Your data matches 55 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000679
Mapping
St000679: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[],[]]
 => 1 = 0 + 1
[[[]]]
 => 1 = 0 + 1
[[],[],[]]
 => 1 = 0 + 1
[[],[[]]]
 => 1 = 0 + 1
[[[]],[]]
 => 1 = 0 + 1
[[[],[]]]
 => 2 = 1 + 1
[[[[]]]]
 => 1 = 0 + 1
[[],[],[],[]]
 => 1 = 0 + 1
[[],[],[[]]]
 => 1 = 0 + 1
[[],[[]],[]]
 => 1 = 0 + 1
[[],[[],[]]]
 => 2 = 1 + 1
[[],[[[]]]]
 => 1 = 0 + 1
[[[]],[],[]]
 => 1 = 0 + 1
[[[]],[[]]]
 => 1 = 0 + 1
[[[],[]],[]]
 => 2 = 1 + 1
[[[[]]],[]]
 => 1 = 0 + 1
[[[],[],[]]]
 => 2 = 1 + 1
[[[],[[]]]]
 => 2 = 1 + 1
[[[[]],[]]]
 => 2 = 1 + 1
[[[[],[]]]]
 => 2 = 1 + 1
[[[[[]]]]]
 => 1 = 0 + 1
[[],[],[],[],[]]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => 1 = 0 + 1
[[],[],[[]],[]]
 => 1 = 0 + 1
[[],[],[[],[]]]
 => 2 = 1 + 1
[[],[],[[[]]]]
 => 1 = 0 + 1
[[],[[]],[],[]]
 => 1 = 0 + 1
[[],[[]],[[]]]
 => 1 = 0 + 1
[[],[[],[]],[]]
 => 2 = 1 + 1
[[],[[[]]],[]]
 => 1 = 0 + 1
[[],[[],[],[]]]
 => 2 = 1 + 1
[[],[[],[[]]]]
 => 2 = 1 + 1
[[],[[[]],[]]]
 => 2 = 1 + 1
[[],[[[],[]]]]
 => 2 = 1 + 1
[[],[[[[]]]]]
 => 1 = 0 + 1
[[[]],[],[],[]]
 => 1 = 0 + 1
[[[]],[],[[]]]
 => 1 = 0 + 1
[[[]],[[]],[]]
 => 1 = 0 + 1
[[[]],[[],[]]]
 => 2 = 1 + 1
[[[]],[[[]]]]
 => 1 = 0 + 1
[[[],[]],[],[]]
 => 2 = 1 + 1
[[[[]]],[],[]]
 => 1 = 0 + 1
[[[],[]],[[]]]
 => 2 = 1 + 1
[[[[]]],[[]]]
 => 1 = 0 + 1
[[[],[],[]],[]]
 => 2 = 1 + 1
[[[],[[]]],[]]
 => 2 = 1 + 1
[[[[]],[]],[]]
 => 2 = 1 + 1
[[[[],[]]],[]]
 => 2 = 1 + 1
[[[[[]]]],[]]
 => 1 = 0 + 1
[[[],[],[],[]]]
 => 2 = 1 + 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 3 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 = 0 + 1
[[[]]]
 => [.,[.,.]]
 => 1 = 0 + 1
[[],[],[]]
 => [[[.,.],.],.]
 => 1 = 0 + 1
[[],[[]]]
 => [[.,.],[.,.]]
 => 2 = 1 + 1
[[[]],[]]
 => [[.,[.,.]],.]
 => 1 = 0 + 1
[[[],[]]]
 => [.,[[.,.],.]]
 => 1 = 0 + 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => 1 = 0 + 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => 1 = 0 + 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => 2 = 1 + 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => 2 = 1 + 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => 1 = 0 + 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => 2 = 1 + 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => 1 = 0 + 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => 1 = 0 + 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => 1 = 0 + 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => 2 = 1 + 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => 1 = 0 + 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => 1 = 0 + 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => 1 = 0 + 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => 2 = 1 + 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => 2 = 1 + 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => 2 = 1 + 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => 2 = 1 + 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => 2 = 1 + 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => 2 = 1 + 1
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => 2 = 1 + 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => 2 = 1 + 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => 2 = 1 + 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => 2 = 1 + 1
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => 2 = 1 + 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => 2 = 1 + 1
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => 2 = 1 + 1
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => 2 = 1 + 1
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => 2 = 1 + 1
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => 2 = 1 + 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => 2 = 1 + 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => 1 = 0 + 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => 2 = 1 + 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => 1 = 0 + 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => 2 = 1 + 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => 1 = 0 + 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => 1 = 0 + 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => 1 = 0 + 1
[[[],[],[],[]]]
 => [.,[[[[.,.],.],.],.]]
 => 1 = 0 + 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 7 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[],[]]
 => [1,0,1,0]
 => 1 = 0 + 1
[[[]]]
 => [1,1,0,0]
 => 1 = 0 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 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: St001174
(load all 11 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[],[]]
 => [1,0,1,0]
 => [1,2] => 0
[[[]]]
 => [1,1,0,0]
 => [2,1] => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1,3] => 0
[[[],[]]]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [3,2,1] => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 0
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 0
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 0
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 0
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 0
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 0
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 0
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 0
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 0
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 0
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 0
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 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
(load all 3 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ 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
[[],[]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[[]]]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[],[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[[[]]]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[],[],[]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 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: St000846
(load all 2 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[],[]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[[]]]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[],[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[[[]]]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[],[],[]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St000397
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 = 0 + 2
[[[]]]
 => [.,[.,.]]
 => [[],[[],[]]]
 => 2 = 0 + 2
[[],[],[]]
 => [[[.,.],.],.]
 => [[[[],[]],[]],[]]
 => 2 = 0 + 2
[[],[[]]]
 => [[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => 3 = 1 + 2
[[[]],[]]
 => [[.,[.,.]],.]
 => [[[],[[],[]]],[]]
 => 2 = 0 + 2
[[[],[]]]
 => [.,[[.,.],.]]
 => [[],[[[],[]],[]]]
 => 2 = 0 + 2
[[[[]]]]
 => [.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 2 = 0 + 2
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [[[[[],[]],[]],[]],[]]
 => 2 = 0 + 2
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => 3 = 1 + 2
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [[[[],[]],[[],[]]],[]]
 => 3 = 1 + 2
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => 3 = 1 + 2
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => 3 = 1 + 2
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [[[[],[[],[]]],[]],[]]
 => 2 = 0 + 2
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [[[],[[],[]]],[[],[]]]
 => 3 = 1 + 2
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [[[],[[[],[]],[]]],[]]
 => 2 = 0 + 2
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [[[],[[],[[],[]]]],[]]
 => 2 = 0 + 2
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => 2 = 0 + 2
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => 3 = 1 + 2
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [[],[[[],[[],[]]],[]]]
 => 2 = 0 + 2
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => 2 = 0 + 2
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => 2 = 0 + 2
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [[[[[[],[]],[]],[]],[]],[]]
 => 2 = 0 + 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 + 2
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [[[[[],[[],[]]],[]],[]],[]]
 => 2 = 0 + 2
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [[[[],[[],[]]],[]],[[],[]]]
 => 3 = 1 + 2
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [[[[],[[],[]]],[[],[]]],[]]
 => 3 = 1 + 2
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [[[],[[],[]]],[[[],[]],[]]]
 => 3 = 1 + 2
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [[[],[[],[]]],[[],[[],[]]]]
 => 3 = 1 + 2
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [[[[],[[[],[]],[]]],[]],[]]
 => 2 = 0 + 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [[[[],[[],[[],[]]]],[]],[]]
 => 2 = 0 + 2
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [[[],[[[],[]],[]]],[[],[]]]
 => 3 = 1 + 2
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [[[],[[],[[],[]]]],[[],[]]]
 => 3 = 1 + 2
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [[[],[[[[],[]],[]],[]]],[]]
 => 2 = 0 + 2
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [[[],[[[],[]],[[],[]]]],[]]
 => 3 = 1 + 2
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [[[],[[[],[[],[]]],[]]],[]]
 => 2 = 0 + 2
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [[[],[[],[[[],[]],[]]]],[]]
 => 2 = 0 + 2
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [[[],[[],[[],[[],[]]]]],[]]
 => 2 = 0 + 2
[[[],[],[],[]]]
 => [.,[[[[.,.],.],.],.]]
 => [[],[[[[[],[]],[]],[]],[]]]
 => 2 = 0 + 2
Description
The Strahler number of a rooted tree.
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
[[],[]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[[]]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[],[],[]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[[],[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[[[]]]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[],[],[],[]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
Description
The rank-width of a graph.
Matching statistic: St000660
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[],[]]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[[[]]]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 0
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
Description
The number of rises of length at least 3 of a Dyck path. The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
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
[[],[]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[[]]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[],[],[]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[[],[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[[[]]]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[],[],[],[]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
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.
The following 45 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001335The cardinality of a minimal cycle-isolating set of a graph. St001393The induced matching number of a graph. St000845The maximal number of elements covered by an element in a poset. St000862The number of parts of the shifted shape of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000640The rank of the largest boolean interval in a poset. St000307The number of rowmotion orbits of 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. St000481The number of upper covers of a partition in dominance order. St000480The number of lower covers of a partition in dominance order. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001820The size of the image of the pop stack sorting operator. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001846The number of elements which do not have a complement in the lattice. St001964The interval resolution global dimension of a poset. St001626The number of maximal proper sublattices of a lattice. St000897The number of different multiplicities of parts of an integer partition. St000031The number of cycles in the cycle decomposition of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000256The number of parts from which one can substract 2 and still get an integer partition. St000761The number of ascents in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St001960The number of descents of a permutation minus one if its first entry is not one. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000805The number of peaks of the associated bargraph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001729The number of visible descents of a permutation.