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Your data matches 848 different statistics following compositions of up to 3 maps.
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Matching statistic: St000344
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Values
([(0,1)],2)
=> 0
([(1,2)],3)
=> 0
([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> 1
([(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> 1
([(3,4)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> 0
([(1,4),(2,3),(3,4)],5)
=> 0
([(0,1),(2,4),(3,4)],5)
=> 0
([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(4,5)],6)
=> 0
([(3,5),(4,5)],6)
=> 0
([(2,5),(3,5),(4,5)],6)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(2,5),(3,4)],6)
=> 0
([(2,5),(3,4),(4,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> 0
([(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> 0
([(0,5),(1,4),(2,3)],6)
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> 0
([(5,6)],7)
=> 0
([(4,6),(5,6)],7)
=> 0
([(3,6),(4,6),(5,6)],7)
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
([(3,6),(4,5)],7)
=> 0
([(3,6),(4,5),(5,6)],7)
=> 0
([(2,3),(4,6),(5,6)],7)
=> 0
([(4,5),(4,6),(5,6)],7)
=> 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> 0
([(1,6),(2,5),(3,4)],7)
=> 0
([(2,6),(3,5),(4,5),(4,6)],7)
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> 0
Description
The number of strongly connected outdegree sequences of a graph.
This is the evaluation of the Tutte polynomial at $x=0$ and $y=1$. According to [1,2], the set of strongly connected outdegree sequences is in bijection with strongly connected minimal orientations and also with external spanning trees.
Matching statistic: St001311
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Values
([(0,1)],2)
=> 0
([(1,2)],3)
=> 0
([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> 1
([(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> 1
([(3,4)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> 0
([(1,4),(2,3),(3,4)],5)
=> 0
([(0,1),(2,4),(3,4)],5)
=> 0
([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(4,5)],6)
=> 0
([(3,5),(4,5)],6)
=> 0
([(2,5),(3,5),(4,5)],6)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(2,5),(3,4)],6)
=> 0
([(2,5),(3,4),(4,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> 0
([(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> 0
([(0,5),(1,4),(2,3)],6)
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> 0
([(5,6)],7)
=> 0
([(4,6),(5,6)],7)
=> 0
([(3,6),(4,6),(5,6)],7)
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
([(3,6),(4,5)],7)
=> 0
([(3,6),(4,5),(5,6)],7)
=> 0
([(2,3),(4,6),(5,6)],7)
=> 0
([(4,5),(4,6),(5,6)],7)
=> 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> 0
([(1,6),(2,5),(3,4)],7)
=> 0
([(2,6),(3,5),(4,5),(4,6)],7)
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> 0
Description
The cyclomatic number of a graph.
This is the minimum number of edges that must be removed from the graph so that the result is a forest. This is also the first Betti number of the graph. It can be computed as $c + m - n$, where $c$ is the number of connected components, $m$ is the number of edges and $n$ is the number of vertices.
Matching statistic: St001317
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Values
([(0,1)],2)
=> 0
([(1,2)],3)
=> 0
([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> 1
([(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> 1
([(3,4)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> 0
([(1,4),(2,3),(3,4)],5)
=> 0
([(0,1),(2,4),(3,4)],5)
=> 0
([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(4,5)],6)
=> 0
([(3,5),(4,5)],6)
=> 0
([(2,5),(3,5),(4,5)],6)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(2,5),(3,4)],6)
=> 0
([(2,5),(3,4),(4,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> 0
([(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> 0
([(0,5),(1,4),(2,3)],6)
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> 0
([(5,6)],7)
=> 0
([(4,6),(5,6)],7)
=> 0
([(3,6),(4,6),(5,6)],7)
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
([(3,6),(4,5)],7)
=> 0
([(3,6),(4,5),(5,6)],7)
=> 0
([(2,3),(4,6),(5,6)],7)
=> 0
([(4,5),(4,6),(5,6)],7)
=> 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> 0
([(1,6),(2,5),(3,4)],7)
=> 0
([(2,6),(3,5),(4,5),(4,6)],7)
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> 0
Description
The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph.
A graph is a forest if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001328
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Values
([(0,1)],2)
=> 0
([(1,2)],3)
=> 0
([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> 1
([(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> 1
([(3,4)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> 0
([(1,4),(2,3),(3,4)],5)
=> 0
([(0,1),(2,4),(3,4)],5)
=> 0
([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(4,5)],6)
=> 0
([(3,5),(4,5)],6)
=> 0
([(2,5),(3,5),(4,5)],6)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(2,5),(3,4)],6)
=> 0
([(2,5),(3,4),(4,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> 0
([(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> 0
([(0,5),(1,4),(2,3)],6)
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> 0
([(5,6)],7)
=> 0
([(4,6),(5,6)],7)
=> 0
([(3,6),(4,6),(5,6)],7)
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
([(3,6),(4,5)],7)
=> 0
([(3,6),(4,5),(5,6)],7)
=> 0
([(2,3),(4,6),(5,6)],7)
=> 0
([(4,5),(4,6),(5,6)],7)
=> 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> 0
([(1,6),(2,5),(3,4)],7)
=> 0
([(2,6),(3,5),(4,5),(4,6)],7)
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> 0
Description
The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph.
A graph is bipartite if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001331
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Values
([(0,1)],2)
=> 0
([(1,2)],3)
=> 0
([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> 1
([(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> 1
([(3,4)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> 0
([(1,4),(2,3),(3,4)],5)
=> 0
([(0,1),(2,4),(3,4)],5)
=> 0
([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(4,5)],6)
=> 0
([(3,5),(4,5)],6)
=> 0
([(2,5),(3,5),(4,5)],6)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(2,5),(3,4)],6)
=> 0
([(2,5),(3,4),(4,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> 0
([(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> 0
([(0,5),(1,4),(2,3)],6)
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> 0
([(5,6)],7)
=> 0
([(4,6),(5,6)],7)
=> 0
([(3,6),(4,6),(5,6)],7)
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
([(3,6),(4,5)],7)
=> 0
([(3,6),(4,5),(5,6)],7)
=> 0
([(2,3),(4,6),(5,6)],7)
=> 0
([(4,5),(4,6),(5,6)],7)
=> 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> 0
([(1,6),(2,5),(3,4)],7)
=> 0
([(2,6),(3,5),(4,5),(4,6)],7)
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> 0
Description
The size of the minimal feedback vertex set.
A feedback vertex set is a set of vertices whose removal results in an acyclic graph.
Matching statistic: St001335
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Values
([(0,1)],2)
=> 0
([(1,2)],3)
=> 0
([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> 1
([(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> 1
([(3,4)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> 0
([(1,4),(2,3),(3,4)],5)
=> 0
([(0,1),(2,4),(3,4)],5)
=> 0
([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(4,5)],6)
=> 0
([(3,5),(4,5)],6)
=> 0
([(2,5),(3,5),(4,5)],6)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(2,5),(3,4)],6)
=> 0
([(2,5),(3,4),(4,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> 0
([(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> 0
([(0,5),(1,4),(2,3)],6)
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> 0
([(5,6)],7)
=> 0
([(4,6),(5,6)],7)
=> 0
([(3,6),(4,6),(5,6)],7)
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
([(3,6),(4,5)],7)
=> 0
([(3,6),(4,5),(5,6)],7)
=> 0
([(2,3),(4,6),(5,6)],7)
=> 0
([(4,5),(4,6),(5,6)],7)
=> 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> 0
([(1,6),(2,5),(3,4)],7)
=> 0
([(2,6),(3,5),(4,5),(4,6)],7)
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> 0
Description
The cardinality of a minimal cycle-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains all cycles.
Matching statistic: St001336
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Values
([(0,1)],2)
=> 0
([(1,2)],3)
=> 0
([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> 1
([(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> 1
([(3,4)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> 0
([(1,4),(2,3),(3,4)],5)
=> 0
([(0,1),(2,4),(3,4)],5)
=> 0
([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(4,5)],6)
=> 0
([(3,5),(4,5)],6)
=> 0
([(2,5),(3,5),(4,5)],6)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(2,5),(3,4)],6)
=> 0
([(2,5),(3,4),(4,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> 0
([(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> 0
([(0,5),(1,4),(2,3)],6)
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> 0
([(5,6)],7)
=> 0
([(4,6),(5,6)],7)
=> 0
([(3,6),(4,6),(5,6)],7)
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
([(3,6),(4,5)],7)
=> 0
([(3,6),(4,5),(5,6)],7)
=> 0
([(2,3),(4,6),(5,6)],7)
=> 0
([(4,5),(4,6),(5,6)],7)
=> 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> 0
([(1,6),(2,5),(3,4)],7)
=> 0
([(2,6),(3,5),(4,5),(4,6)],7)
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> 0
Description
The minimal number of vertices in a graph whose complement is triangle-free.
Matching statistic: St001736
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Values
([(0,1)],2)
=> 0
([(1,2)],3)
=> 0
([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> 1
([(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> 1
([(3,4)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> 0
([(1,4),(2,3),(3,4)],5)
=> 0
([(0,1),(2,4),(3,4)],5)
=> 0
([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(4,5)],6)
=> 0
([(3,5),(4,5)],6)
=> 0
([(2,5),(3,5),(4,5)],6)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(2,5),(3,4)],6)
=> 0
([(2,5),(3,4),(4,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> 0
([(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> 0
([(0,5),(1,4),(2,3)],6)
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> 0
([(5,6)],7)
=> 0
([(4,6),(5,6)],7)
=> 0
([(3,6),(4,6),(5,6)],7)
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
([(3,6),(4,5)],7)
=> 0
([(3,6),(4,5),(5,6)],7)
=> 0
([(2,3),(4,6),(5,6)],7)
=> 0
([(4,5),(4,6),(5,6)],7)
=> 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> 0
([(1,6),(2,5),(3,4)],7)
=> 0
([(2,6),(3,5),(4,5),(4,6)],7)
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> 0
Description
The total number of cycles in a graph.
For example, the complete graph on four vertices has four triangles and three different four-cycles.
Matching statistic: St001797
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Values
([(0,1)],2)
=> 0
([(1,2)],3)
=> 0
([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> 1
([(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> 1
([(3,4)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> 0
([(1,4),(2,3),(3,4)],5)
=> 0
([(0,1),(2,4),(3,4)],5)
=> 0
([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(4,5)],6)
=> 0
([(3,5),(4,5)],6)
=> 0
([(2,5),(3,5),(4,5)],6)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(2,5),(3,4)],6)
=> 0
([(2,5),(3,4),(4,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> 0
([(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> 0
([(0,5),(1,4),(2,3)],6)
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> 0
([(5,6)],7)
=> 0
([(4,6),(5,6)],7)
=> 0
([(3,6),(4,6),(5,6)],7)
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
([(3,6),(4,5)],7)
=> 0
([(3,6),(4,5),(5,6)],7)
=> 0
([(2,3),(4,6),(5,6)],7)
=> 0
([(4,5),(4,6),(5,6)],7)
=> 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> 0
([(1,6),(2,5),(3,4)],7)
=> 0
([(2,6),(3,5),(4,5),(4,6)],7)
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> 0
Description
The number of overfull subgraphs of a graph.
A subgraph $H=(V, E)$ of a graph $G$ is overfull if $|E| > \Delta(G) \lfloor\frac{V}{2}\rfloor$.
This statistic counts the number of subsets $E$ of the edge set of $G$, such that the edge-induced subgraph is overfull.
Matching statistic: St000272
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Values
([(0,1)],2)
=> 1 = 0 + 1
([(1,2)],3)
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(2,3)],4)
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(3,4)],5)
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
([(4,5)],6)
=> 1 = 0 + 1
([(3,5),(4,5)],6)
=> 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(2,5),(3,4)],6)
=> 1 = 0 + 1
([(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
([(1,2),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
=> 1 = 0 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
([(5,6)],7)
=> 1 = 0 + 1
([(4,6),(5,6)],7)
=> 1 = 0 + 1
([(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
([(3,6),(4,5)],7)
=> 1 = 0 + 1
([(3,6),(4,5),(5,6)],7)
=> 1 = 0 + 1
([(2,3),(4,6),(5,6)],7)
=> 1 = 0 + 1
([(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> 1 = 0 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
=> 1 = 0 + 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> 1 = 0 + 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> 1 = 0 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> 1 = 0 + 1
Description
The treewidth of a graph.
A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.
The following 838 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000536The pathwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001592The maximal number of simple paths between any two different vertices of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001029The size of the core of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000387The matching number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000535The rank-width of a graph. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000944The 3-degree of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001071The beta invariant of the graph. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001280The number of parts of an integer partition that are at least two. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001512The minimum rank of a graph. St001638The book thickness of a graph. St001651The Frankl number of a lattice. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000182The number of permutations whose cycle type is the given integer partition. St000268The number of strongly connected orientations of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000378The diagonal inversion number of an integer partition. St000453The number of distinct Laplacian eigenvalues of a graph. St000537The cutwidth of a graph. St001073The number of nowhere zero 3-flows of a graph. St001093The detour number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001270The bandwidth of a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001644The dimension of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001716The 1-improper chromatic number of a graph. St001777The number of weak descents in an integer composition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001962The proper pathwidth of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000668The least common multiple of the parts of the partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001883The mutual visibility number of a graph. St000095The number of triangles of a graph. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000257The number of distinct parts of a partition that occur at least twice. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000386The number of factors DDU in a Dyck path. St000481The number of upper covers of a partition in dominance order. 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. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001092The number of distinct even parts of a partition. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001214The aft of an integer partition. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. 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. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001730The number of times the path corresponding to a binary word crosses the base line. St001910The height of the middle non-run of a Dyck path. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000003The number of standard Young tableaux of the partition. St000006The dinv of a Dyck path. St000053The number of valleys of the Dyck path. St000159The number of distinct parts of the integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000258The burning number of a graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000299The number of nonisomorphic vertex-induced subtrees. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000362The size of a minimal vertex cover of a graph. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000783The side length of the largest staircase partition fitting into a partition. St000935The number of ordered refinements of an integer partition. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000381The largest part of an integer composition. St000382The first part of an integer composition. 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. St000734The last entry in the first row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000759The smallest missing part in an integer partition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. 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:
St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001458The rank of the adjacency matrix of a graph. St001581The achromatic number of a graph. St001654The monophonic hull number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000806The semiperimeter of the associated bargraph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000091The descent variation of a composition. St000137The Grundy value of an integer partition. St000183The side length of the Durfee square of an integer partition. St000185The weighted size of a partition. St000292The number of ascents of a binary word. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000547The number of even non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000877The depth of the binary word interpreted as a path. St000897The number of different multiplicities of parts of an integer partition. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001172The number of 1-rises at odd height of a Dyck path. St001176The size of a partition minus its first part. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001234The number of indecomposable three dimensional modules with projective dimension one. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a 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. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001584The area statistic between a Dyck path and its bounce path. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001961The sum of the greatest common divisors of all pairs of parts. St000005The bounce statistic of a Dyck path. St000012The area of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000047The number of standard immaculate tableaux of a given shape. St000048The multinomial of the parts of a partition. St000079The number of alternating sign matrices for a given Dyck path. St000120The number of left tunnels of a Dyck path. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000306The bounce count of a Dyck path. St000335The difference of lower and upper interactions. St000346The number of coarsenings of a partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000443The number of long tunnels of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000531The leading coefficient of the rook polynomial of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000617The number of global maxima of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000659The number of rises of length at least 2 of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. 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. St000792The Grundy value for the game of ruler on a binary word. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000847The number of standard Young tableaux whose descent set is the binary word. St000874The position of the last double rise in a Dyck path. St000903The number of different parts of an integer composition. St000920The logarithmic height of a Dyck path. St000947The major index east count of a Dyck path. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000984The number of boxes below precisely one peak. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. 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. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. 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. St001480The number of simple summands of the module J^2/J^3. St001481The minimal height of a peak of a Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001884The number of borders of a binary word. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001933The largest multiplicity of a part in an integer partition. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000144The pyramid weight of the Dyck path. St000228The size of a partition. St000383The last part of an integer composition. St000393The number of strictly increasing runs in a binary word. St000420The number of Dyck paths that are weakly above a Dyck path. St000439The position of the first down step of a Dyck path. St000444The length of the maximal rise of a Dyck path. 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. St000479The Ramsey number of a graph. St000636The hull number of a graph. St000708The product of the parts of an integer partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000933The number of multipartitions of sizes given by an integer partition. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000983The length of the longest alternating subword. 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. 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001279The sum of the parts of an integer partition that are at least two. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St001342The number of vertices in the center of a graph. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001437The flex of a binary word. 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. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001808The box weight or horizontal decoration of a Dyck path. 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. St001065Number of indecomposable reflexive modules 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 St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000671The maximin edge-connectivity for choosing a subgraph. St001271The competition number of a graph. St001309The number of four-cliques in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000452The number of distinct eigenvalues of a graph. St000482The (zero)-forcing number of a 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000948The chromatic discriminant of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001642The Prague dimension of a graph. St000741The Colin de Verdière graph invariant. St001746The coalition number of a graph. St001758The number of orbits of promotion on a graph. St001917The order of toric promotion on the set of labellings of a graph. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000379The number of Hamiltonian cycles in a graph. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000454The largest eigenvalue of a graph if it is integral. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001281The normalized isoperimetric number of a graph. St001345The Hamming dimension of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001812The biclique partition number of a graph. St001869The maximum cut size of a graph. St001949The rigidity index of a graph. St000087The number of induced subgraphs. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001316The domatic number 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. St001459The number of zero columns in the nullspace of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001645The pebbling number of a connected 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. St001725The harmonious chromatic number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St001706The number of closed sets in a graph. St001871The number of triconnected components of a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001845The number of join irreducibles minus the rank of a lattice. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000934The 2-degree of an integer partition. 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. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000567The sum of the products of all pairs of parts. St000658The number of rises of length 2 of a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. 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. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000455The second largest eigenvalue of a graph if it is integral. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. 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 $eA$. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000738The first entry in the last row of a standard tableau. St000456The monochromatic index of a connected graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. 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. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. 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. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001177Twice the mean value of the major index among all standard Young tableaux of a 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. 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. 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. 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 $eA$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000089The absolute variation of a composition. St000148The number of odd parts of a partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000680The Grundy value for Hackendot on posets. St000769The major index of a composition regarded as a word. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001057The Grundy value of the game of creating an independent set in a graph. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001383The BG-rank of an integer partition. St001396Number of triples of incomparable elements in a finite poset. St001673The degree of asymmetry of an integer composition. St001675The number of parts equal to the part in the reversed composition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St000008The major index of the composition. St000145The Dyson rank of a partition. St000277The number of ribbon shaped standard tableaux. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000632The jump number of the poset. St000644The number of graphs with given frequency partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000767The number of runs in an integer composition. St000820The number of compositions obtained by rotating the composition. St000899The maximal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. 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. 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. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001286The annihilation number of a graph. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001624The breadth of a lattice. 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. 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. St001742The difference of the maximal and the minimal degree in a graph. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000015The number of peaks of a Dyck path. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000088The row sums of the character table of the symmetric group. St000181The number of connected components of the Hasse diagram for the poset. St000189The number of elements in the poset. St000273The domination number of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000384The maximal part of the shifted composition of an integer partition. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000527The width of the poset. St000544The cop number of a graph. St000553The number of blocks of a graph. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000770The major index of an integer partition when read from bottom to top. St000775The multiplicity of the largest eigenvalue in a graph. St000784The maximum of the length and the largest part of the integer partition. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000815The number of semistandard Young tableaux of partition weight of given shape. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000908The length of the shortest maximal antichain in a poset. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000916The packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000937The number of positive values of the symmetric group character corresponding to the partition. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001268The size of the largest ordinal summand in the poset. St001312Number of parabolic noncrossing partitions indexed by the composition. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001360The number of covering relations in Young's lattice below a partition. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001399The distinguishing number of a poset. St001441The number of non-empty connected induced subgraphs of a graph. St001463The number of distinct columns in the nullspace of a graph. St001486The number of corners of the ribbon associated with an integer composition. St001527The cyclic permutation representation number of an integer partition. St001530The depth of a Dyck path. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001626The number of maximal proper sublattices of a lattice. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001691The number of kings in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001708The number of pairs of vertices of different degree in a graph. St001757The number of orbits of toric promotion on a graph. St001779The order of promotion on the set of linear extensions of a poset. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000180The number of chains of a poset. St000349The number of different adjacency matrices of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St000637The length of the longest cycle in a graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001112The 3-weak dynamic number of a graph. St001182Number of indecomposable injective modules with codominant dimension at least two 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001400The total number of Littlewood-Richardson tableaux of given shape. St001664The number of non-isomorphic subposets of a poset. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001834The number of non-isomorphic minors of a graph. St000656The number of cuts of a poset. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000090The variation of a composition. St000117The number of centered tunnels of a Dyck path. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000289The decimal representation of a binary word. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000297The number of leading ones in a binary word. St000313The number of degree 2 vertices of a graph. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000391The sum of the positions of the ones in a binary word. St000392The length of the longest run of ones in a binary word. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000448The number of pairs of vertices of a graph with distance 2. St000475The number of parts equal to 1 in a partition. St000552The number of cut vertices of a graph. St000628The balance of a binary word. St000682The Grundy value of Welter's game on a binary word. St000753The Grundy value for the game of Kayles on a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001249Sum of the odd parts of a partition. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001323The independence gap of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001350Half of the Albertson index 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. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001423The number of distinct cubes in 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. St001438The number of missing boxes of a skew partition. St001484The number of singletons of an integer partition. St001485The modular major index of a binary word. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001521Half the total irregularity of a graph. St001524The degree of symmetry of a binary word. St001525The number of symmetric hooks on the diagonal of a partition. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001578The minimal number of edges to add or remove to make a graph a line graph. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001689The number of celebrities in a graph. St001764The number of non-convex subsets of vertices in a graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001799The number of proper separations of a graph. St001931The weak major index of an integer composition regarded as a word. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001956The comajor index for set-valued two-row standard Young tableaux. St000026The position of the first return of a Dyck path. St000038The product of the heights of the descending steps of a Dyck path. St000081The number of edges of a graph. St000271The chromatic index of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000331The number of upper interactions of a Dyck path. St000351The determinant of the adjacency matrix of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000418The number of Dyck paths that are weakly below a Dyck path. St000509The diagonal index (content) of a partition. St000626The minimal period of a binary word. St000675The number of centered multitunnels of a Dyck path. St000699The toughness times the least common multiple of 1,. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000744The length of the path to the largest entry in a standard Young tableau. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. 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. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001111The weak 2-dynamic chromatic number of a graph. St001117The game chromatic index of a graph. St001118The acyclic chromatic index of a graph. St001161The major index north count of a Dyck path. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001282The number of graphs with the same chromatic polynomial. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001352The number of internal nodes in the modular decomposition of a graph. St001479The number of bridges of a graph. St001487The number of inner corners of a skew partition. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001531Number of partial orders contained in the poset determined by the Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001595The number of standard Young tableaux of the skew partition. St001649The length of a longest trail in a graph. St001734The lettericity of a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001765The number of connected components of the friends and strangers 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. St001826The maximal number of leaves on a vertex of a graph. St001827The number of two-component spanning forests of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St001959The product of the heights of the peaks of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000086The number of subgraphs. St000312The number of leaves in a graph. St000395The sum of the heights of the peaks of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000468The Hosoya index of a graph. St000507The number of ascents of a standard tableau. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000627The exponent of a binary word. St000657The smallest part of an integer composition. St000674The number of hills of a Dyck path. St000827The decimal representation of a binary word with a leading 1. St000867The sum of the hook lengths in the first row of an integer partition. St000878The number of ones minus the number of zeros of a binary word. St000915The Ore degree of a graph. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St000997The even-odd crank of an integer partition. 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. St001034The area of the parallelogram polyomino associated with the Dyck path. St001060The distinguishing index of a graph. St001127The sum of the squares of the parts of a partition. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001488The number of corners of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001523The degree of symmetry of a Dyck path. St001614The cyclic permutation representation number of a skew partition. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000294The number of distinct factors of a binary word. St000518The number of distinct subsequences in a binary word. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules 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. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St001957The number of Hasse diagrams with a given underlying undirected graph. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001838The number of nonempty primitive factors of a binary word. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St000264The girth of a graph, which is not a tree. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000422The energy of a graph, if it is integral. St000936The number of even values of the symmetric group character corresponding to the partition. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St001964The interval resolution global dimension of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. 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. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001763The Hurwitz number of an integer partition. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001378The product of the cohook lengths of the integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St001943The sum of the squares of the hook lengths of an integer partition.
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