- FindStat

Your data matches 26 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001490
(load all 4 compositions to match this statistic)

Mapping

St001490: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[]]
 => 1
[[2],[]]
 => 1
[[1,1],[]]
 => 1
[[2,1],[1]]
 => 2
[[3],[]]
 => 1
[[2,1],[]]
 => 1
[[3,1],[1]]
 => 2
[[2,2],[1]]
 => 1
[[3,2],[2]]
 => 2
[[1,1,1],[]]
 => 1
[[2,2,1],[1,1]]
 => 2
[[2,1,1],[1]]
 => 2
[[3,2,1],[2,1]]
 => 3
[[4],[]]
 => 1
[[3,1],[]]
 => 1
[[4,1],[1]]
 => 2
[[2,2],[]]
 => 1
[[3,2],[1]]
 => 1
[[4,2],[2]]
 => 2
[[2,1,1],[]]
 => 1
[[3,2,1],[1,1]]
 => 2
[[3,1,1],[1]]
 => 2
[[4,2,1],[2,1]]
 => 3
[[3,3],[2]]
 => 1
[[4,3],[3]]
 => 2
[[2,2,1],[1]]
 => 1
[[3,3,1],[2,1]]
 => 2
[[3,2,1],[2]]
 => 2
[[4,3,1],[3,1]]
 => 3
[[2,2,2],[1,1]]
 => 1
[[3,3,2],[2,2]]
 => 2
[[3,2,2],[2,1]]
 => 2
[[4,3,2],[3,2]]
 => 3
[[1,1,1,1],[]]
 => 1
[[2,2,2,1],[1,1,1]]
 => 2
[[2,2,1,1],[1,1]]
 => 2
[[3,3,2,1],[2,2,1]]
 => 3
[[2,1,1,1],[1]]
 => 2
[[3,2,2,1],[2,1,1]]
 => 3
[[3,2,1,1],[2,1]]
 => 3
[[4,3,2,1],[3,2,1]]
 => 4
[[5],[]]
 => 1
[[4,1],[]]
 => 1
[[5,1],[1]]
 => 2
[[3,2],[]]
 => 1
[[4,2],[1]]
 => 1
[[5,2],[2]]
 => 2
[[3,1,1],[]]
 => 1
[[4,2,1],[1,1]]
 => 2
[[4,1,1],[1]]
 => 2

Description

The number of connected components of a skew partition.

Matching statistic: St000181
(load all 2 compositions to match this statistic)

Mapping

Mp00185: Skew partitions —cell poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of connected components of the Hasse diagram for the poset.

Matching statistic: St000286
(load all 5 compositions to match this statistic)

Mapping

Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000286: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of connected components of the complement of a graph. The complement of a graph is the graph on the same vertex set with complementary edges.

Matching statistic: St000287
(load all 4 compositions to match this statistic)

Mapping

Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000287: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of connected components of a graph.

Matching statistic: St000010

Mapping

Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000383

Mapping

Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The last part of an integer composition.

Matching statistic: St000544
(load all 2 compositions to match this statistic)

Mapping

Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000544: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cop number of a graph. This is the minimal number of cops needed to catch the robber. The algorithm is from [2].

Matching statistic: St000553

Mapping

Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000553: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of blocks of a graph. A cut vertex is a vertex whose deletion increases the number of connected components. A block is a maximal connected subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.

Matching statistic: St000773

Mapping

Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000773: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The multiplicity of the largest Laplacian eigenvalue in a graph.

Matching statistic: St001363
(load all 2 compositions to match this statistic)

Mapping

Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St001363: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Euler characteristic of a graph according to Knill. This is

$\sum_{k\geq 1} (-1)^{k-1} c_k,$ where

$c_k$ is the number of cliques with

$k$ vertices.

The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000914The sum of the values of the Möbius function of a poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000456The monochromatic index of a connected graph. St001545The second Elser number of a connected graph. 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. St001645The pebbling number of a connected graph. 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. St001330The hat guessing number of a graph.