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Your data matches 818 different statistics following compositions of up to 3 maps.
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Matching statistic: St001340
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00046: Ordered trees —to graph⟶ Graphs
St001340: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001340: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
[[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
[[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
[[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[],[],[[[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2
[[],[[[[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2
[[],[[],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[],[[[[]]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[],[[[],[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[],[[[[]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
[[[]],[[[[]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
[[[[[]]]],[],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2
[[[[[]]]],[[]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
[[[],[[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[[]]],[]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[],[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[[]],[]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
[[[],[],[[[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2
[[[],[[[]]],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2
[[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[],[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
[[[],[[[[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2
[[[[]],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[[]]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2
[[[[[]]],[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[[[]],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[[[],[]]],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
[[[[[[]]]],[]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2
[[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[[]],[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 3
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2
[[[[[],[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[[[]],[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
Description
The cardinality of a minimal non-edge 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 only the graph with two isolated vertices.
Matching statistic: St001393
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00046: Ordered trees —to graph⟶ Graphs
St001393: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001393: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
[[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
[[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
[[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[],[],[[[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2
[[],[[[[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2
[[],[[],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[],[[[[]]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[],[[[],[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[],[[[[]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
[[[]],[[[[]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
[[[[[]]]],[],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2
[[[[[]]]],[[]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
[[[],[[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[[]]],[]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[],[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[[]],[]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2
[[[],[],[[[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2
[[[],[[[]]],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2
[[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[],[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
[[[],[[[[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2
[[[[]],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[[]]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2
[[[[[]]],[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[[[]],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[[[],[]]],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
[[[[[[]]]],[]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2
[[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[[]],[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 3
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2
[[[[[],[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
[[[[[[]],[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2
Description
The induced matching number of a graph.
An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.
Matching statistic: St000536
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00046: Ordered trees —to graph⟶ Graphs
St000536: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000536: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 2 - 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 2 - 1
[[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 2 - 1
[[],[],[[[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[],[[[[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[],[[],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[],[[[[]]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[],[[[],[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[],[[[[]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[[[]],[[[[]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[[[[[]]]],[],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[[[[]]]],[[]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[[[],[[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[]]],[]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[],[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[]],[]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[[[],[],[[[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[],[[[]]],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[],[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[[],[[[[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[[[]],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[]]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[]]],[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[[[]],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[[[],[]]],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[[[[[]]]],[]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[]],[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[],[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[[]],[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
Description
The pathwidth of a graph.
Matching statistic: St000772
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00046: Ordered trees —to graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000772: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 2 - 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 2 - 1
[[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 2 - 1
[[],[],[[[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[],[[[[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[],[[],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[],[[[[]]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[],[[[],[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[],[[[[]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[[[]],[[[[]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[[[[[]]]],[],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[[[[]]]],[[]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[[[],[[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[]]],[]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[],[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[]],[]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[[[],[],[[[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[],[[[]]],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[],[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[[],[[[[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[[[]],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[]]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[]]],[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[[[]],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[[[],[]]],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[[[[[]]]],[]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[]],[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[],[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
[[[[[[]],[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 2 - 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St001261
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00046: Ordered trees —to graph⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3 = 2 + 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3 = 2 + 1
[[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3 = 2 + 1
[[],[],[[[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 3 = 2 + 1
[[],[[[[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 3 = 2 + 1
[[],[[],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[],[[[[]]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[],[[[],[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[],[[[[]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 3 = 2 + 1
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 3 = 2 + 1
[[[]],[[[[]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 3 = 2 + 1
[[[[[]]]],[],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 3 = 2 + 1
[[[[[]]]],[[]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 3 = 2 + 1
[[[],[[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[[[[]]],[]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[[[],[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[[[[]],[]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 3 = 2 + 1
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 3 = 2 + 1
[[[],[],[[[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[[],[[[]]],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[[],[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 3 = 2 + 1
[[[],[[[[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 3 = 2 + 1
[[[[]],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[[[[]]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[[[[]]],[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[[[[]],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[[[[],[]]],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 3 = 2 + 1
[[[[[[]]]],[]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 3 = 2 + 1
[[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[[[[]],[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[[[[],[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[[[[[[]],[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
Description
The Castelnuovo-Mumford regularity of a graph.
Matching statistic: St001326
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00046: Ordered trees —to graph⟶ Graphs
St001326: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001326: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 2 - 2
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 0 = 2 - 2
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 0 = 2 - 2
[[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0 = 2 - 2
[[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0 = 2 - 2
[[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0 = 2 - 2
[[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 0 = 2 - 2
[[],[],[[[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 0 = 2 - 2
[[],[[[[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 0 = 2 - 2
[[],[[],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[],[[[[]]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[],[[[],[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[],[[[[]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 0 = 2 - 2
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0 = 2 - 2
[[[]],[[[[]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0 = 2 - 2
[[[[[]]]],[],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 0 = 2 - 2
[[[[[]]]],[[]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0 = 2 - 2
[[[],[[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[[[[]]],[]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[[[],[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[[[[]],[]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 0 = 2 - 2
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0 = 2 - 2
[[[],[],[[[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[[],[[[]]],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 0 = 2 - 2
[[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 0 = 2 - 2
[[[],[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 0 = 2 - 2
[[[],[[[[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 0 = 2 - 2
[[[[]],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[[[[]]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[[[[]]],[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 0 = 2 - 2
[[[[[]],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 0 = 2 - 2
[[[[[],[]]],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 0 = 2 - 2
[[[[[[]]]],[]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 0 = 2 - 2
[[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 0 = 2 - 2
[[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 0 = 2 - 2
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 0 = 2 - 2
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[[[[]],[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 0 = 2 - 2
[[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 0 = 2 - 2
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[[[[],[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0 = 2 - 2
[[[[[[]],[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0 = 2 - 2
Description
The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph.
A graph is an interval graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ is an edge and $(a,b)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St000273
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
[[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[],[],[[[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[],[[[[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[],[[],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[[]]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[],[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[[]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[]],[[[[]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[[[]]]],[],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[[]]]],[[]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[],[[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]]],[]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[],[],[[[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[[]]],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2
[[[],[[[[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[]],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]]],[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[]]],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2
[[[[[[]]]],[]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> 3
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[[]],[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The domination number of a graph.
The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.
Matching statistic: St000916
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Values
[[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[],[],[[[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[],[[[[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[],[[],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[[]]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[],[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[[]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[]],[[[[]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[[[]]]],[],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[[]]]],[[]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[],[[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]]],[]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[],[],[[[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[[]]],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2
[[[],[[[[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[]],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]]],[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[]]],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2
[[[[[[]]]],[]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> 3
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[[]],[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The packing number of a graph.
This is the size of a largest subset of vertices of a graph, such that any two distinct vertices in the subset have disjoint closed neighbourhoods, or, equivalently, have distance greater than two.
Matching statistic: St001322
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Values
[[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[],[],[[[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[],[[[[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[],[[],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[[]]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[],[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[[]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[]],[[[[]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[[[]]]],[],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[[]]]],[[]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[],[[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]]],[]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[],[],[[[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[[]]],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2
[[[],[[[[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[]],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]]],[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[]]],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2
[[[[[[]]]],[]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> 3
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[[]],[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The size of a minimal independent dominating set in a graph.
Matching statistic: St001339
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Values
[[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[],[],[[[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[],[[[[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[],[[],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[[]]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[],[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[[]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[]],[[[[]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[[[]]]],[],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[[]]]],[[]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[],[[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]]],[]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[]]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[]]],[]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[[[],[],[[[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[[]]],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2
[[[],[[[[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[]],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]]],[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[]]],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2
[[[[[[]]]],[]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> 3
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[],[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[[[]],[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The irredundance number of a graph.
A set $S$ of vertices is irredundant, if there is no vertex in $S$, whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of $S$.
The irredundance number is the smallest size of a maximal irredundant set.
The following 808 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. 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. St000805The number of peaks of the associated bargraph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001577The minimal number of edges to add or remove to make a graph a cograph. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000807The sum of the heights of the valleys of the associated bargraph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St000842The breadth of a permutation. St001111The weak 2-dynamic chromatic number of a graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. 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. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001716The 1-improper chromatic number of a graph. St000143The largest repeated part of a partition. St000183The side length of the Durfee square of an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000481The number of upper covers of a partition in dominance order. St000553The number of blocks of a graph. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000785The number of distinct colouring schemes of a graph. St000816The number of standard composition tableaux of the composition. St000903The number of different parts of an integer composition. St000904The maximal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000917The open packing number of a graph. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001672The restrained domination number of a graph. St001743The discrepancy of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001871The number of triconnected components of a graph. St000090The variation of a composition. St000091The descent variation of a composition. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. 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. St000312The number of leaves in a graph. St000552The number of cut vertices of a graph. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000944The 3-degree of an integer partition. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001525The number of symmetric hooks on the diagonal of a partition. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001691The number of kings in a graph. St001777The number of weak descents in an integer composition. St001826The maximal number of leaves on a vertex of a graph. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000258The burning number of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000535The rank-width of a graph. St000636The hull number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000918The 2-limited packing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001654The monophonic hull number of a graph. St001829The common independence number of a graph. St000264The girth of a graph, which is not a tree. St000775The multiplicity of the largest eigenvalue in a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001512The minimum rank of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000351The determinant of the adjacency matrix of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St001057The Grundy value of the game of creating an independent set in a graph. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001797The number of overfull subgraphs of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St001399The distinguishing number of a poset. St001621The number of atoms of a lattice. St001624The breadth 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. St001877Number of indecomposable injective modules with projective dimension 2. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000100The number of linear extensions of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000632The jump number of the poset. St001625The Möbius invariant of a lattice. St001725The harmonious chromatic number of a graph. St001642The Prague dimension of a graph. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000287The number of connected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000908The length of the shortest maximal antichain 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. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. 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. St001333The cardinality of a minimal edge-isolating set of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001510The number of self-evacuating linear extensions of a finite poset. St001518The number of graphs with the same ordinary spectrum as the given graph. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000315The number of isolated vertices of a graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001479The number of bridges of a graph. St001902The number of potential covers of a poset. St001964The interval resolution global dimension of a poset. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St000741The Colin de Verdière graph invariant. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001363The Euler characteristic of a graph according to Knill. St001305The number of induced cycles on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St001717The largest size of an interval in a poset. St000717The number of ordinal summands of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St001718The number of non-empty open intervals in a poset. St000455The second largest eigenvalue of a graph if it is integral. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St000327The number of cover relations in a poset. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000929The constant term of the character polynomial of an integer partition. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000352The Elizalde-Pak rank of a permutation. St000546The number of global descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001116The game chromatic number of a graph. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001271The competition number of a graph. St000234The number of global ascents of a permutation. St000822The Hadwiger number of the graph. St001060The distinguishing index of a graph. St000080The rank of the poset. St000096The number of spanning trees of a graph. St000159The number of distinct parts of the integer partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000286The number of connected components of the complement of a graph. St000310The minimal degree of a vertex of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St001117The game chromatic index of a graph. St001734The lettericity of a graph. St001828The Euler characteristic of a graph. St000095The number of triangles of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. 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. 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. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001309The number of four-cliques in a graph. St001323The independence gap of a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001354The number of series nodes in the modular decomposition of a graph. St001783The number of odd automorphisms of a graph. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St001468The smallest fixpoint of a permutation. St001638The book thickness of a graph. St000210Minimum over maximum difference of elements in cycles. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000623The number of occurrences of the pattern 52341 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001513The number of nested exceedences of a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001128The exponens consonantiae of a partition. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001845The number of join irreducibles minus the rank of a lattice. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001260The permanent of an alternating sign matrix. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000223The number of nestings in the permutation. St000225Difference between largest and smallest parts in a partition. St000237The number of small exceedances. St001330The hat guessing number of a graph. St001597The Frobenius rank of a skew partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000007The number of saliances of the permutation. St000862The number of parts of the shifted shape of a permutation. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001545The second Elser number of a connected graph. St000759The smallest missing part in an integer partition. St000996The number of exclusive left-to-right maxima of a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000405The number of occurrences of the pattern 1324 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001812The biclique partition number of a graph. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000297The number of leading ones in a binary word. St000787The number of flips required to make a perfect matching noncrossing. St000617The number of global maxima of a Dyck path. St000288The number of ones in a binary word. St000054The first entry of the permutation. St001381The fertility of a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000731The number of double exceedences of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001536The number of cyclic misalignments of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000516The number of stretching pairs of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000486The number of cycles of length at least 3 of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001850The number of Hecke atoms of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000069The number of maximal elements of a poset. St000779The tier of a permutation. St001571The Cartan determinant of the integer partition. St000666The number of right tethers of a permutation. 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. St000891The number of distinct diagonal sums of a permutation matrix. St000218The number of occurrences of the pattern 213 in a permutation. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000754The Grundy value for the game of removing nestings in a perfect matching. St000035The number of left outer peaks of a permutation. St000834The number of right outer peaks of a permutation. St000359The number of occurrences of the pattern 23-1. St000534The number of 2-rises of a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000664The number of right ropes of a permutation. St001130The number of two successive successions in a permutation. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001703The villainy of a graph. St001834The number of non-isomorphic minors of a graph. St000451The length of the longest pattern of the form k 1 2. St000454The largest eigenvalue of a graph if it is integral. St000374The number of exclusive right-to-left minima of a permutation. St000667The greatest common divisor of the parts of the partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St001081The number of minimal length factorizations of a permutation into star transpositions. St000002The number of occurrences of the pattern 123 in a permutation. St000422The energy of a graph, if it is integral. St000462The major index minus the number of excedences of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000803The number of occurrences of the vincular pattern |132 in a permutation. 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. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001394The genus of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001535The number of cyclic alignments of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000384The maximal part of the shifted composition of an integer partition. St000479The Ramsey number of a graph. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000098The chromatic number of a graph. St000244The cardinality of the automorphism group of a graph. St000272The treewidth of a graph. St000364The exponent of the automorphism group of a graph. St000469The distinguishing number of a graph. St000544The cop number of a graph. St001029The size of the core of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001277The degeneracy of a graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001358The largest degree of a regular subgraph of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001792The arboricity of a graph. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000172The Grundy number of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000387The matching number of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000694The number of affine bounded permutations that project to a given permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. 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 $c_{n−1}$. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001119The length of a shortest maximal path in a graph. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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 $eA$. 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. 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. 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. St001256Number of simple reflexive modules that are 2-stable reflexive. St001270The bandwidth of a graph. St001316The domatic number of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001546The number of monomials in the Tutte polynomial of a graph. St001590The crossing number of a perfect matching. St001592The maximal number of simple paths between any two different vertices of a graph. St001656The monophonic position number of a graph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001670The connected partition number of a graph. St001765The number of connected components of the friends and strangers graph. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001949The rigidity index of a graph. St001963The tree-depth of a graph. St000221The number of strong fixed points of a permutation. St000268The number of strongly connected orientations of a graph. St000274The number of perfect matchings of a graph. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000637The length of the longest cycle in a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001073The number of nowhere zero 3-flows of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. 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. 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. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001281The normalized isoperimetric number of a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001317The minimal number of occurrences of the forest-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. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001341The number of edges in the center of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001736The total number of cycles in a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001856The number of edges in the reduced word graph of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000062The length of the longest increasing subsequence of the permutation. St000099The number of valleys of a permutation, including the boundary. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000308The height of the tree associated to a permutation. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000485The length of the longest cycle of a permutation. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000630The length of the shortest palindromic decomposition of a binary word. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000733The row containing the largest entry of a standard tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000983The length of the longest alternating subword. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001345The Hamming dimension of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001469The holeyness of a permutation. St001481The minimal height of a peak of a 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. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001649The length of a longest trail in a graph. St001665The number of pure excedances of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. 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). St001928The number of non-overlapping descents in a permutation. St000003The number of standard Young tableaux of the partition. St000023The number of inner peaks of a permutation. St000037The sign of a permutation. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000061The number of nodes on the left branch of a binary tree. St000078The number of alternating sign matrices whose left key is the permutation. St000079The number of alternating sign matrices for a given Dyck path. St000084The number of subtrees. St000146The Andrews-Garvan crank of a partition. St000155The number of exceedances (also excedences) of a permutation. St000160The multiplicity of the smallest part of a partition. St000163The size of the orbit of the set partition under rotation. St000230Sum of the minimal elements of the blocks of a set partition. St000255The number of reduced Kogan faces with the permutation as type. St000271The chromatic index of a graph. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000284The Plancherel distribution on integer partitions. St000314The number of left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000346The number of coarsenings of a partition. St000349The number of different adjacency matrices of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000443The number of long tunnels of a Dyck path. 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. St000482The (zero)-forcing number of a graph. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000547The number of even non-empty partial sums of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000570The Edelman-Greene number of a permutation. St000618The number of self-evacuating tableaux of given shape. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000627The exponent of a binary word. St000654The first descent of a permutation. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000702The number of weak deficiencies of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000756The sum of the positions of the left to right maxima of a permutation. St000783The side length of the largest staircase partition fitting into a partition. 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. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. 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. St000829The Ulam distance of a permutation to the identity permutation. St000843The decomposition number of a perfect matching. St000847The number of standard Young tableaux whose descent set is the binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000958The number of Bruhat factorizations of a permutation. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000991The number of right-to-left minima of a permutation. St000993The multiplicity of the largest part of an integer partition. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in 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. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001112The 3-weak dynamic number of a graph. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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. 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 $eA$. 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001220The width of a permutation. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St001349The number of different graphs obtained from the given graph by removing an edge. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001386The number of prime labellings of a graph. St001423The number of distinct cubes in a binary word. St001432The order dimension of the partition. St001463The number of distinct columns in the nullspace of a graph. St001470The cyclic holeyness of a permutation. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001490The number of connected components of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001520The number of strict 3-descents. St001531Number of partial orders contained in the poset determined by the Dyck path. St001589The nesting number of a perfect matching. St001694The number of maximal dissociation sets in a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001739The number of graphs with the same edge polytope as the given graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. 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. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001917The order of toric promotion on the set of labellings of a graph. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001959The product of the heights of the peaks of a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000042The number of crossings of a perfect matching. St000051The size of the left subtree of a binary tree. St000089The absolute variation of a composition. St000117The number of centered tunnels of a Dyck path. St000133The "bounce" of a permutation. St000185The weighted size of a partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000217The number of occurrences of the pattern 312 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000241The number of cyclical small excedances. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000311The number of vertices of odd degree in a graph. St000313The number of degree 2 vertices of a graph. St000317The cycle descent number of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000347The inversion sum of a binary word. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000461The rix statistic of a permutation. St000475The number of parts equal to 1 in a 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. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000545The number of parabolic double cosets with minimal element being the given permutation. St000549The number of odd partial sums of an integer partition. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000629The defect of a binary word. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000674The number of hills of a Dyck path. St000732The number of double deficiencies of a permutation. 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. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000873The aix statistic of a permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000881The number of short braid edges in the graph of braid moves of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000886The number of permutations with the same antidiagonal sums. St000895The number of ones on the main diagonal of an alternating sign matrix. 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. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. 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. 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. 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. St001022Number of simple modules with projective dimension 3 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. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001082The number of boxed occurrences of 123 in a permutation. 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. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001125The number of simple modules that satisfy the 2-regular condition 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. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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. St001214The aft of an integer partition. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. 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. St001308The number of induced paths on three vertices in a graph. 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. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. St001429The number of negative entries in a signed permutation. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. 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. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001684The reduced word complexity of a permutation. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001708The number of pairs of vertices of different degree in a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001764The number of non-convex subsets of vertices in a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001799The number of proper separations of a graph. St001836The number of occurrences of a 213 pattern in the restricted growth word of a perfect matching. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001948The number of augmented double ascents of a permutation. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001883The mutual visibility number of a graph. St000889The number of alternating sign matrices with the same antidiagonal sums. St000464The Schultz index of a connected graph. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000402Half the size of the symmetry class of a permutation. St001487The number of inner corners of a skew partition. St000648The number of 2-excedences of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001645The pebbling number of a connected graph. St001596The number of two-by-two squares inside a skew partition. St001644The dimension of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001851The number of Hecke atoms of a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1.
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