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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: 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: 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
St000527: 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 width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
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: 96% ●values known / values provided: 96%●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: 96% ●values known / values provided: 96%●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
[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [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
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [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
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [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
[1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [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
[1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [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
[1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [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
[1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [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
[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [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
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,4,9,8,7,6,5,2,1] => ? => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,3,4,9,10,8,7,6,5] => ? => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,3,4,5,8,9,10,7,6] => ? => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,5,6,9,10,8,7] => ? => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,4,5,9,10,8,7,6] => ? => ? = 1 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,2,3,9,10,8,7,6,5,4] => ? => ? = 1 - 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [4,5,9,8,7,6,3,2,1] => ? => ? = 1 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0]
=> [5,6,9,8,7,4,3,2,1] => ? => ? = 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: 92% ●values known / values provided: 92%●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: 92% ●values known / values provided: 92%●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
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [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
[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [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
[1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [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
[1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [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
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [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
[1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [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
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [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
[1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [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
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,3,3,3,3,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,4,4,4,4,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,5,5,5,4,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,6,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,3,3,3,3,3,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,3,3,3,3,2,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [4,4,4,4,4,4,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,4,4,4,3,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [5,5,5,5,5,4,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,5,5,4,4,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [6,6,6,6,5,4,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,6,5,5,4,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [7,7,7,6,5,4,3,2,1]
=> ?
=> ? = 1 - 1
[1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,4,3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,7,6,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
Description
The number of factors DDU in a Dyck path.
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: 91% ●values known / values provided: 91%●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: 91% ●values known / values provided: 91%●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
[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [2,3,7,4,5,6,1,8] => ? => ? => ? = 1 - 1
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,8,7,5,6,4,1] => ? => ? => ? = 1 - 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
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,3,6,1,8] => ? => ? => ? = 1 - 1
[1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [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
[1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [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
[1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [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
[1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [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
[1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [7,3,4,5,2,6,1,8] => ? => ? => ? = 1 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [8,3,4,6,5,2,7,1] => ? => ? => ? = 1 - 1
[1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [8,3,4,7,5,6,2,1] => ? => ? => ? = 1 - 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [6,3,5,4,2,1,7,8] => ? => ? => ? = 1 - 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]
=> [7,3,5,4,2,6,1,8] => [4,5,3,2,6,7,1,8] => ? => ? = 1 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [7,3,6,4,5,2,1,8] => [4,5,6,3,2,7,1,8] => ? => ? = 1 - 1
[1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> [8,3,7,5,4,6,2,1] => [5,4,6,7,3,2,8,1] => ? => ? = 1 - 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [8,3,7,6,5,4,2,1] => [5,6,4,7,3,2,8,1] => ? => ? = 1 - 1
[1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,5,6,1,8] => ? => ? => ? = 1 - 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [7,5,4,3,2,6,1,8] => [4,3,5,2,6,7,1,8] => ? => ? = 1 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,7,4,6,5,3,2,1] => [5,6,4,3,7,2,8,1] => ? => ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [8,6,5,4,3,2,7,1] => [4,5,3,6,2,7,8,1] => ? => ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,5,6,4,3,2,1,10] => [6,5,4,7,3,8,2,9,1,10] => ? => ? = 1 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [8,7,4,6,5,3,2,1,9] => [5,6,4,3,7,2,8,1,9] => ? => ? = 1 - 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [8,7,4,5,6,3,2,1,9] => ? => ? => ? = 1 - 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> [8,7,3,4,5,6,2,1,9] => ? => ? => ? = 1 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1,8,9] => ? => ? => ? = 1 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0,1,0]
=> [8,6,3,4,5,2,7,1,9] => ? => ? => ? = 1 - 1
[1,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,9,8,6,7,5,4,1] => ? => ? => ? = 1 - 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [2,9,4,5,6,7,3,8,1] => ? => ? => ? = 1 - 1
[1,1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [2,9,8,4,5,6,7,3,1] => ? => ? => ? = 1 - 1
[1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,9,8,5,6,7,4,3,1] => ? => ? => ? = 1 - 1
[1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,9,8,7,5,6,4,3,1] => [5,6,7,4,8,3,9,2,1] => ? => ? = 1 - 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,9,8,7,6,5,4,3,1] => [6,5,7,4,8,3,9,2,1] => ? => ? = 1 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> [9,3,4,5,6,2,7,8,1] => ? => ? => ? = 1 - 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [9,3,4,5,6,7,2,8,1] => ? => ? => ? = 1 - 1
[1,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [9,3,4,5,8,6,7,2,1] => ? => ? => ? = 1 - 1
[1,1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [9,3,4,8,7,6,5,2,1] => ? => ? => ? = 1 - 1
[1,1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [9,3,8,5,6,7,4,2,1] => ? => ? => ? = 1 - 1
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [9,3,8,7,6,5,4,2,1] => [6,5,7,4,8,3,2,9,1] => ? => ? = 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: 85% ●values known / values provided: 85%●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: 85% ●values known / values provided: 85%●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
[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [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
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [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
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [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
[1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [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
[1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [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
[1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [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
[1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [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
[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [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
[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
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [6,7,9,8,5,4,3,2,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 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
[1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
=> [5,7,9,8,6,4,3,2,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 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]
=> [1,2,3,4,5,6,7,8,9] => [[1,2,3,4,5,6,7,8,9]]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 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]
=> [1,2,3,4,5,8,9,7,6] => [[1,2,3,4,5,6,7],[8],[9]]
=> ? = 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]
=> [1,2,3,4,5,7,8,9,6] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 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
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,3,4,6,8,9,7,5] => [[1,2,3,4,5,6,7],[8],[9]]
=> ? = 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]
=> [1,2,3,4,6,7,8,9,5] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 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
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 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
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 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
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,2] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 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]
=> [1,7,8,9,6,5,4,3,2] => [[1,2,3,4],[5],[6],[7],[8],[9]]
=> ? = 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]
=> [1,4,5,6,7,8,9,3,2] => [[1,2,3,4,5,6,7],[8],[9]]
=> ? = 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]
=> [1,6,7,8,9,5,4,3,2] => [[1,2,3,4,5],[6],[7],[8],[9]]
=> ? = 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]
=> [1,5,6,7,8,9,4,3,2] => [[1,2,3,4,5,6],[7],[8],[9]]
=> ? = 1 - 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,9,7,6,5,4,3,2,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 1 - 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,8,9,7,6,5,4,3,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 1 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,8,9,7,6,1] => [[1,2,3,4,5,6],[7],[8],[9]]
=> ? = 1 - 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,5,6,8,9,7,1] => [[1,2,3,4,5,6,7],[8],[9]]
=> ? = 1 - 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 1 - 1
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,4,5,7,8,9,6,1] => [[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: 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
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [.,.]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[.,.]],.]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],.]],.]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[.,[.,.]]],.]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 1
[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
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ? = 1
[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
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[.,[.,[[[.,[[.,.],.]],.],.]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> ? = 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [.,[.,[[.,[[[[.,.],.],.],.]],.]]]
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [.,[.,[[[.,[[[.,.],.],.]],.],.]]]
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [.,[.,[[[[.,[[.,.],.]],.],.],.]]]
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> ? = 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]
=> [.,[.,[[.,[.,[[[.,.],.],.]]],.]]]
=> ? = 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]
=> [.,[.,[[.,[[.,[[.,.],.]],.]],.]]]
=> ? = 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
[1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [.,[.,[[.,[.,[.,[[.,.],.]]]],.]]]
=> ? = 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]
=> [.,[.,[[.,[.,[[.,[.,.]],.]]],.]]]
=> ? = 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> ? = 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]
=> [.,[.,[[.,[[.,[.,[.,.]]],.]],.]]]
=> ? = 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]
=> [.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
=> ? = 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
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [.,[.,[[[[.,[.,[.,.]]],.],.],.]]]
=> ? = 1
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [.,[.,[[[.,[.,[.,[.,.]]]],.],.]]]
=> ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> ? = 1
[1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [.,[[[.,[[[[.,.],.],.],.]],.],.]]
=> ? = 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [.,[[.,[.,[[[[.,.],.],.],.]]],.]]
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [.,[[.,[[[.,[[.,.],.]],.],.]],.]]
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[[.,[[[[.,[.,.]],.],.],.]],.]]
=> ? = 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]
=> [.,[[.,[.,[.,[[[.,.],.],.]]]],.]]
=> ? = 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
[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
[1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> ? = 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]
=> [.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> ? = 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> ? = 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]
=> [.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> ? = 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [.,[[.,[[.,[.,[[.,.],.]]],.]],.]]
=> ? = 1
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [.,[[.,[[[.,[.,[.,.]]],.],.]],.]]
=> ? = 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]
=> [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> ? = 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> ? = 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [.,[[[.,[[.,[[.,.],.]],.]],.],.]]
=> ? = 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)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [.,.]
=> ? = 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [.,[.,.]]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [[.,.],.]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [[.,[.,.]],.]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [[[.,.],.],.]
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> 1
[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]
=> [3,4,5,6,7,8,2,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[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]
=> [4,5,6,7,8,3,2,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[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]
=> [5,6,7,8,4,3,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[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]
=> [6,7,8,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 1
[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]
=> [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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,6,8,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 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]
=> [6,7,5,8,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[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]
=> [6,5,7,8,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 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]
=> [7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 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]
=> [5,6,7,4,8,3,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 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]
=> [5,6,4,7,8,3,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 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]
=> [5,4,6,7,8,3,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 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]
=> [6,7,5,4,8,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 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]
=> [6,5,7,4,8,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 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]
=> [7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 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]
=> [6,5,4,7,8,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 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]
=> [4,5,6,7,3,8,2,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[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]
=> [4,5,6,3,7,8,2,1] => ?
=> ? = 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]
=> [4,5,3,6,7,8,2,1] => ?
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [4,3,5,6,7,8,2,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 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]
=> [5,6,7,4,3,8,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 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]
=> [5,6,4,7,3,8,2,1] => ?
=> ? = 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]
=> [5,4,6,7,3,8,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,3,8,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 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]
=> [6,5,7,4,3,8,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,8,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 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]
=> [6,5,4,7,3,8,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 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]
=> [5,6,4,3,7,8,2,1] => ?
=> ? = 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]
=> [5,4,6,3,7,8,2,1] => ?
=> ? = 1
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [5,4,3,6,7,8,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,5,4,3,7,8,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,7,2,8,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,3,2,8,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,3,7,2,8,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,7,2,8,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [4,3,5,6,7,2,8,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 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]
=> [5,6,7,4,3,2,8,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 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]
=> [5,6,4,7,3,2,8,1] => ?
=> ? = 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]
=> [5,4,6,7,3,2,8,1] => ?
=> ? = 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,3,2,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 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]
=> [6,5,7,4,3,2,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 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]
=> [6,5,4,7,3,2,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,2,8,1] => ?
=> ? = 1
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.
Matching statistic: St001037
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
St001037: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 1 - 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
The following 52 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000632The jump number of the poset. St001840The number of descents of a set partition. St000257The number of distinct parts of a partition that occur at least twice. 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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