Your data matches 21 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001335
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001335: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
 => []
 => [1] => ([],1)
 => 0
[[]]
 => [1,0]
 => [2,1] => ([(0,1)],2)
 => 0
[[],[]]
 => [1,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 0
[[[]]]
 => [1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[[],[[]]]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[[[],[]]]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 0
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[],[[],[]]]
 => [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
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 0
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 0
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 0
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[],[[]]]]
 => [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
[[[[]],[]]]
 => [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
[[[[],[]]]]
 => [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
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[[],[],[],[],[]]
 => [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
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 0
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 0
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[[[]]]]
 => [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
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 0
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 0
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[[]]],[]]
 => [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
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[],[[]]]]
 => [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
[[],[[[]],[]]]
 => [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
[[],[[[],[]]]]
 => [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
[[],[[[[]]]]]
 => [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
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 0
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 0
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 0
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => 1
[[[]],[[[]]]]
 => [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
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[[]]],[],[]]
 => [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
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => 1
[[[[]]],[[]]]
 => [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
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[],[[]]],[]]
 => [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
[[[[]],[]],[]]
 => [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
[[[[],[]]],[]]
 => [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
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: St000679
(load all 4 compositions to match this statistic)
Mapping
St000679: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
 => ? = 0 + 1
[[]]
 => 1 = 0 + 1
[[],[]]
 => 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
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 12 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
 => .
 => ? = 0 + 1
[[]]
 => [.,.]
 => 1 = 0 + 1
[[],[]]
 => [.,[.,.]]
 => 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
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: St000298
(load all 5 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
[[]]
 => [.,.]
 => ([],1)
 => 1 = 0 + 1
[[],[]]
 => [.,[.,.]]
 => ([(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
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 7 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ 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
[]
 => .
 => ?
 => ? = 0 + 1
[[]]
 => [.,.]
 => [1,0]
 => 1 = 0 + 1
[[],[]]
 => [.,[.,.]]
 => [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
[[[],[]]]
 => [[.,.],[.,.]]
 => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,1,0,1,0,0]
 => 1 = 0 + 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
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => [1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 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
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 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
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 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
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 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
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 1 + 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => 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: St000535
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ 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)
 => ([],1)
 => 0
[[],[]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[[]]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[],[],[]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[],[[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[[],[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[[[[]]]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(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
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(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),(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),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 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
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(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),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(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),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(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
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(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
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(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),(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),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
Description
The rank-width of a graph.
Matching statistic: St001333
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ 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)
 => ([],1)
 => 0
[[],[]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[[]]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[],[],[]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[],[[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[[],[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[[[[]]]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(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
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(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),(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),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 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
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(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),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(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),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(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
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(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
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(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),(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),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(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.
Matching statistic: St001393
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ 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
[]
 => .
 => ?
 => ?
 => ? = 0
[[]]
 => [.,.]
 => ([],1)
 => ([],1)
 => 0
[[],[]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[[]]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[],[],[]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[],[[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[[],[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[[[[]]]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(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
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(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),(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),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 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
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(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),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(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),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(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
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(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
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(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),(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),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
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: St001261
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ 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
[]
 => .
 => ?
 => ?
 => ? = 0 + 1
[[]]
 => [.,.]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[],[]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 1 = 0 + 1
[[[]]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 1 = 0 + 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1 = 0 + 1
[[],[[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1 = 0 + 1
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1 = 0 + 1
[[[],[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 2 = 1 + 1
[[[[]]]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1 = 0 + 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 0 + 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 0 + 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 0 + 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 2 = 1 + 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 0 + 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 0 + 1
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 0 + 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 0 + 1
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 2 = 1 + 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1 = 0 + 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 0 + 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 2 = 1 + 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 0 + 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 0 + 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 2 = 1 + 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 0 + 1
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 0 + 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 0 + 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1 = 0 + 1
Description
The Castelnuovo-Mumford regularity of a graph.
Matching statistic: St000862
Mapping
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: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[]
 => .
 => ? => ? => ? = 0 + 1
[[]]
 => [.,.]
 => [1] => [1] => 1 = 0 + 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => [1,2] => 1 = 0 + 1
[[[]]]
 => [[.,.],.]
 => [1,2] => [1,2] => 1 = 0 + 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => 1 = 0 + 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => 1 = 0 + 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [3,1,2] => [1,2,3] => 1 = 0 + 1
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => [1,3,2] => 2 = 1 + 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 1 = 0 + 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,3,4] => 1 = 0 + 1
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,2,3,4] => 1 = 0 + 1
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,2,4,3] => 2 = 1 + 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => 1 = 0 + 1
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,2,3,4] => 1 = 0 + 1
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => [1,2,3,4] => 1 = 0 + 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,3,2,4] => 2 = 1 + 1
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,2,3,4] => 1 = 0 + 1
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,4,2,3] => 2 = 1 + 1
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,4,2,3] => 2 = 1 + 1
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,2,4,3] => 2 = 1 + 1
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,3,4,2] => 2 = 1 + 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,2,3,5,4] => 2 = 1 + 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,2,3,4,5] => 1 = 0 + 1
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [1,2,3,4,5] => 1 = 0 + 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,2,4,3,5] => 2 = 1 + 1
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,2,3,4,5] => 1 = 0 + 1
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,2,5,3,4] => 2 = 1 + 1
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,2,5,3,4] => 2 = 1 + 1
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [1,2,3,5,4] => 2 = 1 + 1
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,2,4,5,3] => 2 = 1 + 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,2,3,4,5] => 1 = 0 + 1
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,2,3,4,5] => 1 = 0 + 1
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [1,2,3,4,5] => 1 = 0 + 1
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [1,2,3,4,5] => 1 = 0 + 1
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [1,2,3,5,4] => 2 = 1 + 1
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [1,2,3,4,5] => 1 = 0 + 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [1,3,2,4,5] => 2 = 1 + 1
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,2,3,4,5] => 1 = 0 + 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [1,3,2,4,5] => 2 = 1 + 1
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [1,2,3,4,5] => 1 = 0 + 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [1,4,2,3,5] => 2 = 1 + 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [1,4,2,3,5] => 2 = 1 + 1
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,2,4,3,5] => 2 = 1 + 1
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [1,3,4,2,5] => 2 = 1 + 1
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,2,3,4,5] => 1 = 0 + 1
[[[],[],[],[],[]],[]]
 => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => [7,5,4,3,2,1,6] => [1,6,2,3,4,5,7] => ? = 1 + 1
[[[],[],[[],[],[]]]]
 => [[.,[.,[[.,[.,[.,.]]],.]]],.]
 => [5,4,3,6,2,1,7] => [1,7,2,3,6,4,5] => ? = 1 + 1
[[[],[[],[],[]],[]]]
 => [[.,[[.,[.,[.,.]]],[.,.]]],.]
 => [6,4,3,2,5,1,7] => [1,7,2,5,3,4,6] => ? = 1 + 1
[[[],[[],[],[],[]]]]
 => [[.,[[.,[.,[.,[.,.]]]],.]],.]
 => [5,4,3,2,6,1,7] => [1,7,2,6,3,4,5] => ? = 1 + 1
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.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001741The largest integer such that all patterns of this size are contained in the permutation. St000397The Strahler number of a rooted tree. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000455The second largest eigenvalue of a graph if it is integral. 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. St000264The girth of a graph, which is not a tree. St000805The number of peaks of the associated bargraph.