Your data matches 51 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
Mp00160: Permutations graph of inversionsGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 1
[1,2] => ([],2)
=> [1,1]
=> 1
[2,1] => ([(0,1)],2)
=> [2]
=> 2
[1,2,3] => ([],3)
=> [1,1,1]
=> 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> 2
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> 2
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 3
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 2
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 3
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 4
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 4
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 2
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 4
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 4
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 4
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 4
Description
The largest part of an integer partition.
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00100: Dyck paths touch compositionInteger compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1
[1,2] => [1,0,1,0]
=> [1,1] => 1
[2,1] => [1,1,0,0]
=> [2] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 2
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2
[2,3,1] => [1,1,0,1,0,0]
=> [3] => 3
[3,1,2] => [1,1,1,0,0,0]
=> [3] => 3
[3,2,1] => [1,1,1,0,0,0]
=> [3] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3] => 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1] => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [4] => 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [4] => 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [4] => 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [4] => 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4] => 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 4
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 4
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 4
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 4
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
Description
The largest part of an integer composition.
Matching statistic: St000010
Mp00160: Permutations graph of inversionsGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
Mp00044: Integer partitions conjugateInteger 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)
=> [1,1]
=> [2]
=> 1
[2,1] => ([(0,1)],2)
=> [2]
=> [1,1]
=> 2
[1,2,3] => ([],3)
=> [1,1,1]
=> [3]
=> 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [4]
=> 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [5]
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
Description
The length of the partition.
Matching statistic: St000013
Mp00114: Permutations connectivity setBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => => [1] => [1,0]
=> 1
[1,2] => 1 => [1,1] => [1,0,1,0]
=> 1
[2,1] => 0 => [2] => [1,1,0,0]
=> 2
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
=> 2
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
=> 3
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 3
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 3
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,5,3,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,5,2,3,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,5,2,4,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,5,3,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000097
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00147: Graphs squareGraphs
St000097: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1
[1,2] => [1,2] => ([],2)
=> ([],2)
=> 1
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,2,3] => [1,2,3] => ([],3)
=> ([],3)
=> 1
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],4)
=> 1
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,5,4,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,3,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,5,3,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
Description
The order of the largest clique of the graph. A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00147: Graphs squareGraphs
St000098: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1
[1,2] => [1,2] => ([],2)
=> ([],2)
=> 1
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,2,3] => [1,2,3] => ([],3)
=> ([],3)
=> 1
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],4)
=> 1
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,5,4,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,3,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,5,3,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 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: St000172
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00147: Graphs squareGraphs
St000172: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1
[1,2] => [1,2] => ([],2)
=> ([],2)
=> 1
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,2,3] => [1,2,3] => ([],3)
=> ([],3)
=> 1
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],4)
=> 1
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,5,4,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,3,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,5,3,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 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: St000676
Mp00160: Permutations graph of inversionsGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St000676: Dyck paths ⟶ 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]
=> [1,1,0,0]
=> 1
[2,1] => ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[1,2,3] => ([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000734
Mp00160: Permutations graph of inversionsGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> [[1]]
=> 1
[1,2] => ([],2)
=> [1,1]
=> [[1],[2]]
=> 1
[2,1] => ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2
[1,2,3] => ([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 3
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 3
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
Description
The last entry in the first row of a standard tableau.
Matching statistic: St001029
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00147: Graphs squareGraphs
St001029: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1
[1,2] => [1,2] => ([],2)
=> ([],2)
=> 1
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,2,3] => [1,2,3] => ([],3)
=> ([],3)
=> 1
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],4)
=> 1
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,5,4,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,3,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,5,3,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 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 $G$.
The following 41 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001330The hat guessing number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001883The mutual visibility number of a graph. St000141The maximum drop size of a permutation. St000171The degree of the graph. St000272The treewidth of a graph. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000662The staircase size of the code of a permutation. St001120The length of a longest path in a graph. St001644The dimension of a graph. St000503The maximal difference between two elements in a common block. St000392The length of the longest run of ones in a binary word. St000444The length of the maximal rise of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000730The maximal arc length of a set partition. St001268The size of the largest ordinal summand in the poset. St001645The pebbling number of a connected graph. St001090The number of pop-stack-sorts needed to sort a permutation. St000209Maximum difference of elements in cycles. St000844The size of the largest block in the direct sum decomposition of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000956The maximal displacement of a permutation. St000485The length of the longest cycle of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000822The Hadwiger number of the graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001963The tree-depth of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.