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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St001554
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Mp00061: Permutations —to increasing tree⟶ Binary trees
St001554: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001554: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> 1
[1,2] => [.,[.,.]]
=> 2
[2,1] => [[.,.],.]
=> 2
[1,2,3] => [.,[.,[.,.]]]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> 3
[2,1,3] => [[.,.],[.,.]]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> 3
[3,1,2] => [[.,.],[.,.]]
=> 2
[3,2,1] => [[[.,.],.],.]
=> 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> 4
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> 4
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> 3
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> 3
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> 4
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> 3
[2,4,3,1] => [[.,[[.,.],.]],.]
=> 4
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> 3
[3,1,4,2] => [[.,.],[[.,.],.]]
=> 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> 3
[3,2,4,1] => [[[.,.],[.,.]],.]
=> 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> 3
[3,4,2,1] => [[[.,[.,.]],.],.]
=> 4
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> 3
[4,1,3,2] => [[.,.],[[.,.],.]]
=> 3
[4,2,1,3] => [[[.,.],.],[.,.]]
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> 3
[4,3,1,2] => [[[.,.],.],[.,.]]
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 5
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 4
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 5
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 5
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 4
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> 4
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> 5
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 4
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> 5
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 4
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> 4
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 4
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> 4
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 4
Description
The number of distinct nonempty subtrees of a binary tree.
Matching statistic: St001820
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001820: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 88%
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001820: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 88%
Values
[1] => [.,.]
=> ([],1)
=> ([(0,1)],2)
=> 1
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 2
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 2
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 3
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 4
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 4
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 4
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 4
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 4
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 4
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 4
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 4
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 4
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[4,1,3,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[4,1,5,2,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[4,2,1,5,3] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[4,3,1,5,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[4,5,1,3,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[5,1,3,2,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[5,1,4,2,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[5,2,1,4,3] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[5,3,1,4,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[5,4,1,3,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 5
[1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 5
[1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 4
[1,3,2,5,6,4] => [.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 5
[1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 4
[1,3,2,6,5,4] => [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 5
[1,3,4,2,5,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 4
[1,3,4,2,6,5] => [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 5
[1,3,4,5,2,6] => [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 5
[1,3,4,6,2,5] => [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 5
Description
The size of the image of the pop stack sorting operator.
The pop stack sorting operator is defined by $Pop_L^\downarrow(x) = x\wedge\bigwedge\{y\in L\mid y\lessdot x\}$. This statistic returns the size of $Pop_L^\downarrow(L)\}$.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 62%
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 62%
Values
[1] => [.,.]
=> ([],1)
=> ? = 1
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? = 2
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? = 2
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 3
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 3
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 2
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 3
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 2
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,5,4,2,1] => [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,3,5,6,4] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,4,5,6,3] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,4,6,5,3] => [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,5,6,4,3] => [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,3,4,5,6,2] => [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,3,4,6,5,2] => [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,3,5,6,4,2] => [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,3,6,5,4,2] => [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,4,5,6,3,2] => [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,4,6,5,3,2] => [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,5,6,4,3,2] => [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,3,4,5,6,1] => [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,3,4,6,5,1] => [[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,3,5,6,4,1] => [[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,3,6,5,4,1] => [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,4,5,6,3,1] => [[.,[[.,[.,[.,.]]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,4,6,5,3,1] => [[.,[[.,[[.,.],.]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 62%
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 62%
Values
[1] => [.,.]
=> ([],1)
=> ? = 1 - 1
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? = 2 - 1
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? = 2 - 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 2 - 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 2 - 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4 - 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4 - 1
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3 - 1
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3 - 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 - 1
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 - 1
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 - 1
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 - 1
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,4,2,1] => [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,2,3,5,6,4] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,2,4,5,6,3] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,2,4,6,5,3] => [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,2,5,6,4,3] => [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,3,4,5,6,2] => [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,3,4,6,5,2] => [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,3,5,6,4,2] => [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,3,6,5,4,2] => [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,4,5,6,3,2] => [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,4,6,5,3,2] => [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,5,6,4,3,2] => [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[2,3,4,5,6,1] => [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[2,3,4,6,5,1] => [[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[2,3,5,6,4,1] => [[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[2,3,6,5,4,1] => [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[2,4,5,6,3,1] => [[.,[[.,[.,[.,.]]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[2,4,6,5,3,1] => [[.,[[.,[[.,.],.]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001626
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001626: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 75%
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001626: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 75%
Values
[1] => [.,.]
=> ([],1)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 2 + 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 2 + 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 3 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 3 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 3 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 3 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 3 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 3 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 3 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 3 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 3 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 3 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 3 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 3 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 3 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 3 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 3 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 3 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 4 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 4 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 4 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 1
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 1
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 1
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 4 + 1
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3 + 1
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3 + 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 + 1
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4 + 1
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 + 1
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4 + 1
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 4 + 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 + 1
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4 + 1
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 4 + 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3 + 1
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3 + 1
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4 + 1
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 + 1
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3 + 1
[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 1
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3 + 1
[3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 1
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 + 1
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4 + 1
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 1
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
[3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 + 1
[3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4 + 1
[3,5,2,1,4] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 1
Description
The number of maximal proper sublattices of a lattice.
Matching statistic: St001773
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001773: Signed permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 50%
Mp00223: Permutations —runsort⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001773: Signed permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => [1,2] => 2
[2,1] => [2,1] => [1,2] => [1,2] => 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,3,2] => [3,1,2] => [1,2,3] => [1,2,3] => 3
[2,1,3] => [2,1,3] => [1,3,2] => [1,3,2] => 2
[2,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 3
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,2,4,3] => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 4
[1,3,2,4] => [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 3
[1,3,4,2] => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 4
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 3
[1,4,3,2] => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 4
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 3
[2,1,4,3] => [2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 3
[2,3,1,4] => [2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 3
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4
[2,4,1,3] => [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 3
[2,4,3,1] => [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 4
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 3
[3,1,4,2] => [1,3,4,2] => [1,3,4,2] => [1,3,4,2] => 3
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 3
[3,2,4,1] => [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 3
[3,4,1,2] => [3,1,4,2] => [1,4,2,3] => [1,4,2,3] => 3
[3,4,2,1] => [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 4
[4,1,2,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 3
[4,2,1,3] => [2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 3
[4,2,3,1] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 3
[4,3,1,2] => [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 3
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[1,2,3,5,4] => [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[1,2,4,3,5] => [4,1,2,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 4
[1,2,4,5,3] => [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[1,2,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 4
[1,2,5,4,3] => [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[1,3,2,4,5] => [3,1,2,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 4
[1,3,2,5,4] => [3,5,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 4
[1,3,4,2,5] => [3,4,1,2,5] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 4
[1,3,4,5,2] => [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[1,3,5,2,4] => [5,3,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 4
[1,3,5,4,2] => [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 4
[1,4,2,5,3] => [1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 4
[1,4,3,2,5] => [4,3,1,2,5] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 4
[1,4,3,5,2] => [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 4
[1,4,5,2,3] => [4,1,5,2,3] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 4
[1,4,5,3,2] => [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[1,5,2,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 4
[1,5,2,4,3] => [5,1,4,2,3] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 4
[1,5,3,2,4] => [5,1,3,2,4] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 4
[1,5,3,4,2] => [3,5,4,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 4
[1,5,4,2,3] => [1,5,4,2,3] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 4
[1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 4
[2,1,3,5,4] => [2,5,1,3,4] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 4
[2,1,4,3,5] => [2,4,1,3,5] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 3
[2,1,4,5,3] => [2,4,5,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 4
[2,1,5,3,4] => [5,2,1,3,4] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 3
[2,1,5,4,3] => [5,2,4,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 4
[2,3,1,4,5] => [2,3,1,4,5] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 3
[2,3,1,5,4] => [2,3,5,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 4
[2,3,4,1,5] => [2,3,4,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 4
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[2,3,5,1,4] => [5,2,3,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 4
[2,3,5,4,1] => [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[2,4,1,3,5] => [4,2,1,3,5] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 3
[2,4,1,5,3] => [4,2,5,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 4
[2,4,3,1,5] => [4,2,3,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 4
[2,4,3,5,1] => [4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 4
[2,4,5,1,3] => [4,5,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 4
[2,4,5,3,1] => [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[2,5,1,3,4] => [2,1,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3
[2,5,1,4,3] => [2,5,4,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 4
[2,5,3,1,4] => [2,5,3,1,4] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 4
[2,5,3,4,1] => [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 4
[2,5,4,1,3] => [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 4
[2,5,4,3,1] => [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 4
[3,1,2,5,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 4
Description
The number of minimal elements in Bruhat order not less than the signed permutation.
The minimal elements in question are biGrassmannian, that is both the element and its inverse have at most one descent.
This is the size of the essential set of the signed permutation, see [1].
Matching statistic: St001984
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001984: Binary words ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 50%
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001984: Binary words ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 50%
Values
[1] => [.,.]
=> [1,0]
=> 10 => 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1010 => 2
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1100 => 2
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 101010 => 3
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 101100 => 3
[2,1,3] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 111000 => 2
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 110010 => 3
[3,1,2] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 111000 => 2
[3,2,1] => [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> 110100 => 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 4
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 4
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 3
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 4
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 3
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 4
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 3
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 3
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 3
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 3
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 3
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 5
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 4
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? = 5
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 5
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? = 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 4
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 4
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? = 5
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 4
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => ? = 5
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? = 4
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 4
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 4
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 4
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 4
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => ? = 5
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? = 4
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 4
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 4
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 4
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 4
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 5
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? = 4
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 4
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => ? = 3
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 4
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => ? = 3
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? = 4
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 3
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => ? = 4
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => ? = 4
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? = 5
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => ? = 4
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => ? = 5
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 3
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => ? = 4
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => ? = 4
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 4
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => ? = 4
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => ? = 5
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 3
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => ? = 4
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => ? = 4
[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 4
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => ? = 4
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => ? = 5
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? = 4
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 4
Description
A recursive count of subwords of the form 01, 10 and 11.
We iteratively remove the last two bits, and tally the number of times we see 01, 10 or 11. However, if the last two bits are 00, we drop the last 0 and continue. The statistic is 1 for the word ''1''.
We can recursively define the statistic as
$$
stat(w_1,\dotsc,w_n) =
\begin{cases}
0 & \text{if } w \text{ is the empty word}\\
0 & \text{if } w = (0)\\
1 & \text{if } w=(1)\\
stat(w_1,\dotsc,w_{n-1}) & \text{if } w_n=w_{n-1}=0\\
1+stat(w_1,\dotsc,w_{n-2}) & \text{otherwise.}
\end{cases}
$$
This statistic is Example 4.1 in [1].
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