Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000700
(load all 2 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[],[]]
 => [.,[.,.]]
 => [[],[]]
 => 1
[[[]]]
 => [[.,.],.]
 => [[[]]]
 => 2
[[],[],[]]
 => [.,[.,[.,.]]]
 => [[],[],[]]
 => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [[],[[]]]
 => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [[[],[]]]
 => 2
[[[],[]]]
 => [[.,.],[.,.]]
 => [[[]],[]]
 => 1
[[[[]]]]
 => [[[.,.],.],.]
 => [[[[]]]]
 => 3
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [[],[],[],[]]
 => 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [[],[],[[]]]
 => 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => [[],[[],[]]]
 => 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => [[],[[]],[]]
 => 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [[],[[[]]]]
 => 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => [[[],[],[]]]
 => 2
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => [[[],[[]]]]
 => 2
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [[[],[]],[]]
 => 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => [[[[],[]]]]
 => 3
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => [[[]],[],[]]
 => 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => [[[]],[[]]]
 => 2
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => [[[[]]],[]]
 => 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => [[[[]],[]]]
 => 2
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [[[[[]]]]]
 => 4
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [[],[],[],[],[]]
 => 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [[],[],[],[[]]]
 => 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => [[],[],[[],[]]]
 => 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [[],[],[[]],[]]
 => 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [[],[],[[[]]]]
 => 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => [[],[[],[],[]]]
 => 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => [[],[[],[[]]]]
 => 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [[],[[],[]],[]]
 => 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => [[],[[[],[]]]]
 => 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => [[],[[]],[],[]]
 => 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [[],[[]],[[]]]
 => 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => [[],[[[]]],[]]
 => 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => [[],[[[]],[]]]
 => 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [[],[[[[]]]]]
 => 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [[[],[],[],[]]]
 => 2
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => [[[],[],[[]]]]
 => 2
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [[[],[[],[]]]]
 => 2
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => [[[]],[[]],[]]
 => 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => [[[],[[[]]]]]
 => 2
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [[[],[]],[],[]]
 => 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [[[[],[],[]]]]
 => 3
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [[[],[]],[[]]]
 => 2
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => [[[[],[[]]]]]
 => 3
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [[[],[],[]],[]]
 => 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [[[],[[]]],[]]
 => 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [[[[],[]]],[]]
 => 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [[[[],[]],[]]]
 => 2
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => [[[[[],[]]]]]
 => 4
[[[],[],[],[]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [[[]],[],[],[]]
 => 1
Description
The protection number of an ordered tree. This is the minimal distance from the root to a leaf.
Matching statistic: St000906
(load all 2 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[],[]]
 => [.,[.,.]]
 => [2,1] => ([],2)
 => 1
[[[]]]
 => [[.,.],.]
 => [1,2] => ([(0,1)],2)
 => 2
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => ([],3)
 => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => ([(1,2)],3)
 => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => 2
[[[],[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => ([(1,2)],3)
 => 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([],4)
 => 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(2,3)],4)
 => 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => ([(2,3)],4)
 => 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 3
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => ([(2,3)],4)
 => 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 2
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([],5)
 => 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(3,4)],5)
 => 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => ([(3,4)],5)
 => 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => ([(3,4)],5)
 => 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => 3
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => 2
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 3
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 4
[[[],[],[],[]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => 1
Description
The length of the shortest maximal chain in a poset.
Matching statistic: St001038
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[],[]]
 => [.,[.,.]]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[[[]]]
 => [[.,.],.]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[[],[],[]]
 => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[[[],[]]]
 => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 3
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4
[[[],[],[],[]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001829
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001829: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[[],[]]
 => [.,[.,.]]
 => [2,1] => ([(0,1)],2)
 => 1
[[[]]]
 => [[.,.],.]
 => [1,2] => ([],2)
 => 2
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => ([(1,2)],3)
 => 2
[[[],[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => ([],3)
 => 3
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(2,3)],4)
 => 3
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([],4)
 => 4
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 3
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => ([(3,4)],5)
 => 4
[[[],[],[],[]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[[[]]],[],[]]
 => [.,[.,[[[.,[.,[.,.]]],.],.]]]
 => [5,4,3,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[[[]],[],[]],[]]
 => [.,[[[.,[.,[.,.]]],.],[.,.]]]
 => [7,4,3,2,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[[[]],[],[]],[],[]]
 => [[[.,[.,[.,.]]],.],[.,[.,.]]]
 => [7,6,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[[[]],[],[]],[[]]]
 => [[[.,[.,[.,.]]],.],[[.,.],.]]
 => [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 2
[[[],[[[]]],[],[]]]
 => [[.,.],[[[.,[.,[.,.]]],.],.]]
 => [5,4,3,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 2
Description
The common independence number of a graph. The common independence number of a graph $G$ is the greatest integer $r$ such that every vertex of $G$ belongs to some independent set $X$ of vertices of cardinality at least $r$.
Matching statistic: St001322
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001322: Graphs ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[[],[]]
 => [.,[.,.]]
 => [2,1] => ([(0,1)],2)
 => 1
[[[]]]
 => [[.,.],.]
 => [1,2] => ([],2)
 => 2
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => ([(1,2)],3)
 => 2
[[[],[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => ([],3)
 => 3
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(2,3)],4)
 => 3
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([],4)
 => 4
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 3
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => ([(3,4)],5)
 => 4
[[[],[],[],[]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[],[],[],[],[]]
 => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[],[],[],[],[[]]]
 => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => [6,7,5,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[],[],[],[[]],[]]
 => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
 => [6,5,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[],[],[],[[],[]]]
 => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => [7,5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[],[],[[]],[],[]]
 => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
 => [6,5,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[],[],[[],[]],[]]
 => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [7,5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[],[],[[],[],[]]]
 => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[],[],[[],[[]]]]
 => [.,[.,[.,[[.,.],[[.,.],.]]]]]
 => [6,7,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[],[[]],[],[],[]]
 => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
 => [6,5,4,3,7,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[],[[]],[[],[]]]
 => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[],[[],[]],[],[]]
 => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[],[[],[]],[[]]]
 => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [6,7,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[],[[],[],[]],[]]
 => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [7,5,4,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[],[[],[],[],[]]]
 => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [7,6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[],[[],[],[[]]]]
 => [.,[.,[[.,.],[.,[[.,.],.]]]]]
 => [6,7,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[],[[],[[]],[]]]
 => [.,[.,[[.,.],[[.,[.,.]],.]]]]
 => [6,5,7,3,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[[]],[],[],[],[]]
 => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
 => [6,5,4,3,2,7,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[[]],[],[[],[]]]
 => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [7,5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[[]],[[],[]],[]]
 => [.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [7,5,4,6,2,3,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[[]],[[],[],[]]]
 => [.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [7,6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[[],[]],[],[],[]]
 => [.,[[.,[.,.]],[.,[.,[.,.]]]]]
 => [7,6,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[[],[]],[],[[]]]
 => [.,[[.,[.,.]],[.,[[.,.],.]]]]
 => [6,7,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[[],[]],[[]],[]]
 => [.,[[.,[.,.]],[[.,[.,.]],.]]]
 => [6,5,7,3,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[[],[]],[[],[]]]
 => [.,[[.,[.,.]],[[.,.],[.,.]]]]
 => [7,5,6,3,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[[],[],[]],[],[]]
 => [.,[[.,[.,[.,.]]],[.,[.,.]]]]
 => [7,6,4,3,2,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[[],[],[]],[[]]]
 => [.,[[.,[.,[.,.]]],[[.,.],.]]]
 => [6,7,4,3,2,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1
[[],[[],[],[],[]],[]]
 => [.,[[.,[.,[.,[.,.]]]],[.,.]]]
 => [7,5,4,3,2,6,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[[],[],[],[],[]]]
 => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
 => [7,6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[[],[],[],[[]]]]
 => [.,[[.,.],[.,[.,[[.,.],.]]]]]
 => [6,7,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[[],[],[[]],[]]]
 => [.,[[.,.],[.,[[.,[.,.]],.]]]]
 => [6,5,7,4,2,3,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[],[[],[[]],[],[]]]
 => [.,[[.,.],[[.,[.,[.,.]]],.]]]
 => [6,5,4,7,2,3,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1
[[[]],[],[],[[],[]]]
 => [[.,.],[.,[.,[[.,.],[.,.]]]]]
 => [7,5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[[]],[],[[],[]],[]]
 => [[.,.],[.,[[.,[.,.]],[.,.]]]]
 => [7,5,4,6,3,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[[]],[],[[],[],[]]]
 => [[.,.],[.,[[.,.],[.,[.,.]]]]]
 => [7,6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[[]],[[],[]],[],[]]
 => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => [7,6,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[[]],[[],[]],[[]]]
 => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => [6,7,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 2
[[[]],[[],[],[]],[]]
 => [[.,.],[[.,[.,[.,.]]],[.,.]]]
 => [7,5,4,3,6,1,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1
[[[]],[[],[],[],[]]]
 => [[.,.],[[.,.],[.,[.,[.,.]]]]]
 => [7,6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[[]],[[],[[]],[]]]
 => [[.,.],[[.,.],[[.,[.,.]],.]]]
 => [6,5,7,3,4,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 2
[[[],[]],[],[],[],[]]
 => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
 => [7,6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[[],[]],[],[],[[]]]
 => [[.,[.,.]],[.,[.,[[.,.],.]]]]
 => [6,7,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[[],[]],[],[[]],[]]
 => [[.,[.,.]],[.,[[.,[.,.]],.]]]
 => [6,5,7,4,2,1,3] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[[],[]],[],[[],[]]]
 => [[.,[.,.]],[.,[[.,.],[.,.]]]]
 => [7,5,6,4,2,1,3] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[[],[]],[[]],[],[]]
 => [[.,[.,.]],[[.,[.,[.,.]]],.]]
 => [6,5,4,7,2,1,3] => ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[[[],[]],[[],[]],[]]
 => [[.,[.,.]],[[.,[.,.]],[.,.]]]
 => [7,5,4,6,2,1,3] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[[],[]],[[],[],[]]]
 => [[.,[.,.]],[[.,.],[.,[.,.]]]]
 => [7,6,4,5,2,1,3] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[[],[]],[[],[[]]]]
 => [[.,[.,.]],[[.,.],[[.,.],.]]]
 => [6,7,4,5,2,1,3] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 2
[[[],[],[]],[],[],[]]
 => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
 => [7,6,5,3,2,1,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[[[],[],[]],[],[[]]]
 => [[.,[.,[.,.]]],[.,[[.,.],.]]]
 => [6,7,5,3,2,1,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1
[[[],[],[]],[[]],[]]
 => [[.,[.,[.,.]]],[[.,[.,.]],.]]
 => [6,5,7,3,2,1,4] => ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
Description
The size of a minimal independent dominating set in a graph.
Matching statistic: St001481
(load all 3 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Values
[[],[]]
 => [.,[.,.]]
 => [1,0,1,0]
 => 1
[[[]]]
 => [[.,.],.]
 => [1,1,0,0]
 => 2
[[],[],[]]
 => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 2
[[[],[]]]
 => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 3
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 2
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => 2
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => 3
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 2
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 4
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[[[],[],[],[]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[[],[],[],[],[],[],[]]
 => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[[],[],[],[],[],[[]]]
 => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[[],[],[],[],[[]],[]]
 => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
[[],[],[],[],[[],[]]]
 => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1
[[],[],[],[],[[[]]]]
 => [.,[.,[.,[.,[[[.,.],.],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
[[],[],[],[[]],[],[]]
 => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1
[[],[],[],[[],[]],[]]
 => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => ? = 1
[[],[],[],[[[]]],[]]
 => [.,[.,[.,[[[.,[.,.]],.],.]]]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 1
[[],[],[],[[],[],[]]]
 => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
[[],[],[],[[],[[]]]]
 => [.,[.,[.,[[.,.],[[.,.],.]]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
[[],[],[],[[[]],[]]]
 => [.,[.,[.,[[[.,.],.],[.,.]]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1
[[],[],[],[[[[]]]]]
 => [.,[.,[.,[[[[.,.],.],.],.]]]]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
[[],[],[[]],[],[],[]]
 => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 1
[[],[],[[]],[],[[]]]
 => [.,[.,[[.,[.,[[.,.],.]]],.]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => ? = 1
[[],[],[[]],[[]],[]]
 => [.,[.,[[.,[[.,[.,.]],.]],.]]]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 1
[[],[],[[]],[[],[]]]
 => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
[[],[],[[],[]],[],[]]
 => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 1
[[],[],[[[]]],[],[]]
 => [.,[.,[[[.,[.,[.,.]]],.],.]]]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 1
[[],[],[[],[]],[[]]]
 => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 1
[[],[],[[],[],[]],[]]
 => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 1
[[],[],[[[]],[]],[]]
 => [.,[.,[[[.,[.,.]],.],[.,.]]]]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => ? = 1
[[],[],[[[],[]]],[]]
 => [.,[.,[[[.,[.,.]],[.,.]],.]]]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 1
[[],[],[[[[]]]],[]]
 => [.,[.,[[[[.,[.,.]],.],.],.]]]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 1
[[],[],[[],[],[],[]]]
 => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[[],[],[[],[],[[]]]]
 => [.,[.,[[.,.],[.,[[.,.],.]]]]]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
[[],[],[[],[[]],[]]]
 => [.,[.,[[.,.],[[.,[.,.]],.]]]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 1
[[],[],[[],[[],[]]]]
 => [.,[.,[[.,[[.,.],[.,.]]],.]]]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => ? = 1
[[],[],[[[]],[],[]]]
 => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[[],[],[[[[]]],[]]]
 => [.,[.,[[[[.,.],.],.],[.,.]]]]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1
[[],[],[[[],[],[]]]]
 => [.,[.,[[[.,.],[.,[.,.]]],.]]]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => ? = 1
[[],[[]],[],[],[],[]]
 => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
[[],[[]],[],[],[[]]]
 => [.,[[.,[.,[.,[[.,.],.]]]],.]]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1
[[],[[]],[],[[]],[]]
 => [.,[[.,[.,[[.,[.,.]],.]]],.]]
 => [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 1
[[],[[]],[],[[],[]]]
 => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 1
[[],[[]],[],[[[]]]]
 => [.,[[.,[.,[[[.,.],.],.]]],.]]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 1
[[],[[]],[[]],[],[]]
 => [.,[[.,[[.,[.,[.,.]]],.]],.]]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 1
[[],[[]],[[],[]],[]]
 => [.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 1
[[],[[]],[[[]]],[]]
 => [.,[[.,[[[.,[.,.]],.],.]],.]]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 1
[[],[[]],[[],[],[]]]
 => [.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 1
[[],[[]],[[[[]]]]]
 => [.,[[.,[[[[.,.],.],.],.]],.]]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1
[[],[[],[]],[],[],[]]
 => [.,[[.,[.,.]],[.,[.,[.,.]]]]]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 1
[[],[[[]]],[],[],[]]
 => [.,[[[.,[.,[.,[.,.]]]],.],.]]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 1
[[],[[],[]],[],[[]]]
 => [.,[[.,[.,.]],[.,[[.,.],.]]]]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => ? = 1
[[],[[[]]],[],[[]]]
 => [.,[[[.,[.,[[.,.],.]]],.],.]]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 1
[[],[[],[]],[[]],[]]
 => [.,[[.,[.,.]],[[.,[.,.]],.]]]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 1
[[],[[[]]],[[]],[]]
 => [.,[[[.,[[.,[.,.]],.]],.],.]]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 1
[[],[[],[]],[[],[]]]
 => [.,[[.,[.,.]],[[.,.],[.,.]]]]
 => [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => ? = 1
[[],[[],[]],[[[]]]]
 => [.,[[.,[.,.]],[[[.,.],.],.]]]
 => [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 1
[[],[[],[],[]],[],[]]
 => [.,[[.,[.,[.,.]]],[.,[.,.]]]]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 1
[[],[[[]],[]],[],[]]
 => [.,[[[.,[.,.]],.],[.,[.,.]]]]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => ? = 1
Description
The minimal height of a peak of a Dyck path.
Matching statistic: St000260
(load all 2 compositions to match this statistic)
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 32%●distinct values known / distinct values provided: 17%
Values
[[],[]]
 => [.,[.,.]]
 => [2,1] => ([(0,1)],2)
 => 1
[[[]]]
 => [[.,.],.]
 => [1,2] => ([],2)
 => ? = 2
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [1,3,2] => ([(1,2)],3)
 => ? = 2
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => ([(1,2)],3)
 => ? = 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => ([],3)
 => ? = 3
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 2
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 1
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(2,3)],4)
 => ? = 3
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 2
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [1,3,2,4] => ([(2,3)],4)
 => ? = 1
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(2,3)],4)
 => ? = 2
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([],4)
 => ? = 4
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 3
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? = 2
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? = 3
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 1
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? = 1
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? = 2
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 4
[[[],[],[],[]]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[[[],[],[[]]]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[[[],[[]],[]]]
 => [[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[[[],[[],[]]]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[[[],[[[]]]]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[[[[]],[],[]]]
 => [[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1
[[[[]],[[]]]]
 => [[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? = 2
[[[[],[]],[]]]
 => [[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? = 1
[[[[[]]],[]]]
 => [[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 1
[[[[],[],[]]]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2
[[[[],[[]]]]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? = 3
[[[[[]],[]]]]
 => [[[[.,.],[.,.]],.],.]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ? = 2
[[[[[],[]]]]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ? = 3
[[[[[[]]]]]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => ([],5)
 => ? = 5
[[],[],[],[],[],[]]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[],[],[[]]]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => [5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[],[[]],[]]
 => [.,[.,[.,[[.,.],[.,.]]]]]
 => [4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[],[[],[]]]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[],[[[]]]]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[[]],[],[]]
 => [.,[.,[[.,.],[.,[.,.]]]]]
 => [3,6,5,4,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[[]],[[]]]
 => [.,[.,[[.,.],[[.,.],.]]]]
 => [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[[],[]],[]]
 => [.,[.,[[.,[.,.]],[.,.]]]]
 => [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[[[]]],[]]
 => [.,[.,[[[.,.],.],[.,.]]]]
 => [3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[[],[],[]]]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => [5,4,3,6,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[[],[[]]]]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => [4,5,3,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[[[]],[]]]
 => [.,[.,[[[.,.],[.,.]],.]]]
 => [3,5,4,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[[[],[]]]]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => [4,3,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[[[[]]]]]
 => [.,[.,[[[[.,.],.],.],.]]]
 => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[]],[],[],[]]
 => [.,[[.,.],[.,[.,[.,.]]]]]
 => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[]],[],[[]]]
 => [.,[[.,.],[.,[[.,.],.]]]]
 => [2,5,6,4,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[]],[[]],[]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => [2,4,6,5,3,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[]],[[],[]]]
 => [.,[[.,.],[[.,[.,.]],.]]]
 => [2,5,4,6,3,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[]],[[[]]]]
 => [.,[[.,.],[[[.,.],.],.]]]
 => [2,4,5,6,3,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[],[]],[],[]]
 => [.,[[.,[.,.]],[.,[.,.]]]]
 => [3,2,6,5,4,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[[]]],[],[]]
 => [.,[[[.,.],.],[.,[.,.]]]]
 => [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[],[]],[[]]]
 => [.,[[.,[.,.]],[[.,.],.]]]
 => [3,2,5,6,4,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[[]]],[[]]]
 => [.,[[[.,.],.],[[.,.],.]]]
 => [2,3,5,6,4,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[],[],[]],[]]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => [4,3,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[],[[]]],[]]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => [3,4,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[[]],[]],[]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 1
[[],[[[],[]]],[]]
 => [.,[[[.,[.,.]],.],[.,.]]]
 => [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 1
[[],[[[[]]]],[]]
 => [.,[[[[.,.],.],.],[.,.]]]
 => [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[]],[],[],[],[]]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[[[]],[],[],[[]]]
 => [[.,.],[.,[.,[[.,.],.]]]]
 => [1,5,6,4,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[[[]],[],[[]],[]]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => [1,4,6,5,3,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[[[]],[],[[],[]]]
 => [[.,.],[.,[[.,[.,.]],.]]]
 => [1,5,4,6,3,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[[[]],[],[[[]]]]
 => [[.,.],[.,[[[.,.],.],.]]]
 => [1,4,5,6,3,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[[[]],[[]],[],[]]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => [1,3,6,5,4,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[[[]],[[]],[[]]]
 => [[.,.],[[.,.],[[.,.],.]]]
 => [1,3,5,6,4,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[[[]],[[],[]],[]]
 => [[.,.],[[.,[.,.]],[.,.]]]
 => [1,4,3,6,5,2] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 1
[[[]],[[[]]],[]]
 => [[.,.],[[[.,.],.],[.,.]]]
 => [1,3,4,6,5,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.