Your data matches 239 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001490
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
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
(load all 2 compositions to match this statistic)
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: St001024
(load all 2 compositions to match this statistic)
Mapping
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[[2],[]]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[[1,1],[]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[2,1],[1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[3],[]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[[2,1],[]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[3,1],[1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[[2,2],[1]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[[3,2],[2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[[1,1,1],[]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[2,2,1],[1,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
[[2,1,1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[[3,2,1],[2,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[[4],[]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[3,1],[]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[[4,1],[1]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[[2,2],[]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[[3,2],[1]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[[4,2],[2]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[[2,1,1],[]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[[3,2,1],[1,1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[[3,1,1],[1]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[4,2,1],[2,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[[3,3],[2]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[4,3],[3]]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[2,2,1],[1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
[[3,3,1],[2,1]]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3
[[3,2,1],[2]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[[4,3,1],[3,1]]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[[2,2,2],[1,1]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[[3,3,2],[2,2]]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[[3,2,2],[2,1]]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[4,3,2],[3,2]]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[[1,1,1,1],[]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3
[[5],[]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[[4,1],[]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[[5,1],[1]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2
[[3,2],[]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[[4,2],[1]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[[5,2],[2]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1
[[3,1,1],[]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[4,2,1],[1,1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[[4,1,1],[1]]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2
Description
Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
The following 229 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001363The Euler characteristic of a graph according to Knill. St000932The number of occurrences of the pattern UDU in a Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000914The sum of the values of the Möbius function of a poset. St000022The number of fixed points of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000678The number of up steps after the last double rise of a Dyck path. St000011The number of touch points (or returns) of a Dyck path. St001050The number of terminal closers of a set partition. St001733The number of weak left to right maxima of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000288The number of ones in a binary word. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000013The height of a Dyck path. St000105The number of blocks in the set partition. St000141The maximum drop size of a permutation. St000167The number of leaves of an ordered tree. St000216The absolute length of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000382The first part of an integer composition. St000470The number of runs in a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000653The last descent of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000691The number of changes of a binary word. St000840The number of closers smaller than the largest opener in a perfect matching. St000876The number of factors in the Catalan decomposition of a binary word. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001480The number of simple summands of the module J^2/J^3. St001809The index of the step at the first peak of maximal height in a Dyck path. St000015The number of peaks of a Dyck path. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000166The depth minus 1 of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000381The largest part of an integer composition. St000390The number of runs of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St000991The number of right-to-left minima of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St000738The first entry in the last row of a standard tableau. St001462The number of factors of a standard tableaux under concatenation. St001471The magnitude of a Dyck path. St000006The dinv of a Dyck path. St000505The biggest entry in the block containing the 1. St000746The number of pairs with odd minimum in a perfect matching. St000971The smallest closer of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000143The largest repeated part of a partition. St000702The number of weak deficiencies of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001589The nesting number of a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000740The last entry of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000990The first ascent of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000385The number of vertices with out-degree 1 in a binary tree. St000444The length of the maximal rise of a Dyck path. St000619The number of cyclic descents of a permutation. St000654The first descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000925The number of topologically connected components of a set partition. St001246The maximal difference between two consecutive entries of a permutation. St000485The length of the longest cycle of a permutation. St001330The hat guessing number of a graph. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001488The number of corners of a skew partition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St001062The maximal size of a block of a set partition. St000526The number of posets with combinatorially isomorphic order polytopes. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000144The pyramid weight of the Dyck path. St000384The maximal part of the shifted composition of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001523The degree of symmetry of a Dyck path. St001621The number of atoms of a lattice. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000080The rank of the poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St000937The number of positive values of the symmetric group character corresponding to the partition. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000454The largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001153The number of blocks with even minimum in a set partition. St000075The orbit size of a standard tableau under promotion. St001151The number of blocks with odd minimum. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001568The smallest positive integer that does not appear twice in the partition. St001052The length of the exterior of a permutation. St000460The hook length of the last cell along the main diagonal of an integer partition. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000455The second largest eigenvalue of a graph if it is integral. St000091The descent variation of a composition. St000989The number of final rises of a permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000100The number of linear extensions of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000910The number of maximal chains of minimal length in a poset. St001060The distinguishing index of a graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000420The number of Dyck paths that are weakly above a Dyck path. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St000993The multiplicity of the largest part of an integer partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000307The number of rowmotion orbits of a poset. 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. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St000456The monochromatic index of a connected graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001435The number of missing boxes in the first row. St000264The girth of a graph, which is not a tree. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. 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. 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. St000782The indicator function of whether a given perfect matching is an L & P matching. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001948The number of augmented double ascents of a permutation. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.