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Your data matches 205 different statistics following compositions of up to 3 maps.
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Matching statistic: St000773
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(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(1,2)],3)
=> 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(1,2)],3)
=> 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],6)
=> ([],1)
=> 1
([(4,5)],6)
=> ([(1,2)],3)
=> 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
Description
The multiplicity of the largest Laplacian eigenvalue in a graph.
Matching statistic: St000776
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Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(1,2)],3)
=> 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(1,2)],3)
=> 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],6)
=> ([],1)
=> 1
([(4,5)],6)
=> ([(1,2)],3)
=> 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
Description
The maximal multiplicity of an eigenvalue in a graph.
Matching statistic: St001319
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(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([],1)
=> 0 = 1 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> ([],1)
=> 0 = 1 - 1
([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([],4)
=> ([],1)
=> 0 = 1 - 1
([(2,3)],4)
=> ([(1,2)],3)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([],5)
=> ([],1)
=> 0 = 1 - 1
([(3,4)],5)
=> ([(1,2)],3)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 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 = 2 - 1
([],6)
=> ([],1)
=> 0 = 1 - 1
([(4,5)],6)
=> ([(1,2)],3)
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> 1 = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
Description
The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph.
A graph is a disjoint union of isolated vertices and a star if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ is 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: St000382
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1] => 1
([],2)
=> ([],1)
=> [1] => 1
([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => 1
([],3)
=> ([],1)
=> [1] => 1
([(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
([],4)
=> ([],1)
=> [1] => 1
([(2,3)],4)
=> ([(1,2)],3)
=> [1,2] => 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> [1,2] => 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1,1] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
([],5)
=> ([],1)
=> [1] => 1
([(3,4)],5)
=> ([(1,2)],3)
=> [1,2] => 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [1,2] => 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [1,2] => 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,3] => 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> [2,2] => 2
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> [1,2] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
([],6)
=> ([],1)
=> [1] => 1
([(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> [2,3] => 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> [2,3] => 2
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> [2,2] => 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> [1,2] => 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> [2,2] => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> [1,2] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1,1] => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> [2,2] => 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> [1,1] => 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
Description
The first part of an integer composition.
Matching statistic: St001624
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([],5)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([],6)
=> ([],1)
=> ([],1)
=> 1
([(4,5)],6)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St000377
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> []
=> 0 = 1 - 1
([],2)
=> ([],1)
=> []
=> 0 = 1 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
([],3)
=> ([],1)
=> []
=> 0 = 1 - 1
([(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([],4)
=> ([],1)
=> []
=> 0 = 1 - 1
([(2,3)],4)
=> ([(1,2)],3)
=> [1]
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> [1]
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([],5)
=> ([],1)
=> []
=> 0 = 1 - 1
([(3,4)],5)
=> ([(1,2)],3)
=> [1]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [1]
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [1]
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> [1]
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 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)
=> [3]
=> 1 = 2 - 1
([],6)
=> ([],1)
=> []
=> 0 = 1 - 1
([(4,5)],6)
=> ([(1,2)],3)
=> [1]
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1]
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1]
=> 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1]
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1 = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> [1,1]
=> 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> [1]
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> [1]
=> 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1 = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
Description
The dinv defect of an integer partition.
This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St000469
Values
([],1)
=> ([],1)
=> ([],2)
=> 2 = 1 + 1
([],2)
=> ([],1)
=> ([],2)
=> 2 = 1 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 2 = 1 + 1
([],3)
=> ([],1)
=> ([],2)
=> 2 = 1 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([],4)
=> ([],1)
=> ([],2)
=> 2 = 1 + 1
([(2,3)],4)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([],5)
=> ([],1)
=> ([],2)
=> 2 = 1 + 1
([(3,4)],5)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 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)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([],6)
=> ([],1)
=> ([],2)
=> 2 = 1 + 1
([(4,5)],6)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> 3 = 2 + 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> 3 = 2 + 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> 3 = 2 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> 3 = 2 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
Description
The distinguishing number of a graph.
This is the minimal number of colours needed to colour the vertices of a graph, such that only the trivial automorphism of the graph preserves the colouring.
For connected graphs, this statistic is at most one plus the maximal degree of the graph, with equality attained for complete graphs, complete bipartite graphs and the cycle with five vertices, see Theorem 4.2 of [2].
Matching statistic: St000047
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000047: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000047: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1] => [1] => 1
([],2)
=> ([],1)
=> [1] => [1] => 1
([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => [2] => 1
([],3)
=> ([],1)
=> [1] => [1] => 1
([(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => [1,2] => 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => [2] => 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1] => 2
([],4)
=> ([],1)
=> [1] => [1] => 1
([(2,3)],4)
=> ([(1,2)],3)
=> [1,2] => [1,2] => 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> [1,2] => [1,2] => 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => [2] => 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => [1,2,1] => 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,2,1] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1,1] => [2] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1] => 2
([],5)
=> ([],1)
=> [1] => [1] => 1
([(3,4)],5)
=> ([(1,2)],3)
=> [1,2] => [1,2] => 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [1,2] => [1,2] => 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [1,2] => [1,2] => 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [2] => 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,3] => [1,1,2,1] => 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> [2,2] => [1,2,1] => 2
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,2,1] => 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> [1,2] => [1,2] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,2,1] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => [2] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1] => 2
([],6)
=> ([],1)
=> [1] => [1] => 1
([(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => [1,2] => 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => [1,2] => 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => [1,2] => 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => [1,2] => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> [2,3] => [1,1,2,1] => 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> [2,3] => [1,1,2,1] => 2
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,2,1] => 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> [2,2] => [1,2,1] => 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> [1,2] => [1,2] => 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> [2,2] => [1,2,1] => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,2,1] => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> [1,2] => [1,2] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,2,1] => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1] => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> [2,2] => [1,2,1] => 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,2,1] => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1] => 2
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1] => 2
Description
The number of standard immaculate tableaux of a given shape.
See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.
Matching statistic: St000271
Values
([],1)
=> ([],1)
=> [1] => ([],1)
=> 1
([],2)
=> ([],1)
=> [1] => ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> [1] => ([],1)
=> 1
([(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([],4)
=> ([],1)
=> [1] => ([],1)
=> 1
([(2,3)],4)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([],5)
=> ([],1)
=> [1] => ([],1)
=> 1
([(3,4)],5)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([],6)
=> ([],1)
=> [1] => ([],1)
=> 1
([(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
Description
The chromatic index of a graph.
This is the minimal number of colours needed such that no two adjacent edges have the same colour.
Matching statistic: St000383
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1] => [1] => 1
([],2)
=> ([],1)
=> [1] => [1] => 1
([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 1
([],3)
=> ([],1)
=> [1] => [1] => 1
([(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 2
([],4)
=> ([],1)
=> [1] => [1] => 1
([(2,3)],4)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => [2,2] => 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 2
([],5)
=> ([],1)
=> [1] => [1] => 1
([(3,4)],5)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,3] => [3,2] => 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> [2,2] => [2,2] => 2
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 2
([],6)
=> ([],1)
=> [1] => [1] => 1
([(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> [2,3] => [3,2] => 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> [2,3] => [3,2] => 2
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> [2,2] => [2,2] => 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> [2,2] => [2,2] => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> [2,2] => [2,2] => 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 2
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 2
Description
The last part of an integer composition.
The following 195 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000390The number of runs of ones in a binary word. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001313The number of Dyck paths above the lattice path given by a binary word. St001487The number of inner corners of a skew partition. St001716The 1-improper chromatic number of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St000091The descent variation of a composition. St000292The number of ascents of a binary word. St000313The number of degree 2 vertices of a graph. St000348The non-inversion sum of a binary word. St000448The number of pairs of vertices of a graph with distance 2. St000552The number of cut vertices of a graph. St000682The Grundy value of Welter's game on a binary word. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St001308The number of induced paths on three vertices in a graph. St001323The independence gap of a graph. St001350Half of the Albertson index of a graph. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001689The number of celebrities in a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000258The burning number of a graph. St000482The (zero)-forcing number of a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000918The 2-limited packing number of a graph. 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. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001340The cardinality of a minimal non-edge isolating set of a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000744The length of the path to the largest entry in a standard Young tableau. St001118The acyclic chromatic index of a graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000939The number of characters of the symmetric group whose value on the partition is positive. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001060The distinguishing index of a graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000264The girth of a graph, which is not a tree. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001592The maximal number of simple paths between any two different vertices of a graph. St000741The Colin de Verdière graph invariant. St001877Number of indecomposable injective modules with projective dimension 2. St000456The monochromatic index of a connected graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000455The second largest eigenvalue of a graph if it is integral. St001545The second Elser number of a connected graph. St000464The Schultz index of a connected graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000273The domination number of a graph. St001829The common independence number of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000699The toughness times the least common multiple of 1,. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St001322The size of a minimal independent dominating set in a graph. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. 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. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. 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. St001763The Hurwitz number of an integer 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. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. 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. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. 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. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001389The number of partitions of the same length below the given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001339The irredundance number of a graph. 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 $eA$. St000478Another weight of a partition according to Alladi. St000681The Grundy value of Chomp on Ferrers diagrams. St001281The normalized isoperimetric number of a graph. St000379The number of Hamiltonian cycles in a graph. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000934The 2-degree of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000567The sum of the products of all pairs of parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The 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 vertex labelled trees. 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. St001100The 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 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. St001875The number of simple modules with projective dimension at most 1. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000997The even-odd crank of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
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