searching the database
Your data matches 46 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000071
Mp00043: Integer partitions —to Dyck path⟶ 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]
=> [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> 1
[2]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[[[[[.,.],.],.],.],.],[.,.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[[[[.,[.,.]],.],.],.],[.,.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[[.,[[.,.],.]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[[[[.,.],[.,.]],.],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[[[[[.,[.,.]],.],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[[[[.,.],[.,.]],.],.],[.,.]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> 3
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[[[.,[[.,.],.]],.],.],[.,.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[.,[[[.,.],.],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
Description
The number of maximal chains in a poset.
Matching statistic: St000068
Mp00043: Integer partitions —to Dyck path⟶ 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: 99% ●values known / values provided: 99%●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: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> 1
[2]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[[[[[.,.],.],.],.],.],[.,.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[[[[.,[.,.]],.],.],.],[.,.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[[.,[[.,.],.]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[[[[.,.],[.,.]],.],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[[[[[.,[.,.]],.],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[[[[.,.],[.,.]],.],.],[.,.]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> 3
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[[[.,[[.,.],.]],.],.],[.,.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[.,[[[.,.],.],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
=> ([(0,8),(1,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,7)],9)
=> ? = 2
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [.,[[[[[[[[.,.],[.,.]],.],.],.],.],.],.]]
=> ([(0,9),(1,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6),(9,8)],10)
=> ? = 2
[4,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [.,[[[[[[.,.],[.,[.,.]]],.],.],.],.]]
=> ([(0,8),(1,3),(3,8),(4,6),(5,4),(6,2),(7,5),(8,7)],9)
=> ? = 2
[7,5,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]]
=> ([(0,8),(1,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,7)],9)
=> ? = 2
[7,6,4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]]
=> ([(0,8),(1,3),(3,8),(4,6),(5,4),(6,2),(7,5),(8,7)],9)
=> ? = 2
[7,7,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]]
=> ([(0,8),(1,3),(3,8),(4,6),(5,4),(6,2),(7,5),(8,7)],9)
=> ? = 2
Description
The number of minimal elements in a poset.
Matching statistic: St000291
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●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: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => 0 => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1110 => 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 1 = 2 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0111 => 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 11110 => 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 0110 => 1 = 2 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0011 => 0 = 1 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 01111 => 0 = 1 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => 111110 => 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => 01110 => 1 = 2 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 1010 => 2 = 3 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 0110 => 1 = 2 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1101 => 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0101 => 1 = 2 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1011 => 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0011 => 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 00111 => 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => 011111 => 0 = 1 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => 011110 => 1 = 2 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => 10110 => 2 = 3 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => 01110 => 1 = 2 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1100 => 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 1 = 2 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 1 = 2 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => 01011 => 1 = 2 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 0 = 1 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => 00111 => 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => 001111 => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => 0111111 => 0 = 1 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => 101110 => 2 = 3 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => 011110 => 1 = 2 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => 11010 => 2 = 3 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => 00110 => 1 = 2 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => 01110 => 1 = 2 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => 11101 => 1 = 2 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0100 => 1 = 2 - 1
[3,2,2,1,1,1,1]
=> [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
[3,3,2,1,1,1,1]
=> [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
[4,2,2,2,1,1,1]
=> [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
[2,2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,6,10,9,8,7,5,4,3,2] => ? => ? = 1 - 1
[4,2,2,2,2,1,1]
=> [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
[5,2,2,2,2,1,1]
=> [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
[3,3,2,2,2,2,1]
=> [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
[5,4,2,2,1,1,1]
=> [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
[5,3,3,2,1,1,1]
=> [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
[4,4,3,2,1,1,1]
=> [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
[4,4,2,2,2,1,1]
=> [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
[5,3,2,2,2,2,1]
=> [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
[4,3,3,3,2,1,1]
=> [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
[4,3,3,2,2,2,1]
=> [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
[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]
=> [1,2,5,6,7,8,9,4,3] => ? => ? = 1 - 1
Description
The number of descents of a binary word.
Matching statistic: St000386
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
St000386: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0 = 1 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[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
[3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[4,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[2,2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[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
[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
[4,3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 1 - 1
[2,2,2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[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
[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
[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
[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
[7,6,5,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[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
[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
[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
[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
[7,6,4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[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
[8,7,6,5,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[8,7,6,5,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[8,7,6,5,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[8,7,6,5,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[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
[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
[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
[6,5,4,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 - 1
[7,6,5,4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1 - 1
[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
[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
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000527
Mp00043: Integer partitions —to Dyck path⟶ 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: 93% ●values known / values provided: 93%●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: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> 1
[2]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[[[[[.,.],.],.],.],.],[.,.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[[[[.,[.,.]],.],.],.],[.,.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[[.,[[.,.],.]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[[[[.,.],[.,.]],.],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[[[[[.,[.,.]],.],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[[[[.,.],[.,.]],.],.],[.,.]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> 3
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[[[.,[[.,.],.]],.],.],[.,.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[.,[[[.,.],.],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,1,1,1,1,1,1]
=> [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
[4,1,1,1,1,1,1]
=> [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
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
=> ([(0,8),(1,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,7)],9)
=> ? = 2
[4,2,1,1,1,1,1]
=> [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
[3,3,1,1,1,1,1]
=> [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
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [.,[[[[[[[[.,.],[.,.]],.],.],.],.],.],.]]
=> ([(0,9),(1,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6),(9,8)],10)
=> ? = 2
[4,3,1,1,1,1,1]
=> [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
[4,2,2,1,1,1,1]
=> [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
[4,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [.,[[[[[[.,.],[.,[.,.]]],.],.],.],.]]
=> ([(0,8),(1,3),(3,8),(4,6),(5,4),(6,2),(7,5),(8,7)],9)
=> ? = 2
[4,2,2,2,1,1,1]
=> [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
[5,2,2,2,1,1,1]
=> [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
[4,2,2,2,2,1,1]
=> [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
[5,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [.,[[[.,[[.,[.,.]],[.,.]]],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[5,2,2,2,2,1,1]
=> [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
[4,4,2,2,1,1,1]
=> [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
[4,2,2,2,2,2,1]
=> [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
[5,4,2,2,1,1,1]
=> [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
[5,3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [.,[[[.,[.,[[.,.],[.,.]]]],.],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[5,3,2,2,2,1,1]
=> [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
[5,2,2,2,2,2,1]
=> [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
[4,4,2,2,2,1,1]
=> [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
[5,4,2,2,2,1,1]
=> [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
[5,3,3,2,2,1,1]
=> [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
[5,3,2,2,2,2,1]
=> [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
[4,4,2,2,2,2,1]
=> [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
[7,5,5,4,3,2,1]
=> [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
[7,6,4,4,3,2,1]
=> [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
[6,6,4,4,3,2,1]
=> [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
[7,5,4,4,3,2,1]
=> [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
[7,4,4,4,3,2,1]
=> [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
[6,4,4,4,3,2,1]
=> [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
[6,5,3,3,3,2,1]
=> [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
[6,4,3,3,3,2,1]
=> [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
[6,3,3,3,3,2,1]
=> [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
[7,5,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]]
=> ([(0,8),(1,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,7)],9)
=> ? = 2
[7,6,4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]]
=> ([(0,8),(1,3),(3,8),(4,6),(5,4),(6,2),(7,5),(8,7)],9)
=> ? = 2
[7,7,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]]
=> ([(0,8),(1,3),(3,8),(4,6),(5,4),(6,2),(7,5),(8,7)],9)
=> ? = 2
Description
The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St001712
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 89% ●values known / values provided: 89%●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: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [[1,2]]
=> 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [[1,6],[2],[3],[4],[5]]
=> 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [[1,2,5],[3],[4]]
=> 1 = 2 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 0 = 1 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [[1,7],[2],[3],[4],[5],[6]]
=> 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => [[1,2,6],[3],[4],[5]]
=> 1 = 2 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [[1,3,5],[2],[4]]
=> 2 = 3 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[1,2,5],[3],[4]]
=> 1 = 2 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,2,4],[3],[5]]
=> 1 = 2 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => [[1,2,3],[4],[5],[6]]
=> 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7]]
=> 0 = 1 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => [[1,2,7],[3],[4],[5],[6]]
=> 1 = 2 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => [[1,3,6],[2],[4],[5]]
=> 2 = 3 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => [[1,2,6],[3],[4],[5]]
=> 1 = 2 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 1 = 2 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [[1,2,4],[3],[5],[6]]
=> 1 = 2 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => [[1,2,3],[4],[5],[6]]
=> 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [[1,2,3],[4],[5],[6],[7]]
=> 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8]]
=> 0 = 1 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => [[1,3,7],[2],[4],[5],[6]]
=> 2 = 3 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => [[1,2,7],[3],[4],[5],[6]]
=> 1 = 2 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => [[1,4,6],[2],[3],[5]]
=> 2 = 3 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => [[1,2,3,6],[4],[5]]
=> 1 = 2 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => [[1,2,6],[3],[4],[5]]
=> 1 = 2 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [[1,5],[2,6],[3],[4]]
=> 1 = 2 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 1 - 1
[2,1,1,1,1,1,1,1]
=> [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
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 1 - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [[1,2,4],[3],[5],[6],[7],[8],[9]]
=> ? = 2 - 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,8,6,5,4,3,2] => [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 1 - 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,8,7,6,5,4,3,2] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 1 - 1
[3,2,2,1,1,1,1]
=> [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
[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]
=> [1,7,8,9,6,5,4,3,2] => [[1,2,3,4],[5],[6],[7],[8],[9]]
=> ? = 1 - 1
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9,8,10,7,6,5,4,3,2] => [[1,2,4],[3],[5],[6],[7],[8],[9],[10]]
=> ? = 2 - 1
[3,3,2,1,1,1,1]
=> [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
[3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,10,7,6,5,4,3,2] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]]
=> ? = 1 - 1
[4,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,7,6,8,9,5,4,3,2] => [[1,2,4,5],[3],[6],[7],[8],[9]]
=> ? = 2 - 1
[4,2,2,2,1,1,1]
=> [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
[2,2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,6,10,9,8,7,5,4,3,2] => ?
=> ? = 1 - 1
[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]
=> [1,6,7,8,9,5,4,3,2] => [[1,2,3,4,5],[6],[7],[8],[9]]
=> ? = 1 - 1
[4,2,2,2,2,1,1]
=> [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
[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]
=> [1,3,9,8,7,6,5,4,2] => [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 1 - 1
[5,2,2,2,2,1,1]
=> [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
[4,3,3,2,1,1,1]
=> [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
[4,3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,7,8,9,10,6,5,4,3,2] => [[1,2,3,4,5],[6],[7],[8],[9],[10]]
=> ? = 1 - 1
[3,3,2,2,2,2,1]
=> [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
[2,2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,9,8,7,6,5,4,3] => [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 1 - 1
[5,4,2,2,1,1,1]
=> [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
[5,3,3,2,1,1,1]
=> [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
[4,4,3,2,1,1,1]
=> [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
[4,4,2,2,2,1,1]
=> [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
[2,2,2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,3,10,9,8,7,6,5,4,2] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 1 - 1
[5,3,3,2,2,1,1]
=> [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
[5,3,2,2,2,2,1]
=> [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
[4,3,3,3,2,1,1]
=> [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
[4,3,3,2,2,2,1]
=> [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
[2,2,2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,10,9,8,7,6,5,4,3] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 1 - 1
[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]
=> [1,2,3,4,5,6,7,8,9] => [[1,2,3,4,5,6,7,8,9]]
=> ? = 1 - 1
[7,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 1 - 1
[7,6,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 1 - 1
[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]
=> [1,2,3,4,5,7,8,9,6] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 1 - 1
[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]
=> [1,2,3,4,6,7,8,9,5] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 1 - 1
[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]
=> [1,2,3,5,6,7,8,9,4] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 1 - 1
[7,6,5,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,9,3] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 1 - 1
[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]
=> [1,3,4,5,6,7,8,9,2] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 1 - 1
[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]
=> [1,2,3,4,5,6,7,8,9,10] => [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 1 - 1
[8,8,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => [[1,2,3,4,5,6,7,8,9],[10]]
=> ? = 1 - 1
[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]
=> [1,2,3,4,5,6,9,8,7] => [[1,2,3,4,5,6,7],[8],[9]]
=> ? = 1 - 1
[8,7,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => [[1,2,3,4,5,6,7,8,9],[10]]
=> ? = 1 - 1
[7,5,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,8,7,9,6] => [[1,2,3,4,5,6,8],[7],[9]]
=> ? = 2 - 1
[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]
=> [1,2,3,4,5,6,8,9,10,7] => [[1,2,3,4,5,6,7,8,9],[10]]
=> ? = 1 - 1
[7,6,4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,4,7,6,8,9,5] => [[1,2,3,4,5,7,8],[6],[9]]
=> ? = 2 - 1
[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]
=> [1,2,3,4,5,7,8,9,10,6] => [[1,2,3,4,5,6,7,8,9],[10]]
=> ? = 1 - 1
[8,7,6,5,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,9,10,5] => [[1,2,3,4,5,6,7,8,9],[10]]
=> ? = 1 - 1
[8,7,6,5,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,10,4] => [[1,2,3,4,5,6,7,8,9],[10]]
=> ? = 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 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [[.,.],.]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,[[.,.],.]]],.]
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[.,.],[.,[[.,[.,.]],.]]]
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[.,[.,[[.,.],[.,.]]]],.]
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> 3
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[.,.],[.,[[[.,.],.],.]]]
=> 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[.,.],[[.,[.,[.,.]]],.]]
=> 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,.],[[.,.],.]]]]],.]
=> ? = 2
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,[.,.]],[.,.]]]]],.]
=> ? = 2
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,[[.,.],.]],.]]]],.]
=> ? = 1
[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]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]],.]
=> ? = 1
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]],.]
=> ? = 2
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.]
=> ? = 1
[4,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[[[.,.],.],[.,.]]]]],.]
=> ? = 2
[4,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [[.,[.,[.,[[[.,.],[.,.]],.]]]],.]
=> ? = 2
[3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [[.,[.,[.,[[[.,[.,.]],.],.]]]],.]
=> ? = 1
[3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [[.,[.,[[.,[.,[[.,.],.]]],.]]],.]
=> ? = 1
[3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]],.]
=> ? = 1
[2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> ? = 1
[4,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [[.,[.,[.,[[[[.,.],.],.],.]]]],.]
=> ? = 1
[4,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]],.]
=> ? = 2
[4,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]
=> [[.,[.,[[.,[[.,.],[.,.]]],.]]],.]
=> ? = 2
[3,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [[.,[.,[[.,[[.,[.,.]],.]],.]]],.]
=> ? = 1
[3,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [[.,[[.,[.,[.,[[.,.],.]]]],.]],.]
=> ? = 1
[2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> ? = 1
[2,2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]],.]
=> ? = 1
[5,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> [[.,[.,[[[.,.],[.,[.,.]]],.]]],.]
=> ? = 2
[4,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [[.,[.,[[.,[[[.,.],.],.]],.]]],.]
=> ? = 1
[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]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[[[[.,.],.],.],.]]]]],.]
=> ? = 1
[4,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
=> [[.,[[.,[.,[[.,.],[.,.]]]],.]],.]
=> ? = 2
[3,3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [[.,[.,[[[.,[.,[.,.]]],.],.]]],.]
=> ? = 1
[3,3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> [[.,[[.,[.,[[.,[.,.]],.]]],.]],.]
=> ? = 1
[3,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[.,[[.,.],.]]]]],.],.]
=> ? = 1
[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]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]],.]
=> ? = 1
[5,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
=> [[.,[.,[[[.,.],[[.,.],.]],.]]],.]
=> ? = 2
[5,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
=> [[.,[[.,[[.,.],[.,[.,.]]]],.]],.]
=> ? = 2
[4,4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0,1,0]
=> [[.,[.,[[[.,[.,.]],[.,.]],.]]],.]
=> ? = 2
[4,3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [[.,[.,[[[.,[[.,.],.]],.],.]]],.]
=> ? = 1
[4,3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [[.,[[.,[.,[[[.,.],.],.]]],.]],.]
=> ? = 1
[4,3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]],.]
=> ? = 1
[4,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[[.,.],[.,.]]]]],.],.]
=> ? = 2
[3,3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
=> ? = 1
[3,3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[[.,[.,.]],.]]]],.],.]
=> ? = 1
[2,2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]
=> ? = 1
[5,4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0,1,0]
=> [[.,[.,[[[[.,.],.],[.,.]],.]]],.]
=> ? = 2
[5,3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0,1,0]
=> [[.,[.,[[[[.,.],[.,.]],.],.]]],.]
=> ? = 2
[5,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[[.,[.,[[.,.],[.,[.,.]]]]],.],.]
=> ? = 2
[4,4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [[.,[.,[[[[.,[.,.]],.],.],.]]],.]
=> ? = 1
[4,3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[[[.,.],.],.]]]],.],.]
=> ? = 1
[3,3,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [[.,[[[.,[.,[.,[.,.]]]],.],.]],.]
=> ? = 1
[3,3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [[[.,[.,[[.,[.,[.,.]]],.]]],.],.]
=> ? = 1
[2,2,2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,1,0]
=> [[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]],.]
=> ? = 1
[5,4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [[.,[.,[[[[[.,.],.],.],.],.]]],.]
=> ? = 1
[5,4,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> [[.,[[.,[[[.,.],.],[.,.]]],.]],.]
=> ? = 2
[5,3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> [[.,[[.,[[[.,.],[.,.]],.]],.]],.]
=> ? = 2
[5,3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0,1,0]
=> [[[.,[.,[[.,.],[[.,.],.]]]],.],.]
=> ? = 2
[4,4,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [[.,[[.,[[[.,[.,.]],.],.]],.]],.]
=> ? = 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: St001037
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2 = 3 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [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]
=> 0 = 1 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> 2 = 3 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 2 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 1 - 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 1 - 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 1 - 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1 - 1
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 1 - 1
[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]
=> [1,1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 - 1
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[4,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 2 - 1
[3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1 - 1
[3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1 - 1
[3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 - 1
[2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[4,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[4,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[4,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 2 - 1
[3,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[3,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 1 - 1
[2,2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[5,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 2 - 1
[4,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 1 - 1
[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]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[4,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[3,3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1 - 1
[3,3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 1 - 1
[3,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 1 - 1
[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]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[5,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[5,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 - 1
[4,4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 2 - 1
[4,3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 1 - 1
[4,3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 1 - 1
[4,3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[4,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[3,3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 1 - 1
[3,3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 1 - 1
[2,2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[5,4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[5,3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 2 - 1
[5,3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[4,4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[4,4,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[4,3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 1 - 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000568
Mp00043: Integer partitions —to Dyck path⟶ 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: 78% ●values known / values provided: 78%●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: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [[.,.],.]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [[.,[.,.]],.]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [[[.,.],.],.]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[.,.],[.,[.,[.,[.,.]]]]]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [[[.,.],[.,[.,[.,.]]]],.]
=> 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [[[.,.],.],[.,[.,[.,.]]]]
=> 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [[[.,[.,.]],[.,[.,.]]],.]
=> 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [[[.,.],[.,[.,[.,.]]]],.]
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [[[.,.],[.,.]],[.,[.,.]]]
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [[[.,.],.],[.,[.,[.,.]]]]
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> 3
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [[[.,[.,[.,.]]],[.,.]],.]
=> 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [[[[.,.],.],[.,[.,.]]],.]
=> 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [[[.,.],[.,[.,[.,.]]]],.]
=> 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,.]]
=> 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> 2
[2,1,1,1,1,1,1]
=> [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
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,8,1] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> ? = 2
[2,2,1,1,1,1,1]
=> [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
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [5,2,3,4,6,7,8,1] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? = 2
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,5,6,7,8,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,1] => [[[.,.],[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 2
[2,2,2,1,1,1,1]
=> [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
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,1] => [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [[[.,.],.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,6,7,8,1] => [[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> ? = 2
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [4,5,2,3,6,7,8,1] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> ? = 2
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,3,5,2,6,7,8,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[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]
=> [4,3,2,5,6,7,8,9,1] => [[[[.,.],.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,10,1] => [[[.,.],[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 2
[2,2,2,2,1,1,1]
=> [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
[4,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,2,3,6,7,8,1] => [[[[.,.],[.,.]],.],[.,[.,[.,.]]]]
=> ? = 2
[4,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [5,3,4,2,6,7,8,1] => [[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> ? = 2
[3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,3,2,6,7,8,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [4,3,5,6,2,7,8,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [4,3,2,5,6,7,8,9,10,1] => [[[[.,.],.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1
[2,2,2,2,2,1,1]
=> [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
[4,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,4,3,2,6,7,8,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[4,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [5,4,2,3,6,7,8,9,1] => [[[[.,.],[.,.]],.],[.,[.,[.,[.,.]]]]]
=> ? = 2
[4,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,2,7,8,1] => ?
=> ? = 2
[3,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,2,7,8,1] => ?
=> ? = 1
[3,2,2,2,2,1,1]
=> [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
[2,2,2,2,2,2,1]
=> [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
[2,2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,9,10,1] => [[[.,.],.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[5,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [6,3,4,5,2,7,8,1] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> ? = 2
[4,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [5,4,3,6,2,7,8,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[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]
=> [5,4,3,2,6,7,8,9,1] => [[[[[.,.],.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[4,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,7,2,8,1] => ?
=> ? = 2
[3,3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,3,2,7,8,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[3,3,2,2,2,1,1]
=> [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
[3,2,2,2,2,2,1]
=> [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
[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]
=> [3,4,5,6,7,8,2,9,1] => [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1
[5,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [6,4,3,5,2,7,8,1] => ?
=> ? = 2
[5,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [6,3,4,5,7,2,8,1] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> ? = 2
[4,4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [5,6,3,4,2,7,8,1] => ?
=> ? = 2
[4,3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [5,4,6,3,2,7,8,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[4,3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [5,4,3,6,7,2,8,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[4,3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [5,4,3,2,6,7,8,9,10,1] => [[[[[.,.],.],.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[4,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,7,8,2,1] => [[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> ? = 2
[3,3,3,2,2,1,1]
=> [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
[3,3,2,2,2,2,1]
=> [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
[2,2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,6,7,8,9,2,1] => [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1
[5,4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [6,5,3,4,2,7,8,1] => [[[[[.,.],.],[.,.]],.],[.,[.,.]]]
=> ? = 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.
Matching statistic: St000196
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [.,[.,.]]
=> 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[.,[.,.]],.],[.,.]]
=> 1 = 2 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> 0 = 1 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[[[[[.,.],.],.],.],.],[.,.]]
=> 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[[[.,[.,.]],.],.],[.,.]]
=> 1 = 2 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2 = 3 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> 1 = 2 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 1 = 2 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[[[[.,[.,.]],.],.],.]]
=> 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> 0 = 1 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[[[[.,[.,.]],.],.],.],[.,.]]
=> 1 = 2 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[[[.,.],[.,.]],.],[.,.]]
=> 2 = 3 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[[.,[[.,.],.]],.],[.,.]]
=> 1 = 2 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 1 = 2 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> 1 = 2 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[[[[.,.],[.,.]],.],.]]
=> 1 = 2 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 0 = 1 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[[[.,[[.,.],.]],.],.]]
=> 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[[[[[.,[.,.]],.],.],.],.]]
=> 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 1 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[[[[.,.],[.,.]],.],.],[.,.]]
=> 2 = 3 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[[[.,[[.,.],.]],.],.],[.,.]]
=> 1 = 2 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> 2 = 3 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> 1 = 2 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[.,[[[.,.],.],.]],[.,.]]
=> 1 = 2 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> 1 = 2 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 1 = 2 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 2 = 3 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> ? = 1 - 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> ? = 2 - 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> ? = 1 - 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> ? = 1 - 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> ? = 2 - 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> ? = 1 - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
=> ? = 2 - 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> ? = 1 - 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> ? = 1 - 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> ? = 1 - 1
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [.,[[[[[.,[.,.]],[.,.]],.],.],.]]
=> ? = 2 - 1
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],[[.,.],.]],.],.],.]]
=> ? = 2 - 1
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [.,[[[[.,[[.,[.,.]],.]],.],.],.]]
=> ? = 1 - 1
[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
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [.,[[[[[[[[.,.],[.,.]],.],.],.],.],.],.]]
=> ? = 2 - 1
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [.,[[[.,[[[[.,.],.],.],.]],.],.]]
=> ? = 1 - 1
[4,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [.,[[[[[.,.],[.,[.,.]]],.],.],.]]
=> ? = 2 - 1
[4,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [.,[[[[.,[[.,.],[.,.]]],.],.],.]]
=> ? = 2 - 1
[3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[[[[.,[.,[[.,.],.]]],.],.],.]]
=> ? = 1 - 1
[3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [.,[[[.,[[[.,[.,.]],.],.]],.],.]]
=> ? = 1 - 1
[3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [.,[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.]]
=> ? = 1 - 1
[2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> ? = 1 - 1
[4,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> ? = 1 - 1
[4,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [.,[[[[[[.,.],[.,[.,.]]],.],.],.],.]]
=> ? = 2 - 1
[4,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [.,[[[.,[[[.,.],[.,.]],.]],.],.]]
=> ? = 2 - 1
[3,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [.,[[[.,[[.,[[.,.],.]],.]],.],.]]
=> ? = 1 - 1
[3,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[[.,[[[[.,[.,.]],.],.],.]],.]]
=> ? = 1 - 1
[2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> ? = 1 - 1
[2,2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [.,[[[[[.,[[[[.,.],.],.],.]],.],.],.],.]]
=> ? = 1 - 1
[5,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [.,[[[.,[[[.,.],.],[.,.]]],.],.]]
=> ? = 2 - 1
[4,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [.,[[[.,[[.,[.,[.,.]]],.]],.],.]]
=> ? = 1 - 1
[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
[4,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [.,[[.,[[[[.,.],[.,.]],.],.]],.]]
=> ? = 2 - 1
[3,3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> ? = 1 - 1
[3,3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [.,[[.,[[[.,[[.,.],.]],.],.]],.]]
=> ? = 1 - 1
[3,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> ? = 1 - 1
[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
[5,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [.,[[[.,[[.,[.,.]],[.,.]]],.],.]]
=> ? = 2 - 1
[5,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [.,[[.,[[[[.,.],.],[.,.]],.]],.]]
=> ? = 2 - 1
[4,4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [.,[[[.,[[.,.],[[.,.],.]]],.],.]]
=> ? = 2 - 1
[4,3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [.,[[[.,[.,[[.,[.,.]],.]]],.],.]]
=> ? = 1 - 1
[4,3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [.,[[.,[[[.,[.,[.,.]]],.],.]],.]]
=> ? = 1 - 1
[4,3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [.,[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.]]
=> ? = 1 - 1
[4,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [.,[.,[[[[[.,.],[.,.]],.],.],.]]]
=> ? = 2 - 1
[3,3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> ? = 1 - 1
[3,3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [.,[.,[[[[.,[[.,.],.]],.],.],.]]]
=> ? = 1 - 1
[2,2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]
=> ? = 1 - 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree.
Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
The following 36 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000353The number of inner valleys of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001092The number of distinct even parts of a partition. St000647The number of big descents of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000779The tier of a permutation. St000251The number of nonsingleton blocks of a set partition. St000646The number of big ascents of a permutation. 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. St000360The number of occurrences of the pattern 32-1. St000523The number of 2-protected nodes of a rooted tree. 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. St000660The number of rises of length at least 3 of a Dyck path. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. 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. St001487The number of inner corners of a skew partition.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!