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Your data matches 153 different statistics following compositions of up to 3 maps.
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Matching statistic: St001397
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(load all 6 compositions to match this statistic)
Mp00209: Permutations —pattern poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001397: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> 0
[2,1] => ([(0,1)],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,2,1] => ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
Number of pairs of incomparable elements in a finite poset.
For a finite poset $(P,\leq)$, this is the number of unordered pairs $\{x,y\} \in \binom{P}{2}$ with $x \not\leq y$ and $y \not\leq x$.
Matching statistic: St000081
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
Description
The number of edges of a graph.
Matching statistic: St000987
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(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St001341
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(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
Description
The number of edges in the center of a graph.
The center of a graph is the set of vertices whose maximal distance to any other vertex is minimal. In particular, if the graph is disconnected, all vertices are in the certer.
Matching statistic: St001479
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
Description
The number of bridges of a graph.
A bridge is an edge whose removal increases the number of connected components of the graph.
Matching statistic: St001512
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
Description
The minimum rank of a graph.
The minimum rank of a simple graph G is the smallest possible rank over all symmetric real matrices whose entry in row $i$ and column $j$ (for $i\neq j$) is nonzero whenever $\{i, j\}$ is an edge in
$G$, and zero otherwise.
Matching statistic: St001869
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
Description
The maximum cut size of a graph.
A '''cut''' is a set of edges which connect different sides of a vertex partition $V = A \sqcup B$.
Matching statistic: St000448
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
Description
The number of pairs of vertices of a graph with distance 2.
This is the coefficient of the quadratic term of the Wiener polynomial.
Matching statistic: St001646
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
Description
The number of edges that can be added without increasing the maximal degree of a graph.
This statistic is (except for the degenerate case of two vertices) maximized by the star-graph on $n$ vertices, which has maximal degree $n-1$ and therefore has statistic $\binom{n-1}{2}$.
Matching statistic: St000434
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000434: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000434: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [[1]]
=> [1] => ? = 0
[1,2] => [[1,2]]
=> [[1,2]]
=> [1,2] => 0
[2,1] => [[1],[2]]
=> [[1],[2]]
=> [2,1] => 0
[1,2,3] => [[1,2,3]]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,3,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> [2,1,3] => 1
[2,1,3] => [[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,3,1] => [[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => 1
[3,1,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> [2,1,3] => 1
[3,2,1] => [[1],[2],[3]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,2,3,4] => [[1,2,3,4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,2,4,3] => [[1,2,3],[4]]
=> [[1,2,4],[3]]
=> [3,1,2,4] => 3
[1,4,3,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[2,1,3,4] => [[1,3,4],[2]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 3
[2,1,4,3] => [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,3,4,1] => [[1,3,4],[2]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 3
[3,2,1,4] => [[1,4],[2],[3]]
=> [[1,3],[2],[4]]
=> [4,2,1,3] => 3
[3,4,1,2] => [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 2
[3,4,2,1] => [[1,4],[2],[3]]
=> [[1,3],[2],[4]]
=> [4,2,1,3] => 3
[4,1,2,3] => [[1,2,3],[4]]
=> [[1,2,4],[3]]
=> [3,1,2,4] => 3
[4,3,1,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[4,3,2,1] => [[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
[5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 0
[6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0
Description
The number of occurrences of the pattern 213 or of the pattern 312 in a permutation.
The following 143 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001268The size of the largest ordinal summand in the poset. St001120The length of a longest path in a graph. St001649The length of a longest trail in a graph. St001736The total number of cycles in a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000453The number of distinct Laplacian eigenvalues of a graph. St001093The detour number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001624The breadth of a lattice. St000454The largest eigenvalue of a graph if it is integral. St001877Number of indecomposable injective modules with projective dimension 2. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001645The pebbling number of a connected graph. St001875The number of simple modules with projective dimension at most 1. St000552The number of cut vertices of a graph. St000768The number of peaks in an integer composition. St001323The independence gap of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001470The cyclic holeyness of a permutation. St001793The difference between the clique number and the chromatic number of a graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000785The number of distinct colouring schemes of a graph. St001282The number of graphs with the same chromatic polynomial. St001333The cardinality of a minimal edge-isolating set of a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000098The chromatic number of a graph. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000636The hull number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001691The number of kings in a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001964The interval resolution global dimension of a poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000095The number of triangles 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. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001742The difference of the maximal and the minimal degree in a graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001488The number of corners of a skew partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. 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. St001570The minimal number of edges to add to make a graph Hamiltonian. St000455The second largest eigenvalue of a graph if it is integral. 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. St000181The number of connected components of the Hasse diagram for the poset. St000741The Colin de Verdière graph invariant. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001890The maximum magnitude of the Möbius function of a poset. St000219The number of occurrences of the pattern 231 in a permutation. St000264The girth of a graph, which is not a tree. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001060The distinguishing index of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000284The Plancherel distribution on integer partitions. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St000102The charge of a semistandard tableau. St000101The cocharge of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001625The Möbius invariant of a lattice. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. 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. St001783The number of odd automorphisms of a graph. St001845The number of join irreducibles minus the rank of a lattice. St001856The number of edges in the reduced word graph of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001871The number of triconnected components of a graph. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000310The minimal degree of a vertex of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000477The weight of a partition according to Alladi. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000736The last entry in the first row of a semistandard tableau. St000739The first entry in the last row of a semistandard tableau. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001518The number of graphs with the same ordinary spectrum as the given graph. St001569The maximal modular displacement of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001828The Euler characteristic of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St000822The Hadwiger number of the graph. St001568The smallest positive integer that does not appear twice in the partition. St001734The lettericity of a graph.
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