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Your data matches 56 different statistics following compositions of up to 3 maps.
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Matching statistic: St001613
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
Description
The binary logarithm of the size of the center of a lattice.
An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.
Matching statistic: St001621
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001624
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St001878
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001881
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
Description
The number of factors of a lattice as a Cartesian product of lattices.
Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.
Matching statistic: St000409
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
St000409: Binary trees ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
St000409: Binary trees ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [.,[.,[.,.]]]
=> [[[.,.],.],.]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [[.,[.,.]],.]
=> 1
[2,3,1] => [[.,[.,.]],.]
=> [.,[[.,.],.]]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [.,[.,[.,.]]]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> 1
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> 2
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],.]]
=> 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [.,[[[.,[.,.]],.],.]]
=> 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> 2
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> [.,[[.,[[.,.],.]],.]]
=> 1
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> 2
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> 1
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],.]]
=> 1
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> 1
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> 1
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> 1
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> 2
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> 2
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> 2
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> 1
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> 1
[8,6,7,1,2,3,4,5] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[7,6,8,1,2,3,4,5] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[8,5,6,1,2,3,4,7] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[8,4,5,1,2,3,6,7] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[8,3,4,1,2,5,6,7] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[8,2,3,1,4,5,6,7] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[7,5,6,1,2,3,4,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[6,5,7,1,2,3,4,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[7,4,5,1,2,3,6,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[7,3,4,1,2,5,6,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[7,2,3,1,4,5,6,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[6,2,3,1,4,5,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[4,3,5,1,2,6,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[5,2,3,1,4,6,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[4,2,3,1,5,6,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[3,2,4,1,5,6,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 1
[2,1,7,8,6,5,4,3] => [[.,.],[[[[[.,[.,.]],.],.],.],.]]
=> [[.,[.,[.,[.,[[.,.],.]]]]],[.,.]]
=> ? = 1
[2,1,6,8,7,5,4,3] => [[.,.],[[[[.,[[.,.],.]],.],.],.]]
=> [[.,[.,[.,[[.,[.,.]],.]]]],[.,.]]
=> ? = 1
[2,1,6,7,8,5,4,3] => [[.,.],[[[[.,[.,[.,.]]],.],.],.]]
=> [[.,[.,[.,[[[.,.],.],.]]]],[.,.]]
=> ? = 1
[2,1,5,8,7,6,4,3] => [[.,.],[[[.,[[[.,.],.],.]],.],.]]
=> [[.,[.,[[.,[.,[.,.]]],.]]],[.,.]]
=> ? = 1
[2,1,5,7,8,6,4,3] => [[.,.],[[[.,[[.,[.,.]],.]],.],.]]
=> [[.,[.,[[.,[[.,.],.]],.]]],[.,.]]
=> ? = 1
[2,1,5,6,8,7,4,3] => [[.,.],[[[.,[.,[[.,.],.]]],.],.]]
=> [[.,[.,[[[.,[.,.]],.],.]]],[.,.]]
=> ? = 1
[2,1,5,6,7,8,4,3] => [[.,.],[[[.,[.,[.,[.,.]]]],.],.]]
=> [[.,[.,[[[[.,.],.],.],.]]],[.,.]]
=> ? = 1
[2,1,4,8,7,6,5,3] => [[.,.],[[.,[[[[.,.],.],.],.]],.]]
=> [[.,[[.,[.,[.,[.,.]]]],.]],[.,.]]
=> ? = 1
[2,1,4,7,8,6,5,3] => [[.,.],[[.,[[[.,[.,.]],.],.]],.]]
=> [[.,[[.,[.,[[.,.],.]]],.]],[.,.]]
=> ? = 1
[2,1,4,6,7,8,5,3] => [[.,.],[[.,[[.,[.,[.,.]]],.]],.]]
=> [[.,[[.,[[[.,.],.],.]],.]],[.,.]]
=> ? = 1
[2,1,4,5,8,7,6,3] => [[.,.],[[.,[.,[[[.,.],.],.]]],.]]
=> [[.,[[[.,[.,[.,.]]],.],.]],[.,.]]
=> ? = 1
[2,1,4,5,7,8,6,3] => [[.,.],[[.,[.,[[.,[.,.]],.]]],.]]
=> [[.,[[[.,[[.,.],.]],.],.]],[.,.]]
=> ? = 1
[2,1,4,5,6,8,7,3] => [[.,.],[[.,[.,[.,[[.,.],.]]]],.]]
=> [[.,[[[[.,[.,.]],.],.],.]],[.,.]]
=> ? = 1
[2,1,4,5,6,7,8,3] => [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [[.,[[[[[.,.],.],.],.],.]],[.,.]]
=> ? = 1
[2,1,3,8,7,6,5,4] => [[.,.],[.,[[[[[.,.],.],.],.],.]]]
=> [[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> ? = 1
[2,1,3,7,8,6,5,4] => [[.,.],[.,[[[[.,[.,.]],.],.],.]]]
=> [[[.,[.,[.,[[.,.],.]]]],.],[.,.]]
=> ? = 1
[2,1,3,6,8,7,5,4] => [[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> [[[.,[.,[[.,[.,.]],.]]],.],[.,.]]
=> ? = 1
[2,1,3,5,6,7,8,4] => [[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> [[[.,[[[[.,.],.],.],.]],.],[.,.]]
=> ? = 1
[3,2,4,1,8,7,6,5] => [[[.,.],[.,.]],[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> ? = 1
[3,2,4,1,7,8,6,5] => [[[.,.],[.,.]],[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],[[.,.],[.,.]]]
=> ? = 1
[3,2,4,1,6,7,8,5] => [[[.,.],[.,.]],[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],[[.,.],[.,.]]]
=> ? = 1
[2,1,3,4,8,7,6,5] => [[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> [[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> ? = 1
[2,1,3,4,7,8,6,5] => [[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [[[[.,[.,[[.,.],.]]],.],.],[.,.]]
=> ? = 1
[3,2,4,1,5,8,7,6] => [[[.,.],[.,.]],[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],[[.,.],[.,.]]]
=> ? = 1
[3,2,4,1,5,7,8,6] => [[[.,.],[.,.]],[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],[[.,.],[.,.]]]
=> ? = 1
[5,4,3,2,1,7,6,8] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> ? = 1
[4,5,3,2,1,7,6,8] => [[[[.,[.,.]],.],.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,[.,[[.,.],.]]]]
=> ? = 1
[3,5,4,2,1,7,6,8] => [[[.,[[.,.],.]],.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,[[.,[.,.]],.]]]
=> ? = 1
[3,4,5,2,1,7,6,8] => [[[.,[.,[.,.]]],.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,[[[.,.],.],.]]]
=> ? = 1
[2,4,5,3,1,7,6,8] => [[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,[[.,.],.]],.]]
=> ? = 1
[2,3,4,5,1,7,6,8] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[[[.,.],.],.],.]]
=> ? = 1
[4,2,3,1,5,7,8,6] => [[[.,.],[.,.]],[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],[[.,.],[.,.]]]
=> ? = 1
[4,2,3,1,5,8,7,6] => [[[.,.],[.,.]],[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],[[.,.],[.,.]]]
=> ? = 1
Description
The number of pitchforks in a binary tree.
A pitchfork is a subtree of a complete binary tree with exactly three leaves, see Section 3.2 of [1].
Matching statistic: St001632
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,4,3,7,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,4,5,6,3,7] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,4,5,7,3,6] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,4,6,5,3,7] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,4,6,7,3,5] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,4,7,5,3,6] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,4,7,6,3,5] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,5,3,4,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,5,3,4,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,5,3,6,7,4] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,5,3,7,6,4] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,5,4,3,7,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,5,6,4,3,7] => [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,5,6,7,3,4] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,5,7,3,6,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,5,7,4,3,6] => [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,5,7,6,3,4] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,6,3,4,5,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,6,3,4,7,5] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,6,3,5,7,4] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,6,3,7,5,4] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,6,4,3,5,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,6,4,3,7,5] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,6,5,3,4,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,6,5,3,7,4] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,6,5,4,3,7] => [.,[.,[[[[.,.],.],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,6,7,3,4,5] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,6,7,3,5,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,6,7,4,3,5] => [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,6,7,5,3,4] => [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,7,3,4,5,6] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,7,3,4,6,5] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,7,3,5,6,4] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,7,3,6,5,4] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,7,4,3,5,6] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,7,4,3,6,5] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,2,7,5,3,4,6] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St000454
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00019: Binary trees —right rotate⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00019: Binary trees —right rotate⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[1,2,3] => [.,[.,[.,.]]]
=> [[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> ? = 1
[1,3,2] => [.,[[.,.],.]]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 1
[2,3,1] => [[.,[.,.]],.]
=> [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ? = 1
[3,2,1] => [[[.,.],.],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,5,4,3,7,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,5,7,3,6,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,6,4,3,5,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,6,4,3,7,5] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,6,5,3,4,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,6,5,3,7,4] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,6,7,3,4,5] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,6,7,3,5,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,7,4,3,5,6] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,7,4,3,6,5] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,7,5,3,4,6] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,7,5,3,6,4] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,7,6,3,4,5] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,7,6,3,5,4] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,5,3,6,7,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [[[[.,[.,.]],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,5,3,7,6,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> [[[[.,[.,.]],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,6,3,5,7,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [[[[.,[.,.]],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,6,3,7,5,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> [[[[.,[.,.]],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,7,3,5,6,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [[[[.,[.,.]],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,7,3,6,5,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> [[[[.,[.,.]],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,5,4,3,6,7,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [[[[[.,.],.],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,5,4,3,7,6,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,5,6,3,4,7,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [[[[.,[.,.]],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,5,6,3,7,4,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> [[[[.,[.,.]],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,5,7,3,4,6,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [[[[.,[.,.]],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,5,7,3,6,4,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> [[[[.,[.,.]],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,6,4,3,5,7,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [[[[[.,.],.],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,6,4,3,7,5,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,6,5,3,4,7,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [[[[[.,.],.],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,6,5,3,7,4,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,6,7,3,4,5,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [[[[.,[.,.]],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,6,7,3,5,4,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> [[[[.,[.,.]],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,7,4,3,5,6,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [[[[[.,.],.],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,7,4,3,6,5,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,7,5,3,4,6,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [[[[[.,.],.],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,7,5,3,6,4,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,7,6,3,4,5,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [[[[[.,.],.],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,7,6,3,5,4,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[2,1,4,5,3,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[2,1,4,5,3,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000422
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00019: Binary trees —right rotate⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00019: Binary trees —right rotate⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[1,2,3] => [.,[.,[.,.]]]
=> [[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> ? = 1 + 6
[1,3,2] => [.,[[.,.],.]]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 1 + 6
[2,3,1] => [[.,[.,.]],.]
=> [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ? = 1 + 6
[3,2,1] => [[[.,.],.],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 1 + 6
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 6
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 6
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 6
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 6
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 6
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 6
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 6
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 6
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 6
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 6
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 6
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 6
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 6
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 6
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 6
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 6
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 6
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 6
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 6
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 6
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 6
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 6
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 6
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 6
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 6
[1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,5,4,3,7,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,5,7,3,6,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,6,4,3,5,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,6,4,3,7,5] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,6,5,3,4,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,6,5,3,7,4] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,6,7,3,4,5] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,6,7,3,5,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,7,4,3,5,6] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,7,4,3,6,5] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,7,5,3,4,6] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,7,5,3,6,4] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,7,6,3,4,5] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,2,7,6,3,5,4] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,4,5,3,6,7,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [[[[.,[.,.]],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,4,5,3,7,6,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> [[[[.,[.,.]],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,4,6,3,5,7,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [[[[.,[.,.]],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,4,6,3,7,5,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> [[[[.,[.,.]],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,4,7,3,5,6,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [[[[.,[.,.]],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,4,7,3,6,5,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> [[[[.,[.,.]],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,5,4,3,6,7,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [[[[[.,.],.],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,5,4,3,7,6,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,5,6,3,4,7,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [[[[.,[.,.]],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,5,6,3,7,4,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> [[[[.,[.,.]],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,5,7,3,4,6,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [[[[.,[.,.]],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,5,7,3,6,4,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> [[[[.,[.,.]],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,6,4,3,5,7,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [[[[[.,.],.],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,6,4,3,7,5,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,6,5,3,4,7,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [[[[[.,.],.],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,6,5,3,7,4,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,6,7,3,4,5,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [[[[.,[.,.]],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,6,7,3,5,4,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> [[[[.,[.,.]],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,7,4,3,5,6,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [[[[[.,.],.],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,7,4,3,6,5,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,7,5,3,4,6,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [[[[[.,.],.],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,7,5,3,6,4,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,7,6,3,4,5,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [[[[[.,.],.],[.,[.,.]]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[1,7,6,3,5,4,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[2,1,4,5,3,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
[2,1,4,5,3,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 2 + 6
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 100%
Mp00011: Binary trees —to graph⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,5,2,1,4] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,5,4,3,7,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,5,7,3,6,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,6,4,3,5,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,6,4,3,7,5] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,6,5,3,4,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,6,5,3,7,4] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,6,7,3,4,5] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,6,7,3,5,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,7,4,3,5,6] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,7,4,3,6,5] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,7,5,3,4,6] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,7,5,3,6,4] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,7,6,3,4,5] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,2,7,6,3,5,4] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,4,5,3,6,7,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,4,5,3,7,6,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,4,6,3,5,7,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,4,6,3,7,5,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,4,7,3,5,6,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,4,7,3,6,5,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,5,4,3,6,7,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,5,4,3,7,6,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,5,6,3,4,7,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,5,6,3,7,4,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,5,7,3,4,6,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,5,7,3,6,4,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,6,4,3,5,7,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,6,4,3,7,5,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,6,5,3,4,7,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,6,5,3,7,4,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,6,7,3,4,5,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,6,7,3,5,4,2] => [.,[[[.,[.,.]],[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,7,4,3,5,6,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,7,4,3,6,5,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,7,5,3,4,6,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,7,5,3,6,4,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
The following 46 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001845The number of join irreducibles minus the rank of a lattice. St000068The number of minimal elements in a poset. St000872The number of very big descents of a permutation. St000741The Colin de Verdière graph invariant. St001625The Möbius invariant of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001964The interval resolution global dimension of a poset. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St000850The number of 1/2-balanced pairs in a poset. St001472The permanent of the Coxeter matrix of the poset. St001862The number of crossings of a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001490The number of connected components of a skew partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000098The chromatic number of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St000312The number of leaves in a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000417The size of the automorphism group of the ordered tree. St000679The pruning number of an ordered tree. St001058The breadth of the ordered tree. St001429The number of negative entries in a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001866The nesting alignments of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001892The flag excedance statistic of a signed permutation.
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