Your data matches 133 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000298
(load all 6 compositions to match this statistic)
Mapping
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
[.,[.,.]]
 => ([(0,1)],2)
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
Description
The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St001261
(load all 4 compositions to match this statistic)
Mapping
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
[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 2
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 2
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 2
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 2
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 2
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 2
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 2
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 2
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 2
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2
Description
The Castelnuovo-Mumford regularity of a graph.
Matching statistic: St000535
(load all 4 compositions to match this statistic)
Mapping
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 = 1 - 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0 = 1 - 1
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 1 - 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 1 - 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1 = 2 - 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 1 - 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
Description
The rank-width of a graph.
Matching statistic: St001333
(load all 4 compositions to match this statistic)
Mapping
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 = 1 - 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0 = 1 - 1
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 1 - 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 1 - 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1 = 2 - 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 1 - 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
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
(load all 4 compositions to match this statistic)
Mapping
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 = 1 - 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0 = 1 - 1
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 1 - 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 1 - 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1 = 2 - 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 1 - 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
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: St000159
(load all 3 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [1]
 => 1
[.,[.,.]]
 => [2,1] => [1,2] => [1,1]
 => 1
[[.,.],.]
 => [1,2] => [1,2] => [1,1]
 => 1
[.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => [1,1,1]
 => 1
[.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => [1,1,1]
 => 1
[[.,.],[.,.]]
 => [3,1,2] => [1,2,3] => [1,1,1]
 => 1
[[.,[.,.]],.]
 => [2,1,3] => [1,3,2] => [2,1]
 => 2
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,1,1]
 => 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => [1,1,1,1]
 => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,3,4] => [1,1,1,1]
 => 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,2,3,4] => [1,1,1,1]
 => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,2,4,3] => [2,1,1]
 => 2
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,2,3,4] => [1,1,1,1]
 => 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [1,2,3,4] => [1,1,1,1]
 => 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,3,2,4] => [2,1,1]
 => 2
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,2,3,4] => [1,1,1,1]
 => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,4,2,3] => [2,1,1]
 => 2
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,4,2,3] => [2,1,1]
 => 2
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,2,4,3] => [2,1,1]
 => 2
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,3,4,2] => [2,1,1]
 => 2
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,2,3,5,4] => [2,1,1,1]
 => 2
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,2,4,3,5] => [2,1,1,1]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,2,5,3,4] => [2,1,1,1]
 => 2
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,2,5,3,4] => [2,1,1,1]
 => 2
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [1,2,3,5,4] => [2,1,1,1]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,2,4,5,3] => [2,1,1,1]
 => 2
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [1,2,3,5,4] => [2,1,1,1]
 => 2
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [1,3,2,4,5] => [2,1,1,1]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [1,3,2,4,5] => [2,1,1,1]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [1,4,2,3,5] => [2,1,1,1]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [1,4,2,3,5] => [2,1,1,1]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,2,4,3,5] => [2,1,1,1]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [1,3,4,2,5] => [2,1,1,1]
 => 2
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
Description
The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000396
(load all 6 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [.,.]
 => 1
[.,[.,.]]
 => [2,1] => [1,2] => [.,[.,.]]
 => 1
[[.,.],.]
 => [1,2] => [2,1] => [[.,.],.]
 => 1
[.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => [.,[.,[.,.]]]
 => 1
[.,[[.,.],.]]
 => [2,3,1] => [3,1,2] => [[.,[.,.]],.]
 => 1
[[.,.],[.,.]]
 => [3,1,2] => [1,3,2] => [.,[[.,.],.]]
 => 1
[[.,[.,.]],.]
 => [2,1,3] => [3,2,1] => [[[.,.],.],.]
 => 1
[[[.,.],.],.]
 => [1,2,3] => [2,3,1] => [[.,.],[.,.]]
 => 2
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,1,2] => [[[.,[.,.]],.],.]
 => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => 2
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,1,3,2] => [[.,[[.,.],.]],.]
 => 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => 2
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,4,2,1] => [[[.,.],.],[.,.]]
 => 2
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 2
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [3,2,4,1] => [[[.,.],.],[.,.]]
 => 2
[[[[.,.],.],.],.]
 => [1,2,3,4] => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => 2
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
 => 2
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [5,1,4,2,3] => [[.,[[.,[.,.]],.]],.]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
 => 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => 2
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
 => 2
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
 => 2
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
 => 2
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [5,1,2,4,3] => [[.,[.,[[.,.],.]]],.]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
 => 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
 => 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [4,5,1,3,2] => [[.,[[.,.],.]],[.,.]]
 => 2
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [5,1,4,3,2] => [[.,[[[.,.],.],.]],.]
 => 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => 2
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [5,1,3,4,2] => [[.,[[.,.],[.,.]]],.]
 => 2
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => 2
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => 2
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: St000469
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000469: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(1,2)],3)
 => 2
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(1,2)],3)
 => 2
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(1,2)],3)
 => 2
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(1,2)],3)
 => 2
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 2
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 2
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 2
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 2
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
Description
The distinguishing number of a graph. This is the minimal number of colours needed to colour the vertices of a graph, such that only the trivial automorphism of the graph preserves the colouring. For connected graphs, this statistic is at most one plus the maximal degree of the graph, with equality attained for complete graphs, complete bipartite graphs and the cycle with five vertices, see Theorem 4.2 of [2].
Matching statistic: St000679
(load all 2 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000679: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [.,.]
 => [[]]
 => 1
[.,[.,.]]
 => [2,1] => [[.,.],.]
 => [[[]]]
 => 1
[[.,.],.]
 => [1,2] => [.,[.,.]]
 => [[],[]]
 => 1
[.,[.,[.,.]]]
 => [3,2,1] => [[[.,.],.],.]
 => [[[[]]]]
 => 1
[.,[[.,.],.]]
 => [2,3,1] => [[.,[.,.]],.]
 => [[[],[]]]
 => 2
[[.,.],[.,.]]
 => [3,1,2] => [[.,.],[.,.]]
 => [[[]],[]]
 => 1
[[.,[.,.]],.]
 => [2,1,3] => [[.,.],[.,.]]
 => [[[]],[]]
 => 1
[[[.,.],.],.]
 => [1,2,3] => [.,[.,[.,.]]]
 => [[],[],[]]
 => 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [[[[[]]]]]
 => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[[.,[.,.]],.],.]
 => [[[[],[]]]]
 => 2
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [[[[]],[]]]
 => 2
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[[.,.],[.,.]],.]
 => [[[[]],[]]]
 => 2
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => [[[],[],[]]]
 => 2
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [[[.,.],.],[.,.]]
 => [[[[]]],[]]
 => 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [[[],[]],[]]
 => 2
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [[[[]]],[]]
 => 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => [[[]],[],[]]
 => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [[[[]]],[]]
 => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => [[[],[]],[]]
 => 2
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => [[[]],[],[]]
 => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[[]],[],[]]
 => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[],[],[],[]]
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [[[[[[]]]]]]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
 => [[[[[],[]]]]]
 => 2
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
 => [[[[[]],[]]]]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
 => [[[[[]],[]]]]
 => 2
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
 => [[[[],[],[]]]]
 => 2
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
 => [[[[[]]],[]]]
 => 2
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
 => [[[[],[]],[]]]
 => 2
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => [[[[[]]],[]]]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
 => [[[[]],[],[]]]
 => 2
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
 => [[[[[]]],[]]]
 => 2
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
 => [[[[],[]],[]]]
 => 2
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
 => [[[[]],[],[]]]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
 => [[[[]],[],[]]]
 => 2
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => [[[],[],[],[]]]
 => 2
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
 => [[[[[]]]],[]]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
 => [[[[],[]]],[]]
 => 2
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
 => [[[[]],[]],[]]
 => 2
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
 => [[[[]],[]],[]]
 => 2
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
 => [[[],[],[]],[]]
 => 2
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
 => [[[[[]]]],[]]
 => 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
 => [[[[],[]]],[]]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
 => [[[[]]],[],[]]
 => 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
 => [[[],[]],[],[]]
 => 2
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
 => [[[[[]]]],[]]
 => 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
 => [[[[]],[]],[]]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
 => [[[[]]],[],[]]
 => 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
 => [[[[]]],[],[]]
 => 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
 => [[[]],[],[],[]]
 => 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: St000774
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000774: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(1,2)],3)
 => 2
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(1,2)],3)
 => 2
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(1,2)],3)
 => 2
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(1,2)],3)
 => 2
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 2
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 2
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 2
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 2
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
Description
The maximal multiplicity of a Laplacian eigenvalue in a graph.
The following 123 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000783The side length of the largest staircase partition fitting into a partition. St000916The packing number of a graph. St001432The order dimension of the partition. St001463The number of distinct columns in the nullspace of a graph. St001732The number of peaks visible from the left. St000183The side length of the Durfee square of an integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000660The number of rises of length at least 3 of a Dyck path. St001280The number of parts of an integer partition that are at least two. St001335The cardinality of a minimal cycle-isolating set of a graph. St001743The discrepancy of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000392The length of the longest run of ones in a binary word. St000659The number of rises of length at least 2 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000287The number of connected components of a graph. St001734The lettericity of a graph. St001783The number of odd automorphisms of a graph. St000862The number of parts of the shifted shape of a permutation. St000485The length of the longest cycle of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St000058The order of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. 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. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000260The radius of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000397The Strahler number of a rooted tree. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. St001093The detour number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000259The diameter of a connected graph. St000387The matching number of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001071The beta invariant of the graph. St001271The competition number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001512The minimum rank of a graph. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000307The number of rowmotion orbits of a poset. St001060The distinguishing index of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000100The number of linear extensions of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000640The rank of the largest boolean interval in a poset. St001621The number of atoms of a lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001820The size of the image of the pop stack sorting operator. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000632The jump number of the poset. St001487The number of inner corners of a skew partition. St001846The number of elements which do not have a complement in the lattice. St000031The number of cycles in the cycle decomposition of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001864The number of excedances of a signed permutation. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001964The interval resolution global dimension of a poset. St000807The sum of the heights of the valleys of the associated bargraph. St001877Number of indecomposable injective modules with projective dimension 2. St001875The number of simple modules with projective dimension at most 1. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000381The largest part of an integer composition. St000409The number of pitchforks in a binary tree. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000781The number of proper colouring schemes of a Ferrers diagram. St000805The number of peaks of the associated bargraph. St000808The number of up steps of the associated bargraph. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001729The number of visible descents of a permutation. St000761The number of ascents in an integer composition. St000768The number of peaks in an integer composition. St001673The degree of asymmetry of an integer composition. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.