- FindStat

Your data matches 357 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001326
(load all 20 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St001326: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. A graph is an interval graph if and only if in any linear ordering of its vertices, there are no three vertices

$a < b < c$ such that

$(a,c)$ is an edge and

$(a,b)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

Matching statistic: St001577
(load all 20 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St001577: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The minimal number of edges to add or remove to make a graph a cograph. A cograph is a graph that can be obtained from the one vertex graph by complementation and disjoint union.

Matching statistic: St001703
(load all 21 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St001703: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The villainy of a graph. The villainy of a permutation of a proper coloring

$c$ of a graph is the minimal Hamming distance between

$c$ and a proper coloring. The villainy of a graph is the maximal villainy of a permutation of a proper coloring.

Matching statistic: St001476
(load all 12 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St001476: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1).

Matching statistic: St001642
(load all 12 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St001642: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Prague dimension of a graph. This is the least number of complete graphs such that the graph is an induced subgraph of their (categorical) product. Put differently, this is the least number

$n$ such that the graph can be embedded into

$\mathbb N^n$ , where two points are connected by an edge if and only if they differ in all coordinates.

Matching statistic: St001734
(load all 14 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St001734: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The lettericity of a graph. Let

$D$ be a digraph on

$k$ vertices, possibly with loops and let

$w$ be a word of length

$n$ whose letters are vertices of

$D$ . The letter graph corresponding to

$D$ and

$w$ is the graph with vertex set

$\{1,\dots,n\}$ whose edges are the pairs

$(i,j)$ with

$i < j$ sucht that

$(w_i, w_j)$ is a (directed) edge of

$D$ .

Matching statistic: St000143
(load all 2 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest repeated part of a partition. If the parts of the partition are all distinct, the value of the statistic is defined to be zero.

Matching statistic: St000150

Mapping

Mp00154: Graphs —core⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The floored half-sum of the multiplicities of a partition. This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].

Matching statistic: St000171
(load all 7 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The degree of the graph. This is the maximal vertex degree of a graph.

Matching statistic: St000259
(load all 4 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

The following 347 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000310The minimal degree of a vertex of a graph. St000351The determinant of the adjacency matrix of a graph. St000387The matching number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000778The metric dimension of a graph. St001270The bandwidth of a graph. St001271The competition number of a graph. St001277The degeneracy of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000258The burning number of a graph. St000271The chromatic index of a graph. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000820The number of compositions obtained by rotating the composition. St000822The Hadwiger number of the graph. St000918The 2-limited packing number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001315The dissociation number of a graph. St001478The number of nowhere zero 4-flows of a graph. St001494The Alon-Tarsi number of a graph. St001512The minimum rank of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001655The general position number of a graph. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001883The mutual visibility number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St001093The detour 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. St000008The major index of the composition. St000089The absolute variation of a composition. St000093The cardinality of a maximal independent set of vertices of a graph. St000095The number of triangles of a graph. St000097The order of the largest clique of the graph. St000142The number of even parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000377The dinv defect of an integer partition. St000378The diagonal inversion number of an integer partition. St000403The Szeged index minus the Wiener index of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000995The largest even part of an integer partition. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001214The aft of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. 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. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001592The maximal number of simple paths between any two different vertices of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001657The number of twos in an integer partition. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001777The number of weak descents in an integer composition. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001931The weak major index of an integer composition regarded as a word. St000047The number of standard immaculate tableaux of a given shape. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000390The number of runs of ones in a binary word. St000450The number of edges minus the number of vertices plus 2 of a graph. St000644The number of graphs with given frequency partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000935The number of ordered refinements of an integer partition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. 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. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001282The number of graphs with the same chromatic polynomial. St001286The annihilation number of a graph. St001312Number of parabolic noncrossing partitions indexed by the composition. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001389The number of partitions of the same length below the given integer partition. St001463The number of distinct columns in the nullspace of a graph. St001487The number of inner corners of a skew partition. St001716The 1-improper chromatic number of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001739The number of graphs with the same edge polytope as the given graph. 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. St001746The coalition number of a graph. St001814The number of partitions interlacing the given partition. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000422The energy of a graph, if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St001846The number of elements which do not have a complement in the lattice. St000455The second largest eigenvalue of a graph if it is integral. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001881The number of factors of a lattice as a Cartesian product of lattices. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001621The number of atoms of a lattice. St001622The number of join-irreducible elements of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001616The number of neutral elements in a lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001624The breadth of a lattice. St001625The Möbius invariant of a lattice. St001679The number of subsets of a lattice whose meet is the bottom element. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001720The minimal length of a chain of small intervals in a lattice. St001754The number of tolerances of a finite lattice. St001833The number of linear intervals in a lattice. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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

$eA$ . St001330The hat guessing number of a graph. St001619The number of non-isomorphic sublattices of a lattice. St001620The number of sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001281The normalized isoperimetric number of a graph. St000934The 2-degree of an integer partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000376The bounce deficit of a Dyck path. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000456The monochromatic index of a connected graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000762The sum of the positions of the weak records of an integer composition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001500The global dimension of magnitude 1 Nakayama algebras. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001256Number of simple reflexive modules that are 2-stable reflexive. St000264The girth of a graph, which is not a tree. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000993The multiplicity of the largest part of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. 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. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001204Call 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001248Sum of the even parts of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001118The acyclic chromatic index of a graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001890The maximum magnitude of the Möbius function of a poset. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000069The number of maximal elements of a poset. St001651The Frankl number of a lattice. St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000488The number of cycles of a permutation of length at most 2. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000787The number of flips required to make a perfect matching noncrossing. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in 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. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000068The number of minimal elements in a poset. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001479The number of bridges of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001568The smallest positive integer that does not appear twice in the partition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001060The distinguishing index of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001301The first Betti number of the order complex associated with the poset. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000225Difference between largest and smallest parts in a partition. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001432The order dimension of the partition. 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. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001793The difference between the clique number and the chromatic number of a graph. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000322The skewness of a graph. St001438The number of missing boxes of a skew partition. St001490The number of connected components of a skew partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000096The number of spanning trees of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000315The number of isolated vertices of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St001307The number of induced stars on four vertices in a graph. St000914The sum of the values of the Möbius function of a poset. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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

$eA$ .