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Your data matches 14 different statistics following compositions of up to 3 maps.
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Matching statistic: St001879
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(load all 4 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 9
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,.],[.,[.,[.,.]]]],.]
=> [5,4,3,1,2,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 16
[[[.,.],[.,[[.,.],.]]],.]
=> [4,5,3,1,2,6] => [1,5,4,2,3,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 12
[[[.,.],[[.,.],[.,.]]],.]
=> [5,3,4,1,2,6] => [1,5,2,4,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 10
[[[.,.],[[.,[.,.]],.]],.]
=> [4,3,5,1,2,6] => [1,5,3,2,4,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 10
[[[.,.],[[[.,.],.],.]],.]
=> [3,4,5,1,2,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[[[[.,.],.],[.,[.,.]]],.]
=> [5,4,1,2,3,6] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 10
[[[[.,.],.],[[.,.],.]],.]
=> [4,5,1,2,3,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[[[[.,.],[.,.]],[.,.]],.]
=> [5,3,1,2,4,6] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 7
[[[[[.,.],.],.],[.,.]],.]
=> [5,1,2,3,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[[[[.,.],[.,[.,.]]],.],.]
=> [4,3,1,2,5,6] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 10
[[[[.,.],[[.,.],.]],.],.]
=> [3,4,1,2,5,6] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
[[[[[.,.],.],[.,.]],.],.]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[[[[[.,.],[.,.]],.],.],.]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [6,5,4,3,1,2,7] => [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 25
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [5,6,4,3,1,2,7] => [1,6,5,4,2,3,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> 20
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [6,4,5,3,1,2,7] => [1,6,5,2,4,3,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> 17
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [5,4,6,3,1,2,7] => [1,6,5,3,2,4,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 18
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [4,5,6,3,1,2,7] => [1,6,5,2,3,4,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> 15
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [6,5,3,4,1,2,7] => [1,6,2,5,4,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> 16
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [5,6,3,4,1,2,7] => [1,6,2,5,3,4,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> 13
[[[.,.],[[.,[.,.]],[.,.]]],.]
=> [6,4,3,5,1,2,7] => [1,6,4,2,5,3,7] => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> 14
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [6,3,4,5,1,2,7] => [1,6,2,3,5,4,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> 12
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [5,4,3,6,1,2,7] => [1,6,4,3,2,5,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> 17
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [4,5,3,6,1,2,7] => [1,6,4,2,3,5,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> 13
[[[.,.],[[[.,.],[.,.]],.]],.]
=> [5,3,4,6,1,2,7] => [1,6,2,4,3,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> 11
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [4,3,5,6,1,2,7] => [1,6,3,2,4,5,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> 12
[[[.,.],[[[[.,.],.],.],.]],.]
=> [3,4,5,6,1,2,7] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [6,5,4,1,2,3,7] => [1,2,6,5,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> 17
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [5,6,4,1,2,3,7] => [1,2,6,5,3,4,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> 13
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [6,4,5,1,2,3,7] => [1,2,6,3,5,4,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> 11
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [5,4,6,1,2,3,7] => [1,2,6,4,3,5,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> 11
[[[[.,.],.],[[[.,.],.],.]],.]
=> [4,5,6,1,2,3,7] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 9
[[[[.,.],[.,.]],[.,[.,.]]],.]
=> [6,5,3,1,2,4,7] => [1,5,2,6,4,3,7] => ([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> 12
[[[[.,.],[.,.]],[[.,.],.]],.]
=> [5,6,3,1,2,4,7] => [1,5,2,6,3,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> 9
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [6,5,1,2,3,4,7] => [1,2,3,6,5,4,7] => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> 11
[[[[[.,.],.],.],[[.,.],.]],.]
=> [5,6,1,2,3,4,7] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 8
[[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => [1,5,4,2,6,3,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> 13
[[[[.,.],[[.,.],.]],[.,.]],.]
=> [6,3,4,1,2,5,7] => [1,5,2,3,6,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> 9
[[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => [1,2,5,3,6,4,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 8
[[[[[[.,.],.],.],.],[.,.]],.]
=> [6,1,2,3,4,5,7] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [5,4,3,1,2,6,7] => [1,5,4,3,2,6,7] => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> 17
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: St001846
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 32%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 32%
Values
[[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 3
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 9
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> 6
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 5
[[[[[.,.],.],.],.],.]
=> [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)
=> 4
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,2),(6,3),(6,4),(6,5),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,1)],18)
=> ? = 16
[[[.,.],[.,[[.,.],.]]],.]
=> [1,4,5,3,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,6),(2,10),(2,11),(3,7),(3,9),(4,7),(4,8),(5,2),(5,8),(5,9),(6,3),(6,4),(6,5),(7,12),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13),(13,1)],14)
=> ? = 12
[[[.,.],[[.,.],[.,.]]],.]
=> [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,6),(2,7),(3,8),(3,10),(4,8),(4,9),(5,3),(5,4),(5,7),(6,2),(6,5),(7,9),(7,10),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 10
[[[.,.],[[.,[.,.]],.]],.]
=> [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(2,10),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(6,3),(6,4),(6,5),(7,11),(8,11),(9,2),(9,11),(10,1),(11,10)],12)
=> ? = 10
[[[.,.],[[[.,.],.],.]],.]
=> [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 8
[[[[.,.],.],[.,[.,.]]],.]
=> [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,6),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 10
[[[[.,.],.],[[.,.],.]],.]
=> [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> 7
[[[[.,.],[.,.]],[.,.]],.]
=> [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> 7
[[[[[.,.],.],.],[.,.]],.]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 6
[[[[.,.],[.,[.,.]]],.],.]
=> [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,5)],11)
=> ? = 10
[[[[.,.],[[.,.],.]],.],.]
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> 7
[[[[[.,.],.],[.,.]],.],.]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 6
[[[[[.,.],[.,.]],.],.],.]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 6
[[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 5
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 25
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,5,6,4,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? = 20
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,4,6,5,3,2,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ?
=> ? = 17
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,5,4,6,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 18
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,4,5,6,3,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,7),(2,8),(2,10),(3,8),(3,9),(4,11),(4,12),(5,6),(5,9),(5,10),(6,4),(6,13),(6,14),(7,2),(7,3),(7,5),(8,15),(9,13),(9,15),(10,14),(10,15),(11,17),(12,17),(13,11),(13,16),(14,12),(14,16),(15,16),(16,17),(17,1)],18)
=> ? = 15
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,3,6,5,4,2,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,8),(3,10),(3,11),(3,14),(4,9),(4,11),(4,13),(5,9),(5,10),(5,12),(6,2),(6,7),(7,3),(7,4),(7,5),(7,8),(8,12),(8,13),(8,14),(9,17),(9,18),(10,15),(10,18),(11,16),(11,18),(12,15),(12,17),(13,16),(13,17),(14,15),(14,16),(15,19),(16,19),(17,19),(18,19),(19,1)],20)
=> ? = 16
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,3,5,6,4,2,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,7),(1,9),(3,8),(3,12),(4,10),(4,11),(5,4),(5,8),(5,13),(6,3),(6,5),(6,9),(7,1),(7,6),(8,11),(8,14),(9,12),(9,13),(10,15),(11,15),(12,14),(13,10),(13,14),(14,15),(15,2)],16)
=> ? = 13
[[[.,.],[[.,[.,.]],[.,.]]],.]
=> [1,4,3,6,5,2,7] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(2,10),(2,13),(3,9),(3,13),(4,8),(4,12),(5,8),(5,11),(6,9),(6,10),(7,2),(7,3),(7,6),(8,15),(9,14),(10,14),(11,15),(12,15),(13,4),(13,5),(13,14),(14,11),(14,12),(15,1)],16)
=> ? = 14
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,3,4,6,5,2,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,7),(1,9),(3,8),(3,11),(4,8),(4,10),(5,6),(5,9),(6,3),(6,4),(6,12),(7,1),(7,5),(8,13),(9,12),(10,13),(11,13),(12,10),(12,11),(13,2)],14)
=> ? = 12
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,5,4,3,6,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,7),(1,14),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,11),(6,12),(6,13),(7,3),(7,4),(7,5),(7,6),(8,17),(8,18),(9,15),(9,18),(10,16),(10,18),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,2),(15,19),(16,19),(17,19),(18,1),(18,19),(19,14)],20)
=> ? = 17
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,4,5,3,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> ([(0,7),(1,12),(3,11),(3,13),(4,8),(4,9),(5,8),(5,10),(6,3),(6,9),(6,10),(7,4),(7,5),(7,6),(8,15),(9,13),(9,15),(10,11),(10,15),(11,14),(12,2),(13,1),(13,14),(14,12),(15,14)],16)
=> ? = 13
[[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,3,5,4,6,2,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,7),(2,8),(3,12),(4,10),(4,11),(5,9),(5,11),(6,4),(6,5),(6,8),(7,2),(7,6),(8,9),(8,10),(9,13),(10,13),(11,3),(11,13),(12,1),(13,12)],14)
=> ? = 11
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,4,3,5,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,7),(2,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,11),(7,3),(7,4),(7,5),(8,13),(9,13),(10,6),(10,13),(11,12),(12,1),(13,11)],14)
=> ? = 12
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,7),(2,8),(3,11),(4,5),(4,8),(5,6),(5,10),(6,3),(6,9),(7,2),(7,4),(8,10),(9,11),(10,9),(11,1)],12)
=> ? = 10
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,2,6,5,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,7),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,1)],19)
=> ? = 17
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,2,5,6,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(3,8),(3,10),(4,8),(4,9),(5,2),(5,9),(5,10),(6,7),(7,3),(7,4),(7,5),(8,13),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14),(14,1)],15)
=> ? = 13
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(2,9),(2,11),(3,9),(3,10),(4,8),(5,6),(6,4),(6,7),(7,2),(7,3),(7,8),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 11
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,2,5,4,6,3,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(2,11),(3,9),(3,10),(4,8),(4,10),(5,8),(5,9),(6,7),(7,3),(7,4),(7,5),(8,12),(9,12),(10,2),(10,12),(11,1),(12,11)],13)
=> ? = 11
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,6),(2,10),(3,8),(4,2),(4,9),(5,4),(5,8),(6,7),(7,3),(7,5),(8,9),(9,10),(10,1)],11)
=> ? = 9
[[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,3,2,6,5,4,7] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,11),(2,11),(4,9),(4,10),(5,8),(5,10),(6,8),(6,9),(7,1),(7,2),(8,12),(9,12),(10,12),(11,4),(11,5),(11,6),(12,3)],13)
=> ? = 12
[[[[.,.],[.,.]],[[.,.],.]],.]
=> [1,3,2,5,6,4,7] => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
=> ([(0,7),(1,10),(2,10),(3,9),(5,8),(6,3),(6,8),(7,1),(7,2),(8,9),(9,4),(10,5),(10,6)],11)
=> ? = 9
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,2,3,6,5,4,7] => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,7),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 11
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ([(0,6),(1,9),(3,8),(4,7),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,2)],10)
=> ? = 8
[[[[.,.],[.,[.,.]]],[.,.]],.]
=> [1,4,3,2,6,5,7] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ([(0,7),(2,11),(3,11),(4,9),(4,10),(5,8),(5,10),(6,8),(6,9),(7,4),(7,5),(7,6),(8,12),(9,12),(10,12),(11,1),(12,2),(12,3)],13)
=> ? = 13
[[[[.,.],[[.,.],.]],[.,.]],.]
=> [1,3,4,2,6,5,7] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
=> ([(0,7),(1,9),(2,9),(4,10),(5,8),(6,4),(6,8),(7,5),(7,6),(8,10),(9,3),(10,1),(10,2)],11)
=> ? = 9
[[[[[.,.],.],[.,.]],[.,.]],.]
=> [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ([(0,6),(1,9),(2,9),(3,8),(4,8),(6,7),(7,1),(7,2),(8,5),(9,3),(9,4)],10)
=> ? = 8
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
=> ? = 7
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,1),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,6)],19)
=> ? = 17
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,4,5,3,2,6,7] => ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(0,7),(1,11),(1,12),(3,8),(3,10),(4,8),(4,9),(5,2),(6,1),(6,9),(6,10),(7,3),(7,4),(7,6),(8,14),(9,11),(9,14),(10,12),(10,14),(11,13),(12,13),(13,5),(14,13)],15)
=> ? = 13
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,3,5,4,2,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> ([(0,7),(2,8),(3,9),(3,11),(4,9),(4,10),(5,1),(6,3),(6,4),(6,8),(7,2),(7,6),(8,10),(8,11),(9,12),(10,12),(11,12),(12,5)],13)
=> ? = 11
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,4,3,5,2,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,3),(7,4),(7,5),(8,12),(9,12),(10,6),(11,2),(11,12),(12,10)],13)
=> ? = 11
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ([(0,7),(2,8),(3,9),(4,5),(4,8),(5,3),(5,10),(6,1),(7,2),(7,4),(8,10),(9,6),(10,9)],11)
=> ? = 9
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,7),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,5)],12)
=> ? = 11
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,6),(2,9),(3,8),(4,2),(4,8),(5,1),(6,7),(7,3),(7,4),(8,9),(9,5)],10)
=> ? = 8
[[[[[.,.],[.,.]],[.,.]],.],.]
=> [1,3,2,5,4,6,7] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> ([(0,7),(1,9),(2,9),(4,8),(5,8),(6,3),(7,1),(7,2),(8,6),(9,4),(9,5)],10)
=> ? = 8
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=> 7
[[[[[.,.],[.,[.,.]]],.],.],.]
=> [1,4,3,2,5,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,7),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,6)],12)
=> ? = 11
[[[[[.,.],[[.,.],.]],.],.],.]
=> [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ([(0,7),(1,9),(3,8),(4,2),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,6)],10)
=> ? = 8
[[[[[[.,.],.],[.,.]],.],.],.]
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
=> ? = 7
[[[[[[.,.],[.,.]],.],.],.],.]
=> [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,7),(2,8),(3,8),(4,5),(5,1),(6,4),(7,2),(7,3),(8,6)],9)
=> 7
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 6
Description
The number of elements which do not have a complement in the lattice.
A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.
Matching statistic: St001616
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 32%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 32%
Values
[[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6 = 4 + 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 9 + 2
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> 8 = 6 + 2
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7 = 5 + 2
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7 = 5 + 2
[[[[[.,.],.],.],.],.]
=> [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 = 4 + 2
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,2),(6,3),(6,4),(6,5),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,1)],18)
=> ? = 16 + 2
[[[.,.],[.,[[.,.],.]]],.]
=> [1,4,5,3,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,6),(2,10),(2,11),(3,7),(3,9),(4,7),(4,8),(5,2),(5,8),(5,9),(6,3),(6,4),(6,5),(7,12),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13),(13,1)],14)
=> ? = 12 + 2
[[[.,.],[[.,.],[.,.]]],.]
=> [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,6),(2,7),(3,8),(3,10),(4,8),(4,9),(5,3),(5,4),(5,7),(6,2),(6,5),(7,9),(7,10),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 10 + 2
[[[.,.],[[.,[.,.]],.]],.]
=> [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(2,10),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(6,3),(6,4),(6,5),(7,11),(8,11),(9,2),(9,11),(10,1),(11,10)],12)
=> ? = 10 + 2
[[[.,.],[[[.,.],.],.]],.]
=> [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 8 + 2
[[[[.,.],.],[.,[.,.]]],.]
=> [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,6),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 10 + 2
[[[[.,.],.],[[.,.],.]],.]
=> [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> 9 = 7 + 2
[[[[.,.],[.,.]],[.,.]],.]
=> [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> 9 = 7 + 2
[[[[[.,.],.],.],[.,.]],.]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 8 = 6 + 2
[[[[.,.],[.,[.,.]]],.],.]
=> [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,5)],11)
=> ? = 10 + 2
[[[[.,.],[[.,.],.]],.],.]
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> 9 = 7 + 2
[[[[[.,.],.],[.,.]],.],.]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 8 = 6 + 2
[[[[[.,.],[.,.]],.],.],.]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 8 = 6 + 2
[[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 25 + 2
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,5,6,4,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? = 20 + 2
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,4,6,5,3,2,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ?
=> ? = 17 + 2
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,5,4,6,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 18 + 2
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,4,5,6,3,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,7),(2,8),(2,10),(3,8),(3,9),(4,11),(4,12),(5,6),(5,9),(5,10),(6,4),(6,13),(6,14),(7,2),(7,3),(7,5),(8,15),(9,13),(9,15),(10,14),(10,15),(11,17),(12,17),(13,11),(13,16),(14,12),(14,16),(15,16),(16,17),(17,1)],18)
=> ? = 15 + 2
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,3,6,5,4,2,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,8),(3,10),(3,11),(3,14),(4,9),(4,11),(4,13),(5,9),(5,10),(5,12),(6,2),(6,7),(7,3),(7,4),(7,5),(7,8),(8,12),(8,13),(8,14),(9,17),(9,18),(10,15),(10,18),(11,16),(11,18),(12,15),(12,17),(13,16),(13,17),(14,15),(14,16),(15,19),(16,19),(17,19),(18,19),(19,1)],20)
=> ? = 16 + 2
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,3,5,6,4,2,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,7),(1,9),(3,8),(3,12),(4,10),(4,11),(5,4),(5,8),(5,13),(6,3),(6,5),(6,9),(7,1),(7,6),(8,11),(8,14),(9,12),(9,13),(10,15),(11,15),(12,14),(13,10),(13,14),(14,15),(15,2)],16)
=> ? = 13 + 2
[[[.,.],[[.,[.,.]],[.,.]]],.]
=> [1,4,3,6,5,2,7] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(2,10),(2,13),(3,9),(3,13),(4,8),(4,12),(5,8),(5,11),(6,9),(6,10),(7,2),(7,3),(7,6),(8,15),(9,14),(10,14),(11,15),(12,15),(13,4),(13,5),(13,14),(14,11),(14,12),(15,1)],16)
=> ? = 14 + 2
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,3,4,6,5,2,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,7),(1,9),(3,8),(3,11),(4,8),(4,10),(5,6),(5,9),(6,3),(6,4),(6,12),(7,1),(7,5),(8,13),(9,12),(10,13),(11,13),(12,10),(12,11),(13,2)],14)
=> ? = 12 + 2
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,5,4,3,6,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,7),(1,14),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,11),(6,12),(6,13),(7,3),(7,4),(7,5),(7,6),(8,17),(8,18),(9,15),(9,18),(10,16),(10,18),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,2),(15,19),(16,19),(17,19),(18,1),(18,19),(19,14)],20)
=> ? = 17 + 2
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,4,5,3,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> ([(0,7),(1,12),(3,11),(3,13),(4,8),(4,9),(5,8),(5,10),(6,3),(6,9),(6,10),(7,4),(7,5),(7,6),(8,15),(9,13),(9,15),(10,11),(10,15),(11,14),(12,2),(13,1),(13,14),(14,12),(15,14)],16)
=> ? = 13 + 2
[[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,3,5,4,6,2,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,7),(2,8),(3,12),(4,10),(4,11),(5,9),(5,11),(6,4),(6,5),(6,8),(7,2),(7,6),(8,9),(8,10),(9,13),(10,13),(11,3),(11,13),(12,1),(13,12)],14)
=> ? = 11 + 2
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,4,3,5,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,7),(2,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,11),(7,3),(7,4),(7,5),(8,13),(9,13),(10,6),(10,13),(11,12),(12,1),(13,11)],14)
=> ? = 12 + 2
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,7),(2,8),(3,11),(4,5),(4,8),(5,6),(5,10),(6,3),(6,9),(7,2),(7,4),(8,10),(9,11),(10,9),(11,1)],12)
=> ? = 10 + 2
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,2,6,5,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,7),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,1)],19)
=> ? = 17 + 2
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,2,5,6,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(3,8),(3,10),(4,8),(4,9),(5,2),(5,9),(5,10),(6,7),(7,3),(7,4),(7,5),(8,13),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14),(14,1)],15)
=> ? = 13 + 2
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(2,9),(2,11),(3,9),(3,10),(4,8),(5,6),(6,4),(6,7),(7,2),(7,3),(7,8),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 11 + 2
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,2,5,4,6,3,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(2,11),(3,9),(3,10),(4,8),(4,10),(5,8),(5,9),(6,7),(7,3),(7,4),(7,5),(8,12),(9,12),(10,2),(10,12),(11,1),(12,11)],13)
=> ? = 11 + 2
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,6),(2,10),(3,8),(4,2),(4,9),(5,4),(5,8),(6,7),(7,3),(7,5),(8,9),(9,10),(10,1)],11)
=> ? = 9 + 2
[[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,3,2,6,5,4,7] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,11),(2,11),(4,9),(4,10),(5,8),(5,10),(6,8),(6,9),(7,1),(7,2),(8,12),(9,12),(10,12),(11,4),(11,5),(11,6),(12,3)],13)
=> ? = 12 + 2
[[[[.,.],[.,.]],[[.,.],.]],.]
=> [1,3,2,5,6,4,7] => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
=> ([(0,7),(1,10),(2,10),(3,9),(5,8),(6,3),(6,8),(7,1),(7,2),(8,9),(9,4),(10,5),(10,6)],11)
=> ? = 9 + 2
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,2,3,6,5,4,7] => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,7),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 11 + 2
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ([(0,6),(1,9),(3,8),(4,7),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,2)],10)
=> ? = 8 + 2
[[[[.,.],[.,[.,.]]],[.,.]],.]
=> [1,4,3,2,6,5,7] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ([(0,7),(2,11),(3,11),(4,9),(4,10),(5,8),(5,10),(6,8),(6,9),(7,4),(7,5),(7,6),(8,12),(9,12),(10,12),(11,1),(12,2),(12,3)],13)
=> ? = 13 + 2
[[[[.,.],[[.,.],.]],[.,.]],.]
=> [1,3,4,2,6,5,7] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
=> ([(0,7),(1,9),(2,9),(4,10),(5,8),(6,4),(6,8),(7,5),(7,6),(8,10),(9,3),(10,1),(10,2)],11)
=> ? = 9 + 2
[[[[[.,.],.],[.,.]],[.,.]],.]
=> [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ([(0,6),(1,9),(2,9),(3,8),(4,8),(6,7),(7,1),(7,2),(8,5),(9,3),(9,4)],10)
=> ? = 8 + 2
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
=> ? = 7 + 2
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,1),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,6)],19)
=> ? = 17 + 2
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,4,5,3,2,6,7] => ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(0,7),(1,11),(1,12),(3,8),(3,10),(4,8),(4,9),(5,2),(6,1),(6,9),(6,10),(7,3),(7,4),(7,6),(8,14),(9,11),(9,14),(10,12),(10,14),(11,13),(12,13),(13,5),(14,13)],15)
=> ? = 13 + 2
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,3,5,4,2,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> ([(0,7),(2,8),(3,9),(3,11),(4,9),(4,10),(5,1),(6,3),(6,4),(6,8),(7,2),(7,6),(8,10),(8,11),(9,12),(10,12),(11,12),(12,5)],13)
=> ? = 11 + 2
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,4,3,5,2,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,3),(7,4),(7,5),(8,12),(9,12),(10,6),(11,2),(11,12),(12,10)],13)
=> ? = 11 + 2
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ([(0,7),(2,8),(3,9),(4,5),(4,8),(5,3),(5,10),(6,1),(7,2),(7,4),(8,10),(9,6),(10,9)],11)
=> ? = 9 + 2
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,7),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,5)],12)
=> ? = 11 + 2
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,6),(2,9),(3,8),(4,2),(4,8),(5,1),(6,7),(7,3),(7,4),(8,9),(9,5)],10)
=> ? = 8 + 2
[[[[[.,.],[.,.]],[.,.]],.],.]
=> [1,3,2,5,4,6,7] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> ([(0,7),(1,9),(2,9),(4,8),(5,8),(6,3),(7,1),(7,2),(8,6),(9,4),(9,5)],10)
=> ? = 8 + 2
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=> 9 = 7 + 2
[[[[[.,.],[.,[.,.]]],.],.],.]
=> [1,4,3,2,5,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,7),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,6)],12)
=> ? = 11 + 2
[[[[[.,.],[[.,.],.]],.],.],.]
=> [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ([(0,7),(1,9),(3,8),(4,2),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,6)],10)
=> ? = 8 + 2
[[[[[[.,.],.],[.,.]],.],.],.]
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
=> ? = 7 + 2
[[[[[[.,.],[.,.]],.],.],.],.]
=> [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,7),(2,8),(3,8),(4,5),(5,1),(6,4),(7,2),(7,3),(8,6)],9)
=> 9 = 7 + 2
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 8 = 6 + 2
Description
The number of neutral elements in a lattice.
An element $e$ of the lattice $L$ is neutral if the sublattice generated by $e$, $x$ and $y$ is distributive for all $x, y \in L$.
Matching statistic: St001300
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001300: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 26%
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001300: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 26%
Values
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[[.,.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 9
[[[.,.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6
[[[[.,.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5
[[[[.,.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5
[[[[[.,.],.],.],.],.]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,.],[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 16
[[[.,.],[.,[[.,.],.]]],.]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 12
[[[.,.],[[.,.],[.,.]]],.]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 10
[[[.,.],[[.,[.,.]],.]],.]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 10
[[[.,.],[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 8
[[[[.,.],.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 10
[[[[.,.],.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 7
[[[[.,.],[.,.]],[.,.]],.]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 7
[[[[[.,.],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 6
[[[[.,.],[.,[.,.]]],.],.]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 10
[[[[.,.],[[.,.],.]],.],.]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 7
[[[[[.,.],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 6
[[[[[.,.],[.,.]],.],.],.]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 6
[[[[[[.,.],.],.],.],.],.]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ? = 25
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [.,[[[[.,[.,.]],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,8),(4,3),(4,10),(5,4),(5,7),(6,2),(6,5),(7,10),(8,9),(10,1),(10,8)],11)
=> ? = 20
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [.,[[[[.,.],[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? = 17
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [.,[[[.,[[.,.],.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,6),(1,8),(2,9),(3,10),(4,7),(5,3),(5,9),(6,2),(6,5),(7,8),(9,4),(9,10),(10,1),(10,7)],11)
=> ? = 18
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> ? = 15
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
=> ? = 16
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
=> ? = 13
[[[.,.],[[.,[.,.]],[.,.]]],.]
=> [.,[[[.,.],[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 14
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 12
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [.,[[.,[[[.,.],.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> ? = 17
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,6),(1,8),(2,9),(3,10),(4,7),(5,3),(5,9),(6,2),(6,5),(7,8),(9,4),(9,10),(10,1),(10,7)],11)
=> ? = 13
[[[.,.],[[[.,.],[.,.]],.]],.]
=> [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? = 11
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,8),(4,3),(4,10),(5,4),(5,7),(6,2),(6,5),(7,10),(8,9),(10,1),(10,8)],11)
=> ? = 12
[[[.,.],[[[[.,.],.],.],.]],.]
=> [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ? = 10
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 17
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 13
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? = 11
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 11
[[[[.,.],.],[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 9
[[[[.,.],[.,.]],[.,[.,.]]],.]
=> [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,4),(6,5),(8,9),(9,2),(9,3)],10)
=> ? = 12
[[[[.,.],[.,.]],[[.,.],.]],.]
=> [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,4),(6,5),(8,9),(9,2),(9,3)],10)
=> ? = 9
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 11
[[[[[.,.],.],.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 8
[[[[.,.],[.,[.,.]]],[.,.]],.]
=> [.,[[.,.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
=> ? = 13
[[[[.,.],[[.,.],.]],[.,.]],.]
=> [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
=> ? = 9
[[[[[.,.],.],[.,.]],[.,.]],.]
=> [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 8
[[[[[[.,.],.],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 7
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,4),(3,7),(4,2),(4,9),(5,6),(6,1),(6,3),(7,9),(9,8)],10)
=> ? = 17
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 13
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,9),(2,8),(3,2),(3,7),(4,1),(4,7),(5,6),(6,3),(6,4),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 11
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 11
[[[[.,.],[[[.,.],.],.]],.],.]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,4),(3,7),(4,2),(4,9),(5,6),(6,1),(6,3),(7,9),(9,8)],10)
=> ? = 9
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 11
[[[[[.,.],.],[[.,.],.]],.],.]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 8
[[[[[.,.],[.,.]],[.,.]],.],.]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
=> ? = 8
[[[[[[.,.],.],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 7
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
Matching statistic: St000656
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000656: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 26%
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000656: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 26%
Values
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5 = 4 + 1
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[[.,.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 9 + 1
[[[.,.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 + 1
[[[[.,.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6 = 5 + 1
[[[[.,.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6 = 5 + 1
[[[[[.,.],.],.],.],.]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[[.,.],[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 16 + 1
[[[.,.],[.,[[.,.],.]]],.]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 12 + 1
[[[.,.],[[.,.],[.,.]]],.]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 10 + 1
[[[.,.],[[.,[.,.]],.]],.]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 10 + 1
[[[.,.],[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 8 + 1
[[[[.,.],.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 10 + 1
[[[[.,.],.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 7 + 1
[[[[.,.],[.,.]],[.,.]],.]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 7 + 1
[[[[[.,.],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 6 + 1
[[[[.,.],[.,[.,.]]],.],.]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 10 + 1
[[[[.,.],[[.,.],.]],.],.]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 7 + 1
[[[[[.,.],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7 = 6 + 1
[[[[[.,.],[.,.]],.],.],.]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 6 + 1
[[[[[[.,.],.],.],.],.],.]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ? = 25 + 1
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [.,[[[[.,[.,.]],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,8),(4,3),(4,10),(5,4),(5,7),(6,2),(6,5),(7,10),(8,9),(10,1),(10,8)],11)
=> ? = 20 + 1
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [.,[[[[.,.],[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? = 17 + 1
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [.,[[[.,[[.,.],.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,6),(1,8),(2,9),(3,10),(4,7),(5,3),(5,9),(6,2),(6,5),(7,8),(9,4),(9,10),(10,1),(10,7)],11)
=> ? = 18 + 1
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> ? = 15 + 1
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
=> ? = 16 + 1
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
=> ? = 13 + 1
[[[.,.],[[.,[.,.]],[.,.]]],.]
=> [.,[[[.,.],[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 14 + 1
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 12 + 1
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [.,[[.,[[[.,.],.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> ? = 17 + 1
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,6),(1,8),(2,9),(3,10),(4,7),(5,3),(5,9),(6,2),(6,5),(7,8),(9,4),(9,10),(10,1),(10,7)],11)
=> ? = 13 + 1
[[[.,.],[[[.,.],[.,.]],.]],.]
=> [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? = 11 + 1
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,8),(4,3),(4,10),(5,4),(5,7),(6,2),(6,5),(7,10),(8,9),(10,1),(10,8)],11)
=> ? = 12 + 1
[[[.,.],[[[[.,.],.],.],.]],.]
=> [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ? = 10 + 1
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 17 + 1
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 13 + 1
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? = 11 + 1
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 11 + 1
[[[[.,.],.],[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 9 + 1
[[[[.,.],[.,.]],[.,[.,.]]],.]
=> [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,4),(6,5),(8,9),(9,2),(9,3)],10)
=> ? = 12 + 1
[[[[.,.],[.,.]],[[.,.],.]],.]
=> [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,4),(6,5),(8,9),(9,2),(9,3)],10)
=> ? = 9 + 1
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 11 + 1
[[[[[.,.],.],.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 8 + 1
[[[[.,.],[.,[.,.]]],[.,.]],.]
=> [.,[[.,.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
=> ? = 13 + 1
[[[[.,.],[[.,.],.]],[.,.]],.]
=> [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
=> ? = 9 + 1
[[[[[.,.],.],[.,.]],[.,.]],.]
=> [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 8 + 1
[[[[[[.,.],.],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 7 + 1
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,4),(3,7),(4,2),(4,9),(5,6),(6,1),(6,3),(7,9),(9,8)],10)
=> ? = 17 + 1
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 13 + 1
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,9),(2,8),(3,2),(3,7),(4,1),(4,7),(5,6),(6,3),(6,4),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 11 + 1
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 11 + 1
[[[[.,.],[[[.,.],.],.]],.],.]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,4),(3,7),(4,2),(4,9),(5,6),(6,1),(6,3),(7,9),(9,8)],10)
=> ? = 9 + 1
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 11 + 1
[[[[[.,.],.],[[.,.],.]],.],.]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 8 + 1
[[[[[.,.],[.,.]],[.,.]],.],.]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
=> ? = 8 + 1
[[[[[[.,.],.],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 7 + 1
Description
The number of cuts of a poset.
A cut is a subset $A$ of the poset such that the set of lower bounds of the set of upper bounds of $A$ is exactly $A$.
Matching statistic: St001717
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001717: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 26%
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001717: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 26%
Values
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5 = 4 + 1
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[[.,.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 9 + 1
[[[.,.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 + 1
[[[[.,.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6 = 5 + 1
[[[[.,.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6 = 5 + 1
[[[[[.,.],.],.],.],.]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[[.,.],[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 16 + 1
[[[.,.],[.,[[.,.],.]]],.]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 12 + 1
[[[.,.],[[.,.],[.,.]]],.]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 10 + 1
[[[.,.],[[.,[.,.]],.]],.]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 10 + 1
[[[.,.],[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 8 + 1
[[[[.,.],.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 10 + 1
[[[[.,.],.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 7 + 1
[[[[.,.],[.,.]],[.,.]],.]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 7 + 1
[[[[[.,.],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 6 + 1
[[[[.,.],[.,[.,.]]],.],.]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 10 + 1
[[[[.,.],[[.,.],.]],.],.]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 7 + 1
[[[[[.,.],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7 = 6 + 1
[[[[[.,.],[.,.]],.],.],.]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 6 + 1
[[[[[[.,.],.],.],.],.],.]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ? = 25 + 1
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [.,[[[[.,[.,.]],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,8),(4,3),(4,10),(5,4),(5,7),(6,2),(6,5),(7,10),(8,9),(10,1),(10,8)],11)
=> ? = 20 + 1
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [.,[[[[.,.],[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? = 17 + 1
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [.,[[[.,[[.,.],.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,6),(1,8),(2,9),(3,10),(4,7),(5,3),(5,9),(6,2),(6,5),(7,8),(9,4),(9,10),(10,1),(10,7)],11)
=> ? = 18 + 1
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> ? = 15 + 1
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
=> ? = 16 + 1
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
=> ? = 13 + 1
[[[.,.],[[.,[.,.]],[.,.]]],.]
=> [.,[[[.,.],[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 14 + 1
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 12 + 1
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [.,[[.,[[[.,.],.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> ? = 17 + 1
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,6),(1,8),(2,9),(3,10),(4,7),(5,3),(5,9),(6,2),(6,5),(7,8),(9,4),(9,10),(10,1),(10,7)],11)
=> ? = 13 + 1
[[[.,.],[[[.,.],[.,.]],.]],.]
=> [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? = 11 + 1
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,8),(4,3),(4,10),(5,4),(5,7),(6,2),(6,5),(7,10),(8,9),(10,1),(10,8)],11)
=> ? = 12 + 1
[[[.,.],[[[[.,.],.],.],.]],.]
=> [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ? = 10 + 1
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 17 + 1
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 13 + 1
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? = 11 + 1
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 11 + 1
[[[[.,.],.],[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 9 + 1
[[[[.,.],[.,.]],[.,[.,.]]],.]
=> [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,4),(6,5),(8,9),(9,2),(9,3)],10)
=> ? = 12 + 1
[[[[.,.],[.,.]],[[.,.],.]],.]
=> [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,4),(6,5),(8,9),(9,2),(9,3)],10)
=> ? = 9 + 1
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 11 + 1
[[[[[.,.],.],.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 8 + 1
[[[[.,.],[.,[.,.]]],[.,.]],.]
=> [.,[[.,.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
=> ? = 13 + 1
[[[[.,.],[[.,.],.]],[.,.]],.]
=> [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
=> ? = 9 + 1
[[[[[.,.],.],[.,.]],[.,.]],.]
=> [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 8 + 1
[[[[[[.,.],.],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 7 + 1
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,4),(3,7),(4,2),(4,9),(5,6),(6,1),(6,3),(7,9),(9,8)],10)
=> ? = 17 + 1
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 13 + 1
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,9),(2,8),(3,2),(3,7),(4,1),(4,7),(5,6),(6,3),(6,4),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 11 + 1
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 11 + 1
[[[[.,.],[[[.,.],.],.]],.],.]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,4),(3,7),(4,2),(4,9),(5,6),(6,1),(6,3),(7,9),(9,8)],10)
=> ? = 9 + 1
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 11 + 1
[[[[[.,.],.],[[.,.],.]],.],.]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 8 + 1
[[[[[.,.],[.,.]],[.,.]],.],.]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
=> ? = 8 + 1
[[[[[[.,.],.],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 7 + 1
Description
The largest size of an interval in a poset.
Matching statistic: St000868
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000868: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 32%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000868: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 32%
Values
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 4 = 2 + 2
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 6 = 4 + 2
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 5 = 3 + 2
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 11 = 9 + 2
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 8 = 6 + 2
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 7 = 5 + 2
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 7 = 5 + 2
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 6 = 4 + 2
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ? = 16 + 2
[[[.,.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => ? = 12 + 2
[[[.,.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => ? = 10 + 2
[[[.,.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? = 10 + 2
[[[.,.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => ? = 8 + 2
[[[[.,.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ? = 10 + 2
[[[[.,.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => ? = 7 + 2
[[[[.,.],[.,.]],[.,.]],.]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ? = 7 + 2
[[[[[.,.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ? = 6 + 2
[[[[.,.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => ? = 10 + 2
[[[[.,.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => ? = 7 + 2
[[[[[.,.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 6 + 2
[[[[[.,.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 6 + 2
[[[[[[.,.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 5 + 2
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [8,3,6,5,1,7,2,4] => ? = 25 + 2
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [8,1,4,6,2,7,3,5] => ? = 20 + 2
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [8,1,2,5,3,7,4,6] => ? = 17 + 2
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [8,4,1,6,2,7,3,5] => ? = 18 + 2
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [8,3,5,1,6,2,4,7] => ? = 15 + 2
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [8,3,1,5,7,2,4,6] => ? = 16 + 2
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [2,8,4,6,1,3,5,7] => ? = 13 + 2
[[[.,.],[[.,[.,.]],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [2,5,1,3,8,7,4,6] => ? = 14 + 2
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,8,6,3,5,7] => ? = 12 + 2
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [2,8,4,7,6,1,3,5] => ? = 17 + 2
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,8,5,7,2,4,6] => ? = 13 + 2
[[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [3,1,5,2,8,7,4,6] => ? = 11 + 2
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [2,8,4,1,6,7,3,5] => ? = 12 + 2
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [2,7,4,5,1,3,8,6] => ? = 10 + 2
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [6,8,4,1,2,7,3,5] => ? = 17 + 2
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [8,1,5,2,3,7,4,6] => ? = 13 + 2
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [8,1,4,2,6,3,5,7] => ? = 11 + 2
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [8,3,5,1,7,2,4,6] => ? = 11 + 2
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,8,1,5,3,7,4,6] => ? = 9 + 2
[[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,7,1,3,4,5,8,6] => ? = 12 + 2
[[[[.,.],[.,.]],[[.,.],.]],.]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => ? = 9 + 2
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [2,6,7,1,3,4,8,5] => ? = 11 + 2
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [5,4,1,2,7,3,8,6] => ? = 8 + 2
[[[[.,.],[.,[.,.]]],[.,.]],.]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,8,1,2,3,7,4,6] => ? = 13 + 2
[[[[.,.],[[.,.],.]],[.,.]],.]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [8,4,1,2,6,3,5,7] => ? = 9 + 2
[[[[[.,.],.],[.,.]],[.,.]],.]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [2,8,4,1,3,5,6,7] => ? = 8 + 2
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,3,5,1,8,4,6,7] => ? = 7 + 2
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [7,3,5,1,6,2,8,4] => ? = 17 + 2
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [7,1,4,2,6,3,8,5] => ? = 13 + 2
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,1,6,5,2,8,4,7] => ? = 11 + 2
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [2,8,4,5,1,7,3,6] => ? = 11 + 2
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,1,7,6,3,8,5] => ? = 9 + 2
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [7,4,1,2,6,3,8,5] => ? = 11 + 2
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,1,5,2,8,4,7] => ? = 8 + 2
[[[[[.,.],[.,.]],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => ? = 8 + 2
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [2,3,7,1,4,5,8,6] => ? = 7 + 2
Description
The aid statistic in the sense of Shareshian-Wachs.
This is the number of admissible inversions [[St000866]] plus the number of descents [[St000021]]. This statistic was introduced by John Shareshian and Michelle L. Wachs in [1]. Theorem 4.1 states that the aid statistic together with the descent statistic is Euler-Mahonian.
Matching statistic: St000189
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 21%
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 21%
Values
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5 = 4 + 1
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[[.,.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 9 + 1
[[[.,.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 + 1
[[[[.,.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6 = 5 + 1
[[[[.,.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6 = 5 + 1
[[[[[.,.],.],.],.],.]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[[.,.],[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 16 + 1
[[[.,.],[.,[[.,.],.]]],.]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 12 + 1
[[[.,.],[[.,.],[.,.]]],.]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 10 + 1
[[[.,.],[[.,[.,.]],.]],.]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 10 + 1
[[[.,.],[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 8 + 1
[[[[.,.],.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 10 + 1
[[[[.,.],.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 7 + 1
[[[[.,.],[.,.]],[.,.]],.]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 7 + 1
[[[[[.,.],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 6 + 1
[[[[.,.],[.,[.,.]]],.],.]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 10 + 1
[[[[.,.],[[.,.],.]],.],.]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 7 + 1
[[[[[.,.],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 6 + 1
[[[[[.,.],[.,.]],.],.],.]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 6 + 1
[[[[[[.,.],.],.],.],.],.]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ? = 25 + 1
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [.,[[[[.,[.,.]],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,8),(4,3),(4,10),(5,4),(5,7),(6,2),(6,5),(7,10),(8,9),(10,1),(10,8)],11)
=> ? = 20 + 1
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [.,[[[[.,.],[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? = 17 + 1
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [.,[[[.,[[.,.],.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,6),(1,8),(2,9),(3,10),(4,7),(5,3),(5,9),(6,2),(6,5),(7,8),(9,4),(9,10),(10,1),(10,7)],11)
=> ? = 18 + 1
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> ? = 15 + 1
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
=> ? = 16 + 1
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
=> ? = 13 + 1
[[[.,.],[[.,[.,.]],[.,.]]],.]
=> [.,[[[.,.],[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 14 + 1
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 12 + 1
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [.,[[.,[[[.,.],.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> ? = 17 + 1
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,6),(1,8),(2,9),(3,10),(4,7),(5,3),(5,9),(6,2),(6,5),(7,8),(9,4),(9,10),(10,1),(10,7)],11)
=> ? = 13 + 1
[[[.,.],[[[.,.],[.,.]],.]],.]
=> [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? = 11 + 1
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,8),(4,3),(4,10),(5,4),(5,7),(6,2),(6,5),(7,10),(8,9),(10,1),(10,8)],11)
=> ? = 12 + 1
[[[.,.],[[[[.,.],.],.],.]],.]
=> [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ? = 10 + 1
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 17 + 1
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 13 + 1
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? = 11 + 1
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 11 + 1
[[[[.,.],.],[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 9 + 1
[[[[.,.],[.,.]],[.,[.,.]]],.]
=> [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,4),(6,5),(8,9),(9,2),(9,3)],10)
=> ? = 12 + 1
[[[[.,.],[.,.]],[[.,.],.]],.]
=> [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,4),(6,5),(8,9),(9,2),(9,3)],10)
=> ? = 9 + 1
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 11 + 1
[[[[[.,.],.],.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 8 + 1
[[[[.,.],[.,[.,.]]],[.,.]],.]
=> [.,[[.,.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
=> ? = 13 + 1
[[[[.,.],[[.,.],.]],[.,.]],.]
=> [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
=> ? = 9 + 1
[[[[[.,.],.],[.,.]],[.,.]],.]
=> [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 8 + 1
[[[[[[.,.],.],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 7 + 1
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,4),(3,7),(4,2),(4,9),(5,6),(6,1),(6,3),(7,9),(9,8)],10)
=> ? = 17 + 1
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 13 + 1
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,9),(2,8),(3,2),(3,7),(4,1),(4,7),(5,6),(6,3),(6,4),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 11 + 1
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 11 + 1
[[[[.,.],[[[.,.],.],.]],.],.]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,4),(3,7),(4,2),(4,9),(5,6),(6,1),(6,3),(7,9),(9,8)],10)
=> ? = 9 + 1
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 11 + 1
[[[[[.,.],.],[[.,.],.]],.],.]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 8 + 1
[[[[[.,.],[.,.]],[.,.]],.],.]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
=> ? = 8 + 1
Description
The number of elements in the poset.
Matching statistic: St000550
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St000550: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 21%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St000550: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 21%
Values
[[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6 = 4 + 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 9 + 2
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 6 + 2
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7 = 5 + 2
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7 = 5 + 2
[[[[[.,.],.],.],.],.]
=> [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 = 4 + 2
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,2),(6,3),(6,4),(6,5),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,1)],18)
=> ? = 16 + 2
[[[.,.],[.,[[.,.],.]]],.]
=> [1,4,5,3,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,6),(2,10),(2,11),(3,7),(3,9),(4,7),(4,8),(5,2),(5,8),(5,9),(6,3),(6,4),(6,5),(7,12),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13),(13,1)],14)
=> ? = 12 + 2
[[[.,.],[[.,.],[.,.]]],.]
=> [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,6),(2,7),(3,8),(3,10),(4,8),(4,9),(5,3),(5,4),(5,7),(6,2),(6,5),(7,9),(7,10),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 10 + 2
[[[.,.],[[.,[.,.]],.]],.]
=> [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(2,10),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(6,3),(6,4),(6,5),(7,11),(8,11),(9,2),(9,11),(10,1),(11,10)],12)
=> ? = 10 + 2
[[[.,.],[[[.,.],.],.]],.]
=> [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 8 + 2
[[[[.,.],.],[.,[.,.]]],.]
=> [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,6),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 10 + 2
[[[[.,.],.],[[.,.],.]],.]
=> [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 7 + 2
[[[[.,.],[.,.]],[.,.]],.]
=> [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 7 + 2
[[[[[.,.],.],.],[.,.]],.]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 6 + 2
[[[[.,.],[.,[.,.]]],.],.]
=> [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,5)],11)
=> ? = 10 + 2
[[[[.,.],[[.,.],.]],.],.]
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 7 + 2
[[[[[.,.],.],[.,.]],.],.]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 6 + 2
[[[[[.,.],[.,.]],.],.],.]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 6 + 2
[[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 25 + 2
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,5,6,4,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? = 20 + 2
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,4,6,5,3,2,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ?
=> ? = 17 + 2
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,5,4,6,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 18 + 2
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,4,5,6,3,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,7),(2,8),(2,10),(3,8),(3,9),(4,11),(4,12),(5,6),(5,9),(5,10),(6,4),(6,13),(6,14),(7,2),(7,3),(7,5),(8,15),(9,13),(9,15),(10,14),(10,15),(11,17),(12,17),(13,11),(13,16),(14,12),(14,16),(15,16),(16,17),(17,1)],18)
=> ? = 15 + 2
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,3,6,5,4,2,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,8),(3,10),(3,11),(3,14),(4,9),(4,11),(4,13),(5,9),(5,10),(5,12),(6,2),(6,7),(7,3),(7,4),(7,5),(7,8),(8,12),(8,13),(8,14),(9,17),(9,18),(10,15),(10,18),(11,16),(11,18),(12,15),(12,17),(13,16),(13,17),(14,15),(14,16),(15,19),(16,19),(17,19),(18,19),(19,1)],20)
=> ? = 16 + 2
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,3,5,6,4,2,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,7),(1,9),(3,8),(3,12),(4,10),(4,11),(5,4),(5,8),(5,13),(6,3),(6,5),(6,9),(7,1),(7,6),(8,11),(8,14),(9,12),(9,13),(10,15),(11,15),(12,14),(13,10),(13,14),(14,15),(15,2)],16)
=> ? = 13 + 2
[[[.,.],[[.,[.,.]],[.,.]]],.]
=> [1,4,3,6,5,2,7] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(2,10),(2,13),(3,9),(3,13),(4,8),(4,12),(5,8),(5,11),(6,9),(6,10),(7,2),(7,3),(7,6),(8,15),(9,14),(10,14),(11,15),(12,15),(13,4),(13,5),(13,14),(14,11),(14,12),(15,1)],16)
=> ? = 14 + 2
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,3,4,6,5,2,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,7),(1,9),(3,8),(3,11),(4,8),(4,10),(5,6),(5,9),(6,3),(6,4),(6,12),(7,1),(7,5),(8,13),(9,12),(10,13),(11,13),(12,10),(12,11),(13,2)],14)
=> ? = 12 + 2
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,5,4,3,6,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,7),(1,14),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,11),(6,12),(6,13),(7,3),(7,4),(7,5),(7,6),(8,17),(8,18),(9,15),(9,18),(10,16),(10,18),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,2),(15,19),(16,19),(17,19),(18,1),(18,19),(19,14)],20)
=> ? = 17 + 2
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,4,5,3,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> ([(0,7),(1,12),(3,11),(3,13),(4,8),(4,9),(5,8),(5,10),(6,3),(6,9),(6,10),(7,4),(7,5),(7,6),(8,15),(9,13),(9,15),(10,11),(10,15),(11,14),(12,2),(13,1),(13,14),(14,12),(15,14)],16)
=> ? = 13 + 2
[[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,3,5,4,6,2,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,7),(2,8),(3,12),(4,10),(4,11),(5,9),(5,11),(6,4),(6,5),(6,8),(7,2),(7,6),(8,9),(8,10),(9,13),(10,13),(11,3),(11,13),(12,1),(13,12)],14)
=> ? = 11 + 2
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,4,3,5,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,7),(2,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,11),(7,3),(7,4),(7,5),(8,13),(9,13),(10,6),(10,13),(11,12),(12,1),(13,11)],14)
=> ? = 12 + 2
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,7),(2,8),(3,11),(4,5),(4,8),(5,6),(5,10),(6,3),(6,9),(7,2),(7,4),(8,10),(9,11),(10,9),(11,1)],12)
=> ? = 10 + 2
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,2,6,5,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,7),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,1)],19)
=> ? = 17 + 2
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,2,5,6,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(3,8),(3,10),(4,8),(4,9),(5,2),(5,9),(5,10),(6,7),(7,3),(7,4),(7,5),(8,13),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14),(14,1)],15)
=> ? = 13 + 2
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(2,9),(2,11),(3,9),(3,10),(4,8),(5,6),(6,4),(6,7),(7,2),(7,3),(7,8),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 11 + 2
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,2,5,4,6,3,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(2,11),(3,9),(3,10),(4,8),(4,10),(5,8),(5,9),(6,7),(7,3),(7,4),(7,5),(8,12),(9,12),(10,2),(10,12),(11,1),(12,11)],13)
=> ? = 11 + 2
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,6),(2,10),(3,8),(4,2),(4,9),(5,4),(5,8),(6,7),(7,3),(7,5),(8,9),(9,10),(10,1)],11)
=> ? = 9 + 2
[[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,3,2,6,5,4,7] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,11),(2,11),(4,9),(4,10),(5,8),(5,10),(6,8),(6,9),(7,1),(7,2),(8,12),(9,12),(10,12),(11,4),(11,5),(11,6),(12,3)],13)
=> ? = 12 + 2
[[[[.,.],[.,.]],[[.,.],.]],.]
=> [1,3,2,5,6,4,7] => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
=> ([(0,7),(1,10),(2,10),(3,9),(5,8),(6,3),(6,8),(7,1),(7,2),(8,9),(9,4),(10,5),(10,6)],11)
=> ? = 9 + 2
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,2,3,6,5,4,7] => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,7),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 11 + 2
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ([(0,6),(1,9),(3,8),(4,7),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,2)],10)
=> ? = 8 + 2
[[[[.,.],[.,[.,.]]],[.,.]],.]
=> [1,4,3,2,6,5,7] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ([(0,7),(2,11),(3,11),(4,9),(4,10),(5,8),(5,10),(6,8),(6,9),(7,4),(7,5),(7,6),(8,12),(9,12),(10,12),(11,1),(12,2),(12,3)],13)
=> ? = 13 + 2
[[[[.,.],[[.,.],.]],[.,.]],.]
=> [1,3,4,2,6,5,7] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
=> ([(0,7),(1,9),(2,9),(4,10),(5,8),(6,4),(6,8),(7,5),(7,6),(8,10),(9,3),(10,1),(10,2)],11)
=> ? = 9 + 2
[[[[[.,.],.],[.,.]],[.,.]],.]
=> [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ([(0,6),(1,9),(2,9),(3,8),(4,8),(6,7),(7,1),(7,2),(8,5),(9,3),(9,4)],10)
=> ? = 8 + 2
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
=> ? = 7 + 2
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,1),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,6)],19)
=> ? = 17 + 2
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,4,5,3,2,6,7] => ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(0,7),(1,11),(1,12),(3,8),(3,10),(4,8),(4,9),(5,2),(6,1),(6,9),(6,10),(7,3),(7,4),(7,6),(8,14),(9,11),(9,14),(10,12),(10,14),(11,13),(12,13),(13,5),(14,13)],15)
=> ? = 13 + 2
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,3,5,4,2,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> ([(0,7),(2,8),(3,9),(3,11),(4,9),(4,10),(5,1),(6,3),(6,4),(6,8),(7,2),(7,6),(8,10),(8,11),(9,12),(10,12),(11,12),(12,5)],13)
=> ? = 11 + 2
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,4,3,5,2,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,3),(7,4),(7,5),(8,12),(9,12),(10,6),(11,2),(11,12),(12,10)],13)
=> ? = 11 + 2
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ([(0,7),(2,8),(3,9),(4,5),(4,8),(5,3),(5,10),(6,1),(7,2),(7,4),(8,10),(9,6),(10,9)],11)
=> ? = 9 + 2
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,7),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,5)],12)
=> ? = 11 + 2
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,6),(2,9),(3,8),(4,2),(4,8),(5,1),(6,7),(7,3),(7,4),(8,9),(9,5)],10)
=> ? = 8 + 2
[[[[[.,.],[.,.]],[.,.]],.],.]
=> [1,3,2,5,4,6,7] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> ([(0,7),(1,9),(2,9),(4,8),(5,8),(6,3),(7,1),(7,2),(8,6),(9,4),(9,5)],10)
=> ? = 8 + 2
Description
The number of modular elements of a lattice.
A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$, and is modular if both $(x, y)$ and $(y, x)$ are modular pairs for every $y\in L$.
Matching statistic: St000551
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St000551: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 21%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St000551: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 21%
Values
[[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6 = 4 + 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 9 + 2
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 6 + 2
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7 = 5 + 2
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7 = 5 + 2
[[[[[.,.],.],.],.],.]
=> [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 = 4 + 2
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,2),(6,3),(6,4),(6,5),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,1)],18)
=> ? = 16 + 2
[[[.,.],[.,[[.,.],.]]],.]
=> [1,4,5,3,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,6),(2,10),(2,11),(3,7),(3,9),(4,7),(4,8),(5,2),(5,8),(5,9),(6,3),(6,4),(6,5),(7,12),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13),(13,1)],14)
=> ? = 12 + 2
[[[.,.],[[.,.],[.,.]]],.]
=> [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,6),(2,7),(3,8),(3,10),(4,8),(4,9),(5,3),(5,4),(5,7),(6,2),(6,5),(7,9),(7,10),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 10 + 2
[[[.,.],[[.,[.,.]],.]],.]
=> [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(2,10),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(6,3),(6,4),(6,5),(7,11),(8,11),(9,2),(9,11),(10,1),(11,10)],12)
=> ? = 10 + 2
[[[.,.],[[[.,.],.],.]],.]
=> [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 8 + 2
[[[[.,.],.],[.,[.,.]]],.]
=> [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,6),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 10 + 2
[[[[.,.],.],[[.,.],.]],.]
=> [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 7 + 2
[[[[.,.],[.,.]],[.,.]],.]
=> [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 7 + 2
[[[[[.,.],.],.],[.,.]],.]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 6 + 2
[[[[.,.],[.,[.,.]]],.],.]
=> [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,5)],11)
=> ? = 10 + 2
[[[[.,.],[[.,.],.]],.],.]
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 7 + 2
[[[[[.,.],.],[.,.]],.],.]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 6 + 2
[[[[[.,.],[.,.]],.],.],.]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 6 + 2
[[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 25 + 2
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,5,6,4,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? = 20 + 2
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,4,6,5,3,2,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ?
=> ? = 17 + 2
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,5,4,6,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 18 + 2
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,4,5,6,3,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,7),(2,8),(2,10),(3,8),(3,9),(4,11),(4,12),(5,6),(5,9),(5,10),(6,4),(6,13),(6,14),(7,2),(7,3),(7,5),(8,15),(9,13),(9,15),(10,14),(10,15),(11,17),(12,17),(13,11),(13,16),(14,12),(14,16),(15,16),(16,17),(17,1)],18)
=> ? = 15 + 2
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,3,6,5,4,2,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,8),(3,10),(3,11),(3,14),(4,9),(4,11),(4,13),(5,9),(5,10),(5,12),(6,2),(6,7),(7,3),(7,4),(7,5),(7,8),(8,12),(8,13),(8,14),(9,17),(9,18),(10,15),(10,18),(11,16),(11,18),(12,15),(12,17),(13,16),(13,17),(14,15),(14,16),(15,19),(16,19),(17,19),(18,19),(19,1)],20)
=> ? = 16 + 2
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,3,5,6,4,2,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,7),(1,9),(3,8),(3,12),(4,10),(4,11),(5,4),(5,8),(5,13),(6,3),(6,5),(6,9),(7,1),(7,6),(8,11),(8,14),(9,12),(9,13),(10,15),(11,15),(12,14),(13,10),(13,14),(14,15),(15,2)],16)
=> ? = 13 + 2
[[[.,.],[[.,[.,.]],[.,.]]],.]
=> [1,4,3,6,5,2,7] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(2,10),(2,13),(3,9),(3,13),(4,8),(4,12),(5,8),(5,11),(6,9),(6,10),(7,2),(7,3),(7,6),(8,15),(9,14),(10,14),(11,15),(12,15),(13,4),(13,5),(13,14),(14,11),(14,12),(15,1)],16)
=> ? = 14 + 2
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,3,4,6,5,2,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,7),(1,9),(3,8),(3,11),(4,8),(4,10),(5,6),(5,9),(6,3),(6,4),(6,12),(7,1),(7,5),(8,13),(9,12),(10,13),(11,13),(12,10),(12,11),(13,2)],14)
=> ? = 12 + 2
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,5,4,3,6,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,7),(1,14),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,11),(6,12),(6,13),(7,3),(7,4),(7,5),(7,6),(8,17),(8,18),(9,15),(9,18),(10,16),(10,18),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,2),(15,19),(16,19),(17,19),(18,1),(18,19),(19,14)],20)
=> ? = 17 + 2
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,4,5,3,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> ([(0,7),(1,12),(3,11),(3,13),(4,8),(4,9),(5,8),(5,10),(6,3),(6,9),(6,10),(7,4),(7,5),(7,6),(8,15),(9,13),(9,15),(10,11),(10,15),(11,14),(12,2),(13,1),(13,14),(14,12),(15,14)],16)
=> ? = 13 + 2
[[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,3,5,4,6,2,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,7),(2,8),(3,12),(4,10),(4,11),(5,9),(5,11),(6,4),(6,5),(6,8),(7,2),(7,6),(8,9),(8,10),(9,13),(10,13),(11,3),(11,13),(12,1),(13,12)],14)
=> ? = 11 + 2
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,4,3,5,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,7),(2,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,11),(7,3),(7,4),(7,5),(8,13),(9,13),(10,6),(10,13),(11,12),(12,1),(13,11)],14)
=> ? = 12 + 2
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,7),(2,8),(3,11),(4,5),(4,8),(5,6),(5,10),(6,3),(6,9),(7,2),(7,4),(8,10),(9,11),(10,9),(11,1)],12)
=> ? = 10 + 2
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,2,6,5,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,7),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,1)],19)
=> ? = 17 + 2
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,2,5,6,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(3,8),(3,10),(4,8),(4,9),(5,2),(5,9),(5,10),(6,7),(7,3),(7,4),(7,5),(8,13),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14),(14,1)],15)
=> ? = 13 + 2
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(2,9),(2,11),(3,9),(3,10),(4,8),(5,6),(6,4),(6,7),(7,2),(7,3),(7,8),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 11 + 2
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,2,5,4,6,3,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(2,11),(3,9),(3,10),(4,8),(4,10),(5,8),(5,9),(6,7),(7,3),(7,4),(7,5),(8,12),(9,12),(10,2),(10,12),(11,1),(12,11)],13)
=> ? = 11 + 2
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,6),(2,10),(3,8),(4,2),(4,9),(5,4),(5,8),(6,7),(7,3),(7,5),(8,9),(9,10),(10,1)],11)
=> ? = 9 + 2
[[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,3,2,6,5,4,7] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,11),(2,11),(4,9),(4,10),(5,8),(5,10),(6,8),(6,9),(7,1),(7,2),(8,12),(9,12),(10,12),(11,4),(11,5),(11,6),(12,3)],13)
=> ? = 12 + 2
[[[[.,.],[.,.]],[[.,.],.]],.]
=> [1,3,2,5,6,4,7] => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
=> ([(0,7),(1,10),(2,10),(3,9),(5,8),(6,3),(6,8),(7,1),(7,2),(8,9),(9,4),(10,5),(10,6)],11)
=> ? = 9 + 2
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,2,3,6,5,4,7] => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,7),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 11 + 2
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ([(0,6),(1,9),(3,8),(4,7),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,2)],10)
=> ? = 8 + 2
[[[[.,.],[.,[.,.]]],[.,.]],.]
=> [1,4,3,2,6,5,7] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ([(0,7),(2,11),(3,11),(4,9),(4,10),(5,8),(5,10),(6,8),(6,9),(7,4),(7,5),(7,6),(8,12),(9,12),(10,12),(11,1),(12,2),(12,3)],13)
=> ? = 13 + 2
[[[[.,.],[[.,.],.]],[.,.]],.]
=> [1,3,4,2,6,5,7] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
=> ([(0,7),(1,9),(2,9),(4,10),(5,8),(6,4),(6,8),(7,5),(7,6),(8,10),(9,3),(10,1),(10,2)],11)
=> ? = 9 + 2
[[[[[.,.],.],[.,.]],[.,.]],.]
=> [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ([(0,6),(1,9),(2,9),(3,8),(4,8),(6,7),(7,1),(7,2),(8,5),(9,3),(9,4)],10)
=> ? = 8 + 2
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
=> ? = 7 + 2
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,1),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,6)],19)
=> ? = 17 + 2
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,4,5,3,2,6,7] => ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(0,7),(1,11),(1,12),(3,8),(3,10),(4,8),(4,9),(5,2),(6,1),(6,9),(6,10),(7,3),(7,4),(7,6),(8,14),(9,11),(9,14),(10,12),(10,14),(11,13),(12,13),(13,5),(14,13)],15)
=> ? = 13 + 2
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,3,5,4,2,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> ([(0,7),(2,8),(3,9),(3,11),(4,9),(4,10),(5,1),(6,3),(6,4),(6,8),(7,2),(7,6),(8,10),(8,11),(9,12),(10,12),(11,12),(12,5)],13)
=> ? = 11 + 2
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,4,3,5,2,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,3),(7,4),(7,5),(8,12),(9,12),(10,6),(11,2),(11,12),(12,10)],13)
=> ? = 11 + 2
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ([(0,7),(2,8),(3,9),(4,5),(4,8),(5,3),(5,10),(6,1),(7,2),(7,4),(8,10),(9,6),(10,9)],11)
=> ? = 9 + 2
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,7),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,5)],12)
=> ? = 11 + 2
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,6),(2,9),(3,8),(4,2),(4,8),(5,1),(6,7),(7,3),(7,4),(8,9),(9,5)],10)
=> ? = 8 + 2
[[[[[.,.],[.,.]],[.,.]],.],.]
=> [1,3,2,5,4,6,7] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> ([(0,7),(1,9),(2,9),(4,8),(5,8),(6,3),(7,1),(7,2),(8,6),(9,4),(9,5)],10)
=> ? = 8 + 2
Description
The number of left modular elements of a lattice.
A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$.
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001614The cyclic permutation representation number of a skew partition. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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