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Your data matches 296 different statistics following compositions of up to 3 maps.
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Matching statistic: St000299
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> 1 = 3 - 2
([],2)
=> 1 = 3 - 2
([(0,1)],2)
=> 2 = 4 - 2
([],3)
=> 1 = 3 - 2
([(1,2)],3)
=> 2 = 4 - 2
([(0,2),(1,2)],3)
=> 3 = 5 - 2
([(0,1),(0,2),(1,2)],3)
=> 2 = 4 - 2
([],4)
=> 1 = 3 - 2
([(2,3)],4)
=> 2 = 4 - 2
([(1,3),(2,3)],4)
=> 3 = 5 - 2
([(0,3),(1,3),(2,3)],4)
=> 4 = 6 - 2
([(0,3),(1,2)],4)
=> 2 = 4 - 2
([(0,3),(1,2),(2,3)],4)
=> 4 = 6 - 2
([(1,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 5 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
Description
The number of nonisomorphic vertex-induced subtrees.
Matching statistic: St000453
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
([],1)
=> 1 = 3 - 2
([],2)
=> 1 = 3 - 2
([(0,1)],2)
=> 2 = 4 - 2
([],3)
=> 1 = 3 - 2
([(1,2)],3)
=> 2 = 4 - 2
([(0,2),(1,2)],3)
=> 3 = 5 - 2
([(0,1),(0,2),(1,2)],3)
=> 2 = 4 - 2
([],4)
=> 1 = 3 - 2
([(2,3)],4)
=> 2 = 4 - 2
([(1,3),(2,3)],4)
=> 3 = 5 - 2
([(0,3),(1,3),(2,3)],4)
=> 3 = 5 - 2
([(0,3),(1,2)],4)
=> 2 = 4 - 2
([(0,3),(1,2),(2,3)],4)
=> 4 = 6 - 2
([(1,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 6 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 5 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
Description
The number of distinct Laplacian eigenvalues of a graph.
Matching statistic: St001304
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Values
([],1)
=> 1 = 3 - 2
([],2)
=> 1 = 3 - 2
([(0,1)],2)
=> 2 = 4 - 2
([],3)
=> 1 = 3 - 2
([(1,2)],3)
=> 2 = 4 - 2
([(0,2),(1,2)],3)
=> 2 = 4 - 2
([(0,1),(0,2),(1,2)],3)
=> 3 = 5 - 2
([],4)
=> 1 = 3 - 2
([(2,3)],4)
=> 2 = 4 - 2
([(1,3),(2,3)],4)
=> 2 = 4 - 2
([(0,3),(1,3),(2,3)],4)
=> 2 = 4 - 2
([(0,3),(1,2)],4)
=> 4 = 6 - 2
([(0,3),(1,2),(2,3)],4)
=> 3 = 5 - 2
([(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 4 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 6 - 2
Description
The number of maximally independent sets of vertices of a graph.
An '''independent set''' of vertices of a graph is a set of vertices no two of which are adjacent. If a set of vertices is independent then so is every subset. This statistic counts the number of maximally independent sets of vertices.
Matching statistic: St000777
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Values
([],1)
=> ([(0,1)],2)
=> 2 = 3 - 1
([],2)
=> ([(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000010
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1 = 3 - 2
([],2)
=> [1,1]
=> 2 = 4 - 2
([(0,1)],2)
=> [2]
=> 1 = 3 - 2
([],3)
=> [1,1,1]
=> 3 = 5 - 2
([(1,2)],3)
=> [2,1]
=> 2 = 4 - 2
([(0,2),(1,2)],3)
=> [2,2]
=> 2 = 4 - 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 3 - 2
([],4)
=> [1,1,1,1]
=> 4 = 6 - 2
([(2,3)],4)
=> [2,1,1]
=> 3 = 5 - 2
([(1,3),(2,3)],4)
=> [2,2,1]
=> 3 = 5 - 2
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3 = 5 - 2
([(0,3),(1,2)],4)
=> [2,2]
=> 2 = 4 - 2
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3 = 5 - 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2 = 4 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> 2 = 4 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4 = 6 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> 2 = 4 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 3 - 2
Description
The length of the partition.
Matching statistic: St001318
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1 = 3 - 2
([],2)
=> ([],1)
=> 1 = 3 - 2
([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 4 - 2
([],3)
=> ([],1)
=> 1 = 3 - 2
([(1,2)],3)
=> ([(1,2)],3)
=> 3 = 5 - 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 4 - 2
([],4)
=> ([],1)
=> 1 = 3 - 2
([(2,3)],4)
=> ([(1,2)],3)
=> 3 = 5 - 2
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 3 = 5 - 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 4 = 6 - 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4 = 6 - 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 4 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
Description
The number of vertices of the largest induced subforest with the same number of connected components of a graph.
Matching statistic: St001321
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1 = 3 - 2
([],2)
=> ([],1)
=> 1 = 3 - 2
([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 4 - 2
([],3)
=> ([],1)
=> 1 = 3 - 2
([(1,2)],3)
=> ([(1,2)],3)
=> 3 = 5 - 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 4 - 2
([],4)
=> ([],1)
=> 1 = 3 - 2
([(2,3)],4)
=> ([(1,2)],3)
=> 3 = 5 - 2
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 3 = 5 - 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 4 = 6 - 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4 = 6 - 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 4 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
Description
The number of vertices of the largest induced subforest of a graph.
Matching statistic: St001352
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> ([(0,1)],2)
=> 1 = 3 - 2
([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 4 - 2
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 4 - 2
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 4 - 2
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 4 = 6 - 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 4 - 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 5 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 4 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
Description
The number of internal nodes in the modular decomposition of a graph.
Matching statistic: St001725
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1 = 3 - 2
([],2)
=> ([],1)
=> 1 = 3 - 2
([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 4 - 2
([],3)
=> ([],1)
=> 1 = 3 - 2
([(1,2)],3)
=> ([(1,2)],3)
=> 2 = 4 - 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 5 - 2
([],4)
=> ([],1)
=> 1 = 3 - 2
([(2,3)],4)
=> ([(1,2)],3)
=> 2 = 4 - 2
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2 = 4 - 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 3 = 5 - 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 5 - 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 6 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 4 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 5 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 6 - 2
Description
The harmonious chromatic number of a graph.
A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.
Matching statistic: St000300
Values
([],1)
=> ([(0,1)],2)
=> ([],1)
=> 2 = 3 - 1
([],2)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 3 = 4 - 1
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 2 = 3 - 1
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 4 = 5 - 1
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 3 = 4 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 2 = 3 - 1
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5 = 6 - 1
([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4 = 5 - 1
([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4 = 5 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4 = 5 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 3 = 4 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4 = 5 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 3 = 4 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5 = 6 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 2 = 3 - 1
Description
The number of independent sets of vertices of a graph.
An independent set of vertices of a graph $G$ is a subset $U \subset V(G)$ such that no two vertices in $U$ are adjacent.
This is also the number of vertex covers of $G$ as the map $U \mapsto V(G)\setminus U$ is a bijection between independent sets of vertices and vertex covers.
The size of the largest independent set, also called independence number of $G$, is [[St000093]]
The following 286 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000301The number of facets of the stable set polytope of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001486The number of corners of the ribbon associated with an integer composition. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001746The coalition number of a graph. St001963The tree-depth of a graph. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000087The number of induced subgraphs. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000146The Andrews-Garvan crank of a partition. St000147The largest part of an integer partition. St000171The degree of the graph. St000172The Grundy number of a graph. St000228The size of a partition. St000286The number of connected components of the complement of a graph. St000288The number of ones in a binary word. St000363The number of minimal vertex covers of a graph. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000733The row containing the largest entry of a standard tableau. St000822The Hadwiger number of the graph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000926The clique-coclique number of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001302The number of minimally dominating sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001391The disjunction number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001924The number of cells in an integer partition whose arm and leg length coincide. St000053The number of valleys of the Dyck path. St000157The number of descents of a standard tableau. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000310The minimal degree of a vertex of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000519The largest length of a factor maximising the subword complexity. St000536The pathwidth of a graph. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001345The Hamming dimension of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001812The biclique partition number of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000806The semiperimeter of the associated bargraph. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000439The position of the first down step of a Dyck path. St000638The number of up-down runs of a permutation. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000007The number of saliances of the permutation. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000144The pyramid weight of the Dyck path. St000153The number of adjacent cycles of a permutation. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000443The number of long tunnels of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000542The number of left-to-right-minima of a permutation. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000734The last entry in the first row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000843The decomposition number of a perfect matching. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001058The breadth of the ordered tree. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001267The length of the Lyndon factorization of the binary word. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001372The length of a longest cyclic run of ones of a binary word. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001480The number of simple summands of the module J^2/J^3. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000052The number of valleys of a Dyck path not on the x-axis. St000155The number of exceedances (also excedences) of a permutation. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000224The sorting index of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000245The number of ascents of a permutation. St000292The number of ascents of a binary word. St000317The cycle descent number of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000331The number of upper interactions of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000358The number of occurrences of the pattern 31-2. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000546The number of global descents of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000672The number of minimal elements in Bruhat order not less than the permutation. St000691The number of changes of a binary word. St000732The number of double deficiencies of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001489The maximum of the number of descents and the number of inverse descents. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001712The number of natural descents of a standard Young tableau. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001777The number of weak descents in an integer composition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000061The number of nodes on the left branch of a binary tree. St000444The length of the maximal rise of a Dyck path. St000668The least common multiple of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000925The number of topologically connected components of a set partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000354The number of recoils of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000494The number of inversions of distance at most 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000796The stat' of a permutation. St000809The reduced reflection length of the permutation. St000874The position of the last double rise in a Dyck path. St000957The number of Bruhat lower covers of a permutation. St000984The number of boxes below precisely one peak. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000259The diameter of a connected graph. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000455The second largest eigenvalue of a graph if it is integral. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St000735The last entry on the main diagonal of a standard tableau. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000643The size of the largest orbit of antichains under Panyushev complementation. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001651The Frankl number of a lattice. St000422The energy of a graph, if it is integral. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001881The number of factors of a lattice as a Cartesian product of lattices. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph. St001845The number of join irreducibles minus the rank of a lattice. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001626The number of maximal proper sublattices of a lattice.
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