- FindStat

Your data matches 172 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000527
(load all 4 compositions to match this statistic)

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => 1 = 0 + 1
[1,2] => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,2,1] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 3 = 2 + 1
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 3 = 2 + 1
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 3 = 2 + 1
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 1 + 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 3 = 2 + 1
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 3 = 2 + 1
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 3 = 2 + 1
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 3 = 2 + 1
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 1 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 3 = 2 + 1
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 3 = 2 + 1
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 3 = 2 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1

Description

The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.

Matching statistic: St000272
(load all 2 compositions to match this statistic)

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([(0,1)],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],2)
 => 0
[1,2,3] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,2,1] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0

Description

The treewidth of a graph. A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.

Matching statistic: St000481

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => [1]
 => 0
[1,2] => ([(0,1)],2)
 => [2]
 => 0
[2,1] => ([(0,1)],2)
 => [2]
 => 0
[1,2,3] => ([(0,2),(2,1)],3)
 => [3]
 => 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
[3,2,1] => ([(0,2),(2,1)],3)
 => [3]
 => 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => [4,2,1]
 => 2
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => [4,2,1]
 => 2
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => [4,2,1]
 => 2
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => [4,2,1]
 => 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => [4,2,1]
 => 2
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => [4,2,1]
 => 2
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => [4,2,1]
 => 2
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => [4,2,1]
 => 2
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => [4,2,1]
 => 2
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => [4,2,1]
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => 0
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [7]
 => 0
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [7]
 => 0

Description

The number of upper covers of a partition in dominance order.

Matching statistic: St000536
(load all 2 compositions to match this statistic)

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000536: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([(0,1)],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],2)
 => 0
[1,2,3] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,2,1] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0

Description

The pathwidth of a graph.

Matching statistic: St000537
(load all 2 compositions to match this statistic)

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000537: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([(0,1)],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],2)
 => 0
[1,2,3] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,2,1] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0

Description

The cutwidth of a graph. This is the minimum possible width of a linear ordering of its vertices, where the width of an ordering

$\sigma$ is the maximum, among all the prefixes of

$\sigma$ , of the number of edges that have exactly one vertex in a prefix.

Matching statistic: St001270
(load all 2 compositions to match this statistic)

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001270: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([(0,1)],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],2)
 => 0
[1,2,3] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,2,1] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0

Description

The bandwidth of a graph. The bandwidth of a graph is the smallest number

$k$ such that the vertices of the graph can be ordered as

$v_1,\dots,v_n$ with

$k \cdot d(v_i,v_j) \geq |i-j|$ . We adopt the convention that the singleton graph has bandwidth

$0$ , consistent with the bandwith of the complete graph on

$n$ vertices having bandwidth

$n-1$ , but in contrast to any path graph on more than one vertex having bandwidth

$1$ . The bandwidth of a disconnected graph is the maximum of the bandwidths of the connected components.

Matching statistic: St001277
(load all 2 compositions to match this statistic)

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001277: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([(0,1)],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],2)
 => 0
[1,2,3] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,2,1] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0

Description

The degeneracy of a graph. The degeneracy of a graph

$G$ is the maximum of the minimum degrees of the (vertex induced) subgraphs of

$G$ .

Matching statistic: St001358
(load all 2 compositions to match this statistic)

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001358: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([(0,1)],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],2)
 => 0
[1,2,3] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,2,1] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0

Description

The largest degree of a regular subgraph of a graph. For

$k > 2$ , it is an NP-complete problem to determine whether a graph has a

$k$ -regular subgraph, see [1].

Matching statistic: St001644
(load all 2 compositions to match this statistic)

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([(0,1)],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],2)
 => 0
[1,2,3] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,2,1] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0

Description

The dimension of a graph. The dimension of a graph is the least integer

$n$ such that there exists a representation of the graph in the Euclidean space of dimension

$n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.

Matching statistic: St001792
(load all 2 compositions to match this statistic)

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001792: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([(0,1)],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],2)
 => 0
[1,2,3] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[3,2,1] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 0

Description

The arboricity of a graph. This is the minimum number of forests that covers all edges of the graph.

The following 162 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001962The proper pathwidth of a graph. St000010The length of the partition. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000159The number of distinct parts of the integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000785The number of distinct colouring schemes of a graph. St001029The size of the core of a graph. St001432The order dimension of the partition. St001484The number of singletons of an integer partition. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001883The mutual visibility number of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000480The number of lower covers of a partition in dominance order. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St000759The smallest missing part in an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001592The maximal number of simple paths between any two different vertices of a graph. St000632The jump number of the poset. St001323The independence gap of a graph. St001331The size of the minimal feedback vertex set. St001393The induced matching number of a graph. St001638The book thickness of a graph. St000482The (zero)-forcing number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001388The number of non-attacking neighbors of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits 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. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St000640The rank of the largest boolean interval in a poset. St000454The largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000455The second largest eigenvalue of a graph if it is integral. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000264The girth of a graph, which is not a tree. St000552The number of cut vertices of a graph. St000768The number of peaks in an integer composition. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001470The cyclic holeyness of a permutation. St001793The difference between the clique number and the chromatic number of a graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001282The number of graphs with the same chromatic polynomial. St001333The cardinality of a minimal edge-isolating set of a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000636The hull number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001691The number of kings in a graph. St001964The interval resolution global dimension of a poset. St000095The number of triangles of a graph. 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. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001742The difference of the maximal and the minimal degree in a graph. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001488The number of corners of a skew partition. St000219The number of occurrences of the pattern 231 in a permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001570The minimal number of edges to add to make a graph Hamiltonian. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001645The pebbling number of a connected graph. St000181The number of connected components of the Hasse diagram for the poset. St000741The Colin de Verdière graph invariant. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001890The maximum magnitude of the Möbius function of a poset. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001568The smallest positive integer that does not appear twice in the partition. St000102The charge of a semistandard tableau. St001060The distinguishing index of a graph. St000101The cocharge of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000315The number of isolated vertices of a graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001625The Möbius invariant of a lattice. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001783The number of odd automorphisms of a graph. St001845The number of join irreducibles minus the rank of a lattice. St001856The number of edges in the reduced word graph of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001871The number of triconnected components of a graph. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000310The minimal degree of a vertex of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000736The last entry in the first row of a semistandard tableau. St000739The first entry in the last row of a semistandard tableau. St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001518The number of graphs with the same ordinary spectrum as the given graph. St001569The maximal modular displacement of a permutation. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001828The Euler characteristic of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.