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Your data matches 34 different statistics following compositions of up to 3 maps.
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Matching statistic: St000623
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Mp00080: Set partitions —to permutation⟶ Permutations
St000623: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000623: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => 0
{{1},{2}}
=> [1,2] => 0
{{1,2,3}}
=> [2,3,1] => 0
{{1,2},{3}}
=> [2,1,3] => 0
{{1,3},{2}}
=> [3,2,1] => 0
{{1},{2,3}}
=> [1,3,2] => 0
{{1},{2},{3}}
=> [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => 0
{{1,2},{3,4}}
=> [2,1,4,3] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => 0
Description
The number of occurrences of the pattern 52341 in a permutation.
It is a necessary condition that a permutation $\pi$ avoids this pattern for the Schubert variety associated to $\pi$ to have a complete parabolic bundle structure [1].
Matching statistic: St001719
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 25%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2}}
=> [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1},{2},{3}}
=> [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 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 0 + 1
{{1,2},{3,4}}
=> [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 = 0 + 1
{{1,2},{3},{4}}
=> [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 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> [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 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1},{2},{3,4}}
=> [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 = 0 + 1
{{1},{2},{3},{4}}
=> [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)
=> ? = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(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
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => ([(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)
=> ? = 0 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 1
{{1,2},{3,4},{5}}
=> [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
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => ([(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)
=> ? = 0 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 1
{{1,2},{3},{4,5}}
=> [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
{{1,2},{3},{4},{5}}
=> [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)
=> ? = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 0 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => ([(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)
=> ? = 0 + 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 1
{{1},{2,3},{4,5}}
=> [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},{2,3},{4},{5}}
=> [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)
=> ? = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 0 + 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 0 + 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,3,4,2,5] => ([(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)
=> ? = 0 + 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 0 + 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
{{1},{2},{3,4},{5}}
=> [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)
=> ? = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,4,5,1] => ([(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)
=> ? = 1 + 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
{{1},{2},{3},{4,5}}
=> [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)
=> ? = 0 + 1
{{1},{2},{3},{4},{5}}
=> [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)
=> ? = 0 + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 0 + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [5,1,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 0 + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [5,6,1,2,3,4] => ([(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)
=> ? = 0 + 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [4,1,2,3,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 0 + 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [4,1,2,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,10),(5,11),(5,12),(6,9),(6,13),(8,10),(9,7),(10,9),(11,8),(12,8),(13,7)],14)
=> ? = 0 + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [4,6,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [5,1,2,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 0 + 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,8),(3,7),(4,9),(5,9),(6,7),(6,8),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
=> ? = 0 + 1
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [3,1,2,6,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? = 0 + 1
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [3,1,2,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,10),(8,9),(8,11),(9,12),(10,11),(11,12)],13)
=> ? = 0 + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [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)
=> ? = 0 + 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [3,1,2,5,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? = 0 + 1
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [3,1,2,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,11),(6,9),(6,10),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> ? = 0 + 1
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [5,1,2,4,6,3] => ([(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 = 0 + 1
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [5,3,6,1,2,4] => ([(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 = 0 + 1
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [4,1,2,6,3,5] => ([(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 = 0 + 1
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [3,5,1,2,6,4] => ([(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 = 0 + 1
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [5,1,3,4,6,2] => ([(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 = 0 + 1
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [5,2,6,1,3,4] => ([(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 = 0 + 1
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [4,1,3,6,2,5] => ([(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 = 0 + 1
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [4,1,3,5,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 0 + 1
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [4,1,3,5,6,2] => ([(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 = 0 + 1
{{1,3,5,6},{2,4}}
=> [3,4,5,2,6,1] => [4,2,6,1,3,5] => ([(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 = 0 + 1
{{1,3,5},{2,4,6}}
=> [3,4,5,6,1,2] => [5,1,3,6,2,4] => ([(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 = 0 + 1
{{1,3,5},{2,4},{6}}
=> [3,4,5,2,1,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 0 + 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: St001890
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 50%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
{{1},{2}}
=> [1,2] => [2,1] => ([],2)
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => [3,1,2] => ([(1,2)],3)
=> 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
{{1},{2,3}}
=> [1,3,2] => [2,3,1] => ([(1,2)],3)
=> 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => [3,2,1] => ([],3)
=> 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 1 = 0 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 1 = 0 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 1 = 0 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 1 = 0 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1 = 0 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1 = 0 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
=> 1 = 0 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1 = 0 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 1 = 0 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1 = 0 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> 1 = 0 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 1 = 0 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 1 = 0 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1 = 0 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 1 = 0 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 1 = 0 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1 = 0 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 1 = 0 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 1 = 0 + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ? = 0 + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [6,1,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 0 + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ? = 0 + 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [5,6,1,4,3,2] => ([(0,5),(1,2),(1,3),(1,4)],6)
=> ? = 0 + 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [6,5,1,4,3,2] => ([(2,3),(2,4),(2,5)],6)
=> ? = 0 + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,6,4,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ? = 0 + 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 0 + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [6,1,4,5,3,2] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> ? = 0 + 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,4,5,6,3,2] => ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> ? = 0 + 1
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [4,6,5,1,3,2] => ([(0,4),(0,5),(1,2),(1,3)],6)
=> ? = 0 + 1
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [6,4,5,1,3,2] => ([(1,5),(2,3),(2,4)],6)
=> ? = 0 + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,5,4,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [4,5,6,1,3,2] => ([(0,5),(1,3),(1,4),(5,2)],6)
=> ? = 0 + 1
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [5,6,4,1,3,2] => ([(1,5),(2,3),(2,4)],6)
=> ? = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [6,5,4,1,3,2] => ([(3,4),(3,5)],6)
=> ? = 0 + 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,6,5,3,4,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ? = 0 + 1
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [3,1,5,6,4,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> ? = 0 + 1
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [6,1,5,3,4,2] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> ? = 0 + 1
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [1,3,6,5,4,2] => ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> ? = 0 + 1
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [3,6,1,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
=> ? = 0 + 1
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [6,3,1,5,4,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 0 + 1
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [1,5,6,3,4,2] => ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ? = 0 + 1
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [3,5,1,6,4,2] => ([(0,3),(0,4),(1,2),(1,4),(1,5),(3,5)],6)
=> ? = 0 + 1
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [5,6,1,3,4,2] => ([(0,4),(1,3),(1,5),(5,2)],6)
=> ? = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [6,5,1,3,4,2] => ([(2,3),(2,4),(4,5)],6)
=> ? = 0 + 1
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [1,6,3,4,5,2] => ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> ? = 0 + 1
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [3,1,6,4,5,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> ? = 0 + 1
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [6,1,3,4,5,2] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> ? = 0 + 1
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [1,3,5,4,6,2] => ([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> ? = 0 + 1
{{1,2},{3,4,5,6}}
=> [2,1,4,5,6,3] => [3,6,5,4,1,2] => ([(0,5),(1,2),(1,3),(1,4)],6)
=> ? = 0 + 1
{{1,2},{3,4,5},{6}}
=> [2,1,4,5,3,6] => [6,3,5,4,1,2] => ([(1,5),(2,3),(2,4)],6)
=> ? = 0 + 1
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [1,5,3,4,6,2] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> ? = 0 + 1
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [3,5,6,4,1,2] => ([(0,4),(1,3),(1,5),(5,2)],6)
=> ? = 0 + 1
{{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
=> ? = 0 + 1
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
=> ? = 0 + 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [1,6,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [3,1,4,6,5,2] => ([(0,5),(1,2),(1,5),(5,3),(5,4)],6)
=> ? = 0 + 1
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [4,1,6,3,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
=> ? = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [6,1,4,3,5,2] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1,2,6},{3,5},{4}}
=> [2,6,5,4,3,1] => [1,3,4,5,6,2] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 0 + 1
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [3,6,4,5,1,2] => ([(0,4),(1,3),(1,5),(5,2)],6)
=> ? = 0 + 1
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [4,3,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
=> ? = 0 + 1
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [6,3,4,5,1,2] => ([(1,3),(2,4),(4,5)],6)
=> ? = 0 + 1
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [1,4,5,3,6,2] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> ? = 0 + 1
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ? = 0 + 1
{{1,2},{3},{4,5,6}}
=> [2,1,3,5,6,4] => [4,6,5,3,1,2] => ([(1,5),(2,3),(2,4)],6)
=> ? = 0 + 1
{{1,2},{3},{4,5},{6}}
=> [2,1,3,5,4,6] => [6,4,5,3,1,2] => ([(2,5),(3,4)],6)
=> ? = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [1,5,4,3,6,2] => ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [3,5,4,6,1,2] => ([(0,4),(1,2),(1,3),(2,5),(3,5)],6)
=> ? = 0 + 1
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [4,5,6,3,1,2] => ([(1,3),(2,4),(4,5)],6)
=> ? = 0 + 1
Description
The maximum magnitude of the Möbius function of a poset.
The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
Matching statistic: St001371
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 25%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> 1010 => 0
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 1100 => 0
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 0
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 0
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,5},{2,3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1},{2,3,4,5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => ? = 0
{{1,2,3,4,5,6}}
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 0
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 0
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 0
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 0
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 0
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 0
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001730
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 25%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> 1010 => 0
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 1100 => 0
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 0
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 0
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,5},{2,3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1},{2,3,4,5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => ? = 0
{{1,2,3,4,5,6}}
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 0
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 0
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 0
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 0
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 0
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 0
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001195
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 25%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
{{1,5},{2,3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1},{2,3,4,5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,3,4,5,6}}
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001208
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 25%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 0 + 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1 = 0 + 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
{{1,5},{2,3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
{{1},{2,3,4,5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 1 + 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ? = 0 + 1
{{1,2,3,4,5,6}}
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 0 + 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 0 + 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 0 + 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 0 + 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 0 + 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 0 + 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 0 + 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1 = 0 + 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 + 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 + 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 0 + 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 0 + 1
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 0 + 1
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 0 + 1
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1 = 0 + 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 + 1
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 + 1
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 0 + 1
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1 = 0 + 1
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St001236
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001236: Integer compositions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 25%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001236: Integer compositions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2]
=> 100 => [1,3] => 1 = 0 + 1
{{1},{2}}
=> [1,1]
=> 110 => [1,1,2] => 1 = 0 + 1
{{1,2,3}}
=> [3]
=> 1000 => [1,4] => 1 = 0 + 1
{{1,2},{3}}
=> [2,1]
=> 1010 => [1,2,2] => 1 = 0 + 1
{{1,3},{2}}
=> [2,1]
=> 1010 => [1,2,2] => 1 = 0 + 1
{{1},{2,3}}
=> [2,1]
=> 1010 => [1,2,2] => 1 = 0 + 1
{{1},{2},{3}}
=> [1,1,1]
=> 1110 => [1,1,1,2] => 1 = 0 + 1
{{1,2,3,4}}
=> [4]
=> 10000 => [1,5] => 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> 10010 => [1,3,2] => 1 = 0 + 1
{{1,2,4},{3}}
=> [3,1]
=> 10010 => [1,3,2] => 1 = 0 + 1
{{1,2},{3,4}}
=> [2,2]
=> 1100 => [1,1,3] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
{{1,3,4},{2}}
=> [3,1]
=> 10010 => [1,3,2] => 1 = 0 + 1
{{1,3},{2,4}}
=> [2,2]
=> 1100 => [1,1,3] => 1 = 0 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
{{1,4},{2,3}}
=> [2,2]
=> 1100 => [1,1,3] => 1 = 0 + 1
{{1},{2,3,4}}
=> [3,1]
=> 10010 => [1,3,2] => 1 = 0 + 1
{{1},{2,3},{4}}
=> [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
{{1},{2,4},{3}}
=> [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
{{1},{2},{3,4}}
=> [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 11110 => [1,1,1,1,2] => 1 = 0 + 1
{{1,2,3,4,5}}
=> [5]
=> 100000 => [1,6] => ? = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> 100010 => [1,4,2] => ? = 0 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> 100010 => [1,4,2] => ? = 0 + 1
{{1,2,3},{4,5}}
=> [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> 100010 => [1,4,2] => ? = 0 + 1
{{1,2,4},{3,5}}
=> [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
{{1,2,5},{3,4}}
=> [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
{{1,2},{3,4,5}}
=> [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> 100010 => [1,4,2] => ? = 0 + 1
{{1,3,4},{2,5}}
=> [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
{{1,3,5},{2,4}}
=> [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
{{1,3},{2,4,5}}
=> [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
{{1,4,5},{2,3}}
=> [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
{{1,4},{2,3,5}}
=> [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
{{1,5},{2,3,4}}
=> [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
{{1},{2,3,4,5}}
=> [4,1]
=> 100010 => [1,4,2] => ? = 0 + 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
{{1,5},{2,3},{4}}
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
{{1},{2,3},{4,5}}
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
{{1,4},{2,5},{3}}
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
{{1,4},{2},{3,5}}
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
{{1,5},{2,4},{3}}
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
{{1},{2,4},{3,5}}
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
{{1,5},{2},{3,4}}
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
{{1},{2,5},{3,4}}
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 1 + 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => ? = 0 + 1
{{1,2,3,4,5,6}}
=> [6]
=> 1000000 => [1,7] => ? = 0 + 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> 1000010 => [1,5,2] => ? = 0 + 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> 1000010 => [1,5,2] => ? = 0 + 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> 100100 => [1,3,3] => ? = 0 + 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 0 + 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> 1000010 => [1,5,2] => ? = 0 + 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> 100100 => [1,3,3] => ? = 0 + 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 0 + 1
{{1,2,3,6},{4,5}}
=> [4,2]
=> 100100 => [1,3,3] => ? = 0 + 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> 11000 => [1,1,4] => 1 = 0 + 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> 101010 => [1,2,2,2] => ? = 0 + 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 0 + 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> 101010 => [1,2,2,2] => ? = 0 + 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> 101010 => [1,2,2,2] => ? = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ? = 0 + 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> 1000010 => [1,5,2] => ? = 0 + 1
{{1,2,4,5},{3,6}}
=> [4,2]
=> 100100 => [1,3,3] => ? = 0 + 1
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 0 + 1
{{1,2,4,6},{3,5}}
=> [4,2]
=> 100100 => [1,3,3] => ? = 0 + 1
{{1,2,4},{3,5,6}}
=> [3,3]
=> 11000 => [1,1,4] => 1 = 0 + 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> 101010 => [1,2,2,2] => ? = 0 + 1
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 0 + 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> 101010 => [1,2,2,2] => ? = 0 + 1
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> 101010 => [1,2,2,2] => ? = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ? = 0 + 1
{{1,2,5},{3,4,6}}
=> [3,3]
=> 11000 => [1,1,4] => 1 = 0 + 1
Description
The dominant dimension of the corresponding Comp-Nakayama algebra.
Matching statistic: St001200
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2,1] => [2,1] => [1,1,0,0]
=> ? = 0 + 2
{{1},{2}}
=> [1,2] => [1,2] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2,3}}
=> [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 0 + 2
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1},{2},{3}}
=> [1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,2,3,4}}
=> [2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2 = 0 + 2
{{1,2,3},{4}}
=> [2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2 = 0 + 2
{{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2 = 0 + 2
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2 = 0 + 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2 = 0 + 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 0 + 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,4,5,6}}
=> [2,1,4,5,6,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,4,5},{6}}
=> [2,1,4,5,3,6] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,6},{3,5},{4}}
=> [2,6,5,4,3,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001845
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Values
{{1,2}}
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
{{1},{2}}
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,3}}
=> [2,3,1] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
{{1,2},{3}}
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
{{1,3},{2}}
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
{{1},{2,3}}
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
{{1},{2},{3}}
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 0
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 0
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 0
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 0
{{1},{2,3,4}}
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 0
{{1},{2,3},{4}}
=> [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)
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 0
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> ? = 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> ? = 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
=> ? = 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
=> ? = 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> ? = 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ? = 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> ([(0,2),(0,3),(1,11),(2,1),(2,12),(3,4),(3,5),(3,12),(4,8),(4,10),(5,8),(5,9),(6,14),(7,14),(8,13),(9,6),(9,13),(10,7),(10,13),(11,6),(11,7),(12,9),(12,10),(12,11),(13,14)],15)
=> ? = 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? = 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,8),(2,6),(2,7),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,12),(7,12),(8,12),(9,13),(10,13),(11,1),(11,2),(11,13),(13,7),(13,8)],14)
=> ? = 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,9),(4,1),(4,9),(5,7),(5,8),(6,12),(7,12),(8,12),(9,5),(9,10),(9,11),(10,6),(10,7),(11,6),(11,8)],13)
=> ? = 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> ? = 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> ? = 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> ? = 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => ([(3,4)],5)
=> ?
=> ? = 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> 0
{{1},{2,3,4},{5}}
=> [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)
=> 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> ? = 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> 0
{{1},{2,3},{4},{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)
=> 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? = 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ? = 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,11),(2,10),(3,8),(3,9),(4,7),(4,8),(5,7),(5,9),(7,12),(8,2),(8,12),(9,1),(9,12),(10,6),(11,6),(12,10),(12,11)],13)
=> ? = 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 0
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => ([],5)
=> ?
=> ? = 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(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)
=> ? = 0
{{1},{2,4},{3},{5}}
=> [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)
=> ? = 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
=> ? = 0
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> 0
{{1},{2},{3,4},{5}}
=> [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)
=> 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? = 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 0
{{1},{2},{3},{4},{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)
=> 0
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 0
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => ([(1,4),(4,5),(5,2),(5,3)],6)
=> ([(0,3),(0,5),(1,7),(1,10),(2,7),(2,9),(3,8),(4,6),(4,11),(5,4),(5,8),(6,1),(6,2),(6,12),(7,13),(8,11),(9,13),(10,13),(11,12),(12,9),(12,10)],14)
=> ? = 0
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> ([(0,4),(0,6),(1,10),(2,7),(3,7),(4,8),(5,1),(5,9),(6,5),(6,8),(8,9),(9,10),(10,2),(10,3)],11)
=> ? = 0
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => ([(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,4),(0,5),(1,7),(2,10),(2,12),(3,9),(3,12),(4,8),(5,6),(5,8),(6,2),(6,3),(6,11),(8,11),(9,13),(10,13),(11,9),(11,10),(12,1),(12,13),(13,7)],14)
=> ? = 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
=> ([(0,3),(0,6),(1,9),(2,10),(3,7),(4,5),(4,12),(5,1),(5,8),(6,4),(6,7),(7,12),(8,9),(8,10),(9,11),(10,11),(12,2),(12,8)],13)
=> ? = 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => ([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,4),(0,5),(2,7),(2,10),(3,7),(3,9),(4,8),(5,6),(5,8),(6,2),(6,3),(6,11),(7,12),(8,11),(9,12),(10,12),(11,9),(11,10),(12,1)],13)
=> ? = 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => ([(1,5),(5,2),(5,3),(5,4)],6)
=> ([(0,2),(0,3),(1,4),(1,5),(1,6),(1,14),(2,13),(3,1),(3,13),(4,9),(4,11),(4,12),(5,8),(5,10),(5,12),(6,7),(6,10),(6,11),(7,15),(7,16),(8,15),(8,17),(9,16),(9,17),(10,15),(10,18),(11,16),(11,18),(12,17),(12,18),(13,14),(14,7),(14,8),(14,9),(15,19),(16,19),(17,19),(18,19)],20)
=> ? = 0
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 0
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
=> 0
{{1,2},{3},{4,5},{6}}
=> [2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
=> 0
{{1,2},{3},{4},{5,6}}
=> [2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> 0
{{1},{2,3,4},{5},{6}}
=> [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)
=> 0
{{1},{2,3},{4,5},{6}}
=> [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)
=> 0
{{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=> 0
{{1},{2,3},{4},{5},{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)
=> 0
{{1},{2},{3,4,5},{6}}
=> [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)
=> 0
{{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
=> 0
{{1},{2},{3,4},{5},{6}}
=> [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)
=> 0
{{1},{2},{3},{4,5},{6}}
=> [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)
=> 0
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> 0
{{1},{2},{3},{4},{5},{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)
=> 0
{{1},{2,3},{4},{5},{6},{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)
=> 0
{{1},{2},{3},{4,5},{6},{7}}
=> [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)
=> 0
Description
The number of join irreducibles minus the rank of a lattice.
A lattice is join-extremal, if this statistic is $0$.
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000068The number of minimal elements in a poset. St001862The number of crossings of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001490The number of connected components of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001927Sparre Andersen's number of positives of a signed permutation. St001768The number of reduced words of a signed permutation. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone.
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