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Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St000679
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000679: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> 1
[[],[]]
=> 1
[[[]]]
=> 1
[[],[],[]]
=> 1
[[],[[]]]
=> 1
[[[]],[]]
=> 1
[[[],[]]]
=> 2
[[[[]]]]
=> 1
[[],[],[],[]]
=> 1
[[],[],[[]]]
=> 1
[[],[[]],[]]
=> 1
[[],[[],[]]]
=> 2
[[],[[[]]]]
=> 1
[[[]],[],[]]
=> 1
[[[]],[[]]]
=> 1
[[[],[]],[]]
=> 2
[[[[]]],[]]
=> 1
[[[],[],[]]]
=> 2
[[[],[[]]]]
=> 2
[[[[]],[]]]
=> 2
[[[[],[]]]]
=> 2
[[[[[]]]]]
=> 1
[[],[],[],[],[]]
=> 1
[[],[],[],[[]]]
=> 1
[[],[],[[]],[]]
=> 1
[[],[],[[],[]]]
=> 2
[[],[],[[[]]]]
=> 1
[[],[[]],[],[]]
=> 1
[[],[[]],[[]]]
=> 1
[[],[[],[]],[]]
=> 2
[[],[[[]]],[]]
=> 1
[[],[[],[],[]]]
=> 2
[[],[[],[[]]]]
=> 2
[[],[[[]],[]]]
=> 2
[[],[[[],[]]]]
=> 2
[[],[[[[]]]]]
=> 1
[[[]],[],[],[]]
=> 1
[[[]],[],[[]]]
=> 1
[[[]],[[]],[]]
=> 1
[[[]],[[],[]]]
=> 2
[[[]],[[[]]]]
=> 1
[[[],[]],[],[]]
=> 2
[[[[]]],[],[]]
=> 1
[[[],[]],[[]]]
=> 2
[[[[]]],[[]]]
=> 1
[[[],[],[]],[]]
=> 2
[[[],[[]]],[]]
=> 2
[[[[]],[]],[]]
=> 2
[[[[],[]]],[]]
=> 2
[[[[[]]]],[]]
=> 1
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: St000396
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
St000396: Binary trees ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> 1
[[],[]]
=> [.,[.,.]]
=> 1
[[[]]]
=> [[.,.],.]
=> 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> 1
[[],[[]]]
=> [.,[[.,.],.]]
=> 1
[[[]],[]]
=> [[.,[.,.]],.]
=> 1
[[[],[]]]
=> [[.,.],[.,.]]
=> 2
[[[[]]]]
=> [[[.,.],.],.]
=> 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> 1
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> 1
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> 1
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> 1
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> 2
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> 1
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> 2
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> 2
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> 2
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> 1
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> 1
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> 2
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> 1
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> 2
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> 1
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> 2
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> 2
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> 2
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> 2
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> 1
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> 1
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> 1
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> 1
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> 2
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> 2
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> 2
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> 2
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> 2
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> 2
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> 1
[[[]],[],[],[],[],[],[]]
=> [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ? = 1
[[[]],[],[],[],[],[[]]]
=> [[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> ? = 1
[[[]],[],[],[],[[]],[]]
=> [[.,[.,[.,[.,[[.,[.,.]],.]]]]],.]
=> ? = 1
[[[]],[],[],[],[[[]]]]
=> [[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> ? = 1
[[[]],[],[],[[]],[],[]]
=> [[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> ? = 1
[[[]],[],[],[[]],[[]]]
=> [[.,[.,[.,[[.,[[.,.],.]],.]]]],.]
=> ? = 1
[[[]],[],[],[[[]]],[]]
=> [[.,[.,[.,[[[.,[.,.]],.],.]]]],.]
=> ? = 1
[[[]],[],[],[[[[]]]]]
=> [[.,[.,[.,[[[[.,.],.],.],.]]]],.]
=> ? = 1
[[[]],[],[[]],[],[],[]]
=> [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.]
=> ? = 1
[[[]],[],[[]],[],[[]]]
=> [[.,[.,[[.,[.,[[.,.],.]]],.]]],.]
=> ? = 1
[[[]],[],[[]],[[]],[]]
=> [[.,[.,[[.,[[.,[.,.]],.]],.]]],.]
=> ? = 1
[[[]],[],[[]],[[[]]]]
=> [[.,[.,[[.,[[[.,.],.],.]],.]]],.]
=> ? = 1
[[[]],[],[[[]]],[],[]]
=> [[.,[.,[[[.,[.,[.,.]]],.],.]]],.]
=> ? = 1
[[[]],[],[[[]]],[[]]]
=> [[.,[.,[[[.,[[.,.],.]],.],.]]],.]
=> ? = 1
[[[]],[],[[[[]]]],[]]
=> [[.,[.,[[[[.,[.,.]],.],.],.]]],.]
=> ? = 1
[[[]],[],[[[[[]]]]]]
=> [[.,[.,[[[[[.,.],.],.],.],.]]],.]
=> ? = 1
[[[]],[[]],[],[],[],[]]
=> [[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> ? = 1
[[[]],[[]],[],[],[[]]]
=> [[.,[[.,[.,[.,[[.,.],.]]]],.]],.]
=> ? = 1
[[[]],[[]],[],[[]],[]]
=> [[.,[[.,[.,[[.,[.,.]],.]]],.]],.]
=> ? = 1
[[[]],[[]],[],[[[]]]]
=> [[.,[[.,[.,[[[.,.],.],.]]],.]],.]
=> ? = 1
[[[]],[[]],[[]],[],[]]
=> [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
=> ? = 1
[[[]],[[]],[[]],[[]]]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> ? = 1
[[[]],[[]],[[[]]],[]]
=> [[.,[[.,[[[.,[.,.]],.],.]],.]],.]
=> ? = 1
[[[]],[[]],[[[[]]]]]
=> [[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> ? = 1
[[[]],[[[]]],[],[],[]]
=> [[.,[[[.,[.,[.,[.,.]]]],.],.]],.]
=> ? = 1
[[[]],[[[]]],[],[[]]]
=> [[.,[[[.,[.,[[.,.],.]]],.],.]],.]
=> ? = 1
[[[]],[[[]]],[[]],[]]
=> [[.,[[[.,[[.,[.,.]],.]],.],.]],.]
=> ? = 1
[[[]],[[[]]],[[[]]]]
=> [[.,[[[.,[[[.,.],.],.]],.],.]],.]
=> ? = 1
[[[]],[[[[]]]],[],[]]
=> [[.,[[[[.,[.,[.,.]]],.],.],.]],.]
=> ? = 1
[[[]],[[[[]]]],[[]]]
=> [[.,[[[[.,[[.,.],.]],.],.],.]],.]
=> ? = 1
[[[]],[[[[[]]]]],[]]
=> [[.,[[[[[.,[.,.]],.],.],.],.]],.]
=> ? = 1
[[[]],[[[[[[]]]]]]]
=> [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> ? = 1
[[[[]]],[],[],[],[],[]]
=> [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> ? = 1
[[[[]]],[],[],[],[[]]]
=> [[[.,[.,[.,[.,[[.,.],.]]]]],.],.]
=> ? = 1
[[[[]]],[],[],[[]],[]]
=> [[[.,[.,[.,[[.,[.,.]],.]]]],.],.]
=> ? = 1
[[[[]]],[],[],[[],[]]]
=> [[[.,.],.],[.,[.,[[.,.],[.,.]]]]]
=> ? = 2
[[[[]]],[],[],[[[]]]]
=> [[[.,[.,[.,[[[.,.],.],.]]]],.],.]
=> ? = 1
[[[[]]],[],[[]],[],[]]
=> [[[.,[.,[[.,[.,[.,.]]],.]]],.],.]
=> ? = 1
[[[[]]],[],[[]],[[]]]
=> [[[.,[.,[[.,[[.,.],.]],.]]],.],.]
=> ? = 1
Description
The register function (or Horton-Strahler number) of a binary tree.
This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
Matching statistic: St000920
(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
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1,0]
=> 1
[[],[]]
=> [.,[.,.]]
=> [1,0,1,0]
=> 1
[[[]]]
=> [[.,.],.]
=> [1,1,0,0]
=> 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[[[],[]]]
=> [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> 1
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 2
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> 1
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 2
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> 2
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[[[]],[],[],[],[[],[]]]
=> [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[[[]],[],[],[[],[],[]]]
=> [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[[[]],[],[],[[],[[]]]]
=> [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 2
[[[]],[],[],[[[]],[]]]
=> [[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
[[[]],[],[],[[[],[]]]]
=> [[.,.],[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 2
[[[]],[],[[]],[[],[]]]
=> [[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 2
[[[]],[],[[],[]],[],[]]
=> [[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[[[]],[],[[],[]],[[]]]
=> [[.,.],[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2
[[[]],[],[[[]],[]],[]]
=> [[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 2
[[[]],[],[[[],[]]],[]]
=> [[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 2
[[[]],[],[[],[],[],[]]]
=> [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[[[]],[],[[],[],[[]]]]
=> [[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
[[[]],[],[[],[[]],[]]]
=> [[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2
[[[]],[],[[],[[],[]]]]
=> [[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[[[]],[],[[],[[[]]]]]
=> [[.,.],[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 2
[[[]],[],[[[]],[[]]]]
=> [[.,.],[.,[[[.,.],.],[[.,.],.]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 2
[[[]],[],[[[],[],[]]]]
=> [[.,.],[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 2
[[[]],[],[[[],[[]]]]]
=> [[.,.],[.,[[[.,.],[[.,.],.]],.]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 2
[[[]],[],[[[[]],[]]]]
=> [[.,.],[.,[[[[.,.],.],[.,.]],.]]]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 2
[[[]],[],[[[[],[]]]]]
=> [[.,.],[.,[[[[.,.],[.,.]],.],.]]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 2
[[[]],[[]],[],[[],[]]]
=> [[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[[[]],[[]],[[],[]],[]]
=> [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 2
[[[]],[[]],[[],[],[]]]
=> [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 2
[[[]],[[],[]],[],[],[]]
=> [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2
[[[]],[[],[]],[],[[]]]
=> [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 2
[[[]],[[],[]],[[]],[]]
=> [[.,.],[[.,[.,.]],[[.,[.,.]],.]]]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[]],[[],[]],[[],[]]]
=> [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 2
[[[]],[[],[]],[[[]]]]
=> [[.,.],[[.,[.,.]],[[[.,.],.],.]]]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2
[[[]],[[],[],[]],[],[]]
=> [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[[[]],[[],[[]]],[],[]]
=> [[.,.],[[.,[[.,.],.]],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 2
[[[]],[[[]],[]],[],[]]
=> [[.,.],[[[.,[.,.]],.],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2
[[[]],[[[],[]]],[],[]]
=> [[.,.],[[[.,[.,.]],[.,[.,.]]],.]]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2
[[[]],[[],[],[]],[[]]]
=> [[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2
[[[]],[[],[[]]],[[]]]
=> [[.,.],[[.,[[.,.],.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[[[]],[[[]],[]],[[]]]
=> [[.,.],[[[.,[.,.]],.],[[.,.],.]]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 2
[[[]],[[[],[]]],[[]]]
=> [[.,.],[[[.,[.,.]],[[.,.],.]],.]]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 2
[[[]],[[],[],[[]]],[]]
=> [[.,.],[[.,[.,[[.,.],.]]],[.,.]]]
=> [1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
[[[]],[[],[[]],[]],[]]
=> [[.,.],[[.,[[.,[.,.]],.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 2
[[[]],[[],[[],[]]],[]]
=> [[.,.],[[.,[[.,[.,.]],[.,.]]],.]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2
[[[]],[[],[[[]]]],[]]
=> [[.,.],[[.,[[[.,.],.],.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[[[]],[[[]],[],[]],[]]
=> [[.,.],[[[.,[.,[.,.]]],.],[.,.]]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 2
[[[]],[[[]],[[]]],[]]
=> [[.,.],[[[.,[[.,.],.]],.],[.,.]]]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 2
[[[]],[[[],[]],[]],[]]
=> [[.,.],[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 2
[[[]],[[[[]]],[]],[]]
=> [[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 2
[[[]],[[[],[],[]]],[]]
=> [[.,.],[[[.,[.,[.,.]]],[.,.]],.]]
=> [1,1,1,0,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[[[]],[[[],[[]]]],[]]
=> [[.,.],[[[.,[[.,.],.]],[.,.]],.]]
=> [1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2
[[[]],[[[[]],[]]],[]]
=> [[.,.],[[[[.,[.,.]],.],[.,.]],.]]
=> [1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2
[[[]],[[[[],[]]]],[]]
=> [[.,.],[[[[.,[.,.]],[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
[[[]],[[],[],[],[],[]]]
=> [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2
[[[]],[[],[],[],[[]]]]
=> [[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
Matching statistic: St000862
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => [1] => 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => [1,2] => 1
[[[]]]
=> [[.,.],.]
=> [1,2] => [1,2] => 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => 1
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => 1
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => 2
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => 2
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => 2
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => 2
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,3,4,5] => 1
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => 2
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,3,4,5] => 1
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,2,3,4,5] => 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,2,4,3,5] => 2
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,2,3,4,5] => 1
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => 2
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => 2
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,2,3,5,4] => 2
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => 2
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => 1
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => 1
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => 1
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => 2
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => 2
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => 2
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,4,2,3,5] => 2
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => 2
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => 2
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => 2
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => 1
[[[],[],[],[]],[],[]]
=> [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [7,6,4,3,2,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[],[[]]],[],[]]
=> [[.,[.,[[.,.],.]]],[.,[.,.]]]
=> [7,6,3,4,2,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[[]],[]],[],[]]
=> [[.,[[.,.],[.,.]]],[.,[.,.]]]
=> [7,6,4,2,3,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[[],[]]],[],[]]
=> [[.,[[.,[.,.]],.]],[.,[.,.]]]
=> [7,6,3,2,4,1,5] => [1,5,2,4,3,6,7] => ? = 2
[[[],[[[]]]],[],[]]
=> [[.,[[[.,.],.],.]],[.,[.,.]]]
=> [7,6,2,3,4,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[],[],[]],[[]]]
=> [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [6,7,4,3,2,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[],[[]]],[[]]]
=> [[.,[.,[[.,.],.]]],[[.,.],.]]
=> [6,7,3,4,2,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[[]],[]],[[]]]
=> [[.,[[.,.],[.,.]]],[[.,.],.]]
=> [6,7,4,2,3,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[[],[]]],[[]]]
=> [[.,[[.,[.,.]],.]],[[.,.],.]]
=> [6,7,3,2,4,1,5] => [1,5,2,4,3,6,7] => ? = 2
[[[],[[[]]]],[[]]]
=> [[.,[[[.,.],.],.]],[[.,.],.]]
=> [6,7,2,3,4,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[],[],[],[]],[]]
=> [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [7,5,4,3,2,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[],[],[],[[]]],[]]
=> [[.,[.,[.,[[.,.],.]]]],[.,.]]
=> [7,4,5,3,2,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[],[],[[]],[]],[]]
=> [[.,[.,[[.,.],[.,.]]]],[.,.]]
=> [7,5,3,4,2,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[],[],[[],[]]],[]]
=> [[.,[.,[[.,[.,.]],.]]],[.,.]]
=> [7,4,3,5,2,1,6] => [1,6,2,3,5,4,7] => ? = 2
[[[],[],[[[]]]],[]]
=> [[.,[.,[[[.,.],.],.]]],[.,.]]
=> [7,3,4,5,2,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[],[[]],[],[]],[]]
=> [[.,[[.,.],[.,[.,.]]]],[.,.]]
=> [7,5,4,2,3,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[],[[]],[[]]],[]]
=> [[.,[[.,.],[[.,.],.]]],[.,.]]
=> [7,4,5,2,3,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[],[[],[]],[]],[]]
=> [[.,[[.,[.,.]],[.,.]]],[.,.]]
=> [7,5,3,2,4,1,6] => [1,6,2,4,3,5,7] => ? = 2
[[[],[[[]]],[]],[]]
=> [[.,[[[.,.],.],[.,.]]],[.,.]]
=> [7,5,2,3,4,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[],[[],[],[]]],[]]
=> [[.,[[.,[.,[.,.]]],.]],[.,.]]
=> [7,4,3,2,5,1,6] => [1,6,2,5,3,4,7] => ? = 2
[[[],[[],[[]]]],[]]
=> [[.,[[.,[[.,.],.]],.]],[.,.]]
=> [7,3,4,2,5,1,6] => [1,6,2,5,3,4,7] => ? = 2
[[[],[[[]],[]]],[]]
=> [[.,[[[.,.],[.,.]],.]],[.,.]]
=> [7,4,2,3,5,1,6] => [1,6,2,3,5,4,7] => ? = 2
[[[],[[[],[]]]],[]]
=> [[.,[[[.,[.,.]],.],.]],[.,.]]
=> [7,3,2,4,5,1,6] => [1,6,2,4,5,3,7] => ? = 2
[[[],[[[[]]]]],[]]
=> [[.,[[[[.,.],.],.],.]],[.,.]]
=> [7,2,3,4,5,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[[],[],[],[]]],[]]
=> [[[.,[.,[.,[.,.]]]],.],[.,.]]
=> [7,4,3,2,1,5,6] => [1,5,6,2,3,4,7] => ? = 2
[[[[],[],[[]]]],[]]
=> [[[.,[.,[[.,.],.]]],.],[.,.]]
=> [7,3,4,2,1,5,6] => [1,5,6,2,3,4,7] => ? = 2
[[[[],[[]],[]]],[]]
=> [[[.,[[.,.],[.,.]]],.],[.,.]]
=> [7,4,2,3,1,5,6] => [1,5,6,2,3,4,7] => ? = 2
[[[[],[[],[]]]],[]]
=> [[[.,[[.,[.,.]],.]],.],[.,.]]
=> [7,3,2,4,1,5,6] => [1,5,6,2,4,3,7] => ? = 2
[[[[],[[[]]]]],[]]
=> [[[.,[[[.,.],.],.]],.],[.,.]]
=> [7,2,3,4,1,5,6] => [1,5,6,2,3,4,7] => ? = 2
[[[],[],[],[[],[]]]]
=> [[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [5,4,6,3,2,1,7] => [1,7,2,3,4,6,5] => ? = 2
[[[],[],[[],[]],[]]]
=> [[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [6,4,3,5,2,1,7] => [1,7,2,3,5,4,6] => ? = 2
[[[],[],[[],[],[]]]]
=> [[.,[.,[[.,[.,[.,.]]],.]]],.]
=> [5,4,3,6,2,1,7] => [1,7,2,3,6,4,5] => ? = 2
[[[],[],[[],[[]]]]]
=> [[.,[.,[[.,[[.,.],.]],.]]],.]
=> [4,5,3,6,2,1,7] => [1,7,2,3,6,4,5] => ? = 2
[[[],[],[[[]],[]]]]
=> [[.,[.,[[[.,.],[.,.]],.]]],.]
=> [5,3,4,6,2,1,7] => [1,7,2,3,4,6,5] => ? = 2
[[[],[],[[[],[]]]]]
=> [[.,[.,[[[.,[.,.]],.],.]]],.]
=> [4,3,5,6,2,1,7] => [1,7,2,3,5,6,4] => ? = 2
[[[],[[]],[[],[]]]]
=> [[.,[[.,.],[[.,[.,.]],.]]],.]
=> [5,4,6,2,3,1,7] => [1,7,2,3,4,6,5] => ? = 2
[[[],[[],[]],[],[]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> [6,5,3,2,4,1,7] => [1,7,2,4,3,5,6] => ? = 2
[[[],[[],[]],[[]]]]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> [5,6,3,2,4,1,7] => [1,7,2,4,3,5,6] => ? = 2
[[[],[[],[],[]],[]]]
=> [[.,[[.,[.,[.,.]]],[.,.]]],.]
=> [6,4,3,2,5,1,7] => [1,7,2,5,3,4,6] => ? = 2
[[[],[[],[[]]],[]]]
=> [[.,[[.,[[.,.],.]],[.,.]]],.]
=> [6,3,4,2,5,1,7] => [1,7,2,5,3,4,6] => ? = 2
[[[],[[[]],[]],[]]]
=> [[.,[[[.,.],[.,.]],[.,.]]],.]
=> [6,4,2,3,5,1,7] => [1,7,2,3,5,4,6] => ? = 2
[[[],[[[],[]]],[]]]
=> [[.,[[[.,[.,.]],.],[.,.]]],.]
=> [6,3,2,4,5,1,7] => [1,7,2,4,5,3,6] => ? = 2
[[[],[[],[],[],[]]]]
=> [[.,[[.,[.,[.,[.,.]]]],.]],.]
=> [5,4,3,2,6,1,7] => [1,7,2,6,3,4,5] => ? = 2
[[[],[[],[],[[]]]]]
=> [[.,[[.,[.,[[.,.],.]]],.]],.]
=> [4,5,3,2,6,1,7] => [1,7,2,6,3,4,5] => ? = 2
[[[],[[],[[]],[]]]]
=> [[.,[[.,[[.,.],[.,.]]],.]],.]
=> [5,3,4,2,6,1,7] => [1,7,2,6,3,4,5] => ? = 2
[[[],[[],[[],[]]]]]
=> [[.,[[.,[[.,[.,.]],.]],.]],.]
=> [4,3,5,2,6,1,7] => [1,7,2,6,3,5,4] => ? = 2
[[[],[[],[[[]]]]]]
=> [[.,[[.,[[[.,.],.],.]],.]],.]
=> [3,4,5,2,6,1,7] => [1,7,2,6,3,4,5] => ? = 2
[[[],[[[]],[],[]]]]
=> [[.,[[[.,.],[.,[.,.]]],.]],.]
=> [5,4,2,3,6,1,7] => [1,7,2,3,6,4,5] => ? = 2
[[[],[[[]],[[]]]]]
=> [[.,[[[.,.],[[.,.],.]],.]],.]
=> [4,5,2,3,6,1,7] => [1,7,2,3,6,4,5] => ? = 2
[[[],[[[],[]],[]]]]
=> [[.,[[[.,[.,.]],[.,.]],.]],.]
=> [5,3,2,4,6,1,7] => [1,7,2,4,6,3,5] => ? = 2
Description
The number of parts of the shifted shape of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of parts of the shifted shape.
Matching statistic: St001741
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => [1] => 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => [1,2] => 1
[[[]]]
=> [[.,.],.]
=> [1,2] => [1,2] => 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => 1
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => 1
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => 2
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => 2
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => 2
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => 2
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,3,4,5] => 1
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => 2
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,3,4,5] => 1
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,2,3,4,5] => 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,2,4,3,5] => 2
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,2,3,4,5] => 1
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => 2
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => 2
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,2,3,5,4] => 2
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => 2
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => 1
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => 1
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => 1
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => 2
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => 2
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => 2
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,4,2,3,5] => 2
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => 2
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => 2
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => 2
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => 1
[[[],[],[],[]],[],[]]
=> [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [7,6,4,3,2,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[],[[]]],[],[]]
=> [[.,[.,[[.,.],.]]],[.,[.,.]]]
=> [7,6,3,4,2,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[[]],[]],[],[]]
=> [[.,[[.,.],[.,.]]],[.,[.,.]]]
=> [7,6,4,2,3,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[[],[]]],[],[]]
=> [[.,[[.,[.,.]],.]],[.,[.,.]]]
=> [7,6,3,2,4,1,5] => [1,5,2,4,3,6,7] => ? = 2
[[[],[[[]]]],[],[]]
=> [[.,[[[.,.],.],.]],[.,[.,.]]]
=> [7,6,2,3,4,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[],[],[]],[[]]]
=> [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [6,7,4,3,2,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[],[[]]],[[]]]
=> [[.,[.,[[.,.],.]]],[[.,.],.]]
=> [6,7,3,4,2,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[[]],[]],[[]]]
=> [[.,[[.,.],[.,.]]],[[.,.],.]]
=> [6,7,4,2,3,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[[],[]]],[[]]]
=> [[.,[[.,[.,.]],.]],[[.,.],.]]
=> [6,7,3,2,4,1,5] => [1,5,2,4,3,6,7] => ? = 2
[[[],[[[]]]],[[]]]
=> [[.,[[[.,.],.],.]],[[.,.],.]]
=> [6,7,2,3,4,1,5] => [1,5,2,3,4,6,7] => ? = 2
[[[],[],[],[],[]],[]]
=> [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [7,5,4,3,2,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[],[],[],[[]]],[]]
=> [[.,[.,[.,[[.,.],.]]]],[.,.]]
=> [7,4,5,3,2,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[],[],[[]],[]],[]]
=> [[.,[.,[[.,.],[.,.]]]],[.,.]]
=> [7,5,3,4,2,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[],[],[[],[]]],[]]
=> [[.,[.,[[.,[.,.]],.]]],[.,.]]
=> [7,4,3,5,2,1,6] => [1,6,2,3,5,4,7] => ? = 2
[[[],[],[[[]]]],[]]
=> [[.,[.,[[[.,.],.],.]]],[.,.]]
=> [7,3,4,5,2,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[],[[]],[],[]],[]]
=> [[.,[[.,.],[.,[.,.]]]],[.,.]]
=> [7,5,4,2,3,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[],[[]],[[]]],[]]
=> [[.,[[.,.],[[.,.],.]]],[.,.]]
=> [7,4,5,2,3,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[],[[],[]],[]],[]]
=> [[.,[[.,[.,.]],[.,.]]],[.,.]]
=> [7,5,3,2,4,1,6] => [1,6,2,4,3,5,7] => ? = 2
[[[],[[[]]],[]],[]]
=> [[.,[[[.,.],.],[.,.]]],[.,.]]
=> [7,5,2,3,4,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[],[[],[],[]]],[]]
=> [[.,[[.,[.,[.,.]]],.]],[.,.]]
=> [7,4,3,2,5,1,6] => [1,6,2,5,3,4,7] => ? = 2
[[[],[[],[[]]]],[]]
=> [[.,[[.,[[.,.],.]],.]],[.,.]]
=> [7,3,4,2,5,1,6] => [1,6,2,5,3,4,7] => ? = 2
[[[],[[[]],[]]],[]]
=> [[.,[[[.,.],[.,.]],.]],[.,.]]
=> [7,4,2,3,5,1,6] => [1,6,2,3,5,4,7] => ? = 2
[[[],[[[],[]]]],[]]
=> [[.,[[[.,[.,.]],.],.]],[.,.]]
=> [7,3,2,4,5,1,6] => [1,6,2,4,5,3,7] => ? = 2
[[[],[[[[]]]]],[]]
=> [[.,[[[[.,.],.],.],.]],[.,.]]
=> [7,2,3,4,5,1,6] => [1,6,2,3,4,5,7] => ? = 2
[[[[],[],[],[]]],[]]
=> [[[.,[.,[.,[.,.]]]],.],[.,.]]
=> [7,4,3,2,1,5,6] => [1,5,6,2,3,4,7] => ? = 2
[[[[],[],[[]]]],[]]
=> [[[.,[.,[[.,.],.]]],.],[.,.]]
=> [7,3,4,2,1,5,6] => [1,5,6,2,3,4,7] => ? = 2
[[[[],[[]],[]]],[]]
=> [[[.,[[.,.],[.,.]]],.],[.,.]]
=> [7,4,2,3,1,5,6] => [1,5,6,2,3,4,7] => ? = 2
[[[[],[[],[]]]],[]]
=> [[[.,[[.,[.,.]],.]],.],[.,.]]
=> [7,3,2,4,1,5,6] => [1,5,6,2,4,3,7] => ? = 2
[[[[],[[[]]]]],[]]
=> [[[.,[[[.,.],.],.]],.],[.,.]]
=> [7,2,3,4,1,5,6] => [1,5,6,2,3,4,7] => ? = 2
[[[],[],[],[],[],[]]]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[],[],[],[[]]]]
=> [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [5,6,4,3,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[],[],[[]],[]]]
=> [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [6,4,5,3,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[],[],[[],[]]]]
=> [[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [5,4,6,3,2,1,7] => [1,7,2,3,4,6,5] => ? = 2
[[[],[],[],[[[]]]]]
=> [[.,[.,[.,[[[.,.],.],.]]]],.]
=> [4,5,6,3,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[],[[]],[],[]]]
=> [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [6,5,3,4,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[],[[]],[[]]]]
=> [[.,[.,[[.,.],[[.,.],.]]]],.]
=> [5,6,3,4,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[],[[],[]],[]]]
=> [[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [6,4,3,5,2,1,7] => [1,7,2,3,5,4,6] => ? = 2
[[[],[],[[[]]],[]]]
=> [[.,[.,[[[.,.],.],[.,.]]]],.]
=> [6,3,4,5,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[],[[],[],[]]]]
=> [[.,[.,[[.,[.,[.,.]]],.]]],.]
=> [5,4,3,6,2,1,7] => [1,7,2,3,6,4,5] => ? = 2
[[[],[],[[],[[]]]]]
=> [[.,[.,[[.,[[.,.],.]],.]]],.]
=> [4,5,3,6,2,1,7] => [1,7,2,3,6,4,5] => ? = 2
[[[],[],[[[]],[]]]]
=> [[.,[.,[[[.,.],[.,.]],.]]],.]
=> [5,3,4,6,2,1,7] => [1,7,2,3,4,6,5] => ? = 2
[[[],[],[[[],[]]]]]
=> [[.,[.,[[[.,[.,.]],.],.]]],.]
=> [4,3,5,6,2,1,7] => [1,7,2,3,5,6,4] => ? = 2
[[[],[],[[[[]]]]]]
=> [[.,[.,[[[[.,.],.],.],.]]],.]
=> [3,4,5,6,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[[]],[],[],[]]]
=> [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [6,5,4,2,3,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[[]],[],[[]]]]
=> [[.,[[.,.],[.,[[.,.],.]]]],.]
=> [5,6,4,2,3,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[[]],[[]],[]]]
=> [[.,[[.,.],[[.,.],[.,.]]]],.]
=> [6,4,5,2,3,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[[]],[[],[]]]]
=> [[.,[[.,.],[[.,[.,.]],.]]],.]
=> [5,4,6,2,3,1,7] => [1,7,2,3,4,6,5] => ? = 2
[[[],[[]],[[[]]]]]
=> [[.,[[.,.],[[[.,.],.],.]]],.]
=> [4,5,6,2,3,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[[],[]],[],[]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> [6,5,3,2,4,1,7] => [1,7,2,4,3,5,6] => ? = 2
[[[],[[[]]],[],[]]]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> [6,5,2,3,4,1,7] => [1,7,2,3,4,5,6] => ? = 2
Description
The largest integer such that all patterns of this size are contained in the permutation.
Matching statistic: St000455
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 33%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 33%
Values
[[]]
=> [.,.]
=> ([],1)
=> ([],1)
=> ? = 1 - 2
[[],[]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 2
[[[]]]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 2
[[],[],[]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 2
[[],[[]]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 2
[[[]],[]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 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)
=> ? = 1 - 2
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0 = 2 - 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 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)
=> ? = 1 - 2
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 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
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 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)
=> ? = 1 - 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
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 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)
=> ? = 1 - 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)
=> ? = 1 - 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)
=> ? = 1 - 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)
=> ? = 1 - 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)
=> ? = 1 - 2
[[],[],[],[],[[]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[],[],[[]],[]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 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)
=> ? = 1 - 2
[[],[],[[]],[],[]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[],[[]],[[]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 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)
=> ? = 1 - 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)
=> ? = 1 - 2
[[],[[]],[],[],[]]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[[]],[],[[]]]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[[]],[[]],[]]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 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)
=> ? = 1 - 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)
=> ? = 1 - 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)
=> ? = 1 - 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)
=> ? = 1 - 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: St001335
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001335: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 67%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001335: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 67%
Values
[[]]
=> [1,0]
=> [2,1] => ([(0,1)],2)
=> 0 = 1 - 1
[[],[]]
=> [1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[[]]]
=> [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 0 = 1 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 0 = 1 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 1 - 1
[[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? = 1 - 1
[[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 1 - 1
[[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 - 1
[[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 - 1
[[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,1,2,3,6,8,4,7] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ? = 1 - 1
[[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [7,1,2,3,6,4,8,5] => ([(0,7),(1,7),(2,7),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,1,2,3,8,7,4,6] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6)],8)
=> ? = 2 - 1
[[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [8,1,2,3,6,7,4,5] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ? = 1 - 1
[[],[],[[]],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 - 1
[[],[],[[]],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 1 - 1
[[],[],[[]],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 1 - 1
[[],[],[[]],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,1,2,8,3,7,5,6] => ([(0,6),(1,6),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7)],8)
=> ? = 2 - 1
[[],[],[[]],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 1 - 1
[[],[],[[[]]],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 - 1
[[],[],[[],[]],[[]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,1,2,5,3,4,8,6] => ([(0,7),(1,7),(2,3),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[[[]]],[[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 1 - 1
[[],[],[[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,1,2,5,3,8,4,7] => ([(0,7),(1,7),(2,3),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[[[]],[]],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,8,6,3,5,7] => ([(0,7),(1,4),(2,4),(3,6),(3,7),(4,5),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[[[],[]]],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [8,1,2,5,6,3,4,7] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[[[[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,8,3,7] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 1 - 1
[[],[],[[],[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2 - 1
[[],[],[[],[],[[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [7,1,2,6,3,4,8,5] => ([(0,7),(1,7),(2,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[[],[[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [8,1,2,5,3,7,4,6] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [8,1,2,6,3,7,4,5] => ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [6,1,2,5,3,7,8,4] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[[[]],[],[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => ([(0,4),(1,4),(2,6),(2,7),(3,6),(3,7),(4,5),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[[[]],[[]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,1,2,7,6,3,8,5] => ([(0,7),(1,7),(2,6),(3,4),(3,5),(3,7),(4,5),(4,6),(5,6)],8)
=> ? = 2 - 1
[[],[],[[[],[]],[]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [8,1,2,5,7,3,4,6] => ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[[[[]]],[]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => ([(0,4),(1,4),(2,7),(3,5),(3,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[[[],[[]]]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => ([(0,7),(1,7),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[[[[]],[]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,1,2,8,6,7,3,5] => ([(0,5),(1,5),(2,3),(2,4),(2,6),(3,6),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[[[[],[]]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [8,1,2,5,6,7,3,4] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[],[[[[[]]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 1 - 1
[[],[[]],[],[],[],[]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? = 1 - 1
[[],[[]],[],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 1 - 1
[[],[[]],[],[[]],[]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 1 - 1
[[],[[]],[],[[],[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,1,8,2,4,7,5,6] => ([(0,5),(1,7),(2,5),(2,7),(3,6),(3,7),(4,6),(4,7),(6,7)],8)
=> ? = 2 - 1
[[],[[]],[],[[[]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 1 - 1
[[],[[]],[[]],[],[]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 1 - 1
[[],[[]],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1 - 1
[[],[[]],[[],[]],[]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,8,2,6,4,5,7] => ([(0,5),(1,7),(2,5),(2,7),(3,6),(3,7),(4,6),(4,7),(6,7)],8)
=> ? = 2 - 1
[[],[[]],[[[]]],[]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,5,2,6,8,4,7] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 1 - 1
[[],[[]],[[],[],[]]]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [3,1,8,2,7,4,5,6] => ([(0,5),(1,5),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,7)],8)
=> ? = 2 - 1
[[],[[]],[[],[[]]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,7,2,6,4,8,5] => ([(0,6),(1,4),(2,4),(2,7),(3,5),(3,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[[]],[[[]],[]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [3,1,5,2,8,7,4,6] => ([(0,4),(1,2),(1,4),(2,7),(3,5),(3,6),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[[]],[[[],[]]]]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,1,8,2,6,7,4,5] => ([(0,2),(1,2),(1,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2 - 1
[[],[[]],[[[[]]]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 - 1
[[],[[[]]],[],[],[]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,4,8,2,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ? = 1 - 1
Description
The cardinality of a minimal cycle-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains all cycles.
Matching statistic: St000397
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Values
[[]]
=> [.,.]
=> [[],[]]
=> 2 = 1 + 1
[[],[]]
=> [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[[]]]
=> [[.,.],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> 2 = 1 + 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> 2 = 1 + 1
[[[],[]]]
=> [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 3 = 2 + 1
[[[[]]]]
=> [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 2 = 1 + 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> 2 = 1 + 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> 2 = 1 + 1
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> 2 = 1 + 1
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> 3 = 2 + 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> 2 = 1 + 1
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> 2 = 1 + 1
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> 2 = 1 + 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> 3 = 2 + 1
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> 2 = 1 + 1
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> 3 = 2 + 1
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> 3 = 2 + 1
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> 3 = 2 + 1
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> 3 = 2 + 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> 2 = 1 + 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> 2 = 1 + 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [[],[[],[[],[[[],[]],[]]]]]
=> 2 = 1 + 1
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [[],[[],[[[],[[],[]]],[]]]]
=> 2 = 1 + 1
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> 3 = 2 + 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [[],[[],[[[[],[]],[]],[]]]]
=> 2 = 1 + 1
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [[],[[[],[[],[[],[]]]],[]]]
=> 2 = 1 + 1
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [[],[[[],[[[],[]],[]]],[]]]
=> 2 = 1 + 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> 3 = 2 + 1
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [[],[[[[],[[],[]]],[]],[]]]
=> 2 = 1 + 1
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> 3 = 2 + 1
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> 3 = 2 + 1
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> 3 = 2 + 1
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [[],[[[[],[]],[[],[]]],[]]]
=> 3 = 2 + 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [[],[[[[[],[]],[]],[]],[]]]
=> 2 = 1 + 1
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [[[],[[],[[],[[],[]]]]],[]]
=> 2 = 1 + 1
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [[[],[[],[[[],[]],[]]]],[]]
=> 2 = 1 + 1
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [[[],[[[],[[],[]]],[]]],[]]
=> 2 = 1 + 1
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> 3 = 2 + 1
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [[[],[[[[],[]],[]],[]]],[]]
=> 2 = 1 + 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> 3 = 2 + 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [[[[],[[],[[],[]]]],[]],[]]
=> 2 = 1 + 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> 3 = 2 + 1
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [[[[],[[[],[]],[]]],[]],[]]
=> 2 = 1 + 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> 3 = 2 + 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [[[],[[[],[]],[]]],[[],[]]]
=> 3 = 2 + 1
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> 3 = 2 + 1
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [[[[],[[],[]]],[[],[]]],[]]
=> 3 = 2 + 1
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [[[[[],[[],[]]],[]],[]],[]]
=> 2 = 1 + 1
[[],[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[],[[],[[],[[],[[],[[],[[],[]]]]]]]]
=> ? = 1 + 1
[[],[],[],[],[],[[]]]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [[],[[],[[],[[],[[],[[[],[]],[]]]]]]]
=> ? = 1 + 1
[[],[],[],[],[[]],[]]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [[],[[],[[],[[],[[[],[[],[]]],[]]]]]]
=> ? = 1 + 1
[[],[],[],[],[[],[]]]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [[],[[],[[],[[],[[[],[]],[[],[]]]]]]]
=> ? = 2 + 1
[[],[],[],[],[[[]]]]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [[],[[],[[],[[],[[[[],[]],[]],[]]]]]]
=> ? = 1 + 1
[[],[],[],[[]],[],[]]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [[],[[],[[],[[[],[[],[[],[]]]],[]]]]]
=> ? = 1 + 1
[[],[],[],[[]],[[]]]
=> [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [[],[[],[[],[[[],[[[],[]],[]]],[]]]]]
=> ? = 1 + 1
[[],[],[],[[],[]],[]]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [[],[[],[[],[[[],[[],[]]],[[],[]]]]]]
=> ? = 2 + 1
[[],[],[],[[[]]],[]]
=> [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [[],[[],[[],[[[[],[[],[]]],[]],[]]]]]
=> ? = 1 + 1
[[],[],[],[[],[],[]]]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [[],[[],[[],[[[],[]],[[],[[],[]]]]]]]
=> ? = 2 + 1
[[],[],[],[[],[[]]]]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [[],[[],[[],[[[],[]],[[[],[]],[]]]]]]
=> ? = 2 + 1
[[],[],[],[[[]],[]]]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [[],[[],[[],[[[[],[]],[]],[[],[]]]]]]
=> ? = 2 + 1
[[],[],[],[[[],[]]]]
=> [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [[],[[],[[],[[[[],[]],[[],[]]],[]]]]]
=> ? = 2 + 1
[[],[],[],[[[[]]]]]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [[],[[],[[],[[[[[],[]],[]],[]],[]]]]]
=> ? = 1 + 1
[[],[],[[]],[],[],[]]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [[],[[],[[[],[[],[[],[[],[]]]]],[]]]]
=> ? = 1 + 1
[[],[],[[]],[],[[]]]
=> [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [[],[[],[[[],[[],[[[],[]],[]]]],[]]]]
=> ? = 1 + 1
[[],[],[[]],[[]],[]]
=> [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [[],[[],[[[],[[[],[[],[]]],[]]],[]]]]
=> ? = 1 + 1
[[],[],[[]],[[],[]]]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [[],[[],[[[],[]],[[[],[]],[[],[]]]]]]
=> ? = 2 + 1
[[],[],[[]],[[[]]]]
=> [.,[.,[[.,[[[.,.],.],.]],.]]]
=> [[],[[],[[[],[[[[],[]],[]],[]]],[]]]]
=> ? = 1 + 1
[[],[],[[],[]],[],[]]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[],[[],[[[],[[],[]]],[[],[[],[]]]]]]
=> ? = 2 + 1
[[],[],[[[]]],[],[]]
=> [.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [[],[[],[[[[],[[],[[],[]]]],[]],[]]]]
=> ? = 1 + 1
[[],[],[[],[]],[[]]]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[],[[],[[[],[[],[]]],[[[],[]],[]]]]]
=> ? = 2 + 1
[[],[],[[[]]],[[]]]
=> [.,[.,[[[.,[[.,.],.]],.],.]]]
=> [[],[[],[[[[],[[[],[]],[]]],[]],[]]]]
=> ? = 1 + 1
[[],[],[[],[],[]],[]]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [[],[[],[[[],[[],[[],[]]]],[[],[]]]]]
=> ? = 2 + 1
[[],[],[[],[[]]],[]]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [[],[[],[[[],[[[],[]],[]]],[[],[]]]]]
=> ? = 2 + 1
[[],[],[[[]],[]],[]]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [[],[[],[[[[],[[],[]]],[]],[[],[]]]]]
=> ? = 2 + 1
[[],[],[[[],[]]],[]]
=> [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [[],[[],[[[[],[[],[]]],[[],[]]],[]]]]
=> ? = 2 + 1
[[],[],[[[[]]]],[]]
=> [.,[.,[[[[.,[.,.]],.],.],.]]]
=> [[],[[],[[[[[],[[],[]]],[]],[]],[]]]]
=> ? = 1 + 1
[[],[],[[],[],[],[]]]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [[],[[],[[[],[]],[[],[[],[[],[]]]]]]]
=> ? = 2 + 1
[[],[],[[],[],[[]]]]
=> [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [[],[[],[[[],[]],[[],[[[],[]],[]]]]]]
=> ? = 2 + 1
[[],[],[[],[[]],[]]]
=> [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [[],[[],[[[],[]],[[[],[[],[]]],[]]]]]
=> ? = 2 + 1
[[],[],[[],[[],[]]]]
=> [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [[],[[],[[[],[[[],[]],[[],[]]]],[]]]]
=> ? = 2 + 1
[[],[],[[],[[[]]]]]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [[],[[],[[[],[]],[[[[],[]],[]],[]]]]]
=> ? = 2 + 1
[[],[],[[[]],[],[]]]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[],[[],[[[[],[]],[]],[[],[[],[]]]]]]
=> ? = 2 + 1
[[],[],[[[]],[[]]]]
=> [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[],[[],[[[[],[]],[]],[[[],[]],[]]]]]
=> ? = 2 + 1
[[],[],[[[],[]],[]]]
=> [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [[],[[],[[[[],[]],[[],[]]],[[],[]]]]]
=> ? = 2 + 1
[[],[],[[[[]]],[]]]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [[],[[],[[[[[],[]],[]],[]],[[],[]]]]]
=> ? = 2 + 1
[[],[],[[[],[],[]]]]
=> [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [[],[[],[[[[],[]],[[],[[],[]]]],[]]]]
=> ? = 2 + 1
[[],[],[[[],[[]]]]]
=> [.,[.,[[[.,.],[[.,.],.]],.]]]
=> [[],[[],[[[[],[]],[[[],[]],[]]],[]]]]
=> ? = 2 + 1
[[],[],[[[[]],[]]]]
=> [.,[.,[[[[.,.],.],[.,.]],.]]]
=> [[],[[],[[[[[],[]],[]],[[],[]]],[]]]]
=> ? = 2 + 1
[[],[],[[[[],[]]]]]
=> [.,[.,[[[[.,.],[.,.]],.],.]]]
=> [[],[[],[[[[[],[]],[[],[]]],[]],[]]]]
=> ? = 2 + 1
[[],[],[[[[[]]]]]]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> [[],[[],[[[[[[],[]],[]],[]],[]],[]]]]
=> ? = 1 + 1
[[],[[]],[],[],[],[]]
=> [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [[],[[[],[[],[[],[[],[[],[]]]]]],[]]]
=> ? = 1 + 1
[[],[[]],[],[],[[]]]
=> [.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [[],[[[],[[],[[],[[[],[]],[]]]]],[]]]
=> ? = 1 + 1
[[],[[]],[],[[]],[]]
=> [.,[[.,[.,[[.,[.,.]],.]]],.]]
=> [[],[[[],[[],[[[],[[],[]]],[]]]],[]]]
=> ? = 1 + 1
[[],[[]],[],[[],[]]]
=> [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [[],[[[],[]],[[],[[[],[]],[[],[]]]]]]
=> ? = 2 + 1
[[],[[]],[],[[[]]]]
=> [.,[[.,[.,[[[.,.],.],.]]],.]]
=> [[],[[[],[[],[[[[],[]],[]],[]]]],[]]]
=> ? = 1 + 1
[[],[[]],[[]],[],[]]
=> [.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [[],[[[],[[[],[[],[[],[]]]],[]]],[]]]
=> ? = 1 + 1
[[],[[]],[[]],[[]]]
=> [.,[[.,[[.,[[.,.],.]],.]],.]]
=> [[],[[[],[[[],[[[],[]],[]]],[]]],[]]]
=> ? = 1 + 1
[[],[[]],[[],[]],[]]
=> [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [[],[[[],[]],[[[],[[],[]]],[[],[]]]]]
=> ? = 2 + 1
Description
The Strahler number of a rooted tree.
Matching statistic: St001174
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Values
[[]]
=> [1,0]
=> [1] => ? = 1 - 1
[[],[]]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[[[]]]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 2 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0 = 1 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1 = 2 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1 = 2 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1 = 2 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0 = 1 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0 = 1 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1 = 2 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0 = 1 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0 = 1 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0 = 1 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1 = 2 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0 = 1 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1 = 2 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1 = 2 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 1 = 2 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0 = 1 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0 = 1 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0 = 1 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1 = 2 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 0 = 1 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1 = 2 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 0 = 1 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1 = 2 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 0 = 1 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1 = 2 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 1 = 2 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 1 = 2 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 0 = 1 - 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
[[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 1 - 1
[[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => ? = 1 - 1
[[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 2 - 1
[[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 1 - 1
[[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => ? = 1 - 1
[[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => ? = 1 - 1
[[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => ? = 2 - 1
[[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => ? = 1 - 1
[[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 2 - 1
[[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => ? = 2 - 1
[[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => ? = 2 - 1
[[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => ? = 2 - 1
[[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => ? = 1 - 1
[[],[],[[]],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => ? = 1 - 1
[[],[],[[]],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => ? = 1 - 1
[[],[],[[]],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => ? = 1 - 1
[[],[],[[]],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => ? = 2 - 1
[[],[],[[]],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => ? = 1 - 1
[[],[],[[],[]],[],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => ? = 2 - 1
[[],[],[[[]]],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => ? = 1 - 1
[[],[],[[],[]],[[]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => ? = 2 - 1
[[],[],[[[]]],[[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => ? = 1 - 1
[[],[],[[],[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => ? = 2 - 1
[[],[],[[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => ? = 2 - 1
[[],[],[[[]],[]],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => ? = 2 - 1
[[],[],[[[],[]]],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,6,4,5,3,7] => ? = 2 - 1
[[],[],[[[[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => ? = 1 - 1
[[],[],[[],[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 2 - 1
[[],[],[[],[],[[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => ? = 2 - 1
[[],[],[[],[[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => ? = 2 - 1
[[],[],[[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,7,5,6,3] => ? = 2 - 1
[[],[],[[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => ? = 2 - 1
[[],[],[[[]],[],[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => ? = 2 - 1
[[],[],[[[]],[[]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => ? = 2 - 1
[[],[],[[[],[]],[]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,6,4,5,7,3] => ? = 2 - 1
[[],[],[[[[]]],[]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => ? = 2 - 1
[[],[],[[[],[],[]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => ? = 2 - 1
[[],[],[[[],[[]]]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,7,4,6,5,3] => ? = 2 - 1
[[],[],[[[[]],[]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,7,5,4,6,3] => ? = 2 - 1
[[],[],[[[[],[]]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => ? = 2 - 1
[[],[],[[[[[]]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => ? = 1 - 1
[[],[[]],[],[],[],[]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => ? = 1 - 1
[[],[[]],[],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => ? = 1 - 1
[[],[[]],[],[[]],[]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => ? = 1 - 1
[[],[[]],[],[[],[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => ? = 2 - 1
[[],[[]],[],[[[]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => ? = 1 - 1
[[],[[]],[[]],[],[]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => ? = 1 - 1
[[],[[]],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => ? = 1 - 1
Description
The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St000298
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 67%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 67%
Values
[[]]
=> [1,0]
=> [1,0]
=> ([],1)
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2
[[[],[[]],[]]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 2
[[[],[[[]]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[[[[]],[],[]]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2
[[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 2
[[[[[]]],[]]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[[[[],[],[]]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[[[[],[[]]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2
[[[[[]],[]]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2
[[[[[],[]]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[[],[],[[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2
[[],[],[[[]],[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[[],[],[[[],[]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2
[[],[[]],[[],[]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[[],[[],[]],[],[]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[[],[[],[]],[[]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[[],[[],[],[]],[]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2
[[],[[],[[]]],[]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[[],[[[],[]]],[]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2
[[],[[],[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 2
[[],[[],[],[[]]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2
[[],[[],[[]],[]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 2
[[],[[],[[],[]]]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 2
[[],[[],[[[]]]]]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[[],[[[]],[],[]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2
[[],[[[],[]],[]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 2
[[],[[[[]]],[]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[[],[[[],[],[]]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
[[],[[[],[[]]]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2
[[],[[[[]],[]]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2
[[],[[[[],[]]]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 2
[[[]],[],[[],[]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[[[]],[[],[],[]]]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2
[[[]],[[[]],[]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[[[]],[[[],[]]]]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2
[[[],[]],[],[],[]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2
[[[],[]],[],[[]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2
[[[],[]],[[]],[]]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2
[[[],[]],[[],[]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 2
[[[],[]],[[[]]]]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2
[[[[]]],[[],[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[[[],[],[]],[],[]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[[],[[]]],[],[]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2
[[[[]],[]],[],[]]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[[[[],[]]],[],[]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[[],[],[]],[[]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000307The number of rowmotion orbits of a poset. St001330The hat guessing number of a graph. St000640The rank of the largest boolean interval in a poset. St001621The number of atoms of a lattice. 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. St000805The number of peaks of the associated bargraph.
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