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Your data matches 245 different statistics following compositions of up to 3 maps.
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Matching statistic: St000093
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 2
([(0,1)],2)
=> 1
([],3)
=> 3
([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> 4
([(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> 3
([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],5)
=> 5
([(3,4)],5)
=> 4
([(2,4),(3,4)],5)
=> 4
([(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(1,4),(2,3)],5)
=> 3
([(1,4),(2,3),(3,4)],5)
=> 3
([(0,1),(2,4),(3,4)],5)
=> 3
([(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000097
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 1
([(0,1)],2)
=> 2
([],3)
=> 1
([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> 1
([(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> 1
([(3,4)],5)
=> 2
([(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000786
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 2
([(0,1)],2)
=> 1
([],3)
=> 3
([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> 4
([(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> 3
([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],5)
=> 5
([(3,4)],5)
=> 4
([(2,4),(3,4)],5)
=> 4
([(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(1,4),(2,3)],5)
=> 3
([(1,4),(2,3),(3,4)],5)
=> 3
([(0,1),(2,4),(3,4)],5)
=> 3
([(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Matching statistic: St001337
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 2
([(0,1)],2)
=> 1
([],3)
=> 3
([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> 4
([(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> 3
([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],5)
=> 5
([(3,4)],5)
=> 4
([(2,4),(3,4)],5)
=> 4
([(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(1,4),(2,3)],5)
=> 3
([(1,4),(2,3),(3,4)],5)
=> 3
([(0,1),(2,4),(3,4)],5)
=> 3
([(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
Description
The upper domination number of a graph.
This is the maximum cardinality of a minimal dominating set of $G$.
The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
Matching statistic: St001338
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 2
([(0,1)],2)
=> 1
([],3)
=> 3
([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> 4
([(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> 3
([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],5)
=> 5
([(3,4)],5)
=> 4
([(2,4),(3,4)],5)
=> 4
([(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(1,4),(2,3)],5)
=> 3
([(1,4),(2,3),(3,4)],5)
=> 3
([(0,1),(2,4),(3,4)],5)
=> 3
([(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
Description
The upper irredundance number of a graph.
A set $S$ of vertices is irredundant, if there is no vertex in $S$, whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of $S$.
The upper irredundance number is the largest size of a maximal irredundant set.
The smallest graph with different upper irredundance number and upper domination number [[St001337]] has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [2].
Matching statistic: St000147
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [1,1]
=> 1
([(0,1)],2)
=> [2]
=> 2
([],3)
=> [1,1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> 2
([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
([],4)
=> [1,1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> 2
([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
([],5)
=> [1,1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> 2
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St000381
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => 1
([],2)
=> [2] => 2
([(0,1)],2)
=> [1,1] => 1
([],3)
=> [3] => 3
([(1,2)],3)
=> [2,1] => 2
([(0,2),(1,2)],3)
=> [2,1] => 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([],4)
=> [4] => 4
([(2,3)],4)
=> [3,1] => 3
([(1,3),(2,3)],4)
=> [3,1] => 3
([(0,3),(1,3),(2,3)],4)
=> [3,1] => 3
([(0,3),(1,2)],4)
=> [2,2] => 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
([],5)
=> [5] => 5
([(3,4)],5)
=> [4,1] => 4
([(2,4),(3,4)],5)
=> [4,1] => 4
([(1,4),(2,4),(3,4)],5)
=> [4,1] => 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 4
([(1,4),(2,3)],5)
=> [3,2] => 3
([(1,4),(2,3),(3,4)],5)
=> [3,2] => 3
([(0,1),(2,4),(3,4)],5)
=> [3,2] => 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
Description
The largest part of an integer composition.
Matching statistic: St000382
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => 1
([],2)
=> [2] => 2
([(0,1)],2)
=> [1,1] => 1
([],3)
=> [3] => 3
([(1,2)],3)
=> [2,1] => 2
([(0,2),(1,2)],3)
=> [2,1] => 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([],4)
=> [4] => 4
([(2,3)],4)
=> [3,1] => 3
([(1,3),(2,3)],4)
=> [3,1] => 3
([(0,3),(1,3),(2,3)],4)
=> [3,1] => 3
([(0,3),(1,2)],4)
=> [2,2] => 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
([],5)
=> [5] => 5
([(3,4)],5)
=> [4,1] => 4
([(2,4),(3,4)],5)
=> [4,1] => 4
([(1,4),(2,4),(3,4)],5)
=> [4,1] => 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 4
([(1,4),(2,3)],5)
=> [3,2] => 3
([(1,4),(2,3),(3,4)],5)
=> [3,2] => 3
([(0,1),(2,4),(3,4)],5)
=> [3,2] => 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
Description
The first part of an integer composition.
Matching statistic: St000808
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => 1
([],2)
=> [2] => 2
([(0,1)],2)
=> [1,1] => 1
([],3)
=> [3] => 3
([(1,2)],3)
=> [2,1] => 2
([(0,2),(1,2)],3)
=> [2,1] => 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([],4)
=> [4] => 4
([(2,3)],4)
=> [3,1] => 3
([(1,3),(2,3)],4)
=> [3,1] => 3
([(0,3),(1,3),(2,3)],4)
=> [3,1] => 3
([(0,3),(1,2)],4)
=> [2,2] => 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
([],5)
=> [5] => 5
([(3,4)],5)
=> [4,1] => 4
([(2,4),(3,4)],5)
=> [4,1] => 4
([(1,4),(2,4),(3,4)],5)
=> [4,1] => 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 4
([(1,4),(2,3)],5)
=> [3,2] => 3
([(1,4),(2,3),(3,4)],5)
=> [3,2] => 3
([(0,1),(2,4),(3,4)],5)
=> [3,2] => 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
Description
The number of up steps of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.
Matching statistic: St001316
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
Description
The domatic number of a graph.
This is the maximal size of a partition of the vertices into dominating sets.
The following 235 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001330The hat guessing number of a graph. St001654The monophonic hull number of a graph. St001656The monophonic position number of a graph. St001323The independence gap of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001674The number of vertices of the largest induced star graph in the graph. St000010The length of the partition. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000273The domination number of a graph. St000335The difference of lower and upper interactions. St000383The last part of an integer composition. St000482The (zero)-forcing number of a graph. St000544The cop number of a graph. St000636The hull number of a graph. St000667The greatest common divisor of the parts of the partition. St000676The number of odd rises of a Dyck path. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000734The last entry in the first row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000778The metric dimension of a graph. St000822The Hadwiger number of the graph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000899The maximal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000917The open packing number of a graph. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001571The Cartan determinant of the integer partition. St001580The acyclic chromatic number of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001829The common independence number of a graph. St001949The rigidity index of a graph. St000089The absolute variation of a composition. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000439The position of the first down step of a Dyck path. St000469The distinguishing number of a graph. St000536The pathwidth of a graph. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001277The degeneracy of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001655The general position number of a graph. St001814The number of partitions interlacing the given partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000258The burning number of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000363The number of minimal vertex covers of a graph. St000392The length of the longest run of ones in a binary word. St000548The number of different non-empty partial sums of an integer partition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000876The number of factors in the Catalan decomposition of a binary word. St000916The packing number of a graph. St000918The 2-limited packing number of a graph. St000926The clique-coclique number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001363The Euler characteristic of a graph according to Knill. St001366The maximal multiplicity of a degree of a vertex of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001415The length of the longest palindromic prefix of a binary word. St001530The depth of a Dyck path. St001672The restrained domination number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001933The largest multiplicity of a part in an integer partition. St001963The tree-depth of a graph. St000024The number of double up and double down steps of a Dyck path. St000090The variation of a composition. St000091The descent variation of a composition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000362The size of a minimal vertex cover of a graph. 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 $c_{n-1}$. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001479The number of bridges of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001777The number of weak descents in an integer composition. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St000741The Colin de Verdière graph invariant. St000444The length of the maximal rise of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St001108The 2-dynamic chromatic number of a graph. St000443The number of long tunnels of a Dyck path. St000479The Ramsey number of a graph. St000678The number of up steps after the last double rise of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. 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. St001645The pebbling number of a connected graph. St000157The number of descents of a standard tableau. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000874The position of the last double rise in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001834The number of non-isomorphic minors of a graph. St001644The dimension of a graph. 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. St001480The number of simple summands of the module J^2/J^3. St000144The pyramid weight of the Dyck path. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St001498The normalised height of a Nakayama algebra with magnitude 1. St001638The book thickness of a graph. St000993The multiplicity of the largest part of an integer partition. St001060The distinguishing index of a graph. St000738The first entry in the last row of a standard tableau. St000474Dyson's crank of a partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St000259The diameter of a connected graph. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000260The radius of a connected graph. St000015The number of peaks of a Dyck path. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St001165Number of simple modules with even projective dimension in 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. St001118The acyclic chromatic index of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. 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 $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001621The number of atoms of a lattice. 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 $eA$. St001570The minimal number of edges to add to make a graph Hamiltonian. St000455The second largest eigenvalue of a graph if it is integral. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000294The number of distinct factors of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000518The number of distinct subsequences in a binary word. St000532The total number of rook placements on a Ferrers board. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000983The length of the longest alternating subword. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. 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. 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:
St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. 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. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001267The length of the Lyndon factorization of the binary word. St001400The total number of Littlewood-Richardson tableaux of given shape. St001437The flex of a binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001624The breadth of a lattice. St001578The minimal number of edges to add or remove to make a graph a line graph. St001642The Prague dimension of a graph. St000264The girth of a graph, which is not a tree. St001545The second Elser number of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000910The number of maximal chains of minimal length in a poset. St000939The number of characters of the symmetric group whose value on the partition is positive. St001568The smallest positive integer that does not appear twice in the partition. St001668The number of points of the poset minus the width of the poset. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000422The energy of a graph, if it is integral. St001875The number of simple modules with projective dimension at most 1. St001651The Frankl number of a lattice. 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. 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.
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