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Your data matches 189 different statistics following compositions of up to 3 maps.
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Matching statistic: St001613
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00140: Dyck paths —logarithmic height to pruning number⟶ 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
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 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 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00140: Dyck paths —logarithmic height to pruning number⟶ 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
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 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 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00140: Dyck paths —logarithmic height to pruning number⟶ 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
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 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 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00140: Dyck paths —logarithmic height to pruning number⟶ 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
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 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 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00140: Dyck paths —logarithmic height to pruning number⟶ 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
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 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 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000409: Binary trees ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
St000409: Binary trees ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 1
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> 1
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 1
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> 1
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 1
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> 1
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> 1
[1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [[.,[.,[.,[.,[[.,.],.]]]]],[.,.]]
=> ? = 1
[1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
=> [[.,[.,[.,[[.,[.,.]],.]]]],[.,.]]
=> ? = 1
[1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [[.,[.,[.,[[[.,.],.],.]]]],[.,.]]
=> ? = 1
[1,1,1,0,1,0,1,1,0,0,0,1,1,0,0,0]
=> [[.,[.,[[.,.],.]]],[[.,.],[.,.]]]
=> ? = 1
[1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> [[.,[.,[[.,[.,[.,.]]],.]]],[.,.]]
=> ? = 1
[1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> [[.,[.,[[.,[[.,.],.]],.]]],[.,.]]
=> ? = 1
[1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> [[.,[.,[[[.,[.,.]],.],.]]],[.,.]]
=> ? = 1
[1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [[.,[.,[[[[.,.],.],.],.]]],[.,.]]
=> ? = 1
[1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> [[.,[[.,[.,[.,[.,.]]]],.]],[.,.]]
=> ? = 1
[1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> [[.,[[.,[.,[[.,.],.]]],.]],[.,.]]
=> ? = 1
[1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0]
=> [[.,[[.,[[[.,.],.],.]],.]],[.,.]]
=> ? = 1
[1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> [[.,[[[.,.],.],.]],[[.,.],[.,.]]]
=> ? = 1
[1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
=> [[.,[[[.,[.,[.,.]]],.],.]],[.,.]]
=> ? = 1
[1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
=> [[.,[[[.,[[.,.],.]],.],.]],[.,.]]
=> ? = 1
[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> [[.,[[[[.,[.,.]],.],.],.]],[.,.]]
=> ? = 1
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [[.,[[[[[.,.],.],.],.],.]],[.,.]]
=> ? = 1
[1,1,1,1,0,0,1,0,1,0,0,1,1,0,0,0]
=> [[[.,[.,[.,.]]],.],[[.,.],[.,.]]]
=> ? = 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> ? = 1
[1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [[[.,[.,[.,[[.,.],.]]]],.],[.,.]]
=> ? = 1
[1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,0]
=> [[[.,[.,[[.,[.,.]],.]]],.],[.,.]]
=> ? = 1
[1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [[[.,[.,[[[.,.],.],.]]],.],[.,.]]
=> ? = 1
[1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> [[[.,[[.,.],.]],.],[[.,.],[.,.]]]
=> ? = 1
[1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
=> [[[.,[[.,[.,[.,.]]],.]],.],[.,.]]
=> ? = 1
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [[[.,[[.,[[.,.],.]],.]],.],[.,.]]
=> ? = 1
[1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> [[[.,[[[.,[.,.]],.],.]],.],[.,.]]
=> ? = 1
[1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [[[.,[[[[.,.],.],.],.]],.],[.,.]]
=> ? = 1
[1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> ? = 1
[1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [[[[.,[.,[[.,.],.]]],.],.],[.,.]]
=> ? = 1
[1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0]
=> [[[[.,[[.,[.,.]],.]],.],.],[.,.]]
=> ? = 1
[1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> ? = 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,[.,[[.,.],.]]]]
=> ? = 1
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,[[.,[.,.]],.]]]
=> ? = 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,[[[.,.],.],.]]]
=> ? = 1
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[[.,[.,[.,.]]],.]]
=> ? = 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[[.,[[.,.],.]],.]]
=> ? = 1
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[[[.,[.,.]],.],.]]
=> ? = 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[[[[.,.],.],.],.]]
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [[.,[.,[.,[.,[.,[[.,.],.]]]]]],[.,.]]
=> ? = 1
[1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [[.,[.,[[[[[.,.],.],.],.],.]]],[.,.]]
=> ? = 1
[1,1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [[.,[[.,[.,[.,[.,[.,.]]]]],.]],[.,.]]
=> ? = 1
[1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [[.,[[[[[[.,.],.],.],.],.],.]],[.,.]]
=> ? = 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,[.,[.,[.,.]]]]]],.],[.,.]]
=> ? = 1
[1,1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [[[.,[[[[[.,.],.],.],.],.]],.],[.,.]]
=> ? = 1
[1,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [[[[[.,[.,[.,[.,.]]]],.],.],.],[.,.]]
=> ? = 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[.,.],[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> ? = 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
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [.,[[[.,.],[.,[.,[.,.]]]],.]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [.,[[[.,.],[.,[[.,.],.]]],.]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [.,[[[.,.],[[.,[.,.]],.]],.]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [.,[[[.,.],[[[.,.],.],.]],.]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [.,[[[.,[.,.]],[[.,.],.]],.]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [.,[[[[.,.],.],[.,[.,.]]],.]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [.,[[[[.,.],.],[[.,.],.]],.]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [.,[[[[.,[.,.]],.],[.,.]],.]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [.,[[[[[.,.],.],.],[.,.]],.]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [.,[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,[[.,[.,.]],.]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [.,[[.,.],[.,[[[.,.],.],.]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [.,[[.,.],[[.,[.,[.,.]]],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,[[.,.],.]],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [.,[[.,.],[[[.,[.,.]],.],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [.,[[.,.],[[[[.,.],.],.],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [.,[[.,[.,.]],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [.,[[.,[.,.]],[[[.,.],.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [.,[[.,[.,[.,.]]],[[.,.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [.,[[.,[.,[[.,.],.]]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [.,[[.,[[.,.],.]],[.,[.,.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[.,[[.,.],.]],[[.,.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [.,[[.,[[.,[.,.]],.]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [.,[[.,[[[.,.],.],.]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [.,[[[.,.],.],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [.,[[[.,.],.],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [.,[[[.,.],.],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [.,[[[.,.],.],[[[.,.],.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [.,[[[.,[.,.]],.],[.,[.,.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [.,[[[.,[.,.]],.],[[.,.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],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: St001513
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001513: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001513: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => [1,3,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [3,2,1] => [2,1,3] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,4,3,1] => [1,3,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [1,4,2,3] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,2,4,1] => [2,1,3,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,1,4] => [2,1,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,4,2,1] => [3,1,2,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,4,1,2] => [4,1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,1,5] => [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,5,4,1] => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => [1,2,5,3,4] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,4,3,5,1] => [1,3,2,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => [1,3,2,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,5,3,1] => [1,4,2,3,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,4,5,1,3] => [1,5,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [3,2,4,5,1] => [2,1,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,1,5] => [2,1,3,5,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => [2,1,5,3,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,4,2,5,1] => [3,1,2,4,5] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,4,2,1,5] => [3,1,2,5,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [3,4,5,2,1] => [4,1,2,3,5] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => [5,1,2,3,4] => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [4,3,2,5,1] => [3,2,1,4,5] => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [4,3,2,1,5] => [3,2,1,5,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [4,3,5,2,1] => [4,2,1,3,5] => 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [4,3,5,1,2] => [5,2,1,3,4] => 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [4,5,3,2,1] => [4,3,1,2,5] => 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [4,5,3,1,2] => [5,3,1,2,4] => 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [4,5,1,2,3] => [4,5,1,2,3] => 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [4,1,2,3,5] => [3,4,1,5,2] => 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [5,4,3,1,2] => [5,3,2,1,4] => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [5,4,1,2,3] => [4,5,2,1,3] => 0 = 1 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [5,1,2,3,4] => [3,4,5,1,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [2,3,4,5,1,6] => [1,2,3,4,6,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [2,3,4,6,5,1] => [1,2,3,5,4,6] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [2,3,4,6,1,5] => [1,2,3,6,4,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [2,3,5,4,6,1] => [1,2,4,3,5,6] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [2,3,5,4,1,6] => [1,2,4,3,6,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [2,3,5,6,4,1] => [1,2,5,3,4,6] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [2,3,5,6,1,4] => [1,2,6,3,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [2,4,3,5,6,1] => [1,3,2,4,5,6] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [2,4,3,5,1,6] => [1,3,2,4,6,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => [2,4,5,6,3,7,1] => [1,5,2,3,4,6,7] => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [2,4,5,6,3,1,7] => [1,5,2,3,4,7,6] => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [2,4,5,6,7,1,3] => [1,7,2,3,4,5,6] => ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,4,2,6,7] => [2,4,6,5,3,7,1] => [1,5,2,4,3,6,7] => ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,5,4,2,7,6] => [2,4,6,5,3,1,7] => [1,5,2,4,3,7,6] => ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,4,6,2,7] => [2,4,6,5,7,3,1] => [1,6,2,4,3,5,7] => ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,5,4,6,7,2] => [2,4,6,5,7,1,3] => [1,7,2,4,3,5,6] => ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,5,6,4,2,7] => [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,5,6,4,7,2] => [2,4,6,7,5,1,3] => [1,7,2,5,3,4,6] => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,5,6,7,4,2] => [2,4,6,7,1,3,5] => [1,6,2,7,3,4,5] => ? = 1 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,5,7,6,4,2] => [2,4,6,1,3,5,7] => [1,5,2,6,3,7,4] => ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,6,5,4,2,7] => [2,4,7,6,5,3,1] => [1,6,2,5,4,3,7] => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => [2,4,7,6,5,1,3] => [1,7,2,5,4,3,6] => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,6,5,7,4,2] => [2,4,7,6,1,3,5] => [1,6,2,7,4,3,5] => ? = 1 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,6,7,5,4,2] => [2,4,7,1,3,5,6] => [1,5,2,6,7,3,4] => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,4,3,5,2,6,7] => [2,5,4,6,3,7,1] => [1,5,3,2,4,6,7] => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,4,3,5,2,7,6] => [2,5,4,6,3,1,7] => [1,5,3,2,4,7,6] => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,4,3,5,6,2,7] => [2,5,4,6,7,3,1] => [1,6,3,2,4,5,7] => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,2] => [2,5,4,6,7,1,3] => [1,7,3,2,4,5,6] => ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,4,5,3,2,6,7] => [2,5,6,4,3,7,1] => [1,5,4,2,3,6,7] => ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,4,5,3,2,7,6] => [2,5,6,4,3,1,7] => [1,5,4,2,3,7,6] => ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,4,5,3,6,2,7] => [2,5,6,4,7,3,1] => [1,6,4,2,3,5,7] => ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,5,3,6,7,2] => [2,5,6,4,7,1,3] => [1,7,4,2,3,5,6] => ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,4,5,6,3,2,7] => [2,5,6,7,4,3,1] => [1,6,5,2,3,4,7] => ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,5,6,3,7,2] => [2,5,6,7,4,1,3] => [1,7,5,2,3,4,6] => ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,4,5,6,7,3,2] => [2,5,6,7,1,3,4] => [1,6,7,2,3,4,5] => ? = 1 - 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,4,5,7,6,3,2] => [2,5,6,1,3,4,7] => [1,5,6,2,3,7,4] => ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,4,6,5,3,2,7] => [2,5,7,6,4,3,1] => [1,6,5,2,4,3,7] => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,5,3,7,2] => [2,5,7,6,4,1,3] => [1,7,5,2,4,3,6] => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,5,7,3,2] => [2,5,7,6,1,3,4] => [1,6,7,2,4,3,5] => ? = 1 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => [2,5,7,1,3,4,6] => [1,5,6,2,7,3,4] => ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => [2,6,5,4,3,7,1] => [1,5,4,3,2,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,5,4,3,2,7,6] => [2,6,5,4,3,1,7] => [1,5,4,3,2,7,6] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,5,4,3,6,2,7] => [2,6,5,4,7,3,1] => [1,6,4,3,2,5,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => [2,6,5,4,7,1,3] => [1,7,4,3,2,5,6] => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,4,6,3,2,7] => [2,6,5,7,4,3,1] => [1,6,5,3,2,4,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,4,6,3,7,2] => [2,6,5,7,4,1,3] => [1,7,5,3,2,4,6] => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,4,6,7,3,2] => [2,6,5,7,1,3,4] => [1,6,7,3,2,4,5] => ? = 1 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,4,7,6,3,2] => [2,6,5,1,3,4,7] => [1,5,6,3,2,7,4] => ? = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,5,6,4,3,2,7] => [2,6,7,5,4,3,1] => [1,6,5,4,2,3,7] => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => [2,6,7,5,4,1,3] => [1,7,5,4,2,3,6] => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => [2,6,7,5,1,3,4] => [1,6,7,4,2,3,5] => ? = 1 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => [2,6,7,1,3,4,5] => [1,5,6,7,2,3,4] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => [3,2,4,5,6,7,1] => [2,1,3,4,5,6,7] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => [3,2,4,5,6,1,7] => [2,1,3,4,5,7,6] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => [3,2,4,5,7,6,1] => [2,1,3,4,6,5,7] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => [3,2,4,5,7,1,6] => [2,1,3,4,7,5,6] => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => [3,2,4,6,5,7,1] => [2,1,3,5,4,6,7] => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => [3,2,4,6,5,1,7] => [2,1,3,5,4,7,6] => ? = 1 - 1
Description
The number of nested exceedences of a permutation.
For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see [[St000155]].
Matching statistic: St000535
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,7),(1,2),(1,7),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,7),(1,2),(1,7),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,0,1,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,0,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ? = 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(0,3),(0,9),(1,2),(1,6),(2,8),(3,5),(4,5),(4,7),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,3),(0,9),(1,2),(1,8),(2,6),(3,7),(4,7),(4,8),(5,6),(5,9),(6,8),(7,9),(8,9)],10)
=> ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,3),(0,9),(1,2),(1,6),(2,8),(3,5),(4,5),(4,7),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,3),(0,9),(1,2),(1,6),(2,8),(3,5),(4,5),(4,7),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,3),(0,9),(1,2),(1,8),(2,6),(3,7),(4,7),(4,8),(5,6),(5,9),(6,8),(7,9),(8,9)],10)
=> ? = 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(0,3),(0,9),(1,2),(1,6),(2,8),(3,5),(4,5),(4,7),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ?
=> ? = 2
Description
The rank-width of a graph.
Matching statistic: St001111
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,1,1,0,1,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,1,1,0,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(2,9),(3,8),(4,5),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ? = 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(2,9),(3,8),(4,6),(5,7),(6,8),(7,9),(8,9)],10)
=> ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(2,9),(3,8),(4,5),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ? = 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(2,9),(3,8),(4,5),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ? = 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(2,9),(3,8),(4,6),(5,7),(6,8),(7,9),(8,9)],10)
=> ? = 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(2,9),(3,8),(4,5),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ? = 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ?
=> ? = 2
Description
The weak 2-dynamic chromatic number of a graph.
A $k$-weak-dynamic coloring of a graph $G$ is a (non-proper) coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-weak-dynamic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring.
This statistic records the $2$-weak-dynamic number of a graph.
The following 179 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001112The 3-weak dynamic number of a graph. St001694The number of maximal dissociation sets in a graph. St001716The 1-improper chromatic number of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001306The number of induced paths on four vertices in a graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001350Half of the Albertson index of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001494The Alon-Tarsi number of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St000093The cardinality of a maximal independent set of vertices of a graph. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St000322The skewness of a graph. St000092The number of outer peaks of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000449The number of pairs of vertices of a graph with distance 4. St001330The hat guessing number of a graph. St000649The number of 3-excedences of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. 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. St001533The largest 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. St001570The minimal number of edges to add to make a graph Hamiltonian. St001964The interval resolution global dimension of a poset. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000311The number of vertices of odd degree in a graph. St000315The number of isolated vertices of a graph. St000447The number of pairs of vertices of a graph with distance 3. St001060The distinguishing index of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001827The number of two-component spanning forests of a graph. St001871The number of triconnected components of a graph. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St001261The Castelnuovo-Mumford regularity of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001315The dissociation number of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001356The number of vertices in prime modules of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001479The number of bridges of a graph. St000068The number of minimal elements in a poset. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000478Another weight of a partition according to Alladi. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000741The Colin de Verdière graph invariant. St001625The Möbius invariant of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000007The number of saliances of the permutation. St000022The number of fixed points of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000223The number of nestings in the permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000546The number of global descents of a permutation. St000834The number of right outer peaks of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000647The number of big descents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000862The number of parts of the shifted shape of a permutation. St000648The number of 2-excedences of a permutation. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000878The number of ones minus the number of zeros of a binary word. St001644The dimension of a graph. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St000629The defect of a binary word. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001271The competition number of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000635The number of strictly order preserving maps of a poset into itself. St001545The second Elser number of a connected graph. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000454The largest eigenvalue of a graph if it is integral. St000879The number of long braid edges in the graph of braid moves of a permutation. St001846The number of elements which do not have a complement in the lattice. St000895The number of ones on the main diagonal of an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001118The acyclic chromatic index of a graph. St000422The energy of a graph, if it is integral. St000627The exponent of a binary word. St001851The number of Hecke atoms of a signed permutation. St000296The length of the symmetric border of a binary word. St000768The number of peaks in an integer composition. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000352The Elizalde-Pak rank of a permutation. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St000359The number of occurrences of the pattern 23-1. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000753The Grundy value for the game of Kayles on a binary word. St000787The number of flips required to make a perfect matching noncrossing. St000842The breadth of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation.
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