Möbius function of a poset.
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!