Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001596
(load all 4 compositions to match this statistic)
Mapping
St001596: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => 0
[[2],[]]
 => 0
[[1,1],[]]
 => 0
[[2,1],[1]]
 => 0
[[3],[]]
 => 0
[[2,1],[]]
 => 0
[[3,1],[1]]
 => 0
[[2,2],[1]]
 => 0
[[3,2],[2]]
 => 0
[[1,1,1],[]]
 => 0
[[2,2,1],[1,1]]
 => 0
[[2,1,1],[1]]
 => 0
[[3,2,1],[2,1]]
 => 0
[[4],[]]
 => 0
[[3,1],[]]
 => 0
[[4,1],[1]]
 => 0
[[2,2],[]]
 => 1
[[3,2],[1]]
 => 0
[[4,2],[2]]
 => 0
[[2,1,1],[]]
 => 0
[[3,2,1],[1,1]]
 => 0
[[3,1,1],[1]]
 => 0
[[4,2,1],[2,1]]
 => 0
[[3,3],[2]]
 => 0
[[4,3],[3]]
 => 0
[[2,2,1],[1]]
 => 0
[[3,3,1],[2,1]]
 => 0
[[3,2,1],[2]]
 => 0
[[4,3,1],[3,1]]
 => 0
[[2,2,2],[1,1]]
 => 0
[[3,3,2],[2,2]]
 => 0
[[3,2,2],[2,1]]
 => 0
[[4,3,2],[3,2]]
 => 0
[[1,1,1,1],[]]
 => 0
[[2,2,2,1],[1,1,1]]
 => 0
[[2,2,1,1],[1,1]]
 => 0
[[3,3,2,1],[2,2,1]]
 => 0
[[2,1,1,1],[1]]
 => 0
[[3,2,2,1],[2,1,1]]
 => 0
[[3,2,1,1],[2,1]]
 => 0
[[4,3,2,1],[3,2,1]]
 => 0
[[5],[]]
 => 0
[[4,1],[]]
 => 0
[[5,1],[1]]
 => 0
[[3,2],[]]
 => 1
[[4,2],[1]]
 => 0
[[5,2],[2]]
 => 0
[[3,1,1],[]]
 => 0
[[4,2,1],[1,1]]
 => 0
[[4,1,1],[1]]
 => 0
Description
The number of two-by-two squares inside a skew partition. This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.
Matching statistic: St001305
(load all 2 compositions to match this statistic)
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001305: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => ([],1)
 => ([],1)
 => 0
[[2],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[1,1],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[2,1],[1]]
 => ([],2)
 => ([],2)
 => 0
[[3],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[3,1],[1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[3,2],[2]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[2,1,1],[1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[3,2,1],[2,1]]
 => ([],3)
 => ([],3)
 => 0
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[4,1],[1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[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)
 => 0
[[4,2],[2]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,2,1],[1,1]]
 => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[3,1,1],[1]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[4,2,1],[2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[4,3],[3]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,3,1],[2,1]]
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[3,2,1],[2]]
 => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[4,3,1],[3,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,3,2],[2,2]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[3,2,2],[2,1]]
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[4,3,2],[3,2]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[2,2,2,1],[1,1,1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[2,2,1,1],[1,1]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[3,3,2,1],[2,2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[2,1,1,1],[1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[3,2,2,1],[2,1,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[3,2,1,1],[2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[4,3,2,1],[3,2,1]]
 => ([],4)
 => ([],4)
 => 0
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[5,1],[1]]
 => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[[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)
 => 0
[[5,2],[2]]
 => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
[[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[4,2,1],[1,1]]
 => ([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[[4,1,1],[1]]
 => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
Description
The number of induced cycles on four vertices in a graph.
Matching statistic: St001311
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001311: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => ([],1)
 => ([],1)
 => 0
[[2],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[1,1],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[2,1],[1]]
 => ([],2)
 => ([],2)
 => 0
[[3],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[3,1],[1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[3,2],[2]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[2,1,1],[1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[3,2,1],[2,1]]
 => ([],3)
 => ([],3)
 => 0
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[4,1],[1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[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)
 => 0
[[4,2],[2]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,2,1],[1,1]]
 => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[3,1,1],[1]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[4,2,1],[2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[4,3],[3]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,3,1],[2,1]]
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[3,2,1],[2]]
 => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[4,3,1],[3,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,3,2],[2,2]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[3,2,2],[2,1]]
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[4,3,2],[3,2]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[2,2,2,1],[1,1,1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[2,2,1,1],[1,1]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[3,3,2,1],[2,2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[2,1,1,1],[1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[3,2,2,1],[2,1,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[3,2,1,1],[2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[4,3,2,1],[3,2,1]]
 => ([],4)
 => ([],4)
 => 0
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[5,1],[1]]
 => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[[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)
 => 0
[[5,2],[2]]
 => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
[[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[4,2,1],[1,1]]
 => ([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[[4,1,1],[1]]
 => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
Description
The cyclomatic number of a graph. This is the minimum number of edges that must be removed from the graph so that the result is a forest. This is also the first Betti number of the graph. It can be computed as $c + m - n$, where $c$ is the number of connected components, $m$ is the number of edges and $n$ is the number of vertices.
Matching statistic: St001317
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001317: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => ([],1)
 => ([],1)
 => 0
[[2],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[1,1],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[2,1],[1]]
 => ([],2)
 => ([],2)
 => 0
[[3],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[3,1],[1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[3,2],[2]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[2,1,1],[1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[3,2,1],[2,1]]
 => ([],3)
 => ([],3)
 => 0
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[4,1],[1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[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)
 => 0
[[4,2],[2]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,2,1],[1,1]]
 => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[3,1,1],[1]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[4,2,1],[2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[4,3],[3]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,3,1],[2,1]]
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[3,2,1],[2]]
 => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[4,3,1],[3,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,3,2],[2,2]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[3,2,2],[2,1]]
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[4,3,2],[3,2]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[2,2,2,1],[1,1,1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[2,2,1,1],[1,1]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[3,3,2,1],[2,2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[2,1,1,1],[1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[3,2,2,1],[2,1,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[3,2,1,1],[2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[4,3,2,1],[3,2,1]]
 => ([],4)
 => ([],4)
 => 0
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[5,1],[1]]
 => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[[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)
 => 0
[[5,2],[2]]
 => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
[[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[4,2,1],[1,1]]
 => ([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[[4,1,1],[1]]
 => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
Description
The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. A graph is a forest if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001324
(load all 2 compositions to match this statistic)
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001324: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => ([],1)
 => ([],1)
 => 0
[[2],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[1,1],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[2,1],[1]]
 => ([],2)
 => ([],2)
 => 0
[[3],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[3,1],[1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[3,2],[2]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[2,1,1],[1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[3,2,1],[2,1]]
 => ([],3)
 => ([],3)
 => 0
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[4,1],[1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[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)
 => 0
[[4,2],[2]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,2,1],[1,1]]
 => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[3,1,1],[1]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[4,2,1],[2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[4,3],[3]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,3,1],[2,1]]
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[3,2,1],[2]]
 => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[4,3,1],[3,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,3,2],[2,2]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[3,2,2],[2,1]]
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[4,3,2],[3,2]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[2,2,2,1],[1,1,1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[2,2,1,1],[1,1]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[3,3,2,1],[2,2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[2,1,1,1],[1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[3,2,2,1],[2,1,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[3,2,1,1],[2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[4,3,2,1],[3,2,1]]
 => ([],4)
 => ([],4)
 => 0
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[5,1],[1]]
 => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[[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)
 => 0
[[5,2],[2]]
 => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
[[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[4,2,1],[1,1]]
 => ([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[[4,1,1],[1]]
 => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
Description
The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. A graph is chordal if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges and $(a,b)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001326
(load all 2 compositions to match this statistic)
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001326: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => ([],1)
 => ([],1)
 => 0
[[2],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[1,1],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[2,1],[1]]
 => ([],2)
 => ([],2)
 => 0
[[3],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[3,1],[1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[3,2],[2]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 0
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[2,1,1],[1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[[3,2,1],[2,1]]
 => ([],3)
 => ([],3)
 => 0
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[4,1],[1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[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)
 => 0
[[4,2],[2]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,2,1],[1,1]]
 => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[3,1,1],[1]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[4,2,1],[2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[4,3],[3]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,3,1],[2,1]]
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[3,2,1],[2]]
 => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[4,3,1],[3,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,3,2],[2,2]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[3,2,2],[2,1]]
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[4,3,2],[3,2]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[2,2,2,1],[1,1,1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[2,2,1,1],[1,1]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[[3,3,2,1],[2,2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[2,1,1,1],[1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[[3,2,2,1],[2,1,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[3,2,1,1],[2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[[4,3,2,1],[3,2,1]]
 => ([],4)
 => ([],4)
 => 0
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[5,1],[1]]
 => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[[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)
 => 0
[[5,2],[2]]
 => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
[[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[4,2,1],[1,1]]
 => ([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[[4,1,1],[1]]
 => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
Description
The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. A graph is an interval graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ is an edge and $(a,b)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001633
(load all 2 compositions to match this statistic)
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => ([],1)
 => 0
[[2],[]]
 => ([(0,1)],2)
 => 0
[[1,1],[]]
 => ([(0,1)],2)
 => 0
[[2,1],[1]]
 => ([],2)
 => 0
[[3],[]]
 => ([(0,2),(2,1)],3)
 => 0
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 0
[[3,1],[1]]
 => ([(1,2)],3)
 => 0
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 0
[[3,2],[2]]
 => ([(1,2)],3)
 => 0
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 0
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => 0
[[2,1,1],[1]]
 => ([(1,2)],3)
 => 0
[[3,2,1],[2,1]]
 => ([],3)
 => 0
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 0
[[4,1],[1]]
 => ([(1,2),(2,3)],4)
 => 0
[[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 0
[[4,2],[2]]
 => ([(0,3),(1,2)],4)
 => 0
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 0
[[3,2,1],[1,1]]
 => ([(1,2),(1,3)],4)
 => 0
[[3,1,1],[1]]
 => ([(0,3),(1,2)],4)
 => 0
[[4,2,1],[2,1]]
 => ([(2,3)],4)
 => 0
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[4,3],[3]]
 => ([(1,2),(2,3)],4)
 => 0
[[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 0
[[3,3,1],[2,1]]
 => ([(1,3),(2,3)],4)
 => 0
[[3,2,1],[2]]
 => ([(1,2),(1,3)],4)
 => 0
[[4,3,1],[3,1]]
 => ([(2,3)],4)
 => 0
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[[3,3,2],[2,2]]
 => ([(0,3),(1,2)],4)
 => 0
[[3,2,2],[2,1]]
 => ([(1,3),(2,3)],4)
 => 0
[[4,3,2],[3,2]]
 => ([(2,3)],4)
 => 0
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[2,2,2,1],[1,1,1]]
 => ([(1,2),(2,3)],4)
 => 0
[[2,2,1,1],[1,1]]
 => ([(0,3),(1,2)],4)
 => 0
[[3,3,2,1],[2,2,1]]
 => ([(2,3)],4)
 => 0
[[2,1,1,1],[1]]
 => ([(1,2),(2,3)],4)
 => 0
[[3,2,2,1],[2,1,1]]
 => ([(2,3)],4)
 => 0
[[3,2,1,1],[2,1]]
 => ([(2,3)],4)
 => 0
[[4,3,2,1],[3,2,1]]
 => ([],4)
 => 0
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 0
[[5,1],[1]]
 => ([(1,4),(3,2),(4,3)],5)
 => 0
[[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)
 => 0
[[5,2],[2]]
 => ([(0,3),(1,4),(4,2)],5)
 => 0
[[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 0
[[4,2,1],[1,1]]
 => ([(1,3),(1,4),(4,2)],5)
 => 0
[[4,1,1],[1]]
 => ([(0,3),(1,4),(4,2)],5)
 => 0
[[6,1],[]]
 => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => ? = 0
[[7,1],[1]]
 => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 0
[[5,2],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => ? = 1
[[6,2],[1]]
 => ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
 => ? = 0
[[7,2],[2]]
 => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 0
[[5,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ? = 0
[[6,2,1],[1,1]]
 => ([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
 => ? = 0
[[6,1,1],[1]]
 => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 0
[[7,2,1],[2,1]]
 => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ? = 0
[[4,3],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ? = 2
[[5,3],[1]]
 => ([(0,6),(1,4),(1,6),(3,2),(4,3),(4,5),(6,5)],7)
 => ? = 1
[[6,3],[2]]
 => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ? = 0
[[7,3],[3]]
 => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ? = 0
[[4,2,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ? = 1
[[5,3,1],[1,1]]
 => ([(1,3),(1,5),(3,6),(4,2),(5,4),(5,6)],7)
 => ? = 1
[[5,2,1],[1]]
 => ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7)
 => ? = 0
[[6,3,1],[2,1]]
 => ([(1,6),(2,3),(2,6),(3,5),(5,4)],7)
 => ? = 0
[[6,2,1],[2]]
 => ([(0,6),(1,3),(1,4),(5,2),(6,5)],7)
 => ? = 0
[[7,3,1],[3,1]]
 => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 0
[[5,2,2],[1,1]]
 => ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7)
 => ? = 0
[[6,3,2],[2,2]]
 => ([(0,4),(1,3),(1,6),(5,2),(6,5)],7)
 => ? = 0
[[6,2,2],[2,1]]
 => ([(0,3),(1,6),(2,6),(3,5),(5,4)],7)
 => ? = 0
[[7,3,2],[3,2]]
 => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 0
[[4,1,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
 => ? = 0
[[5,2,2,1],[1,1,1]]
 => ([(1,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ? = 0
[[5,2,1,1],[1,1]]
 => ([(0,4),(1,3),(1,6),(5,2),(6,5)],7)
 => ? = 0
[[6,3,2,1],[2,2,1]]
 => ([(2,4),(2,6),(5,3),(6,5)],7)
 => ? = 0
[[5,1,1,1],[1]]
 => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ? = 0
[[6,2,2,1],[2,1,1]]
 => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 0
[[6,2,1,1],[2,1]]
 => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 0
[[7,3,2,1],[3,2,1]]
 => ([(3,4),(4,6),(6,5)],7)
 => ? = 0
[[4,4],[1]]
 => ([(0,6),(1,3),(1,6),(2,4),(3,2),(3,5),(5,4),(6,5)],7)
 => ? = 2
[[5,4],[2]]
 => ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
 => ? = 1
[[6,4],[3]]
 => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
 => ? = 0
[[7,4],[4]]
 => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ? = 0
[[3,3,1],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ? = 2
[[4,4,1],[1,1]]
 => ([(1,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
 => ? = 2
[[4,3,1],[1]]
 => ([(0,3),(0,6),(1,4),(1,6),(4,2),(4,5),(6,5)],7)
 => ? = 1
[[5,4,1],[2,1]]
 => ([(1,5),(2,3),(2,5),(3,4),(3,6),(5,6)],7)
 => ? = 1
[[5,3,1],[2]]
 => ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7)
 => ? = 0
[[6,4,1],[3,1]]
 => ([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
 => ? = 0
[[6,3,1],[3]]
 => ([(0,6),(1,4),(1,5),(5,3),(6,2)],7)
 => ? = 0
[[7,4,1],[4,1]]
 => ([(1,6),(2,5),(5,3),(6,4)],7)
 => ? = 0
[[3,2,2],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ? = 2
[[4,3,2],[1,1]]
 => ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
 => ? = 1
[[5,4,2],[2,2]]
 => ([(0,4),(1,3),(1,5),(3,6),(5,2),(5,6)],7)
 => ? = 1
[[4,2,2],[1]]
 => ([(0,3),(0,6),(1,4),(1,6),(3,5),(4,2),(6,5)],7)
 => ? = 1
[[5,3,2],[2,1]]
 => ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
 => ? = 0
[[6,4,2],[3,2]]
 => ([(0,6),(1,3),(2,4),(2,6),(4,5)],7)
 => ? = 0
[[5,2,2],[2]]
 => ([(0,5),(1,3),(1,4),(3,6),(4,6),(5,2)],7)
 => ? = 1
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St001271
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Values
[[1],[]]
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 0 + 1
[[2],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[[1,1],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[[2,1],[1]]
 => ([],2)
 => ([],2)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[3],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 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 = 0 + 1
[[3,1],[1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[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 = 0 + 1
[[3,2],[2]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[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 = 0 + 1
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[2,1,1],[1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[3,2,1],[2,1]]
 => ([],3)
 => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[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 = 0 + 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 = 0 + 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)
 => 1 = 0 + 1
[[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)
 => 2 = 1 + 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 = 0 + 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)
 => 1 = 0 + 1
[[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 = 0 + 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)
 => 1 = 0 + 1
[[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)
 => 1 = 0 + 1
[[4,2,1],[2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[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 = 0 + 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)
 => 1 = 0 + 1
[[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 = 0 + 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)
 => 1 = 0 + 1
[[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)
 => 1 = 0 + 1
[[4,3,1],[3,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[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 = 0 + 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)
 => 1 = 0 + 1
[[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)
 => 1 = 0 + 1
[[4,3,2],[3,2]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[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 = 0 + 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)
 => 1 = 0 + 1
[[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)
 => 1 = 0 + 1
[[3,3,2,1],[2,2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[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)
 => 1 = 0 + 1
[[3,2,2,1],[2,1,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[3,2,1,1],[2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[4,3,2,1],[3,2,1]]
 => ([],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[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 = 0 + 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 = 0 + 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)
 => 1 = 0 + 1
[[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 + 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 = 0 + 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)
 => 1 = 0 + 1
[[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 = 0 + 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)
 => 1 = 0 + 1
[[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)
 => 1 = 0 + 1
[[5,2,1],[2,1]]
 => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[[3,3],[1]]
 => ([(0,3),(1,2),(1,3),(2,4),(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 + 1
[[2,2,1],[]]
 => ([(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 + 1
[[3,3,1],[1,1]]
 => ([(1,2),(1,3),(2,4),(3,4)],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 + 1
[[2,2,2],[1]]
 => ([(0,3),(1,2),(1,3),(2,4),(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 + 1
[[3,2,2],[2]]
 => ([(1,2),(1,3),(2,4),(3,4)],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 + 1
[[4,2],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[3,3],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[[4,3],[1]]
 => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[[4,3,1],[1,1]]
 => ([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[4,4],[2]]
 => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[3,3,1],[1]]
 => ([(0,2),(0,4),(1,3),(1,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[4,4,1],[2,1]]
 => ([(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[2,2,2],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[[3,3,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[4,4,2],[2,2]]
 => ([(0,4),(1,2),(1,3),(2,5),(3,5)],6)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[[3,2,2],[1]]
 => ([(0,2),(0,4),(1,3),(1,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[4,2,2],[2]]
 => ([(0,4),(1,2),(1,3),(2,5),(3,5)],6)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[[2,2,1,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[3,3,2,1],[1,1,1]]
 => ([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[3,3,1,1],[1,1]]
 => ([(0,4),(1,2),(1,3),(2,5),(3,5)],6)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[[4,4,2,1],[2,2,1]]
 => ([(2,3),(2,4),(3,5),(4,5)],6)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[[3,3,2],[2]]
 => ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[4,3,2],[3]]
 => ([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[3,3,3],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[[4,3,3],[3,1]]
 => ([(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[2,2,2,1],[1]]
 => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[[3,3,3,1],[2,1,1]]
 => ([(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[3,2,2,1],[2]]
 => ([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[4,3,3,1],[3,1,1]]
 => ([(2,3),(2,4),(3,5),(4,5)],6)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[[2,2,2,2],[1,1]]
 => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[3,3,2,2],[2,2]]
 => ([(0,4),(1,2),(1,3),(2,5),(3,5)],6)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[[3,2,2,2],[2,1]]
 => ([(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[4,3,2,2],[3,2]]
 => ([(2,3),(2,4),(3,5),(4,5)],6)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[[7],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 1
[[6,1],[]]
 => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 1
[[7,1],[1]]
 => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
 => ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 1
[[5,2],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
 => ([(0,5),(0,7),(1,2),(1,3),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
[[6,2],[1]]
 => ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 1
[[7,2],[2]]
 => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,7),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 1
[[5,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 1
[[6,2,1],[1,1]]
 => ([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
 => ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
 => ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 1
[[6,1,1],[1]]
 => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,7),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 1
[[7,2,1],[2,1]]
 => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 1
[[4,3],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,2),(1,4),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[[5,3],[1]]
 => ([(0,6),(1,4),(1,6),(3,2),(4,3),(4,5),(6,5)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
[[6,3],[2]]
 => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 1
[[7,3],[3]]
 => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 1
[[4,2,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
Description
The competition number of a graph. The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x, v)$ and $(y, v)$ are arcs of $D$. For any graph, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ is the smallest number of such isolated vertices.
Matching statistic: St000455
Mapping
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Values
[[1],[]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[2],[]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[1,1],[]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[2,1],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[3],[]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[2,1],[]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[3,1],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[2,2],[1]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[3,2],[2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[1,1,1],[]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[2,2,1],[1,1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[2,1,1],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[3,2,1],[2,1]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0
[[4],[]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[3,1],[]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[4,1],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[2,2],[]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 1
[[3,2],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[4,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0
[[2,1,1],[]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[3,2,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0
[[3,1,1],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[4,2,1],[2,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 0
[[3,3],[2]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[4,3],[3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[2,2,1],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[3,3,1],[2,1]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0
[[3,2,1],[2]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0
[[4,3,1],[3,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 0
[[2,2,2],[1,1]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[3,3,2],[2,2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[3,2,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0
[[4,3,2],[3,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 0
[[1,1,1,1],[]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[2,2,2,1],[1,1,1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0
[[3,3,2,1],[2,2,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 0
[[2,1,1,1],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[3,2,2,1],[2,1,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 0
[[3,2,1,1],[2,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 0
[[4,3,2,1],[3,2,1]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 0
[[5],[]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[4,1],[]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[5,1],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[3,2],[]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 1
[[4,2],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[5,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0
[[3,1,1],[]]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[[4,2,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0
[[4,1,1],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[[5,3,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[6,3,1],[3,1]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[5,3,2],[2,2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[6,3,2],[3,2]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[5,3,2,1],[2,2,1]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[6,3,2,1],[3,2,1]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 0
[[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 0
[[5,4,1],[3,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[5,3,1],[3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[6,4,1],[4,1]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 0
[[5,4,2],[3,2]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[5,3,2],[3,1]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[6,4,2],[4,2]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 0
[[4,3,2,1],[2,1,1]]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(3,6),(4,5)],7)
 => 0
[[4,3,1,1],[2,1]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[5,3,1,1],[3,1]]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => 0
[[5,3,3],[3,2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[6,4,3],[4,3]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 0
[[4,3,3,1],[2,2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[4,3,2,1],[2,2]]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(3,6),(4,5)],7)
 => 0
[[4,2,2,1],[2,1]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[4,3,2,2],[2,2,1]]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(3,6),(4,5)],7)
 => 0
[[5,3,3,2],[3,2,2]]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => 0
[[3,2,2,1,1],[1,1,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 0
[[4,3,3,2,1],[2,2,2,1]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[4,2,2,1,1],[2,1,1]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 0
[[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 0
[[5,5,2],[4,2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[5,4,2],[4,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[6,5,2],[5,2]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[4,4,2,1],[3,1,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 0
[[5,5,2,1],[4,2,1]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[5,4,2,1],[4,1,1]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[6,5,2,1],[5,2,1]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 0
[[5,4,3],[4,2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[6,5,3],[5,3]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[4,4,3,1],[3,2,1]]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(3,6),(4,5)],7)
 => 0
[[4,4,2,1],[3,2]]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(3,6),(4,5)],7)
 => 0
[[4,3,2,1],[3,1]]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(3,6),(4,5)],7)
 => 0
[[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 0
[[4,4,3,2],[3,2,2]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[4,4,2,2],[3,2,1]]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(3,6),(4,5)],7)
 => 0
[[4,3,2,2],[3,1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[3,3,2,2,1],[2,1,1,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 0
[[3,3,2,1,1],[2,1,1]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[4,3,2,2,1],[3,1,1,1]]
 => ([(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)
 => ([(4,5)],6)
 => 0
[[4,3,3,1],[3,2]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 0
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.
Matching statistic: St000456
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => ([],1)
 => ([],1)
 => ? = 0 + 1
[[2],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 0 + 1
[[1,1],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 0 + 1
[[2,1],[1]]
 => ([],2)
 => ([],2)
 => ? = 0 + 1
[[3],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[3,1],[1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0 + 1
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[3,2],[2]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0 + 1
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0 + 1
[[2,1,1],[1]]
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0 + 1
[[3,2,1],[2,1]]
 => ([],3)
 => ([],3)
 => ? = 0 + 1
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[4,1],[1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[4,2],[2]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 0 + 1
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[3,2,1],[1,1]]
 => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[[3,1,1],[1]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 0 + 1
[[4,2,1],[2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 0 + 1
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[4,3],[3]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[3,3,1],[2,1]]
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[[3,2,1],[2]]
 => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[[4,3,1],[3,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 0 + 1
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[3,3,2],[2,2]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 0 + 1
[[3,2,2],[2,1]]
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[[4,3,2],[3,2]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 0 + 1
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[2,2,2,1],[1,1,1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[[2,2,1,1],[1,1]]
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 0 + 1
[[3,3,2,1],[2,2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 0 + 1
[[2,1,1,1],[1]]
 => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[[3,2,2,1],[2,1,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 0 + 1
[[3,2,1,1],[2,1]]
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 0 + 1
[[4,3,2,1],[3,2,1]]
 => ([],4)
 => ([],4)
 => ? = 0 + 1
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[5,1],[1]]
 => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 0 + 1
[[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[5,2],[2]]
 => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 0 + 1
[[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[4,2,1],[1,1]]
 => ([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 0 + 1
[[4,1,1],[1]]
 => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 0 + 1
[[5,2,1],[2,1]]
 => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 0 + 1
[[3,3],[1]]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[5,3],[3]]
 => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 0 + 1
[[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[3,3,1],[1,1]]
 => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 1 + 1
[[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[4,3,1],[2,1]]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 0 + 1
[[4,2,1],[2]]
 => ([(0,4),(1,2),(1,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 0 + 1
[[5,3,1],[3,1]]
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 0 + 1
[[3,2,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[4,3,2],[2,2]]
 => ([(0,4),(1,2),(1,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 0 + 1
[[4,2,2],[2,1]]
 => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 0 + 1
[[5,3,2],[3,2]]
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 0 + 1
[[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[3,2,2,1],[1,1,1]]
 => ([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 0 + 1
[[3,2,1,1],[1,1]]
 => ([(0,4),(1,2),(1,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 0 + 1
[[4,3,2,1],[2,2,1]]
 => ([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 0 + 1
[[3,1,1,1],[1]]
 => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 0 + 1
[[4,2,2,1],[2,1,1]]
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 0 + 1
[[4,2,1,1],[2,1]]
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 0 + 1
[[5,3,2,1],[3,2,1]]
 => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 0 + 1
[[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[5,4],[4]]
 => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 0 + 1
[[3,3,1],[2]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[4,4,1],[3,1]]
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 0 + 1
[[4,3,1],[3]]
 => ([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 0 + 1
[[5,4,1],[4,1]]
 => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 0 + 1
[[2,2,2],[1]]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[2,2,1,1],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[2,2,2,1],[1,1]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[2,2,2,2],[1,1,1]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[[4,2],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[[5,2],[1]]
 => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[[3,3],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => 3 = 2 + 1
[[4,3],[1]]
 => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
[[5,3],[2]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
[[4,2,1],[1]]
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[[4,2,2],[1,1]]
 => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[[4,4],[2]]
 => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[[5,4],[3]]
 => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[[3,3,1],[1]]
 => ([(0,2),(0,4),(1,3),(1,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
The following 1 statistic also match your data. Click on any of them to see the details.
St001871The number of triconnected components of a graph.