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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000698
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [2,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [2,2,1]
=> [2,1]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [2,2,1]
=> [2,1]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,2,1]
=> [2,1]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,2,1]
=> [2,1]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [3,2,1]
=> [2,1]
=> 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [3,2,1]
=> [2,1]
=> 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [2,2,2]
=> [2,2]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [3,2,1]
=> [2,1]
=> 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => [3,2,1]
=> [2,1]
=> 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [3,2,1]
=> [2,1]
=> 0
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => [4,2]
=> [2]
=> 1
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St001613
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Values
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,10),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [3,4,1,2,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [2,1,5,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => [4,5,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,12),(4,12),(5,11),(6,11),(6,13),(8,9),(9,7),(10,7),(11,10),(12,8),(13,9),(13,10),(14,8),(14,13)],15)
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => [3,4,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(6,10),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [2,1,5,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,9),(4,12),(5,12),(6,10),(6,11),(8,7),(9,7),(10,13),(11,13),(12,8),(13,8),(13,9)],14)
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,5,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => [2,1,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => [2,1,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [2,1,6,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,6] => [2,1,5,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,9),(4,12),(5,12),(6,10),(6,11),(8,7),(9,7),(10,13),(11,13),(12,8),(13,8),(13,9)],14)
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,12),(4,12),(5,11),(6,10),(6,13),(8,7),(9,7),(10,8),(11,9),(12,10),(13,8),(13,9),(14,11),(14,13)],15)
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => [3,2,1,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,6,4] => [2,1,3,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,6,4] => [3,2,1,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5] => [2,1,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,3,6,4,5] => [2,1,3,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ? = 0
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => [3,2,1,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,12),(3,13),(4,13),(5,11),(5,14),(6,10),(6,14),(8,7),(9,7),(10,8),(11,9),(12,10),(13,11),(14,8),(14,9)],15)
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => [3,2,1,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => [4,2,3,1,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? = 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [3,2,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,13),(4,14),(5,11),(5,15),(6,12),(6,15),(8,11),(9,7),(10,7),(11,9),(12,10),(13,8),(14,8),(15,9),(15,10)],16)
=> ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,6,2,5] => [5,3,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,6,2,3,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [3,6,1,7,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [4,6,3,1,7,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,5,1,7,3,4,6] => [3,6,1,7,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,5,6,1,7,4] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [3,4,6,1,7,2,5] => [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,5,6,2,7,4] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,7,4,6] => [4,6,3,1,7,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [3,6,1,7,2,4,5] => [5,7,3,6,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,6,2,3,7,5] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [4,1,7,2,3,5,6] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
Description
The binary logarithm of the size of the center of a lattice.
An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.
Matching statistic: St001719
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Values
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,10),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [3,4,1,2,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [2,1,5,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => [4,5,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,12),(4,12),(5,11),(6,11),(6,13),(8,9),(9,7),(10,7),(11,10),(12,8),(13,9),(13,10),(14,8),(14,13)],15)
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => [3,4,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(6,10),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [2,1,5,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,9),(4,12),(5,12),(6,10),(6,11),(8,7),(9,7),(10,13),(11,13),(12,8),(13,8),(13,9)],14)
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,5,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => [2,1,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => [2,1,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [2,1,6,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,6] => [2,1,5,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,9),(4,12),(5,12),(6,10),(6,11),(8,7),(9,7),(10,13),(11,13),(12,8),(13,8),(13,9)],14)
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,12),(4,12),(5,11),(6,10),(6,13),(8,7),(9,7),(10,8),(11,9),(12,10),(13,8),(13,9),(14,11),(14,13)],15)
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => [3,2,1,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,6,4] => [2,1,3,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,6,4] => [3,2,1,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5] => [2,1,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,3,6,4,5] => [2,1,3,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ? = 0
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => [3,2,1,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,12),(3,13),(4,13),(5,11),(5,14),(6,10),(6,14),(8,7),(9,7),(10,8),(11,9),(12,10),(13,11),(14,8),(14,9)],15)
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => [3,2,1,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => [4,2,3,1,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? = 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [3,2,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,13),(4,14),(5,11),(5,15),(6,12),(6,15),(8,11),(9,7),(10,7),(11,9),(12,10),(13,8),(14,8),(15,9),(15,10)],16)
=> ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,6,2,5] => [5,3,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,6,2,3,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [3,6,1,7,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [4,6,3,1,7,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,5,1,7,3,4,6] => [3,6,1,7,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,5,6,1,7,4] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [3,4,6,1,7,2,5] => [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,5,6,2,7,4] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,7,4,6] => [4,6,3,1,7,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [3,6,1,7,2,4,5] => [5,7,3,6,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,6,2,3,7,5] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [4,1,7,2,3,5,6] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001881
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Values
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,10),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [3,4,1,2,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [2,1,5,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => [4,5,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,12),(4,12),(5,11),(6,11),(6,13),(8,9),(9,7),(10,7),(11,10),(12,8),(13,9),(13,10),(14,8),(14,13)],15)
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => [3,4,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(6,10),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [2,1,5,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,9),(4,12),(5,12),(6,10),(6,11),(8,7),(9,7),(10,13),(11,13),(12,8),(13,8),(13,9)],14)
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,5,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => [2,1,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => [2,1,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [2,1,6,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,6] => [2,1,5,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,9),(4,12),(5,12),(6,10),(6,11),(8,7),(9,7),(10,13),(11,13),(12,8),(13,8),(13,9)],14)
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,12),(4,12),(5,11),(6,10),(6,13),(8,7),(9,7),(10,8),(11,9),(12,10),(13,8),(13,9),(14,11),(14,13)],15)
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => [3,2,1,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,6,4] => [2,1,3,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,6,4] => [3,2,1,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5] => [2,1,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,3,6,4,5] => [2,1,3,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ? = 0
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => [3,2,1,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,12),(3,13),(4,13),(5,11),(5,14),(6,10),(6,14),(8,7),(9,7),(10,8),(11,9),(12,10),(13,11),(14,8),(14,9)],15)
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => [3,2,1,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => [4,2,3,1,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? = 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [3,2,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,13),(4,14),(5,11),(5,15),(6,12),(6,15),(8,11),(9,7),(10,7),(11,9),(12,10),(13,8),(14,8),(15,9),(15,10)],16)
=> ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,6,2,5] => [5,3,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,6,2,3,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [3,6,1,7,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [4,6,3,1,7,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,5,1,7,3,4,6] => [3,6,1,7,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,5,6,1,7,4] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [3,4,6,1,7,2,5] => [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,5,6,2,7,4] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,7,4,6] => [4,6,3,1,7,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [3,6,1,7,2,4,5] => [5,7,3,6,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,6,2,3,7,5] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [4,1,7,2,3,5,6] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
Description
The number of factors of a lattice as a Cartesian product of lattices.
Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.
Matching statistic: St001616
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Values
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,10),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [3,4,1,2,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [2,1,5,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => [4,5,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,12),(4,12),(5,11),(6,11),(6,13),(8,9),(9,7),(10,7),(11,10),(12,8),(13,9),(13,10),(14,8),(14,13)],15)
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => [3,4,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(6,10),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [2,1,5,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,9),(4,12),(5,12),(6,10),(6,11),(8,7),(9,7),(10,13),(11,13),(12,8),(13,8),(13,9)],14)
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,5,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => [2,1,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => [2,1,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [2,1,6,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,6] => [2,1,5,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,9),(4,12),(5,12),(6,10),(6,11),(8,7),(9,7),(10,13),(11,13),(12,8),(13,8),(13,9)],14)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,12),(4,12),(5,11),(6,10),(6,13),(8,7),(9,7),(10,8),(11,9),(12,10),(13,8),(13,9),(14,11),(14,13)],15)
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => [3,2,1,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,6,4] => [2,1,3,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,6,4] => [3,2,1,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5] => [2,1,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,3,6,4,5] => [2,1,3,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => [3,2,1,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,12),(3,13),(4,13),(5,11),(5,14),(6,10),(6,14),(8,7),(9,7),(10,8),(11,9),(12,10),(13,11),(14,8),(14,9)],15)
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => [3,2,1,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => [4,2,3,1,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [3,2,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,13),(4,14),(5,11),(5,15),(6,12),(6,15),(8,11),(9,7),(10,7),(11,9),(12,10),(13,8),(14,8),(15,9),(15,10)],16)
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,6,2,5] => [5,3,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,6,2,3,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [3,6,1,7,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [4,6,3,1,7,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,5,1,7,3,4,6] => [3,6,1,7,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,5,6,1,7,4] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [3,4,6,1,7,2,5] => [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,5,6,2,7,4] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,7,4,6] => [4,6,3,1,7,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [3,6,1,7,2,4,5] => [5,7,3,6,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,6,2,3,7,5] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [4,1,7,2,3,5,6] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
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: St001720
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Values
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,10),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [3,4,1,2,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [2,1,5,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => [4,5,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,12),(4,12),(5,11),(6,11),(6,13),(8,9),(9,7),(10,7),(11,10),(12,8),(13,9),(13,10),(14,8),(14,13)],15)
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => [3,4,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(6,10),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [2,1,5,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,9),(4,12),(5,12),(6,10),(6,11),(8,7),(9,7),(10,13),(11,13),(12,8),(13,8),(13,9)],14)
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,5,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => [2,1,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => [2,1,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [2,1,6,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,6] => [2,1,5,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,9),(4,12),(5,12),(6,10),(6,11),(8,7),(9,7),(10,13),(11,13),(12,8),(13,8),(13,9)],14)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,12),(4,12),(5,11),(6,10),(6,13),(8,7),(9,7),(10,8),(11,9),(12,10),(13,8),(13,9),(14,11),(14,13)],15)
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => [3,2,1,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,6,4] => [2,1,3,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,6,4] => [3,2,1,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5] => [2,1,6,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,3,6,4,5] => [2,1,3,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => [3,2,1,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,12),(3,13),(4,13),(5,11),(5,14),(6,10),(6,14),(8,7),(9,7),(10,8),(11,9),(12,10),(13,11),(14,8),(14,9)],15)
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => [3,2,1,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => [4,2,3,1,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,10),(9,12),(10,12),(12,11)],13)
=> ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [3,2,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,13),(4,14),(5,11),(5,15),(6,12),(6,15),(8,11),(9,7),(10,7),(11,9),(12,10),(13,8),(14,8),(15,9),(15,10)],16)
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,6,2,5] => [5,3,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,6,2,3,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [3,6,1,7,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [4,6,3,1,7,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,5,1,7,3,4,6] => [3,6,1,7,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,5,6,1,7,4] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [3,4,6,1,7,2,5] => [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,5,6,2,7,4] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,7,4,6] => [4,6,3,1,7,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [3,6,1,7,2,4,5] => [5,7,3,6,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,6,2,3,7,5] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [4,1,7,2,3,5,6] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 1 + 1
Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
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