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Your data matches 328 different statistics following compositions of up to 3 maps.
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Matching statistic: St000097
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Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> 1
([(0,1)],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> 1
([(1,2)],3)
=> ([],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 1
([],4)
=> ([],4)
=> 1
([(2,3)],4)
=> ([],3)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1
([],5)
=> ([],5)
=> 1
([(3,4)],5)
=> ([],4)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 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: St000256
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([],2)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,1)],2)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([],3)
=> ([],3)
=> [1,1,1]
=> 0 = 1 - 1
([(1,2)],3)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> 0 = 1 - 1
([(2,3)],4)
=> ([],3)
=> [1,1,1]
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0 = 1 - 1
([(3,4)],5)
=> ([],4)
=> [1,1,1,1]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> ([],3)
=> [1,1,1]
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> [7]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St000480
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([],2)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,1)],2)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([],3)
=> ([],3)
=> [1,1,1]
=> 0 = 1 - 1
([(1,2)],3)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> 0 = 1 - 1
([(2,3)],4)
=> ([],3)
=> [1,1,1]
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0 = 1 - 1
([(3,4)],5)
=> ([],4)
=> [1,1,1,1]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> ([],3)
=> [1,1,1]
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> [7]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> 0 = 1 - 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000257
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([],2)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,1)],2)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([],3)
=> ([],3)
=> [1,1,1]
=> [3]
=> 0 = 1 - 1
([(1,2)],3)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> [4]
=> 0 = 1 - 1
([(2,3)],4)
=> ([],3)
=> [1,1,1]
=> [3]
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [5]
=> 0 = 1 - 1
([(3,4)],5)
=> ([],4)
=> [1,1,1,1]
=> [4]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> ([],3)
=> [1,1,1]
=> [3]
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> [3]
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St000481
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([],2)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,1)],2)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([],3)
=> ([],3)
=> [1,1,1]
=> [3]
=> 0 = 1 - 1
([(1,2)],3)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> [4]
=> 0 = 1 - 1
([(2,3)],4)
=> ([],3)
=> [1,1,1]
=> [3]
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [5]
=> 0 = 1 - 1
([(3,4)],5)
=> ([],4)
=> [1,1,1,1]
=> [4]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> ([],3)
=> [1,1,1]
=> [3]
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> [3]
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St001330
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
([],4)
=> ([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([],5)
=> ([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? ∊ {2,2}
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? ∊ {2,2}
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.
Matching statistic: St000098
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
([],4)
=> ([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([],5)
=> ([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? ∊ {2,2,2}
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? ∊ {2,2,2}
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? ∊ {2,2,2}
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: St001029
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
([],4)
=> ([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([],5)
=> ([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? ∊ {2,2,2}
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? ∊ {2,2,2}
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? ∊ {2,2,2}
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: St001494
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
([],4)
=> ([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([],5)
=> ([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? ∊ {2,2,2}
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? ∊ {2,2,2}
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? ∊ {2,2,2}
Description
The Alon-Tarsi number of a graph.
Let $G$ be a graph with vertices $\{1,\dots,n\}$ and edge set $E$. Let $P_G=\prod_{i < j, (i,j)\in E} x_i-x_j$ be its graph polynomial. Then the Alon-Tarsi number is the smallest number $k$ such that $P_G$ contains a monomial with exponents strictly less than $k$.
Matching statistic: St001580
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
([],4)
=> ([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([],5)
=> ([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? ∊ {2,2,2}
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? ∊ {2,2,2}
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? ∊ {2,2,2}
Description
The acyclic chromatic number of a graph.
This is the smallest size of a vertex partition $\{V_1,\dots,V_k\}$ such that each $V_i$ is an independent set and for all $i,j$ the subgraph inducted by $V_i\cup V_j$ does not contain a cycle.
The following 318 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000272The treewidth of a graph. St000535The rank-width of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001826The maximal number of leaves on a vertex of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001716The 1-improper chromatic number of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000767The number of runs in an integer composition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000820The number of compositions obtained by rotating the composition. St000903The number of different parts of an integer composition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St001673The degree of asymmetry of an integer composition. St001777The number of weak descents in an integer composition. St001931The weak major index of an integer composition regarded as a word. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St000822The Hadwiger number of the graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001349The number of different graphs obtained from the given graph by removing an edge. St000086The number of subgraphs. St000087The number of induced subgraphs. St000172The Grundy number of a graph. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000343The number of spanning subgraphs of a graph. St000363The number of minimal vertex covers of a graph. St000364The exponent of the automorphism group of a graph. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000469The distinguishing number of a graph. St000479The Ramsey number of a graph. St000536The pathwidth of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000926The clique-coclique number of a graph. St000972The composition number of a graph. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001316The domatic 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. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001581The achromatic number of a graph. 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. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. 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. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a graph. St001802The number of endomorphisms of a graph. 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. St000081The number of edges of a graph. St000171The degree of the graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000274The number of perfect matchings 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. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000537The cutwidth of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001071The beta invariant of the graph. St001117The game chromatic index of a 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. St001271The competition number of a graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001340The cardinality of a minimal non-edge isolating set of a graph. St001341The number of edges in the center of a graph. St001345The Hamming dimension of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001362The normalized Knill dimension of a graph. St001391The disjunction number of a graph. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001644The dimension of a graph. St001649The length of a longest trail in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001783The number of odd automorphisms of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001812The biclique partition number of a graph. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St001973The Gromov width of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000482The (zero)-forcing number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a 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. St000916The packing number of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001441The number of non-empty connected induced subgraphs of a graph. St001463The number of distinct columns in the nullspace of a graph. St001642The Prague dimension of a graph. St001691The number of kings in a graph. St001765The number of connected components of the friends and strangers graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001799The number of proper separations of a graph. St001834The number of non-isomorphic minors of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000758The length of the longest staircase fitting into an integer composition. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001568The smallest positive integer that does not appear twice in the partition. St001733The number of weak left to right maxima of a Dyck path. St001732The number of peaks visible from the left. St001432The order dimension of 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. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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. St000335The difference of lower and upper interactions. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001024Maximum of dominant dimensions of the 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. 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001481The minimal height of a peak of a Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. 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 $c_{n−1}$. St001592The maximal number of simple paths between any two different vertices of a graph. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000456The monochromatic index of a connected graph. St000464The Schultz index of a connected graph. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. 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. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001845The number of join irreducibles minus the rank of a lattice. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000053The number of valleys of the Dyck path. St000159The number of distinct parts of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000331The number of upper interactions of a Dyck path. St000345The number of refinements of a partition. St000390The number of runs of ones in a binary word. St000517The Kreweras number of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000627The exponent of a binary word. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000759The smallest missing part in an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000983The length of the longest alternating subword. 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. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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 $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. 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. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001471The magnitude of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001597The Frobenius rank of a skew partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St001955The number of natural descents for set-valued two row standard Young tableaux. 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. St001128The exponens consonantiae of a partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001933The largest multiplicity of a part in an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001624The breadth of a lattice. St001570The minimal number of edges to add to make a graph Hamiltonian. 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 $eA$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001626The number of maximal proper sublattices of a lattice. St001060The distinguishing index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001877Number of indecomposable injective modules with projective dimension 2. St000455The second largest eigenvalue of a graph if it is integral. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. 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. St000667The greatest common divisor of the parts of the partition. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001924The number of cells in an integer partition whose arm and leg length coincide.
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