Processing math: 33%

Your data matches 450 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
St000093: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> 1 = 0 + 1
([],2)
=> 2 = 1 + 1
([(0,1)],2)
=> 1 = 0 + 1
([],3)
=> 3 = 2 + 1
([(1,2)],3)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
([],4)
=> 4 = 3 + 1
([(2,3)],4)
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(1,2)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([],5)
=> 5 = 4 + 1
([(3,4)],5)
=> 4 = 3 + 1
([(2,4),(3,4)],5)
=> 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(1,4),(2,3)],5)
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
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 α(G) of G.
St000097: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> 1 = 0 + 1
([],2)
=> 1 = 0 + 1
([(0,1)],2)
=> 2 = 1 + 1
([],3)
=> 1 = 0 + 1
([(1,2)],3)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([],4)
=> 1 = 0 + 1
([(2,3)],4)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([],5)
=> 1 = 0 + 1
([(3,4)],5)
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,1),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
Description
The order of the largest clique of the graph. A clique in a graph G is a subset UV(G) such that any pair of vertices in U are adjacent. I.e. the subgraph induced by U is a complete graph.
St000786: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> 1 = 0 + 1
([],2)
=> 2 = 1 + 1
([(0,1)],2)
=> 1 = 0 + 1
([],3)
=> 3 = 2 + 1
([(1,2)],3)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
([],4)
=> 4 = 3 + 1
([(2,3)],4)
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(1,2)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([],5)
=> 5 = 4 + 1
([(3,4)],5)
=> 4 = 3 + 1
([(2,4),(3,4)],5)
=> 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(1,4),(2,3)],5)
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
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.
St001337: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> 1 = 0 + 1
([],2)
=> 2 = 1 + 1
([(0,1)],2)
=> 1 = 0 + 1
([],3)
=> 3 = 2 + 1
([(1,2)],3)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
([],4)
=> 4 = 3 + 1
([(2,3)],4)
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(1,2)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([],5)
=> 5 = 4 + 1
([(3,4)],5)
=> 4 = 3 + 1
([(2,4),(3,4)],5)
=> 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(1,4),(2,3)],5)
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
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 K4 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].
St001338: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> 1 = 0 + 1
([],2)
=> 2 = 1 + 1
([(0,1)],2)
=> 1 = 0 + 1
([],3)
=> 3 = 2 + 1
([(1,2)],3)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
([],4)
=> 4 = 3 + 1
([(2,3)],4)
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(1,2)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([],5)
=> 5 = 4 + 1
([(3,4)],5)
=> 4 = 3 + 1
([(2,4),(3,4)],5)
=> 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(1,4),(2,3)],5)
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
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 K4 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].
Mp00203: Graphs coneGraphs
St001323: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([(0,1)],2)
=> 0
([],2)
=> ([(0,2),(1,2)],3)
=> 1
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(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)
=> 1
([(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
([(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)
=> 1
([(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)
=> 0
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(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)
=> 3
([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
([(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
([(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
([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(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)
=> 2
([(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)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(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
([(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
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(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)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
Description
The independence gap of a graph. This is the difference between the independence number [[St000093]] and the minimal size of a maximally independent set of a graph. In particular, this statistic is 0 for well covered graphs
Mp00203: Graphs coneGraphs
St001336: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([(0,1)],2)
=> 0
([],2)
=> ([(0,2),(1,2)],3)
=> 0
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(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
([(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
([(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)
=> 2
([(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)
=> 3
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(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
([(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)
=> 1
([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(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
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(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
([(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)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The minimal number of vertices in a graph whose complement is triangle-free.
Mp00251: Graphs clique sizesInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1 = 0 + 1
([],2)
=> [1,1]
=> 1 = 0 + 1
([(0,1)],2)
=> [2]
=> 2 = 1 + 1
([],3)
=> [1,1,1]
=> 1 = 0 + 1
([(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> [2,2]
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3 = 2 + 1
([],4)
=> [1,1,1,1]
=> 1 = 0 + 1
([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4 = 3 + 1
([],5)
=> [1,1,1,1,1]
=> 1 = 0 + 1
([(3,4)],5)
=> [2,1,1,1]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 2 = 1 + 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 2 = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 3 = 2 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 4 = 3 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> 3 = 2 + 1
Description
The largest part of an integer partition.
Mp00324: Graphs chromatic difference sequenceInteger compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => 1 = 0 + 1
([],2)
=> [2] => 2 = 1 + 1
([(0,1)],2)
=> [1,1] => 1 = 0 + 1
([],3)
=> [3] => 3 = 2 + 1
([(1,2)],3)
=> [2,1] => 2 = 1 + 1
([(0,2),(1,2)],3)
=> [2,1] => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1 = 0 + 1
([],4)
=> [4] => 4 = 3 + 1
([(2,3)],4)
=> [3,1] => 3 = 2 + 1
([(1,3),(2,3)],4)
=> [3,1] => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => 3 = 2 + 1
([(0,3),(1,2)],4)
=> [2,2] => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> [2,2] => 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1 = 0 + 1
([],5)
=> [5] => 5 = 4 + 1
([(3,4)],5)
=> [4,1] => 4 = 3 + 1
([(2,4),(3,4)],5)
=> [4,1] => 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 4 = 3 + 1
([(1,4),(2,3)],5)
=> [3,2] => 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [3,2] => 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> [3,2] => 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 1 + 1
Description
The largest part of an integer composition.
Mp00324: Graphs chromatic difference sequenceInteger compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => 1 = 0 + 1
([],2)
=> [2] => 2 = 1 + 1
([(0,1)],2)
=> [1,1] => 1 = 0 + 1
([],3)
=> [3] => 3 = 2 + 1
([(1,2)],3)
=> [2,1] => 2 = 1 + 1
([(0,2),(1,2)],3)
=> [2,1] => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1 = 0 + 1
([],4)
=> [4] => 4 = 3 + 1
([(2,3)],4)
=> [3,1] => 3 = 2 + 1
([(1,3),(2,3)],4)
=> [3,1] => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => 3 = 2 + 1
([(0,3),(1,2)],4)
=> [2,2] => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> [2,2] => 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1 = 0 + 1
([],5)
=> [5] => 5 = 4 + 1
([(3,4)],5)
=> [4,1] => 4 = 3 + 1
([(2,4),(3,4)],5)
=> [4,1] => 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 4 = 3 + 1
([(1,4),(2,3)],5)
=> [3,2] => 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [3,2] => 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> [3,2] => 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 1 + 1
Description
The first part of an integer composition.
The following 440 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000808The number of up steps of the associated bargraph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001654The monophonic hull number of a graph. St001656The monophonic position number of a graph. St001674The number of vertices of the largest induced star graph in the 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. St000536The pathwidth of a graph. St000778The metric dimension of a graph. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. 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. 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. 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. St000439The position of the first down step of a Dyck path. St000469The distinguishing number of a graph. St001655The general position number of a graph. St001814The number of partitions interlacing the given partition. 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. St000392The length of the longest run of ones in a binary word. 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. 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. St001340The cardinality of a minimal non-edge isolating set of a graph. 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. 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. 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. 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 S0 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. 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. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) [c0,c1,...,cn1] by adding c0 to cn1. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St000741The Colin de Verdière graph invariant. St000442The maximal area to the right of an up step of a Dyck path. 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. St001108The 2-dynamic chromatic number of a 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. 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. 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. St001638The book thickness of a graph. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001570The minimal number of edges to add to make a graph Hamiltonian. St000299The number of nonisomorphic vertex-induced subtrees. St001093The detour number of a graph. St000738The first entry in the last row of a standard tableau. St000474Dyson's crank of a partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000053The number of valleys of the Dyck path. St000259The diameter of a connected graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000260The radius of a connected graph. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between eiJ and ejJ (the radical of the indecomposable projective modules). St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules S with grade inf at least two in the Nakayama algebra A 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001271The competition number of a graph. 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. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St000455The second largest eigenvalue of a graph if it is integral. St001118The acyclic chromatic index of a graph. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000456The monochromatic index of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001432The order dimension of the partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000003The number of standard Young tableaux of the partition. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000015The number of peaks of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000075The orbit size of a standard tableau under promotion. St000088The row sums of the character table of the symmetric group. St000108The number of partitions contained in the given partition. St000120The number of left tunnels of a Dyck path. St000159The number of distinct parts of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000228The size of a partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000306The bounce count of a Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000390The number of runs of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000459The hook length of the base cell of a partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000517The Kreweras number of an integer partition. St000519The largest length of a factor maximising the subword complexity. St000529The number of permutations whose descent word is the given binary word. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000753The Grundy value for the game of Kayles on a binary word. St000759The smallest missing part in an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000847The number of standard Young tableaux whose descent set is the binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000922The minimal number such that all substrings of this length are unique. St000935The number of ordered refinements of an integer partition. St000947The major index east count of a Dyck path. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has Ext^i(D(A),A)>0 for i>0 for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St000993The multiplicity of the largest part of an integer partition. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in 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. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001129The product of the squares of the parts of a partition. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001161The major index north count of a Dyck path. St001165Number of simple modules with even projective dimension 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. St001183The maximum of projdim(S)+injdim(S) over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001191Number of simple modules S with Ext_A^i(S,A)=0 for all i=0,1,...,g-1 in the corresponding Nakayama algebra A with global dimension g. St001192The maximal dimension of Ext_A^2(S,A) for a simple module S over the corresponding Nakayama algebra A. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001196The global dimension of A minus the global dimension of eAe for the corresponding Nakayama algebra with minimal faithful projective-injective module eA. St001197The global dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. 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: St001205The number of non-simple indecomposable projective-injective modules of the algebra eAe in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by \tau \Omega^1 composed with its inverse in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. 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. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001437The flex of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001471The magnitude of a Dyck path. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001485The modular major index of a binary word. St001487The number of inner corners of a skew partition. St001488The number of corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001614The cyclic permutation representation number of a skew partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001721The degree of a binary word. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001838The number of nonempty primitive factors of a binary word. St001884The number of borders of a binary word. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001955The number of natural descents for set-valued two row standard Young tableaux. St001959The product of the heights of the peaks of a Dyck path. St001281The normalized isoperimetric number of a graph. St001060The distinguishing index of a 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. St001578The minimal number of edges to add or remove to make a graph a line graph. St001624The breadth of a lattice. St001642The Prague dimension of a graph. St001378The product of the cohook lengths of the integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001563The value of the power-sum symmetric function evaluated at 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001545The second Elser number of a connected graph. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St001128The exponens consonantiae of a partition. St001877Number of indecomposable injective modules with projective dimension 2. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000640The rank of the largest boolean interval in a poset. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000770The major index of an integer partition when read from bottom to top. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001632The number of indecomposable injective modules I with dim Ext^1(I,A)=1 for the incidence algebra A of a poset. St001637The number of (upper) dissectors of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000934The 2-degree of an integer partition. St000707The product of the factorials of the parts. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000422The energy of a graph, if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. 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. 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. 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. St001651The Frankl number of a 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. St000264The girth of a graph, which is not a tree.