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Your data matches 37 different statistics following compositions of up to 3 maps.
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Matching statistic: St001320
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Mp00011: Binary trees —to graph⟶ Graphs
St001320: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001320: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 0
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> 0
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
Description
The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph.
A graph is a disjoint union of paths 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. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001328
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(load all 2 compositions to match this statistic)
Values
[.,.]
=> ([],1)
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([],1)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> ([],1)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
Description
The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph.
A graph is bipartite if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001334
Values
[.,.]
=> ([],1)
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
Description
The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph.
A graph is $3$-colourable if and only if in any linear ordering of its vertices, there are no four vertices $a < b < c < d$ such that $(a,b), (b,c)$ and $(c,d)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St000252
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St000252: Binary trees ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> 0
[.,[.,.]]
=> 0
[[.,.],.]
=> 0
[.,[.,[.,.]]]
=> 0
[.,[[.,.],.]]
=> 0
[[.,.],[.,.]]
=> 0
[[.,[.,.]],.]
=> 0
[[[.,.],.],.]
=> 0
[.,[.,[.,[.,.]]]]
=> 0
[.,[.,[[.,.],.]]]
=> 0
[.,[[.,.],[.,.]]]
=> 1
[.,[[.,[.,.]],.]]
=> 0
[.,[[[.,.],.],.]]
=> 0
[[.,.],[.,[.,.]]]
=> 0
[[.,.],[[.,.],.]]
=> 0
[[.,[.,.]],[.,.]]
=> 0
[[[.,.],.],[.,.]]
=> 0
[[.,[.,[.,.]]],.]
=> 0
[[.,[[.,.],.]],.]
=> 0
[[[.,.],[.,.]],.]
=> 1
[[[.,[.,.]],.],.]
=> 0
[[[[.,.],.],.],.]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> 0
[.,[.,[[[.,.],.],.]]]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> 1
[.,[[.,.],[[.,.],.]]]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> 1
[.,[[[.,.],.],[.,.]]]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> 0
[.,[[.,[[.,.],.]],.]]
=> 0
[.,[[[.,.],[.,.]],.]]
=> 1
[.,[[[.,[.,.]],.],.]]
=> 0
[.,[[[[.,.],.],.],.]]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> 1
[[.,.],[[.,[.,.]],.]]
=> 0
[[.,.],[[[.,.],.],.]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> 0
[[.,[.,.]],[[.,.],.]]
=> 0
[[[.,.],.],[.,[.,.]]]
=> 0
[[[.,.],.],[[.,.],.]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> 0
[[.,[[.,.],.]],[.,.]]
=> 0
[[[.,.],[.,.]],[.,.]]
=> 1
[[[.,[.,.]],.],[.,.]]
=> 0
[[[[.,.],.],.],[.,.]]
=> 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ? = 0
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ? = 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 0
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ? = 0
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ? = 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ? = 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ? = 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ? = 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> ? = 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> ? = 0
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ? = 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> ? = 0
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> ? = 0
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ? = 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ? = 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ? = 2
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ? = 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> ? = 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ? = 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ? = 2
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> ? = 2
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ? = 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ? = 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ? = 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ? = 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> ? = 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> ? = 0
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ? = 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> ? = 0
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> ? = 0
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> ? = 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> ? = 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> ? = 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> ? = 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> ? = 0
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> ? = 0
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> ? = 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> ? = 0
[.,[.,[[[[[.,.],.],.],.],.]]]
=> ? = 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ? = 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> ? = 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ? = 2
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> ? = 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> ? = 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ? = 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ? = 2
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ? = 2
Description
The number of nodes of degree 3 of a binary tree.
Equivalently, the number of internal triangles in the associated triangulation of an $(n+2)$-gon.
Matching statistic: St001964
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
St001964: Posets ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 0
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? = 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? = 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ? = 2
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ? = 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? = 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? = 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ? = 2
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 2
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 2
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St001307
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[.,.]
=> ([],1)
=> ([],2)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> 0
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> 0
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,4),(2,4),(3,4)],5)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,4),(2,4),(3,4)],5)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 2
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ? = 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ? = 2
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ? = 2
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 2
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 2
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 2
Description
The number of induced stars on four vertices in a graph.
Matching statistic: St001329
Values
[.,.]
=> ([],1)
=> ([(0,1)],2)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
Description
The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph.
A graph is outerplanar if and only if in any linear ordering of its vertices, there are no four vertices $a < b < c < d$ such that $(a,c)$ and $(b,d)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001871
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[.,.]
=> ([],1)
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
Description
The number of triconnected components of a graph.
A connected graph is '''triconnected''' or '''3-vertex connected''' if it cannot be disconnected by removing two or fewer vertices. An arbitrary connected graph can be decomposed as a union of biconnected (2-vertex connected) graphs, known as '''blocks''', and each biconnected graph can be decomposed as a union of components with are either a cycle (type "S"), a cocyle (type "P"), or triconnected (type "R"). The decomposition of a biconnected graph into these components is known as the '''SPQR-tree''' of the graph.
Matching statistic: St000259
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
The following 27 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St000482The (zero)-forcing number of a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001734The lettericity of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000327The number of cover relations in a poset. St000741The Colin de Verdière graph invariant. St000455The second largest eigenvalue of a graph if it is integral. St001282The number of graphs with the same chromatic polynomial. St001845The number of join irreducibles minus the rank of a lattice. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset.
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