searching the database
Your data matches 16 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001176
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> ([],1)
=> [1]
=> 0
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> 0
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> [2]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 0
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 0
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 0
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000272
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Values
[1] => [.,.]
=> ([],1)
=> ([],1)
=> 0
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 0
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,5,6,7,8,3,2,1] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,4,5,6,7,8,2,1] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[6,8,7,5,4,3,2,1] => [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[5,8,7,6,4,3,2,1] => [[[[[.,[[[.,.],.],.]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[5,7,8,6,4,3,2,1] => [[[[[.,[[.,[.,.]],.]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[5,6,8,7,4,3,2,1] => [[[[[.,[.,[[.,.],.]]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,8,7,6,5,3,2,1] => [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,7,8,6,5,3,2,1] => [[[[.,[[[.,[.,.]],.],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,6,8,7,5,3,2,1] => [[[[.,[[.,[[.,.],.]],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,5,8,7,6,3,2,1] => [[[[.,[.,[[[.,.],.],.]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,5,6,8,7,3,2,1] => [[[[.,[.,[.,[[.,.],.]]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,8,7,6,5,4,2,1] => [[[.,[[[[[.,.],.],.],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,7,8,6,5,4,2,1] => [[[.,[[[[.,[.,.]],.],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,6,8,7,5,4,2,1] => [[[.,[[[.,[[.,.],.]],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,5,8,7,6,4,2,1] => [[[.,[[.,[[[.,.],.],.]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,5,7,8,6,4,2,1] => [[[.,[[.,[[.,[.,.]],.]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,5,6,8,7,4,2,1] => [[[.,[[.,[.,[[.,.],.]]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,5,6,7,8,4,2,1] => [[[.,[[.,[.,[.,[.,.]]]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,4,6,7,8,5,2,1] => [[[.,[.,[[.,[.,[.,.]]],.]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,4,5,7,8,6,2,1] => [[[.,[.,[.,[[.,[.,.]],.]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,4,5,6,8,7,2,1] => [[[.,[.,[.,[.,[[.,.],.]]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,7,8,6,5,4,3,1] => [[.,[[[[[.,[.,.]],.],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,6,8,7,5,4,3,1] => [[.,[[[[.,[[.,.],.]],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,6,7,8,5,4,3,1] => [[.,[[[[.,[.,[.,.]]],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,5,8,7,6,4,3,1] => [[.,[[[.,[[[.,.],.],.]],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,5,6,7,8,4,3,1] => [[.,[[[.,[.,[.,[.,.]]]],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,8,7,6,5,3,1] => [[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,7,8,6,5,3,1] => [[.,[[.,[[[.,[.,.]],.],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,6,8,7,5,3,1] => [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,6,7,8,5,3,1] => [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,5,7,8,6,3,1] => [[.,[[.,[.,[[.,[.,.]],.]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,5,6,8,7,3,1] => [[.,[[.,[.,[.,[[.,.],.]]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,5,6,7,8,3,1] => [[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,8,7,6,5,4,1] => [[.,[.,[[[[[.,.],.],.],.],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,6,7,8,5,4,1] => [[.,[.,[[[.,[.,[.,.]]],.],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,5,8,7,6,4,1] => [[.,[.,[[.,[[[.,.],.],.]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,5,7,8,6,4,1] => [[.,[.,[[.,[[.,[.,.]],.]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,5,6,8,7,4,1] => [[.,[.,[[.,[.,[[.,.],.]]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,5,6,7,8,4,1] => [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,4,8,7,6,5,1] => [[.,[.,[.,[[[[.,.],.],.],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,4,6,8,7,5,1] => [[.,[.,[.,[[.,[[.,.],.]],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,4,6,7,8,5,1] => [[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,4,5,8,7,6,1] => [[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
Description
The treewidth of a graph.
A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.
Matching statistic: St000362
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Values
[1] => [.,.]
=> ([],1)
=> ([],1)
=> 0
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 0
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,5,6,7,8,3,2,1] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,4,5,6,7,8,2,1] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[6,8,7,5,4,3,2,1] => [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[5,8,7,6,4,3,2,1] => [[[[[.,[[[.,.],.],.]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[5,7,8,6,4,3,2,1] => [[[[[.,[[.,[.,.]],.]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[5,6,8,7,4,3,2,1] => [[[[[.,[.,[[.,.],.]]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,8,7,6,5,3,2,1] => [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,7,8,6,5,3,2,1] => [[[[.,[[[.,[.,.]],.],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,6,8,7,5,3,2,1] => [[[[.,[[.,[[.,.],.]],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,5,8,7,6,3,2,1] => [[[[.,[.,[[[.,.],.],.]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,5,6,8,7,3,2,1] => [[[[.,[.,[.,[[.,.],.]]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,8,7,6,5,4,2,1] => [[[.,[[[[[.,.],.],.],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,7,8,6,5,4,2,1] => [[[.,[[[[.,[.,.]],.],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,6,8,7,5,4,2,1] => [[[.,[[[.,[[.,.],.]],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,5,8,7,6,4,2,1] => [[[.,[[.,[[[.,.],.],.]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,5,7,8,6,4,2,1] => [[[.,[[.,[[.,[.,.]],.]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,5,6,8,7,4,2,1] => [[[.,[[.,[.,[[.,.],.]]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,5,6,7,8,4,2,1] => [[[.,[[.,[.,[.,[.,.]]]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,4,6,7,8,5,2,1] => [[[.,[.,[[.,[.,[.,.]]],.]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,4,5,7,8,6,2,1] => [[[.,[.,[.,[[.,[.,.]],.]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,4,5,6,8,7,2,1] => [[[.,[.,[.,[.,[[.,.],.]]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,7,8,6,5,4,3,1] => [[.,[[[[[.,[.,.]],.],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,6,8,7,5,4,3,1] => [[.,[[[[.,[[.,.],.]],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,6,7,8,5,4,3,1] => [[.,[[[[.,[.,[.,.]]],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,5,8,7,6,4,3,1] => [[.,[[[.,[[[.,.],.],.]],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,5,6,7,8,4,3,1] => [[.,[[[.,[.,[.,[.,.]]]],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,8,7,6,5,3,1] => [[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,7,8,6,5,3,1] => [[.,[[.,[[[.,[.,.]],.],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,6,8,7,5,3,1] => [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,6,7,8,5,3,1] => [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,5,7,8,6,3,1] => [[.,[[.,[.,[[.,[.,.]],.]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,5,6,8,7,3,1] => [[.,[[.,[.,[.,[[.,.],.]]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,5,6,7,8,3,1] => [[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,8,7,6,5,4,1] => [[.,[.,[[[[[.,.],.],.],.],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,6,7,8,5,4,1] => [[.,[.,[[[.,[.,[.,.]]],.],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,5,8,7,6,4,1] => [[.,[.,[[.,[[[.,.],.],.]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,5,7,8,6,4,1] => [[.,[.,[[.,[[.,[.,.]],.]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,5,6,8,7,4,1] => [[.,[.,[[.,[.,[[.,.],.]]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,5,6,7,8,4,1] => [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,4,8,7,6,5,1] => [[.,[.,[.,[[[[.,.],.],.],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,4,6,8,7,5,1] => [[.,[.,[.,[[.,[[.,.],.]],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,4,6,7,8,5,1] => [[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,4,5,8,7,6,1] => [[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
Description
The size of a minimal vertex cover of a graph.
A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. Finding a minimal vertex cover is an NP-hard optimization problem.
Matching statistic: St000536
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000536: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000536: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Values
[1] => [.,.]
=> ([],1)
=> ([],1)
=> 0
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 0
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,5,6,7,8,3,2,1] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,4,5,6,7,8,2,1] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[6,8,7,5,4,3,2,1] => [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[5,8,7,6,4,3,2,1] => [[[[[.,[[[.,.],.],.]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[5,7,8,6,4,3,2,1] => [[[[[.,[[.,[.,.]],.]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[5,6,8,7,4,3,2,1] => [[[[[.,[.,[[.,.],.]]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,8,7,6,5,3,2,1] => [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,7,8,6,5,3,2,1] => [[[[.,[[[.,[.,.]],.],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,6,8,7,5,3,2,1] => [[[[.,[[.,[[.,.],.]],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,5,8,7,6,3,2,1] => [[[[.,[.,[[[.,.],.],.]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[4,5,6,8,7,3,2,1] => [[[[.,[.,[.,[[.,.],.]]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,8,7,6,5,4,2,1] => [[[.,[[[[[.,.],.],.],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,7,8,6,5,4,2,1] => [[[.,[[[[.,[.,.]],.],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,6,8,7,5,4,2,1] => [[[.,[[[.,[[.,.],.]],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,5,8,7,6,4,2,1] => [[[.,[[.,[[[.,.],.],.]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,5,7,8,6,4,2,1] => [[[.,[[.,[[.,[.,.]],.]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,5,6,8,7,4,2,1] => [[[.,[[.,[.,[[.,.],.]]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,5,6,7,8,4,2,1] => [[[.,[[.,[.,[.,[.,.]]]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,4,6,7,8,5,2,1] => [[[.,[.,[[.,[.,[.,.]]],.]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,4,5,7,8,6,2,1] => [[[.,[.,[.,[[.,[.,.]],.]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[3,4,5,6,8,7,2,1] => [[[.,[.,[.,[.,[[.,.],.]]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,7,8,6,5,4,3,1] => [[.,[[[[[.,[.,.]],.],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,6,8,7,5,4,3,1] => [[.,[[[[.,[[.,.],.]],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,6,7,8,5,4,3,1] => [[.,[[[[.,[.,[.,.]]],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,5,8,7,6,4,3,1] => [[.,[[[.,[[[.,.],.],.]],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,5,6,7,8,4,3,1] => [[.,[[[.,[.,[.,[.,.]]]],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,8,7,6,5,3,1] => [[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,7,8,6,5,3,1] => [[.,[[.,[[[.,[.,.]],.],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,6,8,7,5,3,1] => [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,6,7,8,5,3,1] => [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,5,7,8,6,3,1] => [[.,[[.,[.,[[.,[.,.]],.]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,5,6,8,7,3,1] => [[.,[[.,[.,[.,[[.,.],.]]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,4,5,6,7,8,3,1] => [[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,8,7,6,5,4,1] => [[.,[.,[[[[[.,.],.],.],.],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,6,7,8,5,4,1] => [[.,[.,[[[.,[.,[.,.]]],.],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,5,8,7,6,4,1] => [[.,[.,[[.,[[[.,.],.],.]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,5,7,8,6,4,1] => [[.,[.,[[.,[[.,[.,.]],.]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,5,6,8,7,4,1] => [[.,[.,[[.,[.,[[.,.],.]]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,5,6,7,8,4,1] => [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,4,8,7,6,5,1] => [[.,[.,[.,[[[[.,.],.],.],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,4,6,8,7,5,1] => [[.,[.,[.,[[.,[[.,.],.]],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,4,6,7,8,5,1] => [[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
[2,3,4,5,8,7,6,1] => [[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0
Description
The pathwidth of a graph.
Matching statistic: St001580
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001580: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001580: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Values
[1] => [.,.]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,5,6,7,8,3,2,1] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,4,5,6,7,8,2,1] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[6,8,7,5,4,3,2,1] => [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[5,8,7,6,4,3,2,1] => [[[[[.,[[[.,.],.],.]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[5,7,8,6,4,3,2,1] => [[[[[.,[[.,[.,.]],.]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[5,6,8,7,4,3,2,1] => [[[[[.,[.,[[.,.],.]]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,8,7,6,5,3,2,1] => [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,7,8,6,5,3,2,1] => [[[[.,[[[.,[.,.]],.],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,6,8,7,5,3,2,1] => [[[[.,[[.,[[.,.],.]],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,5,8,7,6,3,2,1] => [[[[.,[.,[[[.,.],.],.]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,5,6,8,7,3,2,1] => [[[[.,[.,[.,[[.,.],.]]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,8,7,6,5,4,2,1] => [[[.,[[[[[.,.],.],.],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,7,8,6,5,4,2,1] => [[[.,[[[[.,[.,.]],.],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,6,8,7,5,4,2,1] => [[[.,[[[.,[[.,.],.]],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,5,8,7,6,4,2,1] => [[[.,[[.,[[[.,.],.],.]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,5,7,8,6,4,2,1] => [[[.,[[.,[[.,[.,.]],.]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,5,6,8,7,4,2,1] => [[[.,[[.,[.,[[.,.],.]]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,5,6,7,8,4,2,1] => [[[.,[[.,[.,[.,[.,.]]]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,4,6,7,8,5,2,1] => [[[.,[.,[[.,[.,[.,.]]],.]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,4,5,7,8,6,2,1] => [[[.,[.,[.,[[.,[.,.]],.]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,4,5,6,8,7,2,1] => [[[.,[.,[.,[.,[[.,.],.]]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,7,8,6,5,4,3,1] => [[.,[[[[[.,[.,.]],.],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,6,8,7,5,4,3,1] => [[.,[[[[.,[[.,.],.]],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,6,7,8,5,4,3,1] => [[.,[[[[.,[.,[.,.]]],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,5,8,7,6,4,3,1] => [[.,[[[.,[[[.,.],.],.]],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,5,6,7,8,4,3,1] => [[.,[[[.,[.,[.,[.,.]]]],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,8,7,6,5,3,1] => [[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,7,8,6,5,3,1] => [[.,[[.,[[[.,[.,.]],.],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,6,8,7,5,3,1] => [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,6,7,8,5,3,1] => [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,5,7,8,6,3,1] => [[.,[[.,[.,[[.,[.,.]],.]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,5,6,8,7,3,1] => [[.,[[.,[.,[.,[[.,.],.]]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,5,6,7,8,3,1] => [[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,8,7,6,5,4,1] => [[.,[.,[[[[[.,.],.],.],.],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,6,7,8,5,4,1] => [[.,[.,[[[.,[.,[.,.]]],.],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,5,8,7,6,4,1] => [[.,[.,[[.,[[[.,.],.],.]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,5,7,8,6,4,1] => [[.,[.,[[.,[[.,[.,.]],.]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,5,6,8,7,4,1] => [[.,[.,[[.,[.,[[.,.],.]]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,5,6,7,8,4,1] => [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,4,8,7,6,5,1] => [[.,[.,[.,[[[[.,.],.],.],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,4,6,8,7,5,1] => [[.,[.,[.,[[.,[[.,.],.]],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,4,6,7,8,5,1] => [[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,4,5,8,7,6,1] => [[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
Description
The acyclic chromatic number of a graph.
This is the smallest size of a vertex partition $\{V_1,\dots,V_k\}$ such that each $V_i$ is an independent set and for all $i,j$ the subgraph inducted by $V_i\cup V_j$ does not contain a cycle.
Matching statistic: St001670
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001670: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001670: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Values
[1] => [.,.]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,5,6,7,8,3,2,1] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,4,5,6,7,8,2,1] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[6,8,7,5,4,3,2,1] => [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[5,8,7,6,4,3,2,1] => [[[[[.,[[[.,.],.],.]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[5,7,8,6,4,3,2,1] => [[[[[.,[[.,[.,.]],.]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[5,6,8,7,4,3,2,1] => [[[[[.,[.,[[.,.],.]]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,8,7,6,5,3,2,1] => [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,7,8,6,5,3,2,1] => [[[[.,[[[.,[.,.]],.],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,6,8,7,5,3,2,1] => [[[[.,[[.,[[.,.],.]],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,5,8,7,6,3,2,1] => [[[[.,[.,[[[.,.],.],.]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,5,6,8,7,3,2,1] => [[[[.,[.,[.,[[.,.],.]]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,8,7,6,5,4,2,1] => [[[.,[[[[[.,.],.],.],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,7,8,6,5,4,2,1] => [[[.,[[[[.,[.,.]],.],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,6,8,7,5,4,2,1] => [[[.,[[[.,[[.,.],.]],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,5,8,7,6,4,2,1] => [[[.,[[.,[[[.,.],.],.]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,5,7,8,6,4,2,1] => [[[.,[[.,[[.,[.,.]],.]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,5,6,8,7,4,2,1] => [[[.,[[.,[.,[[.,.],.]]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,5,6,7,8,4,2,1] => [[[.,[[.,[.,[.,[.,.]]]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,4,6,7,8,5,2,1] => [[[.,[.,[[.,[.,[.,.]]],.]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,4,5,7,8,6,2,1] => [[[.,[.,[.,[[.,[.,.]],.]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,4,5,6,8,7,2,1] => [[[.,[.,[.,[.,[[.,.],.]]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,7,8,6,5,4,3,1] => [[.,[[[[[.,[.,.]],.],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,6,8,7,5,4,3,1] => [[.,[[[[.,[[.,.],.]],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,6,7,8,5,4,3,1] => [[.,[[[[.,[.,[.,.]]],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,5,8,7,6,4,3,1] => [[.,[[[.,[[[.,.],.],.]],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,5,6,7,8,4,3,1] => [[.,[[[.,[.,[.,[.,.]]]],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,8,7,6,5,3,1] => [[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,7,8,6,5,3,1] => [[.,[[.,[[[.,[.,.]],.],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,6,8,7,5,3,1] => [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,6,7,8,5,3,1] => [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,5,7,8,6,3,1] => [[.,[[.,[.,[[.,[.,.]],.]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,5,6,8,7,3,1] => [[.,[[.,[.,[.,[[.,.],.]]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,5,6,7,8,3,1] => [[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,8,7,6,5,4,1] => [[.,[.,[[[[[.,.],.],.],.],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,6,7,8,5,4,1] => [[.,[.,[[[.,[.,[.,.]]],.],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,5,8,7,6,4,1] => [[.,[.,[[.,[[[.,.],.],.]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,5,7,8,6,4,1] => [[.,[.,[[.,[[.,[.,.]],.]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,5,6,8,7,4,1] => [[.,[.,[[.,[.,[[.,.],.]]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,5,6,7,8,4,1] => [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,4,8,7,6,5,1] => [[.,[.,[.,[[[[.,.],.],.],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,4,6,8,7,5,1] => [[.,[.,[.,[[.,[[.,.],.]],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,4,6,7,8,5,1] => [[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,4,5,8,7,6,1] => [[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
Description
The connected partition number of a graph.
This is the maximal number of blocks of a set partition $P$ of the set of vertices of a graph such that contracting each block of $P$ to a single vertex yields a clique.
Also called the pseudoachromatic number of a graph. This is the largest $n$ such that there exists a (not necessarily proper) $n$-coloring of the graph so that every two distinct colors are adjacent.
Matching statistic: St001963
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001963: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001963: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%
Values
[1] => [.,.]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,5,6,7,8,3,2,1] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,4,5,6,7,8,2,1] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[6,8,7,5,4,3,2,1] => [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[5,8,7,6,4,3,2,1] => [[[[[.,[[[.,.],.],.]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[5,7,8,6,4,3,2,1] => [[[[[.,[[.,[.,.]],.]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[5,6,8,7,4,3,2,1] => [[[[[.,[.,[[.,.],.]]],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,8,7,6,5,3,2,1] => [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,7,8,6,5,3,2,1] => [[[[.,[[[.,[.,.]],.],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,6,8,7,5,3,2,1] => [[[[.,[[.,[[.,.],.]],.]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,5,8,7,6,3,2,1] => [[[[.,[.,[[[.,.],.],.]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[4,5,6,8,7,3,2,1] => [[[[.,[.,[.,[[.,.],.]]]],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,8,7,6,5,4,2,1] => [[[.,[[[[[.,.],.],.],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,7,8,6,5,4,2,1] => [[[.,[[[[.,[.,.]],.],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,6,8,7,5,4,2,1] => [[[.,[[[.,[[.,.],.]],.],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,5,8,7,6,4,2,1] => [[[.,[[.,[[[.,.],.],.]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,5,7,8,6,4,2,1] => [[[.,[[.,[[.,[.,.]],.]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,5,6,8,7,4,2,1] => [[[.,[[.,[.,[[.,.],.]]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,5,6,7,8,4,2,1] => [[[.,[[.,[.,[.,[.,.]]]],.]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,4,6,7,8,5,2,1] => [[[.,[.,[[.,[.,[.,.]]],.]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,4,5,7,8,6,2,1] => [[[.,[.,[.,[[.,[.,.]],.]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[3,4,5,6,8,7,2,1] => [[[.,[.,[.,[.,[[.,.],.]]]]],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,7,8,6,5,4,3,1] => [[.,[[[[[.,[.,.]],.],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,6,8,7,5,4,3,1] => [[.,[[[[.,[[.,.],.]],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,6,7,8,5,4,3,1] => [[.,[[[[.,[.,[.,.]]],.],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,5,8,7,6,4,3,1] => [[.,[[[.,[[[.,.],.],.]],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,5,6,7,8,4,3,1] => [[.,[[[.,[.,[.,[.,.]]]],.],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,8,7,6,5,3,1] => [[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,7,8,6,5,3,1] => [[.,[[.,[[[.,[.,.]],.],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,6,8,7,5,3,1] => [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,6,7,8,5,3,1] => [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,5,7,8,6,3,1] => [[.,[[.,[.,[[.,[.,.]],.]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,5,6,8,7,3,1] => [[.,[[.,[.,[.,[[.,.],.]]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,4,5,6,7,8,3,1] => [[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,8,7,6,5,4,1] => [[.,[.,[[[[[.,.],.],.],.],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,6,7,8,5,4,1] => [[.,[.,[[[.,[.,[.,.]]],.],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,5,8,7,6,4,1] => [[.,[.,[[.,[[[.,.],.],.]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,5,7,8,6,4,1] => [[.,[.,[[.,[[.,[.,.]],.]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,5,6,8,7,4,1] => [[.,[.,[[.,[.,[[.,.],.]]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,5,6,7,8,4,1] => [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,4,8,7,6,5,1] => [[.,[.,[.,[[[[.,.],.],.],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,4,6,8,7,5,1] => [[.,[.,[.,[[.,[[.,.],.]],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,4,6,7,8,5,1] => [[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
[2,3,4,5,8,7,6,1] => [[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 0 + 1
Description
The tree-depth of a graph.
The tree-depth $\operatorname{td}(G)$ of a graph $G$ whose connected components are $G_1,\ldots,G_p$ is recursively defined as
$$\operatorname{td}(G)=\begin{cases} 1, & \text{if }|G|=1\\ 1 + \min_{v\in V} \operatorname{td}(G-v), & \text{if } p=1 \text{ and } |G| > 1\\ \max_{i=1}^p \operatorname{td}(G_i), & \text{otherwise} \end{cases}$$
Nešetřil and Ossona de Mendez [2] proved that the tree-depth of a connected graph is equal to its minimum elimination tree height and its centered chromatic number (fewest colors needed for a vertex coloring where every connected induced subgraph has a color that appears exactly once).
Tree-depth is strictly greater than [[St000536|pathwidth]]. A [[St001120|longest path]] in $G$ has at least $\operatorname{td}(G)$ vertices [3].
Matching statistic: St000624
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000624: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000624: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [2,1] => [2,1] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [3,1,2] => [3,1,2] => [2,1,3] => 0
[2,1,3] => [2,3,1] => [1,3,2] => [2,3,1] => 1
[2,3,1] => [2,1,3] => [2,1,3] => [3,1,2] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [4,3,1,2] => [4,3,1,2] => [2,1,3,4] => 0
[1,3,2,4] => [4,2,3,1] => [4,1,3,2] => [2,3,1,4] => 1
[1,3,4,2] => [4,2,1,3] => [4,2,1,3] => [3,1,2,4] => 0
[1,4,2,3] => [4,1,3,2] => [4,1,3,2] => [2,3,1,4] => 1
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => [3,2,1,4] => 0
[2,1,3,4] => [3,4,2,1] => [1,4,3,2] => [2,3,4,1] => 1
[2,1,4,3] => [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[2,3,1,4] => [3,2,4,1] => [2,1,4,3] => [3,4,1,2] => 1
[2,3,4,1] => [3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 0
[2,4,1,3] => [3,1,4,2] => [2,1,4,3] => [3,4,1,2] => 1
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => [4,2,1,3] => 0
[3,1,2,4] => [2,4,3,1] => [1,4,3,2] => [2,3,4,1] => 1
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[3,2,1,4] => [2,3,4,1] => [1,2,4,3] => [3,4,2,1] => 1
[3,2,4,1] => [2,3,1,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 0
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 1
[4,2,1,3] => [1,3,4,2] => [1,2,4,3] => [3,4,2,1] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [5,4,3,1,2] => [5,4,3,1,2] => [2,1,3,4,5] => 0
[1,2,4,3,5] => [5,4,2,3,1] => [5,4,1,3,2] => [2,3,1,4,5] => 1
[1,2,4,5,3] => [5,4,2,1,3] => [5,4,2,1,3] => [3,1,2,4,5] => 0
[1,2,5,3,4] => [5,4,1,3,2] => [5,4,1,3,2] => [2,3,1,4,5] => 1
[1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => [3,2,1,4,5] => 0
[1,3,2,4,5] => [5,3,4,2,1] => [5,1,4,3,2] => [2,3,4,1,5] => 1
[1,3,2,5,4] => [5,3,4,1,2] => [5,2,4,1,3] => [3,1,4,2,5] => 1
[1,3,4,2,5] => [5,3,2,4,1] => [5,2,1,4,3] => [3,4,1,2,5] => 1
[1,3,4,5,2] => [5,3,2,1,4] => [5,3,2,1,4] => [4,1,2,3,5] => 0
[1,3,5,2,4] => [5,3,1,4,2] => [5,2,1,4,3] => [3,4,1,2,5] => 1
[1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,2,4] => [4,2,1,3,5] => 0
[1,4,2,3,5] => [5,2,4,3,1] => [5,1,4,3,2] => [2,3,4,1,5] => 1
[1,4,2,5,3] => [5,2,4,1,3] => [5,2,4,1,3] => [3,1,4,2,5] => 1
[1,4,3,2,5] => [5,2,3,4,1] => [5,1,2,4,3] => [3,4,2,1,5] => 1
[1,4,3,5,2] => [5,2,3,1,4] => [5,1,3,2,4] => [4,2,3,1,5] => 1
[1,4,5,2,3] => [5,2,1,4,3] => [5,2,1,4,3] => [3,4,1,2,5] => 1
[1,4,5,3,2] => [5,2,1,3,4] => [5,2,1,3,4] => [4,3,1,2,5] => 0
[1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [2,1,3,4,5,6,7] => ? = 0
[1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [7,6,5,4,1,3,2] => [2,3,1,4,5,6,7] => ? = 1
[1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => [3,1,2,4,5,6,7] => ? = 0
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => [2,3,1,4,5,6,7] => ? = 1
[1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => [3,2,1,4,5,6,7] => ? = 0
[1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [7,6,5,1,4,3,2] => [2,3,4,1,5,6,7] => ? = 1
[1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [7,6,5,2,4,1,3] => [3,1,4,2,5,6,7] => ? = 1
[1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => [7,6,5,2,1,4,3] => [3,4,1,2,5,6,7] => ? = 1
[1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => [7,6,5,3,2,1,4] => [4,1,2,3,5,6,7] => ? = 0
[1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => [7,6,5,2,1,4,3] => [3,4,1,2,5,6,7] => ? = 1
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => [7,6,5,3,1,2,4] => [4,2,1,3,5,6,7] => ? = 0
[1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [7,6,5,1,4,3,2] => [2,3,4,1,5,6,7] => ? = 1
[1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => [7,6,5,2,4,1,3] => [3,1,4,2,5,6,7] => ? = 1
[1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => [7,6,5,1,2,4,3] => [3,4,2,1,5,6,7] => ? = 1
[1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => [7,6,5,1,3,2,4] => [4,2,3,1,5,6,7] => ? = 1
[1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => [7,6,5,2,1,4,3] => [3,4,1,2,5,6,7] => ? = 1
[1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => [7,6,5,2,1,3,4] => [4,3,1,2,5,6,7] => ? = 0
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [7,6,5,1,4,3,2] => [2,3,4,1,5,6,7] => ? = 1
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => [7,6,5,1,4,2,3] => [3,2,4,1,5,6,7] => ? = 1
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => [7,6,5,1,2,4,3] => [3,4,2,1,5,6,7] => ? = 1
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => [7,6,5,1,3,2,4] => [4,2,3,1,5,6,7] => ? = 1
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => [7,6,5,1,2,4,3] => [3,4,2,1,5,6,7] => ? = 1
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => ? = 0
[1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => [7,6,1,5,4,3,2] => [2,3,4,5,1,6,7] => ? = 1
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => [7,6,2,5,4,1,3] => [3,1,4,5,2,6,7] => ? = 1
[1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => [7,6,2,5,1,4,3] => [3,4,1,5,2,6,7] => ? = 2
[1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => [7,6,3,5,2,1,4] => [4,1,2,5,3,6,7] => ? = 1
[1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => [7,6,2,5,1,4,3] => [3,4,1,5,2,6,7] => ? = 2
[1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [7,6,3,5,1,2,4] => [4,2,1,5,3,6,7] => ? = 1
[1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => [7,6,2,1,5,4,3] => [3,4,5,1,2,6,7] => ? = 2
[1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => [7,6,3,2,5,1,4] => [4,1,5,2,3,6,7] => ? = 2
[1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => [7,6,3,2,1,5,4] => [4,5,1,2,3,6,7] => ? = 1
[1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => [7,6,4,3,2,1,5] => [5,1,2,3,4,6,7] => ? = 0
[1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => [7,6,3,2,1,5,4] => [4,5,1,2,3,6,7] => ? = 1
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => [7,6,4,3,1,2,5] => [5,2,1,3,4,6,7] => ? = 0
[1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => [7,6,2,1,5,4,3] => [3,4,5,1,2,6,7] => ? = 2
[1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => [7,6,3,2,5,1,4] => [4,1,5,2,3,6,7] => ? = 2
[1,2,4,6,5,3,7] => [7,6,4,2,3,5,1] => [7,6,3,1,2,5,4] => [4,5,2,1,3,6,7] => ? = 1
[1,2,4,6,5,7,3] => [7,6,4,2,3,1,5] => [7,6,4,1,3,2,5] => [5,2,3,1,4,6,7] => ? = 1
[1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => [7,6,3,2,1,5,4] => [4,5,1,2,3,6,7] => ? = 1
[1,2,4,6,7,5,3] => [7,6,4,2,1,3,5] => [7,6,4,2,1,3,5] => [5,3,1,2,4,6,7] => ? = 0
[1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => [7,6,2,1,5,4,3] => [3,4,5,1,2,6,7] => ? = 2
[1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => [7,6,3,1,5,2,4] => [4,2,5,1,3,6,7] => ? = 2
[1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => [7,6,3,1,2,5,4] => [4,5,2,1,3,6,7] => ? = 1
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => [7,6,4,1,3,2,5] => [5,2,3,1,4,6,7] => ? = 1
[1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => [7,6,3,1,2,5,4] => [4,5,2,1,3,6,7] => ? = 1
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => [7,6,4,1,2,3,5] => [5,3,2,1,4,6,7] => ? = 0
[1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => [7,6,1,5,4,3,2] => [2,3,4,5,1,6,7] => ? = 1
[1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => [7,6,2,5,4,1,3] => [3,1,4,5,2,6,7] => ? = 1
Description
The normalized sum of the minimal distances to a greater element.
Set $\pi_0 = \pi_{n+1} = n+1$, then this statistic is
$$
\sum_{i=1}^n \min_d(\pi_{i-1-d}>\pi_i\text{ or }\pi_{i+1+d}>\pi_i)
$$
A closely related statistic appears in [1].
The generating function for the sequence of maximal values attained on $\mathfrak S_r$, $r\geq 0$ apparently satisfies the functional equation
$$
(x-1)^2 (x+1)^3 f(x^2) - (x-1)^2 (x+1) f(x) + x^3 = 0.
$$
Matching statistic: St000822
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000822: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000822: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Values
[1] => [.,.]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ? = 1 + 1
[1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ? = 1 + 1
[1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,5,6,4,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,3,5,7,4,6] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,5,7,6,4] => [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,3,6,4,5,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,6,4,7,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,6,5,4,7] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,6,5,7,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ? = 1 + 1
[1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,6,7,5,4] => [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,3,7,4,5,6] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,7,4,6,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,7,5,4,6] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,7,5,6,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ? = 1 + 1
[1,2,3,7,6,4,5] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,3,6,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,3,7,5,6] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,2,4,3,7,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,2,4,5,6,3,7] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,5,6,7,3] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,4,5,7,3,6] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,5,7,6,3] => [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,2,4,6,5,3,7] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,6,5,7,3] => [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ? = 1 + 1
[1,2,4,6,7,3,5] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,6,7,5,3] => [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,2,4,7,5,3,6] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,7,5,6,3] => [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ? = 1 + 1
[1,2,4,7,6,3,5] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,7,6,5,3] => [.,[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,5,3,4,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,5,3,4,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
Description
The Hadwiger number of the graph.
Also known as clique contraction number, this is the size of the largest complete minor.
Matching statistic: St001330
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Values
[1] => [.,.]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,1,3,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,1,5,2,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,1,3,2,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,1,4,2,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,5,4,6,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,6,3,4,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,6,3,5,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,6,4,5,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,3,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,3,6,4,5] => [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,4,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,4,3,6,5] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,4,5,3,6] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,4,6,3,5] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,5,3,4,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,5,3,6,4] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,5,4,3,6] => [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,5,4,6,3] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,5,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,6,3,4,5] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,6,3,5,4] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,6,4,3,5] => [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,6,4,5,3] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,6,5,3,4] => [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,3,1,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 + 1
[2,3,1,4,6,5] => [[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001823The Stasinski-Voll length of a signed permutation. 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.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!