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Your data matches 427 different statistics following compositions of up to 3 maps.
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Matching statistic: St001814
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St001814: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001814: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 1
([],2)
=> []
=> 1
([(0,1)],2)
=> [1]
=> 2
([],3)
=> []
=> 1
([(1,2)],3)
=> [1]
=> 2
([(0,2),(1,2)],3)
=> [2]
=> 3
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4
([],4)
=> []
=> 1
([(2,3)],4)
=> [1]
=> 2
([(1,3),(2,3)],4)
=> [2]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 4
([(0,3),(1,2)],4)
=> [1,1]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4
([],5)
=> []
=> 1
([(3,4)],5)
=> [1]
=> 2
([(2,4),(3,4)],5)
=> [2]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 5
([(1,4),(2,3)],5)
=> [1,1]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4
([],6)
=> []
=> 1
([(4,5)],6)
=> [1]
=> 2
([(3,5),(4,5)],6)
=> [2]
=> 3
([(2,5),(3,5),(4,5)],6)
=> [3]
=> 4
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> 5
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> 6
([(2,5),(3,4)],6)
=> [1,1]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 4
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> 2
([],7)
=> []
=> 1
([(5,6)],7)
=> [1]
=> 2
([(4,6),(5,6)],7)
=> [2]
=> 3
([(3,6),(4,6),(5,6)],7)
=> [3]
=> 4
([(2,6),(3,6),(4,6),(5,6)],7)
=> [4]
=> 5
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [5]
=> 6
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [6]
=> 7
([(3,6),(4,5)],7)
=> [1,1]
=> 2
([(4,5),(4,6),(5,6)],7)
=> [3]
=> 4
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> 2
Description
The number of partitions interlacing the given partition.
Matching statistic: St000097
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],0)
=> 0 = 1 - 1
([],2)
=> ([],0)
=> 0 = 1 - 1
([(0,1)],2)
=> ([],1)
=> 1 = 2 - 1
([],3)
=> ([],0)
=> 0 = 1 - 1
([(1,2)],3)
=> ([],1)
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([],4)
=> ([],0)
=> 0 = 1 - 1
([(2,3)],4)
=> ([],1)
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> ([],2)
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([],5)
=> ([],0)
=> 0 = 1 - 1
([(3,4)],5)
=> ([],1)
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(1,4),(2,3)],5)
=> ([],2)
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([],6)
=> ([],0)
=> 0 = 1 - 1
([(4,5)],6)
=> ([],1)
=> 1 = 2 - 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(2,5),(3,4)],6)
=> ([],2)
=> 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> 1 = 2 - 1
([],7)
=> ([],0)
=> 0 = 1 - 1
([(5,6)],7)
=> ([],1)
=> 1 = 2 - 1
([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
([(3,6),(4,5)],7)
=> ([],2)
=> 1 = 2 - 1
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(1,6),(2,5),(3,4)],7)
=> ([],3)
=> 1 = 2 - 1
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],0)
=> 0 = 1 - 1
([],2)
=> ([],0)
=> 0 = 1 - 1
([(0,1)],2)
=> ([],1)
=> 1 = 2 - 1
([],3)
=> ([],0)
=> 0 = 1 - 1
([(1,2)],3)
=> ([],1)
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([],4)
=> ([],0)
=> 0 = 1 - 1
([(2,3)],4)
=> ([],1)
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> ([],2)
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([],5)
=> ([],0)
=> 0 = 1 - 1
([(3,4)],5)
=> ([],1)
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(1,4),(2,3)],5)
=> ([],2)
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([],6)
=> ([],0)
=> 0 = 1 - 1
([(4,5)],6)
=> ([],1)
=> 1 = 2 - 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(2,5),(3,4)],6)
=> ([],2)
=> 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> 1 = 2 - 1
([],7)
=> ([],0)
=> 0 = 1 - 1
([(5,6)],7)
=> ([],1)
=> 1 = 2 - 1
([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
([(3,6),(4,5)],7)
=> ([],2)
=> 1 = 2 - 1
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(1,6),(2,5),(3,4)],7)
=> ([],3)
=> 1 = 2 - 1
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000147
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 0 = 1 - 1
([],2)
=> []
=> 0 = 1 - 1
([(0,1)],2)
=> [1]
=> 1 = 2 - 1
([],3)
=> []
=> 0 = 1 - 1
([(1,2)],3)
=> [1]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [2]
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3 = 4 - 1
([],4)
=> []
=> 0 = 1 - 1
([(2,3)],4)
=> [1]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3 = 4 - 1
([],5)
=> []
=> 0 = 1 - 1
([(3,4)],5)
=> [1]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [2]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 4 = 5 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3 = 4 - 1
([],6)
=> []
=> 0 = 1 - 1
([(4,5)],6)
=> [1]
=> 1 = 2 - 1
([(3,5),(4,5)],6)
=> [2]
=> 2 = 3 - 1
([(2,5),(3,5),(4,5)],6)
=> [3]
=> 3 = 4 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> 4 = 5 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> 5 = 6 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 3 = 4 - 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> 1 = 2 - 1
([],7)
=> []
=> 0 = 1 - 1
([(5,6)],7)
=> [1]
=> 1 = 2 - 1
([(4,6),(5,6)],7)
=> [2]
=> 2 = 3 - 1
([(3,6),(4,6),(5,6)],7)
=> [3]
=> 3 = 4 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> [4]
=> 4 = 5 - 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [5]
=> 5 = 6 - 1
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [6]
=> 6 = 7 - 1
([(3,6),(4,5)],7)
=> [1,1]
=> 1 = 2 - 1
([(4,5),(4,6),(5,6)],7)
=> [3]
=> 3 = 4 - 1
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> 1 = 2 - 1
Description
The largest part of an integer partition.
Matching statistic: St000172
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],0)
=> ([],1)
=> 1
([],2)
=> ([],0)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],0)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],4)
=> ([],0)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> ([],0)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],6)
=> ([],0)
=> ([],1)
=> 1
([(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([],7)
=> ([],0)
=> ([],1)
=> 1
([(5,6)],7)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
([(3,6),(4,5)],7)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(1,6),(2,5),(3,4)],7)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
Description
The Grundy number of a graph.
The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring.
In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).
Matching statistic: St000300
Values
([],1)
=> ([],0)
=> ([],0)
=> 1
([],2)
=> ([],0)
=> ([],0)
=> 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 2
([],3)
=> ([],0)
=> ([],0)
=> 1
([(1,2)],3)
=> ([],1)
=> ([],1)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([],4)
=> ([],0)
=> ([],0)
=> 1
([(2,3)],4)
=> ([],1)
=> ([],1)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([],5)
=> ([],0)
=> ([],0)
=> 1
([(3,4)],5)
=> ([],1)
=> ([],1)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
([(1,4),(2,3)],5)
=> ([],2)
=> ([],1)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([],6)
=> ([],0)
=> ([],0)
=> 1
([(4,5)],6)
=> ([],1)
=> ([],1)
=> 2
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 3
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
([(2,5),(3,4)],6)
=> ([],2)
=> ([],1)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> 2
([],7)
=> ([],0)
=> ([],0)
=> 1
([(5,6)],7)
=> ([],1)
=> ([],1)
=> 2
([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 3
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
([(3,6),(4,5)],7)
=> ([],2)
=> ([],1)
=> 2
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(1,6),(2,5),(3,4)],7)
=> ([],3)
=> ([],1)
=> 2
Description
The number of independent sets of vertices of a graph.
An independent set of vertices of a graph $G$ is a subset $U \subset V(G)$ such that no two vertices in $U$ are adjacent.
This is also the number of vertex covers of $G$ as the map $U \mapsto V(G)\setminus U$ is a bijection between independent sets of vertices and vertex covers.
The size of the largest independent set, also called independence number of $G$, is [[St000093]]
Matching statistic: St000722
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],0)
=> ([],1)
=> 1
([],2)
=> ([],0)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],0)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],4)
=> ([],0)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> ([],0)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],6)
=> ([],0)
=> ([],1)
=> 1
([(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([],7)
=> ([],0)
=> ([],1)
=> 1
([(5,6)],7)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
([(3,6),(4,5)],7)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(1,6),(2,5),(3,4)],7)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
Description
The number of different neighbourhoods in a graph.
Matching statistic: St001029
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],0)
=> ([],1)
=> 1
([],2)
=> ([],0)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],0)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],4)
=> ([],0)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> ([],0)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],6)
=> ([],0)
=> ([],1)
=> 1
([(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([],7)
=> ([],0)
=> ([],1)
=> 1
([(5,6)],7)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
([(3,6),(4,5)],7)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(1,6),(2,5),(3,4)],7)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
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: St001116
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],0)
=> ([],1)
=> 1
([],2)
=> ([],0)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],0)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],4)
=> ([],0)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> ([],0)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],6)
=> ([],0)
=> ([],1)
=> 1
([(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([],7)
=> ([],0)
=> ([],1)
=> 1
([(5,6)],7)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
([(3,6),(4,5)],7)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(1,6),(2,5),(3,4)],7)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
Description
The game chromatic number of a graph.
Two players, Alice and Bob, take turns colouring properly any uncolored vertex of the graph. Alice begins. If it is not possible for either player to colour a vertex, then Bob wins. If the graph is completely colored, Alice wins.
The game chromatic number is the smallest number of colours such that Alice has a winning strategy.
Matching statistic: St001330
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],1)
=> ([],0)
=> ([],1)
=> 1
([],2)
=> ([],0)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],0)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],4)
=> ([],0)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> ([],0)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],6)
=> ([],0)
=> ([],1)
=> 1
([(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([],7)
=> ([],0)
=> ([],1)
=> 1
([(5,6)],7)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
([(3,6),(4,5)],7)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(1,6),(2,5),(3,4)],7)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
The following 417 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000160The multiplicity of the smallest part of a partition. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000507The number of ascents of a standard tableau. St000536The pathwidth of a graph. St000548The number of different non-empty partial sums of an integer partition. St000676The number of odd rises of a Dyck path. St000013The height of a Dyck path. St000479The Ramsey number of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000286The number of connected components of the complement of a graph. St000822The Hadwiger number of the graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001963The tree-depth of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001812The biclique partition number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001118The acyclic chromatic index of a graph. St000363The number of minimal vertex covers of a graph. St000667The greatest common divisor of the parts of the partition. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001725The harmonious chromatic number of a graph. St001883The mutual visibility number of a graph. St000171The degree of the graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001962The proper pathwidth of a graph. St000301The number of facets of the stable set polytope of a graph. St000439The position of the first down step of a Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000026The position of the first return of a Dyck path. St000087The number of induced subgraphs. St000273The domination number of a graph. St000287The number of connected components of a graph. St000469The distinguishing number of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000544The cop number of a graph. St000553The number of blocks of a graph. St000636The hull number of a graph. St000685The dominant dimension of the LNakayama algebra associated to 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. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000775The multiplicity of the largest eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000916The packing number of a graph. St000926The clique-coclique number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000947The major index east count of a Dyck path. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001161The major index north count of a 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001286The annihilation number of a graph. St001312Number of parabolic noncrossing partitions indexed by the composition. 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. St001342The number of vertices in the center of a graph. St001363The Euler characteristic of a graph according to Knill. 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. 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. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001675The number of parts equal to the part in the reversed composition. 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. St001746The coalition number of a graph. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001933The largest multiplicity of a part in an integer partition. St000008The major index of the composition. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000877The depth of the binary word interpreted as a path. St001176The size of a partition minus its first part. St001391The disjunction number of a graph. St001644The dimension of a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001777The number of weak descents in an integer composition. St001949The rigidity index of a graph. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000326The position of the first one in a binary word after appending a 1 at the end. St000420The number of Dyck paths that are weakly above a Dyck path. 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. St001371The length of the longest Yamanouchi prefix of a binary word. 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. St001808The box weight or horizontal decoration of a Dyck path. St000012The area of a Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000258The burning number of a graph. St000288The number of ones in a binary word. St000290The major index of a binary word. St000297The number of leading ones in a binary word. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000335The difference of lower and upper interactions. St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000443The number of long tunnels of a Dyck path. St000482The (zero)-forcing number of a graph. St000532The total number of rook placements on a Ferrers board. St000655The length of the minimal rise of a Dyck path. St000657The smallest part of an integer composition. St000678The number of up steps after the last double rise of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000984The number of boxes below precisely one peak. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. 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. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001313The number of Dyck paths above the lattice path given by a binary word. 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. St001372The length of a longest cyclic run of ones of a binary word. St001400The total number of Littlewood-Richardson tableaux of given shape. St001415The length of the longest palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001441The number of non-empty connected induced subgraphs of a graph. St001481The minimal height of a peak of a Dyck path. St001485The modular major index of a binary word. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001672The restrained domination number of a graph. 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. St001828The Euler characteristic of a graph. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001955The number of natural descents for set-valued two row standard Young tableaux. St000148The number of odd parts of a partition. St000228The size of a partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000384The maximal part of the shifted composition of an integer partition. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000674The number of hills of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000784The maximum of the length and the largest part of the integer partition. St000867The sum of the hook lengths in the first row of an integer partition. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001127The sum of the squares of the parts of a partition. St001172The number of 1-rises at odd height of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001340The cardinality of a minimal non-edge isolating set of a graph. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001584The area statistic between a Dyck path and its bounce path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000005The bounce statistic of a Dyck path. St000120The number of left tunnels of a Dyck path. St000144The pyramid weight of the Dyck path. St001834The number of non-isomorphic minors of a graph. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001273The projective dimension of the first term in an injective coresolution of the regular module. St001480The number of simple summands of the module J^2/J^3. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000967The value p(1) for the Coxeterpolynomial p 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. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001117The game chromatic index of a graph. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000271The chromatic index of a graph. 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. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000444The length of the maximal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. 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. St001360The number of covering relations in Young's lattice below a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001570The minimal number of edges to add to make a graph Hamiltonian. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000478Another weight of a partition according to Alladi. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001060The distinguishing index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001651The Frankl number of a lattice. St000699The toughness times the least common multiple of 1,. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St000422The energy of a graph, if it is integral. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001568The smallest positive integer that does not appear twice in the partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000929The constant term of the character polynomial of an integer partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000456The monochromatic index of a connected graph. St001545The second Elser number of a connected graph. St000464The Schultz index of a connected graph. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001943The sum of the squares of the hook lengths of an integer partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001383The BG-rank of an integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000145The Dyson rank of a partition. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000474Dyson's crank of a partition. St000997The even-odd crank of an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001845The number of join irreducibles minus the rank of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000379The number of Hamiltonian cycles in a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001626The number of maximal proper sublattices of a lattice.
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