Your data matches 27 different statistics following compositions of up to 3 maps.
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Matching statistic: St000288
(load all 34 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 1 = 2 - 1
[2]
 => 0 => 0 = 1 - 1
[1,1]
 => 11 => 2 = 3 - 1
[3]
 => 1 => 1 = 2 - 1
[2,1]
 => 01 => 1 = 2 - 1
[1,1,1]
 => 111 => 3 = 4 - 1
[4]
 => 0 => 0 = 1 - 1
[3,1]
 => 11 => 2 = 3 - 1
[2,2]
 => 00 => 0 = 1 - 1
[2,1,1]
 => 011 => 2 = 3 - 1
[1,1,1,1]
 => 1111 => 4 = 5 - 1
[5]
 => 1 => 1 = 2 - 1
[4,1]
 => 01 => 1 = 2 - 1
[3,2]
 => 10 => 1 = 2 - 1
[3,1,1]
 => 111 => 3 = 4 - 1
[2,2,1]
 => 001 => 1 = 2 - 1
[2,1,1,1]
 => 0111 => 3 = 4 - 1
[1,1,1,1,1]
 => 11111 => 5 = 6 - 1
[6]
 => 0 => 0 = 1 - 1
[5,1]
 => 11 => 2 = 3 - 1
[4,2]
 => 00 => 0 = 1 - 1
[4,1,1]
 => 011 => 2 = 3 - 1
[3,3]
 => 11 => 2 = 3 - 1
[3,2,1]
 => 101 => 2 = 3 - 1
[3,1,1,1]
 => 1111 => 4 = 5 - 1
[2,2,2]
 => 000 => 0 = 1 - 1
[2,2,1,1]
 => 0011 => 2 = 3 - 1
[2,1,1,1,1]
 => 01111 => 4 = 5 - 1
[1,1,1,1,1,1]
 => 111111 => 6 = 7 - 1
[7]
 => 1 => 1 = 2 - 1
[6,1]
 => 01 => 1 = 2 - 1
[5,2]
 => 10 => 1 = 2 - 1
[5,1,1]
 => 111 => 3 = 4 - 1
[4,3]
 => 01 => 1 = 2 - 1
[4,2,1]
 => 001 => 1 = 2 - 1
[4,1,1,1]
 => 0111 => 3 = 4 - 1
[3,3,1]
 => 111 => 3 = 4 - 1
[3,2,2]
 => 100 => 1 = 2 - 1
[3,2,1,1]
 => 1011 => 3 = 4 - 1
[3,1,1,1,1]
 => 11111 => 5 = 6 - 1
[2,2,2,1]
 => 0001 => 1 = 2 - 1
[2,2,1,1,1]
 => 00111 => 3 = 4 - 1
[2,1,1,1,1,1]
 => 011111 => 5 = 6 - 1
[8]
 => 0 => 0 = 1 - 1
[7,1]
 => 11 => 2 = 3 - 1
[6,2]
 => 00 => 0 = 1 - 1
[6,1,1]
 => 011 => 2 = 3 - 1
[5,3]
 => 11 => 2 = 3 - 1
[5,2,1]
 => 101 => 2 = 3 - 1
[5,1,1,1]
 => 1111 => 4 = 5 - 1
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000010
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => [1,1] => [1,1]
 => 2
[2]
 => 0 => [2] => [2]
 => 1
[1,1]
 => 11 => [1,1,1] => [1,1,1]
 => 3
[3]
 => 1 => [1,1] => [1,1]
 => 2
[2,1]
 => 01 => [2,1] => [2,1]
 => 2
[1,1,1]
 => 111 => [1,1,1,1] => [1,1,1,1]
 => 4
[4]
 => 0 => [2] => [2]
 => 1
[3,1]
 => 11 => [1,1,1] => [1,1,1]
 => 3
[2,2]
 => 00 => [3] => [3]
 => 1
[2,1,1]
 => 011 => [2,1,1] => [2,1,1]
 => 3
[1,1,1,1]
 => 1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 5
[5]
 => 1 => [1,1] => [1,1]
 => 2
[4,1]
 => 01 => [2,1] => [2,1]
 => 2
[3,2]
 => 10 => [1,2] => [2,1]
 => 2
[3,1,1]
 => 111 => [1,1,1,1] => [1,1,1,1]
 => 4
[2,2,1]
 => 001 => [3,1] => [3,1]
 => 2
[2,1,1,1]
 => 0111 => [2,1,1,1] => [2,1,1,1]
 => 4
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 6
[6]
 => 0 => [2] => [2]
 => 1
[5,1]
 => 11 => [1,1,1] => [1,1,1]
 => 3
[4,2]
 => 00 => [3] => [3]
 => 1
[4,1,1]
 => 011 => [2,1,1] => [2,1,1]
 => 3
[3,3]
 => 11 => [1,1,1] => [1,1,1]
 => 3
[3,2,1]
 => 101 => [1,2,1] => [2,1,1]
 => 3
[3,1,1,1]
 => 1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 5
[2,2,2]
 => 000 => [4] => [4]
 => 1
[2,2,1,1]
 => 0011 => [3,1,1] => [3,1,1]
 => 3
[2,1,1,1,1]
 => 01111 => [2,1,1,1,1] => [2,1,1,1,1]
 => 5
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
 => 7
[7]
 => 1 => [1,1] => [1,1]
 => 2
[6,1]
 => 01 => [2,1] => [2,1]
 => 2
[5,2]
 => 10 => [1,2] => [2,1]
 => 2
[5,1,1]
 => 111 => [1,1,1,1] => [1,1,1,1]
 => 4
[4,3]
 => 01 => [2,1] => [2,1]
 => 2
[4,2,1]
 => 001 => [3,1] => [3,1]
 => 2
[4,1,1,1]
 => 0111 => [2,1,1,1] => [2,1,1,1]
 => 4
[3,3,1]
 => 111 => [1,1,1,1] => [1,1,1,1]
 => 4
[3,2,2]
 => 100 => [1,3] => [3,1]
 => 2
[3,2,1,1]
 => 1011 => [1,2,1,1] => [2,1,1,1]
 => 4
[3,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 6
[2,2,2,1]
 => 0001 => [4,1] => [4,1]
 => 2
[2,2,1,1,1]
 => 00111 => [3,1,1,1] => [3,1,1,1]
 => 4
[2,1,1,1,1,1]
 => 011111 => [2,1,1,1,1,1] => [2,1,1,1,1,1]
 => 6
[8]
 => 0 => [2] => [2]
 => 1
[7,1]
 => 11 => [1,1,1] => [1,1,1]
 => 3
[6,2]
 => 00 => [3] => [3]
 => 1
[6,1,1]
 => 011 => [2,1,1] => [2,1,1]
 => 3
[5,3]
 => 11 => [1,1,1] => [1,1,1]
 => 3
[5,2,1]
 => 101 => [1,2,1] => [2,1,1]
 => 3
[5,1,1,1]
 => 1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 5
Description
The length of the partition.
Matching statistic: St000097
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2]
 => 0 => [2] => ([],2)
 => 1
[1,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4]
 => 0 => [2] => ([],2)
 => 1
[3,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,2]
 => 00 => [3] => ([],3)
 => 1
[2,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[4,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[3,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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
[6]
 => 0 => [2] => ([],2)
 => 1
[5,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[4,2]
 => 00 => [3] => ([],3)
 => 1
[4,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,2,2]
 => 000 => [4] => ([],4)
 => 1
[2,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,1,1,1]
 => 01111 => [2,1,1,1,1] => ([(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)
 => 5
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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
[7]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[6,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[5,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[5,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[4,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,3,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,2,2]
 => 100 => [1,3] => ([(2,3)],4)
 => 2
[3,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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
[2,2,2,1]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,2,1,1,1]
 => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,1,1,1,1,1]
 => 011111 => [2,1,1,1,1,1] => ([(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)
 => 6
[8]
 => 0 => [2] => ([],2)
 => 1
[7,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[6,2]
 => 00 => [3] => ([],3)
 => 1
[6,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[5,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
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: St001581
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2]
 => 0 => [2] => ([],2)
 => 1
[1,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4]
 => 0 => [2] => ([],2)
 => 1
[3,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,2]
 => 00 => [3] => ([],3)
 => 1
[2,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[4,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[3,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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
[6]
 => 0 => [2] => ([],2)
 => 1
[5,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[4,2]
 => 00 => [3] => ([],3)
 => 1
[4,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,2,2]
 => 000 => [4] => ([],4)
 => 1
[2,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,1,1,1]
 => 01111 => [2,1,1,1,1] => ([(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)
 => 5
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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
[7]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[6,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[5,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[5,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[4,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,3,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,2,2]
 => 100 => [1,3] => ([(2,3)],4)
 => 2
[3,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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
[2,2,2,1]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,2,1,1,1]
 => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,1,1,1,1,1]
 => 011111 => [2,1,1,1,1,1] => ([(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)
 => 6
[8]
 => 0 => [2] => ([],2)
 => 1
[7,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[6,2]
 => 00 => [3] => ([],3)
 => 1
[6,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[5,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
Description
The achromatic number of a graph. This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.
Matching statistic: St000098
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2]
 => 0 => [2] => ([],2)
 => 1
[1,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4]
 => 0 => [2] => ([],2)
 => 1
[3,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,2]
 => 00 => [3] => ([],3)
 => 1
[2,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[4,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[3,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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
[6]
 => 0 => [2] => ([],2)
 => 1
[5,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[4,2]
 => 00 => [3] => ([],3)
 => 1
[4,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,2,2]
 => 000 => [4] => ([],4)
 => 1
[2,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,1,1,1]
 => 01111 => [2,1,1,1,1] => ([(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)
 => 5
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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
[7]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[6,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[5,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[5,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[4,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,3,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,2,2]
 => 100 => [1,3] => ([(2,3)],4)
 => 2
[3,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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
[2,2,2,1]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,2,1,1,1]
 => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,1,1,1,1,1]
 => 011111 => [2,1,1,1,1,1] => ([(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)
 => 6
[8]
 => 0 => [2] => ([],2)
 => 1
[7,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[6,2]
 => 00 => [3] => ([],3)
 => 1
[6,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[5,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,2,2,2,2,2,2]
 => 0000000 => [8] => ([],8)
 => ? = 1
[3,2,2,2,2,2,2]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 2
[4,2,2,2,2,2,2]
 => 0000000 => [8] => ([],8)
 => ? = 1
[3,3,2,2,2,2,2]
 => 1100000 => [1,1,6] => ([(5,6),(5,7),(6,7)],8)
 => ? = 3
[5,2,2,2,2,2,2]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 2
[4,3,2,2,2,2,2]
 => 0100000 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2
[3,3,3,2,2,2,2]
 => 1110000 => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
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: St000148
(load all 2 compositions to match this statistic)
Mapping
St000148: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 = 2 - 1
[2]
 => 0 = 1 - 1
[1,1]
 => 2 = 3 - 1
[3]
 => 1 = 2 - 1
[2,1]
 => 1 = 2 - 1
[1,1,1]
 => 3 = 4 - 1
[4]
 => 0 = 1 - 1
[3,1]
 => 2 = 3 - 1
[2,2]
 => 0 = 1 - 1
[2,1,1]
 => 2 = 3 - 1
[1,1,1,1]
 => 4 = 5 - 1
[5]
 => 1 = 2 - 1
[4,1]
 => 1 = 2 - 1
[3,2]
 => 1 = 2 - 1
[3,1,1]
 => 3 = 4 - 1
[2,2,1]
 => 1 = 2 - 1
[2,1,1,1]
 => 3 = 4 - 1
[1,1,1,1,1]
 => 5 = 6 - 1
[6]
 => 0 = 1 - 1
[5,1]
 => 2 = 3 - 1
[4,2]
 => 0 = 1 - 1
[4,1,1]
 => 2 = 3 - 1
[3,3]
 => 2 = 3 - 1
[3,2,1]
 => 2 = 3 - 1
[3,1,1,1]
 => 4 = 5 - 1
[2,2,2]
 => 0 = 1 - 1
[2,2,1,1]
 => 2 = 3 - 1
[2,1,1,1,1]
 => 4 = 5 - 1
[1,1,1,1,1,1]
 => 6 = 7 - 1
[7]
 => 1 = 2 - 1
[6,1]
 => 1 = 2 - 1
[5,2]
 => 1 = 2 - 1
[5,1,1]
 => 3 = 4 - 1
[4,3]
 => 1 = 2 - 1
[4,2,1]
 => 1 = 2 - 1
[4,1,1,1]
 => 3 = 4 - 1
[3,3,1]
 => 3 = 4 - 1
[3,2,2]
 => 1 = 2 - 1
[3,2,1,1]
 => 3 = 4 - 1
[3,1,1,1,1]
 => 5 = 6 - 1
[2,2,2,1]
 => 1 = 2 - 1
[2,2,1,1,1]
 => 3 = 4 - 1
[2,1,1,1,1,1]
 => 5 = 6 - 1
[8]
 => 0 = 1 - 1
[7,1]
 => 2 = 3 - 1
[6,2]
 => 0 = 1 - 1
[6,1,1]
 => 2 = 3 - 1
[5,3]
 => 2 = 3 - 1
[5,2,1]
 => 2 = 3 - 1
[5,1,1,1]
 => 4 = 5 - 1
[5,5,2,2,2,1]
 => ? = 4 - 1
[5,4,4,4]
 => ? = 2 - 1
[5,4,4,3,1]
 => ? = 4 - 1
[5,4,4,2,2]
 => ? = 2 - 1
[5,4,4,2,1,1]
 => ? = 4 - 1
[5,4,3,3,2]
 => ? = 4 - 1
[5,4,3,3,1,1]
 => ? = 6 - 1
[5,4,3,2,2,1]
 => ? = 4 - 1
[5,4,2,2,2,2]
 => ? = 2 - 1
[5,4,2,2,2,1,1]
 => ? = 4 - 1
[5,3,3,3,3]
 => ? = 6 - 1
[5,3,3,3,2,1]
 => ? = 6 - 1
[5,3,3,2,2,2]
 => ? = 4 - 1
[5,3,2,2,2,2,1]
 => ? = 4 - 1
[5,2,2,2,2,2,2]
 => ? = 2 - 1
[4,4,4,4,1]
 => ? = 2 - 1
[4,4,4,3,2]
 => ? = 2 - 1
[4,4,4,3,1,1]
 => ? = 4 - 1
[4,4,4,2,2,1]
 => ? = 2 - 1
[4,4,4,2,1,1,1]
 => ? = 4 - 1
[4,4,3,3,3]
 => ? = 4 - 1
[4,4,3,3,2,1]
 => ? = 4 - 1
[4,4,3,2,2,2]
 => ? = 2 - 1
[4,4,3,2,2,1,1]
 => ? = 4 - 1
[4,4,2,2,2,2,1]
 => ? = 2 - 1
[4,3,3,3,3,1]
 => ? = 6 - 1
[4,3,3,3,2,2]
 => ? = 4 - 1
[4,3,3,2,2,2,1]
 => ? = 4 - 1
[4,3,2,2,2,2,2]
 => ? = 2 - 1
[3,3,3,3,3,2]
 => ? = 6 - 1
[3,3,3,2,2,2,2]
 => ? = 4 - 1
Description
The number of odd parts of a partition.
Matching statistic: St001494
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001494: Graphs ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2]
 => 0 => [2] => ([],2)
 => 1
[1,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4]
 => 0 => [2] => ([],2)
 => 1
[3,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,2]
 => 00 => [3] => ([],3)
 => 1
[2,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[4,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[3,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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
[6]
 => 0 => [2] => ([],2)
 => 1
[5,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[4,2]
 => 00 => [3] => ([],3)
 => 1
[4,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,2,2]
 => 000 => [4] => ([],4)
 => 1
[2,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,1,1,1]
 => 01111 => [2,1,1,1,1] => ([(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)
 => 5
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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
[7]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[6,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[5,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[5,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[4,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,3,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,2,2]
 => 100 => [1,3] => ([(2,3)],4)
 => 2
[3,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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
[2,2,2,1]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,2,1,1,1]
 => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,1,1,1,1,1]
 => 011111 => [2,1,1,1,1,1] => ([(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)
 => 6
[8]
 => 0 => [2] => ([],2)
 => 1
[7,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[6,2]
 => 00 => [3] => ([],3)
 => 1
[6,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[5,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[2,2,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[3,2,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[2,2,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[4,2,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,3,2,2,2,1,1]
 => 1100011 => [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[3,2,2,2,2,2,1]
 => 1000001 => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[2,2,2,2,2,2,2]
 => 0000000 => [8] => ([],8)
 => ? = 1
[6,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[5,2,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,4,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,3,2,2,2,1,1]
 => 0100011 => [2,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,2,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,3,2,2,2,2,1]
 => 1100001 => [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[3,2,2,2,2,2,2]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 2
[6,2,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[5,3,2,2,2,1,1]
 => 1100011 => [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[5,2,2,2,2,2,1]
 => 1000001 => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,4,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,3,2,2,2,2,1]
 => 0100001 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,2,2,2,2,2,2]
 => 0000000 => [8] => ([],8)
 => ? = 1
[3,3,3,2,2,2,1]
 => 1110001 => [1,1,1,4,1] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[3,3,2,2,2,2,2]
 => 1100000 => [1,1,6] => ([(5,6),(5,7),(6,7)],8)
 => ? = 3
[8,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[7,2,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[6,4,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[6,3,2,2,2,1,1]
 => 0100011 => [2,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[6,2,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[5,4,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[5,3,2,2,2,2,1]
 => 1100001 => [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[5,2,2,2,2,2,2]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 2
[4,4,4,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,4,3,2,2,1,1]
 => 0010011 => [3,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,4,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[4,3,3,2,2,2,1]
 => 0110001 => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,3,2,2,2,2,2]
 => 0100000 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2
[3,3,3,2,2,2,2]
 => 1110000 => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
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: St000172
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2]
 => 0 => [2] => ([],2)
 => 1
[1,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4]
 => 0 => [2] => ([],2)
 => 1
[3,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,2]
 => 00 => [3] => ([],3)
 => 1
[2,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[4,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[3,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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
[6]
 => 0 => [2] => ([],2)
 => 1
[5,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[4,2]
 => 00 => [3] => ([],3)
 => 1
[4,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,2,2]
 => 000 => [4] => ([],4)
 => 1
[2,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,1,1,1]
 => 01111 => [2,1,1,1,1] => ([(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)
 => 5
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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
[7]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[6,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[5,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[5,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[4,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,3,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,2,2]
 => 100 => [1,3] => ([(2,3)],4)
 => 2
[3,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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
[2,2,2,1]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,2,1,1,1]
 => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,1,1,1,1,1]
 => 011111 => [2,1,1,1,1,1] => ([(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)
 => 6
[8]
 => 0 => [2] => ([],2)
 => 1
[7,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[6,2]
 => 00 => [3] => ([],3)
 => 1
[6,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[5,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[3,3,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(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)
 => ? = 5
[2,2,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[3,2,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[2,2,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[5,3,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(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)
 => ? = 5
[4,2,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,3,2,2,2,1,1]
 => 1100011 => [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[3,2,2,2,2,2,1]
 => 1000001 => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[2,2,2,2,2,2,2]
 => 0000000 => [8] => ([],8)
 => ? = 1
[6,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[5,2,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,4,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,3,2,2,2,1,1]
 => 0100011 => [2,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,2,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,3,2,2,2,2,1]
 => 1100001 => [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[3,2,2,2,2,2,2]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 2
[7,3,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(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)
 => ? = 5
[6,2,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[5,5,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(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)
 => ? = 5
[5,3,2,2,2,1,1]
 => 1100011 => [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[5,2,2,2,2,2,1]
 => 1000001 => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,4,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,3,2,2,2,2,1]
 => 0100001 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,2,2,2,2,2,2]
 => 0000000 => [8] => ([],8)
 => ? = 1
[3,3,3,2,2,2,1]
 => 1110001 => [1,1,1,4,1] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[3,3,2,2,2,2,2]
 => 1100000 => [1,1,6] => ([(5,6),(5,7),(6,7)],8)
 => ? = 3
[8,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[7,2,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[6,4,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[6,3,2,2,2,1,1]
 => 0100011 => [2,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[6,2,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[5,4,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[5,3,2,2,2,2,1]
 => 1100001 => [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[5,2,2,2,2,2,2]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 2
[4,4,4,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,4,3,2,2,1,1]
 => 0010011 => [3,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,4,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[4,3,3,2,2,2,1]
 => 0110001 => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,3,2,2,2,2,2]
 => 0100000 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2
[3,3,3,2,2,2,2]
 => 1110000 => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
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: St001029
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001029: Graphs ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2]
 => 0 => [2] => ([],2)
 => 1
[1,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4]
 => 0 => [2] => ([],2)
 => 1
[3,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,2]
 => 00 => [3] => ([],3)
 => 1
[2,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[4,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[3,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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
[6]
 => 0 => [2] => ([],2)
 => 1
[5,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[4,2]
 => 00 => [3] => ([],3)
 => 1
[4,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,2,2]
 => 000 => [4] => ([],4)
 => 1
[2,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,1,1,1]
 => 01111 => [2,1,1,1,1] => ([(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)
 => 5
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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
[7]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[6,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[5,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[5,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[4,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,3,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,2,2]
 => 100 => [1,3] => ([(2,3)],4)
 => 2
[3,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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
[2,2,2,1]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,2,1,1,1]
 => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,1,1,1,1,1]
 => 011111 => [2,1,1,1,1,1] => ([(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)
 => 6
[8]
 => 0 => [2] => ([],2)
 => 1
[7,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[6,2]
 => 00 => [3] => ([],3)
 => 1
[6,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[5,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[3,3,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(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)
 => ? = 5
[2,2,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[3,2,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[2,2,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[5,3,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(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)
 => ? = 5
[4,2,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,3,2,2,2,1,1]
 => 1100011 => [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[3,2,2,2,2,2,1]
 => 1000001 => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[2,2,2,2,2,2,2]
 => 0000000 => [8] => ([],8)
 => ? = 1
[6,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[5,2,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,4,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,3,2,2,2,1,1]
 => 0100011 => [2,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,2,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,3,2,2,2,2,1]
 => 1100001 => [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[3,2,2,2,2,2,2]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 2
[7,3,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(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)
 => ? = 5
[6,2,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[5,5,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(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)
 => ? = 5
[5,3,2,2,2,1,1]
 => 1100011 => [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[5,2,2,2,2,2,1]
 => 1000001 => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,4,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,3,2,2,2,2,1]
 => 0100001 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,2,2,2,2,2,2]
 => 0000000 => [8] => ([],8)
 => ? = 1
[3,3,3,2,2,2,1]
 => 1110001 => [1,1,1,4,1] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[3,3,2,2,2,2,2]
 => 1100000 => [1,1,6] => ([(5,6),(5,7),(6,7)],8)
 => ? = 3
[8,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[7,2,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[6,4,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[6,3,2,2,2,1,1]
 => 0100011 => [2,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[6,2,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[5,4,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[5,3,2,2,2,2,1]
 => 1100001 => [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[5,2,2,2,2,2,2]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 2
[4,4,4,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,4,3,2,2,1,1]
 => 0010011 => [3,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,4,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[4,3,3,2,2,2,1]
 => 0110001 => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,3,2,2,2,2,2]
 => 0100000 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2
[3,3,3,2,2,2,2]
 => 1110000 => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
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: St001580
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001580: Graphs ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2]
 => 0 => [2] => ([],2)
 => 1
[1,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4]
 => 0 => [2] => ([],2)
 => 1
[3,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,2]
 => 00 => [3] => ([],3)
 => 1
[2,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[4,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[3,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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
[6]
 => 0 => [2] => ([],2)
 => 1
[5,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[4,2]
 => 00 => [3] => ([],3)
 => 1
[4,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,2,2]
 => 000 => [4] => ([],4)
 => 1
[2,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,1,1,1]
 => 01111 => [2,1,1,1,1] => ([(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)
 => 5
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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
[7]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[6,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[5,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[5,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[4,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,3,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,2,2]
 => 100 => [1,3] => ([(2,3)],4)
 => 2
[3,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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
[2,2,2,1]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,2,1,1,1]
 => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,1,1,1,1,1]
 => 011111 => [2,1,1,1,1,1] => ([(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)
 => 6
[8]
 => 0 => [2] => ([],2)
 => 1
[7,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[6,2]
 => 00 => [3] => ([],3)
 => 1
[6,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[5,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[3,3,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(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)
 => ? = 5
[2,2,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[3,2,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[2,2,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[5,3,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(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)
 => ? = 5
[4,2,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,3,2,2,2,1,1]
 => 1100011 => [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[3,2,2,2,2,2,1]
 => 1000001 => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[2,2,2,2,2,2,2]
 => 0000000 => [8] => ([],8)
 => ? = 1
[6,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[5,2,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,4,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,3,2,2,2,1,1]
 => 0100011 => [2,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,2,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,3,2,2,2,2,1]
 => 1100001 => [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[3,2,2,2,2,2,2]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 2
[7,3,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(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)
 => ? = 5
[6,2,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[5,5,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(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)
 => ? = 5
[5,3,2,2,2,1,1]
 => 1100011 => [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[5,2,2,2,2,2,1]
 => 1000001 => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,4,2,2,2,1,1]
 => 0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,3,2,2,2,2,1]
 => 0100001 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[4,2,2,2,2,2,2]
 => 0000000 => [8] => ([],8)
 => ? = 1
[3,3,3,2,2,2,1]
 => 1110001 => [1,1,1,4,1] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[3,3,2,2,2,2,2]
 => 1100000 => [1,1,6] => ([(5,6),(5,7),(6,7)],8)
 => ? = 3
[8,2,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[7,2,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[6,4,2,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[6,3,2,2,2,1,1]
 => 0100011 => [2,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[6,2,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[5,4,2,2,2,1,1]
 => 1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[5,3,2,2,2,2,1]
 => 1100001 => [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[5,2,2,2,2,2,2]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 2
[4,4,4,2,1,1,1]
 => 0000111 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,4,3,2,2,1,1]
 => 0010011 => [3,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,4,2,2,2,2,1]
 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[4,3,3,2,2,2,1]
 => 0110001 => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[4,3,2,2,2,2,2]
 => 0100000 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2
[3,3,3,2,2,2,2]
 => 1110000 => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
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 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth 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. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000822The Hadwiger number of the graph. St001812The biclique partition number of a graph. St000992The alternating sum of the parts of an integer partition. St001330The hat guessing number of a graph. St000022The number of fixed points of a permutation. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St000895The number of ones on the main diagonal of an alternating sign matrix. St000696The number of cycles in the breakpoint graph of a permutation.