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Your data matches 62 different statistics following compositions of up to 3 maps.
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Matching statistic: St000068
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(load all 4 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [.,.]
=> ([],1)
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [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]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
Description
The number of minimal elements in a poset.
Matching statistic: St000071
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [.,.]
=> ([],1)
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [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]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
Description
The number of maximal chains in a poset.
Matching statistic: St000291
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => => ? = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1 => 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0 => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 11 => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 01 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 101 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 011 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 011 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1111 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1110 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1101 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1100 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 1101 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1011 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 1011 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 1010 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 1011 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1001 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0111 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0011 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0011 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 0111 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,4,6,5,8,7,3] => ? => ? = 2 - 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,6,8,7,5,4,3] => ? => ? = 1 - 1
[1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,2,6,7,5,8,4,3] => ? => ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,2,4,5,7,6,8,3] => ? => ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,5,8,3] => ? => ? = 2 - 1
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,2,5,7,8,6,4,3] => ? => ? = 1 - 1
[1,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,3,7,6,5,8,4,2] => ? => ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,6,5,8,7,4,2] => ? => ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,3,7,6,8,5,4,2] => ? => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,4,7,6,8,5,3,2] => ? => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,4,6,5,7,8,3,2] => ? => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,4,6,5,8,7,2] => ? => ? = 2 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,6,8,7,5,2] => ? => ? = 1 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,4,7,8,6,5,2] => ? => ? = 1 - 1
[1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,4,5,7,6,8,3,2] => ? => ? = 2 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,5,6,8,7,4,3,2] => ? => ? = 1 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,5,7,8,6,4,3,2] => ? => ? = 1 - 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,4,5,6,8,7,3,2] => ? => ? = 1 - 1
[1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,7,6,5,8,4,3,1] => ? => ? = 2 - 1
[1,1,1,0,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,6,5,8,7,4,3,1] => ? => ? = 2 - 1
[1,1,1,0,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> [3,6,5,8,7,4,2,1] => ? => ? = 2 - 1
[1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,6,5,7,8,4,2,1] => ? => ? = 2 - 1
[1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,7,6,8,5,4,1] => ? => ? = 2 - 1
[1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,6,5,7,8,4,1] => ? => ? = 2 - 1
[1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [2,4,7,6,8,5,3,1] => ? => ? = 2 - 1
[1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [2,5,7,6,8,4,3,1] => ? => ? = 2 - 1
[1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,5,8,7,6,4,1] => ? => ? = 1 - 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,6,8,7,5,4,1] => ? => ? = 1 - 1
[1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,7,8,6,5,4,1] => ? => ? = 1 - 1
[1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,3,6,7,5,8,4,1] => ? => ? = 2 - 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [2,4,5,8,7,6,3,1] => ? => ? = 1 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [2,4,5,7,6,8,3,1] => ? => ? = 2 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,5,6,8,7,4,3,1] => ? => ? = 1 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,5,7,8,6,4,3,1] => ? => ? = 1 - 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,4,6,5,8,7,2,1] => ? => ? = 2 - 1
[1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,4,7,6,8,5,2,1] => ? => ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,4,6,8,7,5,2,1] => ? => ? = 1 - 1
[1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,4,7,8,6,5,2,1] => ? => ? = 1 - 1
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,6,7,8,5,4,2,1] => ? => ? = 1 - 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,6,7,9,8,2,1] => ? => ? = 1 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,6,9,8,7,2,1] => ? => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,4,8,9,7,6,5] => ? => ? = 1 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,3,8,9,7,6,5,4] => ? => ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,4,8,9,7,6,5,3] => ? => ? = 1 - 1
[1,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,8,9,4,3] => ? => ? = 1 - 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,5,6,8,9,7,2] => ? => ? = 1 - 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> [6,8,9,7,5,4,3,2,1] => ? => ? = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0]
=> [5,7,8,9,6,4,3,2,1] => ? => ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,6,7,8,9,5,3,2,1] => ? => ? = 1 - 1
Description
The number of descents of a binary word.
Matching statistic: St000386
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> []
=> []
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,2,2,2,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,3,2,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,3,3,3,2,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,3,2,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,3,3,2,2,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,2,2,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,3,3,2,2,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,2,2,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,3,2,2,2,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,2,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,2,2,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,2,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,2,2,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,2,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,3,2,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,3,3,2,1,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,2,1,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,3,3,2,1,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,2,1,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,2,1,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,2,1,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,1,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,2,1,1]
=> ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,2,1,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,3,2,1,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,2,1,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [6,5,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,6,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [5,4,4,4,4,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [6,5,5,4,4,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [4,3,3,3,3,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [4,3,3,3,3,2,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,5,4,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [6,5,5,4,3,2,1,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[1,1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,5,4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,4,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000527
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [.,.]
=> ([],1)
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [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]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [[[[[[.,.],[[.,.],.]],.],.],.],.]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],[[.,.],.]],.],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [.,[[[[[.,.],[.,[.,.]]],.],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [.,[.,[[[[[.,.],.],[.,.]],.],.]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [.,[.,[[[[.,.],[[.,.],.]],.],.]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [.,[.,[[[[[.,.],[.,.]],.],.],.]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [.,[.,[[[[.,.],[.,[.,.]]],.],.]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[.,[.,[[[[.,.],[.,.]],.],.]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [.,[.,[[.,[[.,.],[[.,.],.]]],.]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[.,[[[.,.],[.,.]],.]],.]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [.,[.,[[[.,[[.,.],[.,.]]],.],.]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [.,[.,[[[[.,[.,.]],[.,.]],.],.]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [.,[.,[[.,[[.,[.,.]],[.,.]]],.]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [.,[[.,[[[[.,.],.],[.,.]],.]],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [.,[[.,[[[.,.],[[.,.],.]],.]],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [.,[[.,[[[[.,.],[.,.]],.],.]],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [.,[[[.,[[[.,.],.],[.,.]]],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [.,[[[.,[[.,.],[[.,.],.]]],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [.,[[[.,[[[.,.],[.,.]],.]],.],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [.,[[[.,[[.,.],[.,[.,.]]]],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [.,[[[[.,[[.,.],[.,.]]],.],.],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [.,[[[[[.,[.,.]],[.,.]],.],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [.,[[.,[.,[[[.,.],.],[.,.]]]],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [.,[[.,[.,[[.,.],[[.,.],.]]]],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [.,[[.,[.,[[[.,.],[.,.]],.]]],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [.,[[.,[.,[[.,.],[.,[.,.]]]]],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [.,[[.,[.,[.,[[.,.],[.,.]]]]],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[.,[[.,[.,.]],[.,.]]]],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
Description
The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000292
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => => ? = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => 0 => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => 1 => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 00 => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 01 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 10 => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 11 => 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 10 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 001 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 011 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 010 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 100 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 110 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => 111 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,2,1] => 110 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => 100 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => 101 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => 100 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => 110 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 0011 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => 0010 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 0110 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 0111 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,3,2] => 0110 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => 0100 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,4,2] => 0101 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => 0100 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => 0110 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1011 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => 1010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 1100 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 1101 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => 1110 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => 1111 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,3,2,1] => 1110 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,2,1,5] => 1100 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,4,2,1] => 1101 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,4,5,2,1] => 1100 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,2,1] => 1110 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,3,1,4,5] => 1000 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [2,1,7,4,5,6,3,8] => ? => ? => ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,1,4,8,6,7,5] => ? => ? => ? = 2 - 1
[1,1,0,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,4,1,8,7,6] => [4,5,3,2,1,7,8,6] => ? => ? = 2 - 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [2,3,5,4,7,6,1,8] => ? => ? => ? = 2 - 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [2,3,7,4,5,6,1,8] => ? => ? => ? = 1 - 1
[1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [2,3,8,4,6,5,7,1] => ? => ? => ? = 2 - 1
[1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,8,4,7,6,5,1] => ? => ? => ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,4,8,7,1] => ? => ? => ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [2,3,8,5,4,7,6,1] => ? => ? => ? = 2 - 1
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,8,7,5,6,4,1] => ? => ? => ? = 1 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [2,4,3,7,5,6,1,8] => ? => ? => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [2,7,3,5,4,6,1,8] => ? => ? => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [2,8,3,5,4,6,7,1] => ? => ? => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,8,3,5,7,6,4,1] => [3,6,7,5,4,8,2,1] => ? => ? = 2 - 1
[1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [2,8,3,6,5,4,7,1] => ? => ? => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,5,4,3,1,8,7,6] => [4,3,5,2,1,7,8,6] => ? => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [2,5,4,3,7,6,1,8] => ? => ? => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [2,5,4,3,8,7,6,1] => [4,3,7,6,8,5,2,1] => ? => ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [2,6,4,3,5,1,7,8] => ? => ? => ? = 1 - 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [2,7,4,3,5,6,1,8] => ? => ? => ? = 1 - 1
[1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,8,4,3,7,6,5,1] => [4,3,6,7,5,8,2,1] => ? => ? = 2 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,3,6,1,8] => ? => ? => ? = 1 - 1
[1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [2,8,4,5,3,7,6,1] => ? => ? => ? = 2 - 1
[1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,8,4,5,7,6,3,1] => ? => ? => ? = 1 - 1
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [2,8,4,6,5,3,7,1] => [5,6,4,3,7,8,2,1] => ? => ? = 1 - 1
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,8,4,7,5,6,3,1] => ? => ? => ? = 1 - 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,6,5,4,3,1,7,8] => ? => ? => ? = 1 - 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [2,7,5,4,3,6,1,8] => ? => ? => ? = 1 - 1
[1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [2,8,5,4,3,6,7,1] => [4,5,3,6,7,8,2,1] => ? => ? = 1 - 1
[1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [2,8,5,4,6,3,7,1] => ? => ? => ? = 2 - 1
[1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,8,5,4,7,6,3,1] => [4,6,7,5,3,8,2,1] => ? => ? = 2 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [2,7,6,4,5,3,1,8] => ? => ? => ? = 1 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [2,8,6,4,5,3,7,1] => ? => ? => ? = 1 - 1
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,8,6,5,4,3,7,1] => ? => ? => ? = 1 - 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,8,7,6,5,4,3,1] => [5,6,4,7,3,8,2,1] => ? => ? = 1 - 1
[1,1,1,0,1,1,0,0,1,0,1,0,0,0,1,0]
=> [7,2,4,3,5,6,1,8] => ? => ? => ? = 2 - 1
[1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> [8,2,4,3,5,6,7,1] => ? => ? => ? = 2 - 1
[1,1,1,1,0,0,0,1,1,0,1,0,0,0,1,0]
=> [4,3,2,7,5,6,1,8] => ? => ? => ? = 2 - 1
[1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [4,3,2,8,7,6,5,1] => [3,2,6,7,5,8,4,1] => ? => ? = 2 - 1
[1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [5,3,2,4,8,6,7,1] => ? => ? => ? = 2 - 1
[1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [6,3,2,4,5,1,7,8] => ? => ? => ? = 1 - 1
[1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [7,3,2,4,5,6,1,8] => ? => ? => ? = 1 - 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,0,1,0]
=> [7,3,2,5,4,6,1,8] => [3,2,5,4,6,7,1,8] => ? => ? = 2 - 1
[1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
=> [8,3,2,5,4,6,7,1] => ? => ? => ? = 2 - 1
[1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> [8,3,2,6,5,4,7,1] => ? => ? => ? = 2 - 1
[1,1,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> [5,3,4,2,8,6,7,1] => [4,3,2,6,7,8,5,1] => ? => ? = 2 - 1
[1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> [5,3,4,2,8,7,6,1] => ? => ? => ? = 2 - 1
[1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [6,3,4,2,5,1,7,8] => ? => ? => ? = 1 - 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [7,3,4,2,5,6,1,8] => ? => ? => ? = 1 - 1
Description
The number of ascents of a binary word.
Matching statistic: St001712
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [[1]]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [[1],[2]]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [[1,2]]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [[1],[2],[3]]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[1,2],[3],[4]]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [[1,3],[2,4],[5]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [[1,3,5],[2],[4]]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,2,4],[3],[5]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,4,6,5,8,7,3] => ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,2,4,7,6,8,5,3] => ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,2,4,6,5,7,8,3] => ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,6,8,7,5,4,3] => ?
=> ? = 1 - 1
[1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,2,6,7,5,8,4,3] => ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,2,4,5,7,6,8,3] => ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,5,8,3] => ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,2,5,7,8,6,4,3] => ?
=> ? = 1 - 1
[1,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,3,7,6,5,8,4,2] => ?
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,6,5,8,7,4,2] => ?
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,3,7,6,8,5,4,2] => ?
=> ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,4,7,6,8,5,3,2] => ?
=> ? = 2 - 1
[1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,4,6,5,7,8,3,2] => ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,4,6,5,8,7,2] => ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,3,5,7,6,8,4,2] => ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,6,8,7,5,2] => ?
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,4,7,8,6,5,2] => ?
=> ? = 1 - 1
[1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,4,5,7,6,8,3,2] => ?
=> ? = 2 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,5,6,8,7,4,3,2] => ?
=> ? = 1 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,5,7,8,6,4,3,2] => ?
=> ? = 1 - 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,4,5,6,8,7,3,2] => ?
=> ? = 1 - 1
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,4,5,7,8,6,3,2] => ?
=> ? = 1 - 1
[1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,7,6,5,8,4,3,1] => ?
=> ? = 2 - 1
[1,1,1,0,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,6,5,8,7,4,3,1] => ?
=> ? = 2 - 1
[1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [2,6,5,7,8,4,3,1] => ?
=> ? = 2 - 1
[1,1,1,0,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> [3,6,5,8,7,4,2,1] => ?
=> ? = 2 - 1
[1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,6,5,7,8,4,2,1] => ?
=> ? = 2 - 1
[1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,7,6,8,5,4,1] => ?
=> ? = 2 - 1
[1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,6,5,7,8,4,1] => ?
=> ? = 2 - 1
[1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [2,4,7,6,8,5,3,1] => ?
=> ? = 2 - 1
[1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [2,5,7,6,8,4,3,1] => ?
=> ? = 2 - 1
[1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,5,8,7,6,4,1] => ?
=> ? = 1 - 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,6,8,7,5,4,1] => ?
=> ? = 1 - 1
[1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,7,8,6,5,4,1] => ?
=> ? = 1 - 1
[1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,3,6,7,5,8,4,1] => ?
=> ? = 2 - 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [2,4,5,8,7,6,3,1] => ?
=> ? = 1 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [2,4,5,7,6,8,3,1] => ?
=> ? = 2 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,5,6,8,7,4,3,1] => ?
=> ? = 1 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,5,7,8,6,4,3,1] => ?
=> ? = 1 - 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,4,6,5,8,7,2,1] => ?
=> ? = 2 - 1
[1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,4,7,6,8,5,2,1] => ?
=> ? = 2 - 1
[1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,6,7,5,8,4,2,1] => ?
=> ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,4,6,8,7,5,2,1] => ?
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,4,7,8,6,5,2,1] => ?
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,6,7,8,5,4,2,1] => ?
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,6,7,9,8,3,2,1] => [[1,2,3,4,5],[6],[7],[8],[9]]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [5,6,7,8,10,9,4,3,2,1] => [[1,2,3,4,5],[6],[7],[8],[9],[10]]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [5,6,7,9,8,4,3,2,1] => [[1,2,3,4],[5],[6],[7],[8],[9]]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,6,7,8,10,9,3,2,1] => [[1,2,3,4,5,6],[7],[8],[9],[10]]
=> ? = 1 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [3,5,6,7,9,8,4,2,1] => [[1,2,3,4,5],[6],[7],[8],[9]]
=> ? = 1 - 1
Description
The number of natural descents of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Matching statistic: St000257
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> ? = 3 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2,1]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,2,1]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,2,1]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,2,1]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,2,1]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> ? = 3 - 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,1,1]
=> ? = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ? = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,1,1]
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> ? = 3 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,1,1]
=> ? = 3 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [5,4,4,1,1,1]
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,1,1,1]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,4,1,1,1,1]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1,1,1]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [5,5,3,1,1,1]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1,1,1]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3,3,1,1,1]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,3,1,1,1]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1,1,1]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,4,3,1,1,1]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,3,1,1,1]
=> ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,1,1]
=> ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,1,1]
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,1,1]
=> ? = 4 - 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,1,1]
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,1,1]
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,1,1]
=> ? = 4 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,1,1]
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,1,1]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,1,1]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1,1]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,1,1]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1,1]
=> ? = 3 - 1
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St000201
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [.,.]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [[.,.],.]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [.,[.,.]]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 1
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[[.,.],[[[[[.,.],.],.],.],.]],.]
=> ? = 2
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [[.,.],[.,[[[[[.,.],.],.],.],.]]]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> ? = 1
[1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[[.,[.,.]],.],[[[[.,.],.],.],.]]
=> ? = 2
[1,1,0,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [[[.,[.,.]],[[[[.,.],.],.],.]],.]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> ? = 1
[1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
=> [[[.,[.,[.,.]]],.],[[[.,.],.],.]]
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> [[[.,[.,[.,.]]],[[[.,.],.],.]],.]
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],[[[[.,.],.],.],.]]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> ? = 1
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> [[[.,[.,[.,[.,.]]]],.],[[.,.],.]]
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
=> [[[.,[.,[.,[.,.]]]],[[.,.],.]],.]
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],[[[.,.],.],.]]
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [[.,[.,[.,[.,.]]]],[.,[[.,.],.]]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [.,[[[.,[.,[.,[.,[.,.]]]]],.],.]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [[[.,[[.,[.,[.,[.,.]]]],.]],.],.]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [[.,[[[.,[.,[.,[.,.]]]],.],.]],.]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> [.,[[.,[.,[.,[.,.]]]],[[.,.],.]]]
=> ? = 2
[1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.]
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [.,[.,[[[.,[.,[.,[.,.]]]],.],.]]]
=> ? = 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [[[[.,[[.,[.,[.,.]]],.]],.],.],.]
=> ? = 1
[1,1,0,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,0]
=> [[.,[[.,[.,[.,.]]],.]],[[.,.],.]]
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0,1,0]
=> [[.,[[[.,[.,[.,.]]],[.,.]],.]],.]
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> [.,[[[[.,[.,[.,.]]],[.,.]],.],.]]
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> [.,[[.,[.,[.,.]]],[[.,[.,.]],.]]]
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [[[.,[[[.,[.,[.,.]]],.],.]],.],.]
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> [[.,[[[[.,[.,[.,.]]],.],.],.]],.]
=> ? = 1
[1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,1,0,0]
=> [.,[[[.,[.,[.,.]]],[[.,.],.]],.]]
=> ? = 2
[1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> [.,[[.,[.,[.,.]]],[[[.,.],.],.]]]
=> ? = 2
[1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [[[.,[.,[[.,[.,[.,.]]],.]]],.],.]
=> ? = 1
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> ? = 2
[1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
=> ? = 1
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> [.,[[[.,[[.,[.,[.,.]]],.]],.],.]]
=> ? = 1
Description
The number of leaf nodes in a binary tree.
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2].
The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000568
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [.,.]
=> ? = 1
[1,0,1,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 1
[1,1,0,0]
=> [2,1] => [2,1] => [[.,.],.]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [[[.,.],.],.]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => [[.,.],[.,.]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [2,1,7,4,5,6,3,8] => ? => ?
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,1,4,8,6,7,5] => ? => ?
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [2,3,1,7,5,6,4,8] => [3,2,1,5,6,7,4,8] => ?
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,1,8,5,6,7,4] => [3,2,1,5,6,7,8,4] => [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> ? = 2
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,8,7,6,5,4] => [3,2,1,6,7,5,8,4] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,4,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,8,7,6] => [4,3,2,1,5,7,8,6] => [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,7,6,5,8] => [4,3,2,1,6,7,5,8] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,8,6,7,5] => [4,3,2,1,6,7,8,5] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,8,7,6,5] => [4,3,2,1,7,6,8,5] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,4,5,1,6,7,8] => [5,4,3,2,1,6,7,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6,8,7] => [5,4,3,2,1,6,8,7] => [[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6,8] => [5,4,3,2,1,7,6,8] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,4,5,1,7,8,6] => [5,4,3,2,1,8,7,6] => [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,5,1,8,7,6] => [5,4,3,2,1,7,8,6] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,6,1,7,8] => [6,5,4,3,2,1,7,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,5,6,1,8,7] => [6,5,4,3,2,1,8,7] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,1,8] => [7,6,5,4,3,2,1,8] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,8,7,1] => [7,8,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,4,5,7,6,1,8] => [6,7,5,4,3,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,4,5,7,6,8,1] => [6,8,7,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,5,8,6,7,1] => [6,7,8,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,5,8,7,6,1] => [7,6,8,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,4,6,5,1,7,8] => [5,6,4,3,2,1,7,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,4,6,5,8,7,1] => [5,7,8,6,4,3,2,1] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [2,3,4,7,5,6,1,8] => [5,6,7,4,3,2,1,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,4,8,5,6,7,1] => [5,6,7,8,4,3,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,4,8,5,7,6,1] => [5,7,6,8,4,3,2,1] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 2
[1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,4,7,6,5,1,8] => [6,5,7,4,3,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,4,8,6,5,7,1] => [6,5,7,8,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,4,8,6,7,5,1] => [7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,8,7,6,5,1] => [6,7,5,8,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,5,4,1,6,7,8] => [4,5,3,2,1,6,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[1,1,0,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,4,1,8,7,6] => [4,5,3,2,1,7,8,6] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [2,3,5,4,7,6,1,8] => ? => ?
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [2,3,5,4,8,6,7,1] => [4,6,7,8,5,3,2,1] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,3,5,4,8,7,6,1] => [4,7,6,8,5,3,2,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [2,3,6,4,5,1,7,8] => [4,5,6,3,2,1,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [2,3,7,4,5,6,1,8] => ? => ?
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,3,8,4,5,6,7,1] => [4,5,6,7,8,3,2,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [2,3,8,4,6,5,7,1] => ? => ?
=> ? = 2
[1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,8,4,7,6,5,1] => ? => ?
=> ? = 2
[1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,6,5,4,1,7,8] => [5,4,6,3,2,1,7,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,4,8,7,1] => ? => ?
=> ? = 2
[1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [2,3,7,5,4,6,1,8] => [5,4,6,7,3,2,1,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,8,5,4,6,7,1] => [5,4,6,7,8,3,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [2,3,8,5,4,7,6,1] => ? => ?
=> ? = 2
Description
The hook number of a binary tree.
A hook of a binary tree is a vertex together with is left- and its right-most branch. Then there is a unique decomposition of the tree into hooks and the hook number is the number of hooks in this decomposition.
The following 52 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000632The jump number of the poset. St001840The number of descents of a set partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000159The number of distinct parts of the integer partition. St000390The number of runs of ones in a binary word. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000353The number of inner valleys of a permutation. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000647The number of big descents of a permutation. St000053The number of valleys of the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000779The tier of a permutation. St000069The number of maximal elements of a poset. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000251The number of nonsingleton blocks of a set partition. St000360The number of occurrences of the pattern 32-1. St001728The number of invisible descents of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000659The number of rises of length at least 2 of a Dyck path. St000023The number of inner peaks of a permutation. St000523The number of 2-protected nodes of a rooted tree. St000646The number of big ascents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000021The number of descents of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000252The number of nodes of degree 3 of a binary tree. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001487The number of inner corners of a skew partition. St001896The number of right descents of a signed permutations. St000805The number of peaks of the associated bargraph.
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