Your data matches 18 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000396
(load all 11 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => 1 = 2 - 1
[[],[]]
 => [[.,.],.]
 => 1 = 2 - 1
[[[]]]
 => [.,[.,.]]
 => 1 = 2 - 1
[[],[],[]]
 => [[[.,.],.],.]
 => 1 = 2 - 1
[[],[[]]]
 => [[.,.],[.,.]]
 => 2 = 3 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => 1 = 2 - 1
[[[],[]]]
 => [.,[[.,.],.]]
 => 1 = 2 - 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => 1 = 2 - 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => 1 = 2 - 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => 2 = 3 - 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => 2 = 3 - 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => 1 = 2 - 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => 1 = 2 - 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => 1 = 2 - 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => 1 = 2 - 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => 2 = 3 - 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => 1 = 2 - 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => 1 = 2 - 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => 1 = 2 - 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => 1 = 2 - 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => 2 = 3 - 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => 2 = 3 - 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => 2 = 3 - 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => 2 = 3 - 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => 2 = 3 - 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => 2 = 3 - 1
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => 2 = 3 - 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => 2 = 3 - 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => 2 = 3 - 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => 2 = 3 - 1
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => 2 = 3 - 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => 2 = 3 - 1
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => 1 = 2 - 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => 2 = 3 - 1
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => 2 = 3 - 1
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => 2 = 3 - 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => 2 = 3 - 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => 1 = 2 - 1
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => 2 = 3 - 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => 2 = 3 - 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => 2 = 3 - 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => 1 = 2 - 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => 1 = 2 - 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => 1 = 2 - 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: St000679
(load all 6 compositions to match this statistic)
Mapping
Mp00328: Ordered trees —DeBruijn-Morselt plane tree automorphism⟶ Ordered trees
St000679: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [[]]
 => 1 = 2 - 1
[[],[]]
 => [[[]]]
 => 1 = 2 - 1
[[[]]]
 => [[],[]]
 => 1 = 2 - 1
[[],[],[]]
 => [[[[]]]]
 => 1 = 2 - 1
[[],[[]]]
 => [[[],[]]]
 => 2 = 3 - 1
[[[]],[]]
 => [[],[[]]]
 => 1 = 2 - 1
[[[],[]]]
 => [[[]],[]]
 => 1 = 2 - 1
[[[[]]]]
 => [[],[],[]]
 => 1 = 2 - 1
[[],[],[],[]]
 => [[[[[]]]]]
 => 1 = 2 - 1
[[],[],[[]]]
 => [[[[],[]]]]
 => 2 = 3 - 1
[[],[[]],[]]
 => [[[],[[]]]]
 => 2 = 3 - 1
[[],[[],[]]]
 => [[[[]],[]]]
 => 2 = 3 - 1
[[],[[[]]]]
 => [[[],[],[]]]
 => 2 = 3 - 1
[[[]],[],[]]
 => [[],[[[]]]]
 => 1 = 2 - 1
[[[]],[[]]]
 => [[],[[],[]]]
 => 2 = 3 - 1
[[[],[]],[]]
 => [[[]],[[]]]
 => 1 = 2 - 1
[[[[]]],[]]
 => [[],[],[[]]]
 => 1 = 2 - 1
[[[],[],[]]]
 => [[[[]]],[]]
 => 1 = 2 - 1
[[[],[[]]]]
 => [[[],[]],[]]
 => 2 = 3 - 1
[[[[]],[]]]
 => [[],[[]],[]]
 => 1 = 2 - 1
[[[[],[]]]]
 => [[[]],[],[]]
 => 1 = 2 - 1
[[[[[]]]]]
 => [[],[],[],[]]
 => 1 = 2 - 1
[[],[],[],[],[]]
 => [[[[[[]]]]]]
 => 1 = 2 - 1
[[],[],[],[[]]]
 => [[[[[],[]]]]]
 => 2 = 3 - 1
[[],[],[[]],[]]
 => [[[[],[[]]]]]
 => 2 = 3 - 1
[[],[],[[],[]]]
 => [[[[[]],[]]]]
 => 2 = 3 - 1
[[],[],[[[]]]]
 => [[[[],[],[]]]]
 => 2 = 3 - 1
[[],[[]],[],[]]
 => [[[],[[[]]]]]
 => 2 = 3 - 1
[[],[[]],[[]]]
 => [[[],[[],[]]]]
 => 2 = 3 - 1
[[],[[],[]],[]]
 => [[[[]],[[]]]]
 => 2 = 3 - 1
[[],[[[]]],[]]
 => [[[],[],[[]]]]
 => 2 = 3 - 1
[[],[[],[],[]]]
 => [[[[[]]],[]]]
 => 2 = 3 - 1
[[],[[],[[]]]]
 => [[[[],[]],[]]]
 => 2 = 3 - 1
[[],[[[]],[]]]
 => [[[],[[]],[]]]
 => 2 = 3 - 1
[[],[[[],[]]]]
 => [[[[]],[],[]]]
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [[[],[],[],[]]]
 => 2 = 3 - 1
[[[]],[],[],[]]
 => [[],[[[[]]]]]
 => 1 = 2 - 1
[[[]],[],[[]]]
 => [[],[[[],[]]]]
 => 2 = 3 - 1
[[[]],[[]],[]]
 => [[],[[],[[]]]]
 => 2 = 3 - 1
[[[]],[[],[]]]
 => [[],[[[]],[]]]
 => 2 = 3 - 1
[[[]],[[[]]]]
 => [[],[[],[],[]]]
 => 2 = 3 - 1
[[[],[]],[],[]]
 => [[[]],[[[]]]]
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [[],[],[[[]]]]
 => 1 = 2 - 1
[[[],[]],[[]]]
 => [[[]],[[],[]]]
 => 2 = 3 - 1
[[[[]]],[[]]]
 => [[],[],[[],[]]]
 => 2 = 3 - 1
[[[],[],[]],[]]
 => [[[[]]],[[]]]
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [[[],[]],[[]]]
 => 2 = 3 - 1
[[[[]],[]],[]]
 => [[],[[]],[[]]]
 => 1 = 2 - 1
[[[[],[]]],[]]
 => [[[]],[],[[]]]
 => 1 = 2 - 1
[[[[[]]]],[]]
 => [[],[],[],[[]]]
 => 1 = 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: St000397
(load all 3 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [[],[]]
 => 2
[[],[]]
 => [[.,.],.]
 => [[[],[]],[]]
 => 2
[[[]]]
 => [.,[.,.]]
 => [[],[[],[]]]
 => 2
[[],[],[]]
 => [[[.,.],.],.]
 => [[[[],[]],[]],[]]
 => 2
[[],[[]]]
 => [[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => 3
[[[]],[]]
 => [[.,[.,.]],.]
 => [[[],[[],[]]],[]]
 => 2
[[[],[]]]
 => [.,[[.,.],.]]
 => [[],[[[],[]],[]]]
 => 2
[[[[]]]]
 => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [[[[[],[]],[]],[]],[]]
 => 2
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => 3
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [[[[],[]],[[],[]]],[]]
 => 3
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => 3
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => 3
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [[[[],[[],[]]],[]],[]]
 => 2
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [[[],[[],[]]],[[],[]]]
 => 3
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [[[],[[[],[]],[]]],[]]
 => 2
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [[[],[[],[[],[]]]],[]]
 => 2
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => 2
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => 3
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [[],[[[],[[],[]]],[]]]
 => 2
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => 2
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => 2
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [[[[[[],[]],[]],[]],[]],[]]
 => 2
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [[[[[],[]],[]],[]],[[],[]]]
 => 3
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [[[[[],[]],[]],[[],[]]],[]]
 => 3
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [[[[],[]],[]],[[[],[]],[]]]
 => 3
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [[[[],[]],[]],[[],[[],[]]]]
 => 3
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [[[[[],[]],[[],[]]],[]],[]]
 => 3
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [[[[],[]],[[],[]]],[[],[]]]
 => 3
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [[[[],[]],[[[],[]],[]]],[]]
 => 3
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [[[[],[]],[[],[[],[]]]],[]]
 => 3
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [[[],[]],[[[[],[]],[]],[]]]
 => 3
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [[[],[]],[[[],[]],[[],[]]]]
 => 3
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [[[],[]],[[[],[[],[]]],[]]]
 => 3
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [[[],[]],[[],[[[],[]],[]]]]
 => 3
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [[[],[]],[[],[[],[[],[]]]]]
 => 3
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [[[[[],[[],[]]],[]],[]],[]]
 => 2
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [[[[],[[],[]]],[]],[[],[]]]
 => 3
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [[[[],[[],[]]],[[],[]]],[]]
 => 3
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [[[],[[],[]]],[[[],[]],[]]]
 => 3
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [[[],[[],[]]],[[],[[],[]]]]
 => 3
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [[[[],[[[],[]],[]]],[]],[]]
 => 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [[[[],[[],[[],[]]]],[]],[]]
 => 2
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [[[],[[[],[]],[]]],[[],[]]]
 => 3
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [[[],[[],[[],[]]]],[[],[]]]
 => 3
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [[[],[[[[],[]],[]],[]]],[]]
 => 2
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [[[],[[[],[]],[[],[]]]],[]]
 => 3
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [[[],[[[],[[],[]]],[]]],[]]
 => 2
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [[[],[[],[[[],[]],[]]]],[]]
 => 2
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [[[],[[],[[],[[],[]]]]],[]]
 => 2
Description
The Strahler number of a rooted tree.
Matching statistic: St000298
(load all 3 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => ([],1)
 => 1 = 2 - 1
[[],[]]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1 = 2 - 1
[[[]]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 2 - 1
[[],[],[]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1 = 2 - 1
[[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1 = 2 - 1
[[[],[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 2 - 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 2 - 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 2 - 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 3 - 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 3 - 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 3 - 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 3 - 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 2 - 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 3 - 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 2 - 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 2 - 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 2 - 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 3 - 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 2 - 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 2 - 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 2 - 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 2 - 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 3 - 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 3 - 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 3 - 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2 = 3 - 1
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 3 - 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 3 - 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 3 - 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2 = 3 - 1
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 3 - 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 3 - 1
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 2 - 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 3 - 1
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 3 - 1
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 2 - 1
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 3 - 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 3 - 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 3 - 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 2 - 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 2 - 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 2 - 1
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.
Matching statistic: St000920
(load all 10 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1,0]
 => 1 = 2 - 1
[[],[]]
 => [[.,.],.]
 => [1,1,0,0]
 => 1 = 2 - 1
[[[]]]
 => [.,[.,.]]
 => [1,0,1,0]
 => 1 = 2 - 1
[[],[],[]]
 => [[[.,.],.],.]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[],[[]]]
 => [[.,.],[.,.]]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[[[],[]]]
 => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [1,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 3 - 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0]
 => 2 = 3 - 1
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 3 - 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 3 - 1
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2 = 3 - 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 3 - 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 3 - 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
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: St001261
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
[[],[]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 1 = 2 - 1
[[[]]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 1 = 2 - 1
[[],[],[]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1 = 2 - 1
[[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 2 = 3 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1 = 2 - 1
[[[],[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1 = 2 - 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1 = 2 - 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 2 - 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 2 = 3 - 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 2 - 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 2 - 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 2 - 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 2 - 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 2 = 3 - 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 2 - 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 2 - 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 2 - 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 2 - 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 3 - 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 3 - 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 2 = 3 - 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 2 - 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 3 - 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 3 - 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 2 - 1
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 2 = 3 - 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 2 - 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 2 - 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 2 - 1
Description
The Castelnuovo-Mumford regularity of a graph.
Matching statistic: St000535
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000535: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => ([],1)
 => ([],1)
 => 0 = 2 - 2
[[],[]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0 = 2 - 2
[[[]]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0 = 2 - 2
[[],[],[]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 2 - 2
[[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1 = 3 - 2
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 2 - 2
[[[],[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 2 - 2
[[[[]]]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 2 - 2
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1 = 3 - 2
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1 = 3 - 2
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 3 - 2
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 3 - 2
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1 = 3 - 2
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 3 - 2
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 3 - 2
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1 = 3 - 2
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
Description
The rank-width of a graph.
Matching statistic: St001333
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => ([],1)
 => ([],1)
 => 0 = 2 - 2
[[],[]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0 = 2 - 2
[[[]]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0 = 2 - 2
[[],[],[]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 2 - 2
[[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1 = 3 - 2
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 2 - 2
[[[],[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 2 - 2
[[[[]]]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 2 - 2
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1 = 3 - 2
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1 = 3 - 2
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 3 - 2
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 3 - 2
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1 = 3 - 2
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 3 - 2
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 3 - 2
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1 = 3 - 2
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
Description
The cardinality of a minimal 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 one edge.
Matching statistic: St001393
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001393: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => ([],1)
 => ([],1)
 => 0 = 2 - 2
[[],[]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0 = 2 - 2
[[[]]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0 = 2 - 2
[[],[],[]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 2 - 2
[[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1 = 3 - 2
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 2 - 2
[[[],[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 2 - 2
[[[[]]]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 2 - 2
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1 = 3 - 2
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1 = 3 - 2
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 2 - 2
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 3 - 2
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 3 - 2
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1 = 3 - 2
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 3 - 2
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 3 - 2
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1 = 3 - 2
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 2 - 2
Description
The induced matching number of a graph. An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.
Matching statistic: St001174
(load all 17 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1] => ? = 2 - 2
[[],[]]
 => [.,[.,.]]
 => [2,1] => 0 = 2 - 2
[[[]]]
 => [[.,.],.]
 => [1,2] => 0 = 2 - 2
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => 0 = 2 - 2
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => 1 = 3 - 2
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => 0 = 2 - 2
[[[],[]]]
 => [[.,.],[.,.]]
 => [1,3,2] => 0 = 2 - 2
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => 0 = 2 - 2
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 0 = 2 - 2
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 1 = 3 - 2
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 1 = 3 - 2
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => 1 = 3 - 2
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1 = 3 - 2
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 0 = 2 - 2
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1 = 3 - 2
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0 = 2 - 2
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 0 = 2 - 2
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => 0 = 2 - 2
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1 = 3 - 2
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0 = 2 - 2
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 0 = 2 - 2
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => 0 = 2 - 2
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => 0 = 2 - 2
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => 1 = 3 - 2
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => 1 = 3 - 2
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => 1 = 3 - 2
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => 1 = 3 - 2
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => 1 = 3 - 2
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => 1 = 3 - 2
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => 1 = 3 - 2
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => 1 = 3 - 2
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => 1 = 3 - 2
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => 1 = 3 - 2
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => 1 = 3 - 2
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => 1 = 3 - 2
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1 = 3 - 2
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => 0 = 2 - 2
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => 1 = 3 - 2
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => 1 = 3 - 2
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 1 = 3 - 2
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => 1 = 3 - 2
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => 0 = 2 - 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => 0 = 2 - 2
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => 1 = 3 - 2
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => 1 = 3 - 2
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => 0 = 2 - 2
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => 1 = 3 - 2
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => 0 = 2 - 2
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => 0 = 2 - 2
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => 0 = 2 - 2
[[[],[],[],[]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => 0 = 2 - 2
Description
The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000455The second largest eigenvalue of a graph if it is integral. 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. St000862The number of parts of the shifted shape of a permutation. St000805The number of peaks of the associated bargraph. St001729The number of visible descents of a permutation. St000807The sum of the heights of the valleys of the associated bargraph.