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Your data matches 125 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [1,1]
=> 1
[1,2,3] => [1,2,3] => [3]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> 1
[2,1,3] => [2,1,3] => [2,1]
=> 1
[1,2,3,4] => [1,2,3,4] => [4]
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1]
=> 1
[1,3,2,4] => [1,3,2,4] => [3,1]
=> 1
[2,1,3,4] => [2,1,3,4] => [3,1]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[2,4,1,3] => [3,4,1,2] => [2,2]
=> 2
[3,1,4,2] => [4,2,3,1] => [2,1,1]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> 2
[1,3,5,2,4] => [1,4,5,2,3] => [3,2]
=> 2
[1,4,2,5,3] => [1,5,3,4,2] => [3,1,1]
=> 2
[2,1,3,4,5] => [2,1,3,4,5] => [4,1]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [3,2]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [3,2]
=> 2
[2,4,1,3,5] => [3,4,1,2,5] => [3,2]
=> 2
[2,4,1,5,3] => [3,5,1,4,2] => [2,2,1]
=> 3
[3,1,4,2,5] => [4,2,3,1,5] => [3,1,1]
=> 2
[3,1,5,2,4] => [4,2,5,1,3] => [2,2,1]
=> 3
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6]
=> 0
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [5,1]
=> 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1]
=> 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [5,1]
=> 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
=> 2
[1,2,4,6,3,5] => [1,2,5,6,3,4] => [4,2]
=> 2
[1,2,5,3,6,4] => [1,2,6,4,5,3] => [4,1,1]
=> 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [5,1]
=> 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [4,2]
=> 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [4,2]
=> 2
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [4,2]
=> 2
[1,3,5,2,6,4] => [1,4,6,2,5,3] => [3,2,1]
=> 3
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [4,1,1]
=> 2
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [3,2,1]
=> 3
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [5,1]
=> 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => [4,2]
=> 2
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [4,2]
=> 2
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [4,2]
=> 2
[2,1,4,6,3,5] => [2,1,5,6,3,4] => [3,3]
=> 3
[2,1,5,3,6,4] => [2,1,6,4,5,3] => [3,2,1]
=> 3
[2,4,1,3,5,6] => [3,4,1,2,5,6] => [4,2]
=> 2
[2,4,1,3,6,5] => [3,4,1,2,6,5] => [3,3]
=> 3
[2,4,1,5,3,6] => [3,5,1,4,2,6] => [3,2,1]
=> 3
[2,4,6,1,3,5] => [4,5,6,1,2,3] => [3,3]
=> 3
[2,5,1,3,6,4] => [3,6,1,4,5,2] => [3,2,1]
=> 3
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001876
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Values
[2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 3
[3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 3
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 1
[1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1
[1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> 2
[1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2
[1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2
[1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,3,5,2,6,4] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 3
[1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 3
[2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1
[2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> 2
[2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> 2
[2,1,4,6,3,5] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> 3
[2,1,5,3,6,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> 3
[2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[2,4,1,3,6,5] => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> 3
[2,4,1,5,3,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 3
[2,4,6,1,3,5] => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 3
[2,5,1,3,6,4] => ([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 3
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St000228
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [1,1]
=> [1]
=> 1
[1,2,3] => [1,2,3] => [3]
=> []
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 1
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 1
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 1
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,3,4] => [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,4,1,3] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,1,4,2] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[1,3,5,2,4] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[1,4,2,5,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 2
[2,1,3,4,5] => [2,1,3,4,5] => [4,1]
=> [1]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [3,2]
=> [2]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [3,2]
=> [2]
=> 2
[2,4,1,3,5] => [3,4,1,2,5] => [3,2]
=> [2]
=> 2
[2,4,1,5,3] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 3
[3,1,4,2,5] => [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 2
[3,1,5,2,4] => [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 3
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6]
=> []
=> 0
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [5,1]
=> [1]
=> 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
=> [2]
=> 2
[1,2,4,6,3,5] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 2
[1,2,5,3,6,4] => [1,2,6,4,5,3] => [4,1,1]
=> [1,1]
=> 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [5,1]
=> [1]
=> 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [4,2]
=> [2]
=> 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [4,2]
=> [2]
=> 2
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [4,2]
=> [2]
=> 2
[1,3,5,2,6,4] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 3
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [4,1,1]
=> [1,1]
=> 2
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [3,2,1]
=> [2,1]
=> 3
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [5,1]
=> [1]
=> 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => [4,2]
=> [2]
=> 2
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [4,2]
=> [2]
=> 2
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [4,2]
=> [2]
=> 2
[2,1,4,6,3,5] => [2,1,5,6,3,4] => [3,3]
=> [3]
=> 3
[2,1,5,3,6,4] => [2,1,6,4,5,3] => [3,2,1]
=> [2,1]
=> 3
[2,4,1,3,5,6] => [3,4,1,2,5,6] => [4,2]
=> [2]
=> 2
[2,4,1,3,6,5] => [3,4,1,2,6,5] => [3,3]
=> [3]
=> 3
[2,4,1,5,3,6] => [3,5,1,4,2,6] => [3,2,1]
=> [2,1]
=> 3
[2,4,6,1,3,5] => [4,5,6,1,2,3] => [3,3]
=> [3]
=> 3
[2,5,1,3,6,4] => [3,6,1,4,5,2] => [3,2,1]
=> [2,1]
=> 3
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000738
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,4,1,3] => [3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,1,4,2] => [4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> [[1,2,3,4,5]]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
[1,3,5,2,4] => [1,4,5,2,3] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
[1,4,2,5,3] => [1,5,3,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
[2,4,1,3,5] => [3,4,1,2,5] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
[2,4,1,5,3] => [3,5,1,4,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
[3,1,4,2,5] => [4,2,3,1,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[3,1,5,2,4] => [4,2,5,1,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6]
=> [[1,2,3,4,5,6]]
=> 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [5,1]
=> [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1]
=> [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [5,1]
=> [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
[1,2,4,6,3,5] => [1,2,5,6,3,4] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
[1,2,5,3,6,4] => [1,2,6,4,5,3] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 3 = 2 + 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [5,1]
=> [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
[1,3,5,2,6,4] => [1,4,6,2,5,3] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 4 = 3 + 1
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 3 = 2 + 1
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 4 = 3 + 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [5,1]
=> [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
[2,1,4,6,3,5] => [2,1,5,6,3,4] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 4 = 3 + 1
[2,1,5,3,6,4] => [2,1,6,4,5,3] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 4 = 3 + 1
[2,4,1,3,5,6] => [3,4,1,2,5,6] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
[2,4,1,3,6,5] => [3,4,1,2,6,5] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 4 = 3 + 1
[2,4,1,5,3,6] => [3,5,1,4,2,6] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 4 = 3 + 1
[2,4,6,1,3,5] => [4,5,6,1,2,3] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 4 = 3 + 1
[2,5,1,3,6,4] => [3,6,1,4,5,2] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 4 = 3 + 1
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000362
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => ([],3)
=> 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,3,5,2,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,4,2,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2
[2,4,1,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,4,2,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 2
[1,2,4,6,3,5] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,2,5,3,6,4] => [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(4,5)],6)
=> 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 2
[1,3,5,2,4,6] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,3,5,2,6,4] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[1,4,2,5,3,6] => [1,5,3,4,2,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,2,6,3,5] => [1,5,3,6,2,4] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
=> 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 2
[2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 2
[2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 2
[2,1,4,6,3,5] => [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[2,1,5,3,6,4] => [2,1,6,4,5,3] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,4,1,3,5,6] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[2,4,1,3,6,5] => [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[2,4,1,5,3,6] => [3,5,1,4,2,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[2,4,6,1,3,5] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 3
[2,5,1,3,6,4] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,3,5,7,4,6,8] => [1,2,3,6,7,4,5,8] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,2,3,6,4,7,5,8] => [1,2,3,7,5,6,4,8] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
Description
The size of a minimal vertex cover of a graph.
A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. Finding a minimal vertex cover is an NP-hard optimization problem.
Matching statistic: St001725
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St001725: Graphs ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
St001725: Graphs ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => ([],3)
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => ([],4)
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => ([(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 3 = 2 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 3 = 2 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[2,1,3,4,5] => ([(3,4)],5)
=> 2 = 1 + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 3 = 2 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 3 = 2 + 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4 = 3 + 1
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4 = 3 + 1
[1,2,3,4,5,6] => ([],6)
=> 1 = 0 + 1
[1,2,3,4,6,5] => ([(4,5)],6)
=> 2 = 1 + 1
[1,2,3,5,4,6] => ([(4,5)],6)
=> 2 = 1 + 1
[1,2,4,3,5,6] => ([(4,5)],6)
=> 2 = 1 + 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 3 = 2 + 1
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,3,2,4,5,6] => ([(4,5)],6)
=> 2 = 1 + 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 3 = 2 + 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 3 = 2 + 1
[1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 4 = 3 + 1
[1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 4 = 3 + 1
[2,1,3,4,5,6] => ([(4,5)],6)
=> 2 = 1 + 1
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 3 = 2 + 1
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 3 = 2 + 1
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 3 = 2 + 1
[2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> 3 = 2 + 1
[2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 4 = 3 + 1
[2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,4,6,2,7,3,5] => ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 4 + 1
[3,5,1,6,2,4,7] => ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 4 + 1
[3,6,1,4,7,2,5] => ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[1,2,3,5,7,4,6,8] => ([(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[1,2,3,6,4,7,5,8] => ([(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
Description
The harmonious chromatic number of a graph.
A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.
Matching statistic: St000918
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[2,1] => ([(0,1)],2)
=> ([],1)
=> 1
[1,2,3] => ([],3)
=> ([],0)
=> ? = 0
[1,3,2] => ([(1,2)],3)
=> ([],1)
=> 1
[2,1,3] => ([(1,2)],3)
=> ([],1)
=> 1
[1,2,3,4] => ([],4)
=> ([],0)
=> ? = 0
[1,2,4,3] => ([(2,3)],4)
=> ([],1)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> ([],1)
=> 1
[2,1,3,4] => ([(2,3)],4)
=> ([],1)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,3,4,5] => ([],5)
=> ([],0)
=> ? = 0
[1,2,3,5,4] => ([(3,4)],5)
=> ([],1)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([],1)
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([],1)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],2)
=> 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,1,3,4,5] => ([(3,4)],5)
=> ([],1)
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],2)
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],2)
=> 2
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,2,3,4,5,6] => ([],6)
=> ([],0)
=> ? = 0
[1,2,3,4,6,5] => ([(4,5)],6)
=> ([],1)
=> 1
[1,2,3,5,4,6] => ([(4,5)],6)
=> ([],1)
=> 1
[1,2,4,3,5,6] => ([(4,5)],6)
=> ([],1)
=> 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> 2
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[1,3,2,4,5,6] => ([(4,5)],6)
=> ([],1)
=> 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> 2
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> 2
[1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,1,3,4,5,6] => ([(4,5)],6)
=> ([],1)
=> 1
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> 2
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> 2
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> 2
[2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 3
[2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 3
[2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 3
[2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,4,2,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 3
[3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,5,2,4,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,2,3,4,5,6,7] => ([],7)
=> ([],0)
=> ? = 0
[3,6,1,4,7,2,5] => ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,6),(4,8),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 4
[1,2,3,5,7,4,6,8] => ([(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 2
[1,2,3,6,4,7,5,8] => ([(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 2
Description
The 2-limited packing number of a graph.
A subset $B$ of the set of vertices of a graph is a $k$-limited packing set if its intersection with the (closed) neighbourhood of any vertex is at most $k$. The $k$-limited packing number is the largest number of vertices in a $k$-limited packing set.
Matching statistic: St000172
Values
[2,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
[1,2,3] => ([],3)
=> ([],0)
=> ([],0)
=> ? = 0
[1,3,2] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
[2,1,3] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
[1,2,3,4] => ([],4)
=> ([],0)
=> ([],0)
=> ? = 0
[1,2,4,3] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[2,1,3,4] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,2,3,4,5] => ([],5)
=> ([],0)
=> ([],0)
=> ? = 0
[1,2,3,5,4] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,1,3,4,5] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,2,3,4,5,6] => ([],6)
=> ([],0)
=> ([],0)
=> ? = 0
[1,2,3,4,6,5] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[1,2,3,5,4,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[1,2,4,3,5,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,3,2,4,5,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,1,3,4,5,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> 3
[2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,1,4,2,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,5,2,4,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,2,3,4,5,6,7] => ([],7)
=> ([],0)
=> ([],0)
=> ? = 0
[3,6,1,4,7,2,5] => ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,6),(4,8),(5,7),(5,8),(6,8),(7,8)],9)
=> ?
=> ? = 4
[1,2,3,5,7,4,6,8] => ([(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[1,2,3,6,4,7,5,8] => ([(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
Description
The Grundy number of a graph.
The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring.
In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).
Matching statistic: St001581
Values
[2,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
[1,2,3] => ([],3)
=> ([],0)
=> ([],0)
=> ? = 0
[1,3,2] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
[2,1,3] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
[1,2,3,4] => ([],4)
=> ([],0)
=> ([],0)
=> ? = 0
[1,2,4,3] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[2,1,3,4] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,2,3,4,5] => ([],5)
=> ([],0)
=> ([],0)
=> ? = 0
[1,2,3,5,4] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,1,3,4,5] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,2,3,4,5,6] => ([],6)
=> ([],0)
=> ([],0)
=> ? = 0
[1,2,3,4,6,5] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[1,2,3,5,4,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[1,2,4,3,5,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,3,2,4,5,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,1,3,4,5,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> 3
[2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,1,4,2,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,5,2,4,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,2,3,4,5,6,7] => ([],7)
=> ([],0)
=> ([],0)
=> ? = 0
[3,6,1,4,7,2,5] => ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,6),(4,8),(5,7),(5,8),(6,8),(7,8)],9)
=> ?
=> ? = 4
[1,2,3,5,7,4,6,8] => ([(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[1,2,3,6,4,7,5,8] => ([(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
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: St000985
Values
[2,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,2,3] => ([],3)
=> ([],0)
=> ([],0)
=> ? = 0 - 1
[1,3,2] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[2,1,3] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,2,3,4] => ([],4)
=> ([],0)
=> ([],0)
=> ? = 0 - 1
[1,2,4,3] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,3,2,4] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[2,1,3,4] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,2,3,4,5] => ([],5)
=> ([],0)
=> ([],0)
=> ? = 0 - 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[2,1,3,4,5] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
[1,2,3,4,5,6] => ([],6)
=> ([],0)
=> ([],0)
=> ? = 0 - 1
[1,2,3,4,6,5] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,2,3,5,4,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,2,4,3,5,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,3,2,4,5,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
[1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
[2,1,3,4,5,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
[2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[3,1,4,2,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,5,2,4,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
[1,2,3,4,5,6,7] => ([],7)
=> ([],0)
=> ([],0)
=> ? = 0 - 1
[3,6,1,4,7,2,5] => ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,6),(4,8),(5,7),(5,8),(6,8),(7,8)],9)
=> ?
=> ? = 4 - 1
[1,2,3,5,7,4,6,8] => ([(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[1,2,3,6,4,7,5,8] => ([(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
Description
The number of positive eigenvalues of the adjacency matrix of the graph.
The following 115 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001812The biclique partition number of a graph. St000636The hull number of a graph. St000670The reversal length of a permutation. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001656The monophonic position number of a graph. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St001875The number of simple modules with projective dimension at most 1. St000783The side length of the largest staircase partition fitting into a partition. St000454The largest eigenvalue of a graph if it is integral. St000672The number of minimal elements in Bruhat order not less than the permutation. St000653The last descent of a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000260The radius of a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001060The distinguishing index of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000080The rank of the poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001668The number of points of the poset minus the width of the poset. St000039The number of crossings of a permutation. St000100The number of linear extensions of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001637The number of (upper) dissectors of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001330The hat guessing number of a graph. St000456The monochromatic index of a connected graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000934The 2-degree of an integer partition. St001545The second Elser number of a connected graph. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St000264The girth of a graph, which is not a tree. St001864The number of excedances of a signed permutation. St001769The reflection length of a signed permutation. St001896The number of right descents of a signed permutations. St001866The nesting alignments of a signed permutation. St001863The number of weak excedances of a signed permutation. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001597The Frobenius rank of a skew partition. St001892The flag excedance statistic of a signed permutation. St000632The jump number of the poset. St001893The flag descent of a signed permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001905The number of preferred parking spots in a parking function less than the index of the car. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001946The number of descents in a parking function. St000116The major index of a semistandard tableau obtained by standardizing. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001397Number of pairs of incomparable elements in a finite poset. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001712The number of natural descents of a standard Young tableau. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000746The number of pairs with odd minimum in a perfect matching. St001645The pebbling number of a connected graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St001267The length of the Lyndon factorization of the binary word. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000443The number of long tunnels of a Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra.
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