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Your data matches 79 different statistics following compositions of up to 3 maps.
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Matching statistic: St000397
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(load all 3 compositions to match this statistic)
St000397: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> 1
[[],[]]
=> 2
[[[]]]
=> 1
[[],[],[]]
=> 2
[[],[[]]]
=> 2
[[[]],[]]
=> 2
[[[],[]]]
=> 2
[[[[]]]]
=> 1
[[],[],[],[]]
=> 2
[[],[],[[]]]
=> 2
[[],[[]],[]]
=> 2
[[],[[],[]]]
=> 2
[[],[[[]]]]
=> 2
[[[]],[],[]]
=> 2
[[[]],[[]]]
=> 2
[[[],[]],[]]
=> 2
[[[[]]],[]]
=> 2
[[[],[],[]]]
=> 2
[[[],[[]]]]
=> 2
[[[[]],[]]]
=> 2
[[[[],[]]]]
=> 2
[[[[[]]]]]
=> 1
[[],[],[],[],[]]
=> 2
[[],[],[],[[]]]
=> 2
[[],[],[[]],[]]
=> 2
[[],[],[[],[]]]
=> 2
[[],[],[[[]]]]
=> 2
[[],[[]],[],[]]
=> 2
[[],[[]],[[]]]
=> 2
[[],[[],[]],[]]
=> 2
[[],[[[]]],[]]
=> 2
[[],[[],[],[]]]
=> 2
[[],[[],[[]]]]
=> 2
[[],[[[]],[]]]
=> 2
[[],[[[],[]]]]
=> 2
[[],[[[[]]]]]
=> 2
[[[]],[],[],[]]
=> 2
[[[]],[],[[]]]
=> 2
[[[]],[[]],[]]
=> 2
[[[]],[[],[]]]
=> 2
[[[]],[[[]]]]
=> 2
[[[],[]],[],[]]
=> 2
[[[[]]],[],[]]
=> 2
[[[],[]],[[]]]
=> 2
[[[[]]],[[]]]
=> 2
[[[],[],[]],[]]
=> 2
[[[],[[]]],[]]
=> 2
[[[[]],[]],[]]
=> 2
[[[[],[]]],[]]
=> 2
[[[[[]]]],[]]
=> 2
Description
The Strahler number of a rooted tree.
Matching statistic: St000455
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[[]]
=> [.,.]
=> ([],1)
=> ([],1)
=> ? = 1 - 2
[[],[]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 2 - 2
[[[]]]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 2
[[],[],[]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 2 - 2
[[],[[]]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 2 - 2
[[[]],[]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 2 - 2
[[[],[]]]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 0 = 2 - 2
[[[[]]]]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 2
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 2
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 2
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 2
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0 = 2 - 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 2
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 2
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 2
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0 = 2 - 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[[[],[],[],[]]]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[],[],[[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[],[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[],[[],[]]]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[[[],[[[]]]]]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[[]],[],[]]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[[[[]],[[]]]]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[[[[],[]],[]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[[[[]]],[]]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[[],[],[]]]]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[[],[[]]]]]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[[[]],[]]]]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[[[],[]]]]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[[[[[[]]]]]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[[],[],[],[],[[]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[[],[],[],[[]],[]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[[],[],[],[[],[]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 0 = 2 - 2
[[],[],[],[[[]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[[],[],[[]],[],[]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[[],[],[[]],[[]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[[],[],[[],[]],[]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[],[[[]]],[]]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[[],[],[[],[],[]]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[],[[],[[]]]]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[],[[[]],[]]]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[],[[[],[]]]]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 0 = 2 - 2
[[],[],[[[[]]]]]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[[],[[]],[],[],[]]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[[],[[]],[],[[]]]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[[],[[]],[[]],[]]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[[],[[]],[[],[]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[],[[]],[[[]]]]
=> [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[[],[[],[]],[],[]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 2 - 2
[[],[[[]]],[],[]]
=> [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[[],[[],[]],[[]]]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 2 - 2
[[],[[[]]],[[]]]
=> [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[[],[[],[],[]],[]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[],[[]]],[]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[[]],[]],[]]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[[],[]]],[]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[[[]]]],[]]
=> [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[[],[[],[],[],[]]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[],[],[[]]]]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[],[[]],[]]]
=> [.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[],[[],[]]]]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 0 = 2 - 2
[[],[[],[[[]]]]]
=> [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[[]],[],[]]]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 2 - 2
[[],[[[]],[[]]]]
=> [.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 2 - 2
[[],[[[],[]],[]]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000258
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[[]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[]]]
=> ([(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)
=> 2
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2
[[[],[]],[]]
=> ([(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)
=> 2
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(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),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(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),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[]],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[[]]]]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[]],[[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]]],[[]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]]]],[]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[]],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[],[]]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[]],[]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]]],[]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]],[]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[],[]]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,4)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[[]]]]]]
=> ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[]],[[]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[],[]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
Description
The burning number of a graph.
This is the minimum number of rounds needed to burn all vertices of a graph. In each round, the neighbours of all burned vertices are burnt. Additionally, an unburned vertex may be chosen to be burned.
Matching statistic: St000273
Values
[[]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[]]]
=> ([(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)
=> 2
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2
[[[],[]],[]]
=> ([(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)
=> 2
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(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),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(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),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[]],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[[]]]]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[]],[[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]]],[[]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]]]],[]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[]],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[],[]]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[]],[]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]]],[]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]],[]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[],[]]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,4)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[[]]]]]]
=> ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[]],[[]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[],[]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 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: St000544
Values
[[]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[]]]
=> ([(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)
=> 2
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2
[[[],[]],[]]
=> ([(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)
=> 2
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(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),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(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),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[]],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[[]]]]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[]],[[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]]],[[]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]]]],[]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[]],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[],[]]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[]],[]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]]],[]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]],[]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[],[]]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,4)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[[]]]]]]
=> ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[]],[[]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[],[]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
Description
The cop number of a graph.
This is the minimal number of cops needed to catch the robber. The algorithm is from [2].
Matching statistic: St000679
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000679: Ordered trees ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000679: Ordered trees ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,1,0,0]
=> [[[]]]
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> 2
[[[]]]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[[]],[[]]]]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[[],[],[]]]]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[]]]
=> 2
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[[],[],[],[[]]]]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [[[],[],[[]],[]]]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[[],[],[[],[]]]]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[[],[],[[[]]]]]
=> 2
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [[[],[[]],[],[]]]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [[[],[[]],[[]]]]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [[[],[[],[]],[]]]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [[[],[[[]]],[]]]
=> 2
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[[],[[],[],[]]]]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[[],[[]]]]]
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[[],[[[]],[]]]]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[[],[[[],[]]]]]
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[[[[]]]]]]
=> 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[[]],[],[],[]]]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [[[[]],[],[[]]]]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [[[[]],[[]],[]]]
=> 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [[[[]],[[],[]]]]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[[[]],[[[]]]]]
=> 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[[[],[]],[],[]]]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[[[[]]],[],[]]]
=> 2
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [[[[],[]],[[]]]]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[[[[]]],[[]]]]
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[[[],[],[]],[]]]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[[[],[[]]],[]]]
=> 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [[[[[]],[]],[]]]
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [[[[[],[]]],[]]]
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [[[[[[]]]],[]]]
=> 2
[[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[],[],[]]]
=> ? = 2
[[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[[],[],[],[],[],[[]]]]
=> ? = 2
[[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [[[],[],[],[],[[]],[]]]
=> ? = 2
[[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[[],[],[],[],[[],[]]]]
=> ? = 2
[[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[[],[],[],[],[[[]]]]]
=> ? = 2
[[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [[[],[],[],[[]],[],[]]]
=> ? = 2
[[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [[[],[],[],[[]],[[]]]]
=> ? = 2
[[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [[[],[],[],[[],[]],[]]]
=> ? = 2
[[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [[[],[],[],[[[]]],[]]]
=> ? = 2
[[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[[],[],[],[[],[],[]]]]
=> ? = 2
[[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[],[],[[],[[]]]]]
=> ? = 2
[[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [[[],[],[],[[[]],[]]]]
=> ? = 2
[[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [[[],[],[],[[[],[]]]]]
=> ? = 2
[[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[],[],[[[[]]]]]]
=> ? = 2
[[],[],[[]],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [[[],[],[[]],[],[],[]]]
=> ? = 2
[[],[],[[]],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [[[],[],[[]],[],[[]]]]
=> ? = 2
[[],[],[[]],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [[[],[],[[]],[[]],[]]]
=> ? = 2
[[],[],[[]],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [[[],[],[[]],[[],[]]]]
=> ? = 2
[[],[],[[]],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [[[],[],[[]],[[[]]]]]
=> ? = 2
[[],[],[[],[]],[],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [[[],[],[[],[]],[],[]]]
=> ? = 2
[[],[],[[[]]],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [[[],[],[[[]]],[],[]]]
=> ? = 2
[[],[],[[],[]],[[]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [[[],[],[[],[]],[[]]]]
=> ? = 2
[[],[],[[[]]],[[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [[[],[],[[[]]],[[]]]]
=> ? = 2
[[],[],[[],[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [[[],[],[[],[],[]],[]]]
=> ? = 2
[[],[],[[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [[[],[],[[],[[]]],[]]]
=> ? = 2
[[],[],[[[]],[]],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [[[],[],[[[]],[]],[]]]
=> ? = 2
[[],[],[[[],[]]],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [[[],[],[[[],[]]],[]]]
=> ? = 2
[[],[],[[[[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [[[],[],[[[[]]]],[]]]
=> ? = 2
[[],[],[[],[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[[],[],[[],[],[],[]]]]
=> ? = 2
[[],[],[[],[],[[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [[[],[],[[],[],[[]]]]]
=> ? = 2
[[],[],[[],[[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [[[],[],[[],[[]],[]]]]
=> ? = 2
[[],[],[[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [[[],[],[[],[[],[]]]]]
=> ? = 2
[[],[],[[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [[[],[],[[],[[[]]]]]]
=> ? = 2
[[],[],[[[]],[],[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [[[],[],[[[]],[],[]]]]
=> ? = 2
[[],[],[[[]],[[]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [[[],[],[[[]],[[]]]]]
=> ? = 2
[[],[],[[[],[]],[]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [[[],[],[[[],[]],[]]]]
=> ? = 2
[[],[],[[[[]]],[]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [[[],[],[[[[]]],[]]]]
=> ? = 2
[[],[],[[[],[],[]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [[[],[],[[[],[],[]]]]]
=> ? = 2
[[],[],[[[],[[]]]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [[[],[],[[[],[[]]]]]]
=> ? = 2
[[],[],[[[[]],[]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [[[],[],[[[[]],[]]]]]
=> ? = 2
[[],[],[[[[],[]]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [[[],[],[[[[],[]]]]]]
=> ? = 2
[[],[],[[[[[]]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[],[],[[[[[]]]]]]]
=> ? = 2
[[],[[]],[],[],[],[]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[[],[[]],[],[],[],[]]]
=> ? = 2
[[],[[]],[],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [[[],[[]],[],[],[[]]]]
=> ? = 2
[[],[[]],[],[[]],[]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[],[[]],[]]]
=> ? = 2
[[],[[]],[],[[],[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [[[],[[]],[],[[],[]]]]
=> ? = 2
[[],[[]],[],[[[]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [[[],[[]],[],[[[]]]]]
=> ? = 2
[[],[[]],[[]],[],[]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [[[],[[]],[[]],[],[]]]
=> ? = 2
[[],[[]],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [[[],[[]],[[]],[[]]]]
=> ? = 2
[[],[[]],[[],[]],[]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [[[],[[]],[[],[]],[]]]
=> ? = 2
Description
The pruning number of an ordered tree.
A hanging branch of an ordered tree is a proper factor of the form $[^r]^r$ for some $r\geq 1$. A hanging branch is a maximal hanging branch if it is not a proper factor of another hanging branch.
A pruning of an ordered tree is the act of deleting all its maximal hanging branches. The pruning order of an ordered tree is the number of prunings required to reduce it to $[]$.
Matching statistic: St001322
Values
[[]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[]]]
=> ([(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)
=> 2
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2
[[[],[]],[]]
=> ([(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)
=> 2
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(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),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(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),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[]],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[[]]]]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[]],[[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]]],[[]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]]]],[]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[]],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[],[]]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[]],[]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]]],[]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]],[]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[],[]]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,4)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[[]]]]]]
=> ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[]],[[]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[],[]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
Description
The size of a minimal independent dominating set in a graph.
Matching statistic: St001829
Values
[[]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[]]]
=> ([(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)
=> 2
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2
[[[],[]],[]]
=> ([(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)
=> 2
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(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),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(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),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[]],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[[]]]]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[]],[[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]]],[[]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]]]],[]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[]],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[],[]]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[]],[]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]]],[]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]],[]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[],[]]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,4)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[[]]]]]]
=> ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[]],[[]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[],[]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
Description
The common independence number of a graph.
The common independence number of a graph $G$ is the greatest integer $r$ such that every vertex of $G$ belongs to some independent set $X$ of vertices of cardinality at least $r$.
Matching statistic: St001339
Values
[[]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[]]]
=> ([(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)
=> 2
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2
[[[],[]],[]]
=> ([(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)
=> 2
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(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),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(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),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[[],[],[],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[]],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[],[[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[],[[[[]]]]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[]],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[]],[[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]]],[[]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]]]],[]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[],[[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[]],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[],[]]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[],[[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[]],[[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[]],[]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]]],[]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[],[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[]],[]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[],[]]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,4)],8)
=> ?
=> ?
=> ? = 2
[[],[],[[[[[]]]]]]
=> ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
[[],[[]],[[]],[[]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 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.
Matching statistic: St001340
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[[]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 0 = 1 - 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0 = 1 - 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0 = 1 - 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[],[[],[]]]
=> ([(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 = 2 - 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[[],[]],[]]
=> ([(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 = 2 - 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0 = 1 - 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[],[],[],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[],[[]],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[],[[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[],[[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[],[[],[[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[],[[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[],[[[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[],[[[[]]]]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[]],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[]],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[]],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[]],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[],[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[[]]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[],[]],[[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[[]]],[[]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[],[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[],[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[[]],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[[],[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[[[]]]],[]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[],[],[[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[],[[]],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[],[[],[]]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[],[[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[[]],[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[[]],[[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[[],[]],[]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[[[]]],[]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[[],[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[[],[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[[[]],[]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[[[],[]]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,4)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[],[[[[[]]]]]]
=> ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[[]],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[[]],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[[]],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[[]],[],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[[]],[],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[[]],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
[[],[[]],[[]],[[]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
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.
The following 69 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000287The number of connected components of a graph. St001734The lettericity of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001335The cardinality of a minimal cycle-isolating set of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St000374The number of exclusive right-to-left minima of a permutation. St000451The length of the longest pattern of the form k 1 2. St001665The number of pure excedances of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000299The number of nonisomorphic vertex-induced subtrees. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001093The detour number of a graph. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001471The magnitude of a Dyck path. St001530The depth of a Dyck path. St001674The number of vertices of the largest induced star graph in the graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000486The number of cycles of length at least 3 of a permutation. St000535The rank-width of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001271The competition number of a graph. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001512The minimum rank of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St001498The normalised height of a Nakayama algebra with magnitude 1. St000929The constant term of the character polynomial of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001060The distinguishing index of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001488The number of corners of a skew partition. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths.
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