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Your data matches 103 different statistics following compositions of up to 3 maps.
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Matching statistic: St001880
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(load all 16 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
{{1},{2,3,4},{5,6}}
=> [1,3,4,2,6,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
{{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
{{1},{2,4},{3},{5,6}}
=> [1,4,3,2,6,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
{{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
{{1},{2,3,4,5},{6,7}}
=> [1,3,4,5,2,7,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
{{1},{2,3,4},{5},{6,7}}
=> [1,3,4,2,5,7,6] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
{{1},{2,3,5},{4},{6,7}}
=> [1,3,5,4,2,7,6] => [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 2
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 2
{{1},{2,3},{4},{5},{6,7}}
=> [1,3,2,4,5,7,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
{{1},{2,4,5},{3},{6,7}}
=> [1,4,3,5,2,7,6] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,4},{3,5},{6,7}}
=> [1,4,5,2,3,7,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
{{1},{2,4},{3},{5},{6,7}}
=> [1,4,3,2,5,7,6] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 2
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 2
{{1},{2,5},{3,4},{6,7}}
=> [1,5,4,3,2,7,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
{{1},{2},{3,4,5},{6,7}}
=> [1,2,4,5,3,7,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
{{1},{2},{3,4},{5},{6,7}}
=> [1,2,4,3,5,7,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
{{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 1
{{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 1
{{1},{2},{3,5},{4},{6,7}}
=> [1,2,5,4,3,7,6] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 2
{{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 2
{{1},{2},{3},{4,5},{6,7}}
=> [1,2,3,5,4,7,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
{{1},{2},{3},{4,5},{6},{7}}
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
{{1},{2},{3},{4},{5},{6,7}}
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
{{1},{2},{3,5},{4,6},{7},{8}}
=> [1,2,5,6,3,4,7,8] => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> 5
{{1},{2,4},{3,5,6},{7,8}}
=> [1,4,5,2,6,3,8,7] => [1,4,5,2,6,3,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> 6
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000672
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 71%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 71%
Values
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 3
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 4
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 4
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 4
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 4
{{1},{2,3,4},{5,6}}
=> [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => ? = 4
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ? = 4
{{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 5
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 5
{{1},{2,4},{3},{5,6}}
=> [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 1
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 1
{{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 5
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 5
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 5
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => 5
{{1},{2,3,4,5},{6,7}}
=> [1,3,4,5,2,7,6] => [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,1,5,2,3,4,8,6] => ? = 4
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [8,1,5,2,3,4,6,7] => ? = 4
{{1},{2,3,4},{5},{6,7}}
=> [1,3,4,2,5,7,6] => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [7,1,4,2,3,5,8,6] => ? = 5
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [8,1,4,2,3,5,6,7] => ? = 5
{{1},{2,3,5},{4},{6,7}}
=> [1,3,5,4,2,7,6] => [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,1,4,2,7,3,8,6] => ? = 2
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,1,4,2,8,3,6,7] => ? = 2
{{1},{2,3},{4},{5},{6,7}}
=> [1,3,2,4,5,7,6] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => ? = 6
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => ? = 6
{{1},{2,4,5},{3},{6,7}}
=> [1,4,3,5,2,7,6] => [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,7,5,2,4,8,6] => ? = 2
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,8,5,2,4,6,7] => ? = 2
{{1},{2,4},{3,5},{6,7}}
=> [1,4,5,2,3,7,6] => [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,1,4,5,2,3,8,6] => ? = 4
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [8,1,4,5,2,3,6,7] => ? = 4
{{1},{2,4},{3},{5},{6,7}}
=> [1,4,3,2,5,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => ? = 2
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,4,8,2,5,6,7] => ? = 2
{{1},{2,5},{3,4},{6,7}}
=> [1,5,4,3,2,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ? = 1
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,4,5,8,2,6,7] => ? = 1
{{1},{2},{3,4,5},{6,7}}
=> [1,2,4,5,3,7,6] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,1,2,5,3,4,8,6] => ? = 5
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [8,1,2,5,3,4,6,7] => ? = 5
{{1},{2},{3,4},{5},{6,7}}
=> [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => ? = 6
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => ? = 6
{{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ? = 1
{{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,4,5,8,2,6,7] => ? = 1
{{1},{2},{3,5},{4},{6,7}}
=> [1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => ? = 2
{{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => ? = 2
{{1},{2},{3},{4,5},{6,7}}
=> [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => ? = 6
{{1},{2},{3},{4,5},{6},{7}}
=> [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => ? = 6
{{1},{2},{3},{4},{5},{6,7}}
=> [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => 6
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 6
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => 7
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => 7
{{1},{2},{3,5},{4,6},{7},{8}}
=> [1,2,5,6,3,4,7,8] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [9,1,2,5,6,3,4,7,8] => ? = 5
{{1},{2,4},{3,5,6},{7,8}}
=> [1,4,5,2,6,3,8,7] => [1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [8,1,4,6,2,3,5,9,7] => ? = 6
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000018
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 71%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 71%
Values
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 4 = 3 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 4 = 3 + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 5 = 4 + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 5 = 4 + 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 5 = 4 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 5 = 4 + 1
{{1},{2,3,4},{5,6}}
=> [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => ? = 4 + 1
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ? = 4 + 1
{{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 5 + 1
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 5 + 1
{{1},{2,4},{3},{5,6}}
=> [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 1 + 1
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 1 + 1
{{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 5 + 1
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 5 + 1
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 6 = 5 + 1
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => 6 = 5 + 1
{{1},{2,3,4,5},{6,7}}
=> [1,3,4,5,2,7,6] => [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,1,5,2,3,4,8,6] => ? = 4 + 1
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [8,1,5,2,3,4,6,7] => ? = 4 + 1
{{1},{2,3,4},{5},{6,7}}
=> [1,3,4,2,5,7,6] => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [7,1,4,2,3,5,8,6] => ? = 5 + 1
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [8,1,4,2,3,5,6,7] => ? = 5 + 1
{{1},{2,3,5},{4},{6,7}}
=> [1,3,5,4,2,7,6] => [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,1,4,2,7,3,8,6] => ? = 2 + 1
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,1,4,2,8,3,6,7] => ? = 2 + 1
{{1},{2,3},{4},{5},{6,7}}
=> [1,3,2,4,5,7,6] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => ? = 6 + 1
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => ? = 6 + 1
{{1},{2,4,5},{3},{6,7}}
=> [1,4,3,5,2,7,6] => [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,7,5,2,4,8,6] => ? = 2 + 1
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,8,5,2,4,6,7] => ? = 2 + 1
{{1},{2,4},{3,5},{6,7}}
=> [1,4,5,2,3,7,6] => [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,1,4,5,2,3,8,6] => ? = 4 + 1
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [8,1,4,5,2,3,6,7] => ? = 4 + 1
{{1},{2,4},{3},{5},{6,7}}
=> [1,4,3,2,5,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => ? = 2 + 1
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,4,8,2,5,6,7] => ? = 2 + 1
{{1},{2,5},{3,4},{6,7}}
=> [1,5,4,3,2,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ? = 1 + 1
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,4,5,8,2,6,7] => ? = 1 + 1
{{1},{2},{3,4,5},{6,7}}
=> [1,2,4,5,3,7,6] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,1,2,5,3,4,8,6] => ? = 5 + 1
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [8,1,2,5,3,4,6,7] => ? = 5 + 1
{{1},{2},{3,4},{5},{6,7}}
=> [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => ? = 6 + 1
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => ? = 6 + 1
{{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ? = 1 + 1
{{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,4,5,8,2,6,7] => ? = 1 + 1
{{1},{2},{3,5},{4},{6,7}}
=> [1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => ? = 2 + 1
{{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => ? = 2 + 1
{{1},{2},{3},{4,5},{6,7}}
=> [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => ? = 6 + 1
{{1},{2},{3},{4,5},{6},{7}}
=> [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => ? = 6 + 1
{{1},{2},{3},{4},{5},{6,7}}
=> [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => 7 = 6 + 1
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 7 = 6 + 1
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => 8 = 7 + 1
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => 8 = 7 + 1
{{1},{2},{3,5},{4,6},{7},{8}}
=> [1,2,5,6,3,4,7,8] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [9,1,2,5,6,3,4,7,8] => ? = 5 + 1
{{1},{2,4},{3,5,6},{7,8}}
=> [1,4,5,2,6,3,8,7] => [1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [8,1,4,6,2,3,5,9,7] => ? = 6 + 1
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000019
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 71%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 71%
Values
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 4 = 3 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 4 = 3 + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 5 = 4 + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 5 = 4 + 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 5 = 4 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 5 = 4 + 1
{{1},{2,3,4},{5,6}}
=> [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => ? = 4 + 1
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ? = 4 + 1
{{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 5 + 1
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 5 + 1
{{1},{2,4},{3},{5,6}}
=> [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 1 + 1
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 1 + 1
{{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 5 + 1
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 5 + 1
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 6 = 5 + 1
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => 6 = 5 + 1
{{1},{2,3,4,5},{6,7}}
=> [1,3,4,5,2,7,6] => [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,1,5,2,3,4,8,6] => ? = 4 + 1
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [8,1,5,2,3,4,6,7] => ? = 4 + 1
{{1},{2,3,4},{5},{6,7}}
=> [1,3,4,2,5,7,6] => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [7,1,4,2,3,5,8,6] => ? = 5 + 1
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [8,1,4,2,3,5,6,7] => ? = 5 + 1
{{1},{2,3,5},{4},{6,7}}
=> [1,3,5,4,2,7,6] => [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,1,4,2,7,3,8,6] => ? = 2 + 1
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,1,4,2,8,3,6,7] => ? = 2 + 1
{{1},{2,3},{4},{5},{6,7}}
=> [1,3,2,4,5,7,6] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => ? = 6 + 1
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => ? = 6 + 1
{{1},{2,4,5},{3},{6,7}}
=> [1,4,3,5,2,7,6] => [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,7,5,2,4,8,6] => ? = 2 + 1
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,8,5,2,4,6,7] => ? = 2 + 1
{{1},{2,4},{3,5},{6,7}}
=> [1,4,5,2,3,7,6] => [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,1,4,5,2,3,8,6] => ? = 4 + 1
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [8,1,4,5,2,3,6,7] => ? = 4 + 1
{{1},{2,4},{3},{5},{6,7}}
=> [1,4,3,2,5,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => ? = 2 + 1
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,4,8,2,5,6,7] => ? = 2 + 1
{{1},{2,5},{3,4},{6,7}}
=> [1,5,4,3,2,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ? = 1 + 1
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,4,5,8,2,6,7] => ? = 1 + 1
{{1},{2},{3,4,5},{6,7}}
=> [1,2,4,5,3,7,6] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,1,2,5,3,4,8,6] => ? = 5 + 1
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [8,1,2,5,3,4,6,7] => ? = 5 + 1
{{1},{2},{3,4},{5},{6,7}}
=> [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => ? = 6 + 1
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => ? = 6 + 1
{{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ? = 1 + 1
{{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,4,5,8,2,6,7] => ? = 1 + 1
{{1},{2},{3,5},{4},{6,7}}
=> [1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => ? = 2 + 1
{{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => ? = 2 + 1
{{1},{2},{3},{4,5},{6,7}}
=> [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => ? = 6 + 1
{{1},{2},{3},{4,5},{6},{7}}
=> [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => ? = 6 + 1
{{1},{2},{3},{4},{5},{6,7}}
=> [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => 7 = 6 + 1
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 7 = 6 + 1
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => 8 = 7 + 1
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => 8 = 7 + 1
{{1},{2},{3,5},{4,6},{7},{8}}
=> [1,2,5,6,3,4,7,8] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [9,1,2,5,6,3,4,7,8] => ? = 5 + 1
{{1},{2,4},{3,5,6},{7,8}}
=> [1,4,5,2,6,3,8,7] => [1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [8,1,4,6,2,3,5,9,7] => ? = 6 + 1
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
Matching statistic: St001004
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 71%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 71%
Values
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 5 = 3 + 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 5 = 3 + 2
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 6 = 4 + 2
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 6 = 4 + 2
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 6 = 4 + 2
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 6 = 4 + 2
{{1},{2,3,4},{5,6}}
=> [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => ? = 4 + 2
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ? = 4 + 2
{{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 5 + 2
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 5 + 2
{{1},{2,4},{3},{5,6}}
=> [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 1 + 2
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 1 + 2
{{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 5 + 2
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 5 + 2
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 7 = 5 + 2
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => 7 = 5 + 2
{{1},{2,3,4,5},{6,7}}
=> [1,3,4,5,2,7,6] => [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,1,5,2,3,4,8,6] => ? = 4 + 2
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [8,1,5,2,3,4,6,7] => ? = 4 + 2
{{1},{2,3,4},{5},{6,7}}
=> [1,3,4,2,5,7,6] => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [7,1,4,2,3,5,8,6] => ? = 5 + 2
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [8,1,4,2,3,5,6,7] => ? = 5 + 2
{{1},{2,3,5},{4},{6,7}}
=> [1,3,5,4,2,7,6] => [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,1,4,2,7,3,8,6] => ? = 2 + 2
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,1,4,2,8,3,6,7] => ? = 2 + 2
{{1},{2,3},{4},{5},{6,7}}
=> [1,3,2,4,5,7,6] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => ? = 6 + 2
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => ? = 6 + 2
{{1},{2,4,5},{3},{6,7}}
=> [1,4,3,5,2,7,6] => [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,7,5,2,4,8,6] => ? = 2 + 2
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,8,5,2,4,6,7] => ? = 2 + 2
{{1},{2,4},{3,5},{6,7}}
=> [1,4,5,2,3,7,6] => [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,1,4,5,2,3,8,6] => ? = 4 + 2
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [8,1,4,5,2,3,6,7] => ? = 4 + 2
{{1},{2,4},{3},{5},{6,7}}
=> [1,4,3,2,5,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => ? = 2 + 2
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,4,8,2,5,6,7] => ? = 2 + 2
{{1},{2,5},{3,4},{6,7}}
=> [1,5,4,3,2,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ? = 1 + 2
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,4,5,8,2,6,7] => ? = 1 + 2
{{1},{2},{3,4,5},{6,7}}
=> [1,2,4,5,3,7,6] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,1,2,5,3,4,8,6] => ? = 5 + 2
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [8,1,2,5,3,4,6,7] => ? = 5 + 2
{{1},{2},{3,4},{5},{6,7}}
=> [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => ? = 6 + 2
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => ? = 6 + 2
{{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ? = 1 + 2
{{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,4,5,8,2,6,7] => ? = 1 + 2
{{1},{2},{3,5},{4},{6,7}}
=> [1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => ? = 2 + 2
{{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => ? = 2 + 2
{{1},{2},{3},{4,5},{6,7}}
=> [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => ? = 6 + 2
{{1},{2},{3},{4,5},{6},{7}}
=> [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => ? = 6 + 2
{{1},{2},{3},{4},{5},{6,7}}
=> [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => 8 = 6 + 2
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 8 = 6 + 2
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => 9 = 7 + 2
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => 9 = 7 + 2
{{1},{2},{3,5},{4,6},{7},{8}}
=> [1,2,5,6,3,4,7,8] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [9,1,2,5,6,3,4,7,8] => ? = 5 + 2
{{1},{2,4},{3,5,6},{7,8}}
=> [1,4,5,2,6,3,8,7] => [1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [8,1,4,6,2,3,5,9,7] => ? = 6 + 2
Description
The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
Matching statistic: St000924
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000924: Perfect matchings ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 43%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000924: Perfect matchings ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 43%
Values
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 4 = 3 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 4 = 3 + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 5 = 4 + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 5 = 4 + 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 5 = 4 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 5 = 4 + 1
{{1},{2,3,4},{5,6}}
=> [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> ? = 4 + 1
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> ? = 4 + 1
{{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> ? = 5 + 1
{{1},{2,4},{3},{5,6}}
=> [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 1 + 1
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> ? = 1 + 1
{{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 6 = 5 + 1
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 6 = 5 + 1
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> 6 = 5 + 1
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> 6 = 5 + 1
{{1},{2,3,4,5},{6,7}}
=> [1,3,4,5,2,7,6] => [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,14),(12,13)]
=> ? = 4 + 1
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12),(13,14)]
=> ? = 4 + 1
{{1},{2,3,4},{5},{6,7}}
=> [1,3,4,2,5,7,6] => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,14),(12,13)]
=> ? = 5 + 1
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12),(13,14)]
=> ? = 5 + 1
{{1},{2,3,5},{4},{6,7}}
=> [1,3,5,4,2,7,6] => [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2 + 1
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12),(13,14)]
=> ? = 2 + 1
{{1},{2,3},{4},{5},{6,7}}
=> [1,3,2,4,5,7,6] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,14),(12,13)]
=> ? = 6 + 1
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6 + 1
{{1},{2,4,5},{3},{6,7}}
=> [1,4,3,5,2,7,6] => [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9),(11,14),(12,13)]
=> ? = 2 + 1
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9),(11,12),(13,14)]
=> ? = 2 + 1
{{1},{2,4},{3,5},{6,7}}
=> [1,4,5,2,3,7,6] => [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8),(11,14),(12,13)]
=> ? = 4 + 1
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8),(11,12),(13,14)]
=> ? = 4 + 1
{{1},{2,4},{3},{5},{6,7}}
=> [1,4,3,2,5,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,14),(12,13)]
=> ? = 2 + 1
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12),(13,14)]
=> ? = 2 + 1
{{1},{2,5},{3,4},{6,7}}
=> [1,5,4,3,2,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,14),(12,13)]
=> ? = 1 + 1
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12),(13,14)]
=> ? = 1 + 1
{{1},{2},{3,4,5},{6,7}}
=> [1,2,4,5,3,7,6] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,14),(12,13)]
=> ? = 5 + 1
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12),(13,14)]
=> ? = 5 + 1
{{1},{2},{3,4},{5},{6,7}}
=> [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> ? = 6 + 1
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12),(13,14)]
=> ? = 6 + 1
{{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,14),(12,13)]
=> ? = 1 + 1
{{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12),(13,14)]
=> ? = 1 + 1
{{1},{2},{3,5},{4},{6,7}}
=> [1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2 + 1
{{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> ? = 2 + 1
{{1},{2},{3},{4,5},{6,7}}
=> [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,14),(12,13)]
=> ? = 6 + 1
{{1},{2},{3},{4,5},{6},{7}}
=> [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 6 + 1
{{1},{2},{3},{4},{5},{6,7}}
=> [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> ? = 6 + 1
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6 + 1
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
=> ? = 7 + 1
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> ? = 7 + 1
{{1},{2},{3,5},{4,6},{7},{8}}
=> [1,2,5,6,3,4,7,8] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10),(13,14),(15,16)]
=> ? = 5 + 1
{{1},{2,4},{3,5,6},{7,8}}
=> [1,4,5,2,6,3,8,7] => [1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11),(13,16),(14,15)]
=> ? = 6 + 1
Description
The number of topologically connected components of a perfect matching.
For example, the perfect matching $\{\{1,4\},\{2,3\}\}$ has the two connected components $\{1,4\}$ and $\{2,3\}$.
The number of perfect matchings with only one block is [[oeis:A000699]].
Matching statistic: St001645
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Values
{{1},{2},{3,4}}
=> [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
{{1},{2,3},{4,5}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
{{1},{2,3,4},{5,6}}
=> [1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
{{1},{2,3,4},{5},{6}}
=> [1,3,1,1] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
{{1},{2,3},{4},{5,6}}
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
{{1},{2,3},{4},{5},{6}}
=> [1,2,1,1,1] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
{{1},{2,4},{3},{5,6}}
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1},{2,4},{3},{5},{6}}
=> [1,2,1,1,1] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1},{2},{3,4},{5,6}}
=> [1,1,2,2] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
{{1},{2},{3,4},{5},{6}}
=> [1,1,2,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
{{1},{2},{3},{4},{5,6}}
=> [1,1,1,1,2] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
{{1},{2,3,4,5},{6,7}}
=> [1,4,2] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
{{1},{2,3,4,5},{6},{7}}
=> [1,4,1,1] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
{{1},{2,3,4},{5},{6,7}}
=> [1,3,1,2] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,1,1,1] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
{{1},{2,3,5},{4},{6,7}}
=> [1,3,1,2] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,1,1,1] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
{{1},{2,3},{4},{5},{6,7}}
=> [1,2,1,1,2] => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
{{1},{2,3},{4},{5},{6},{7}}
=> [1,2,1,1,1,1] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
{{1},{2,4,5},{3},{6,7}}
=> [1,3,1,2] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
{{1},{2,4,5},{3},{6},{7}}
=> [1,3,1,1,1] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
{{1},{2,4},{3,5},{6,7}}
=> [1,2,2,2] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
{{1},{2,4},{3,5},{6},{7}}
=> [1,2,2,1,1] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
{{1},{2,4},{3},{5},{6,7}}
=> [1,2,1,1,2] => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
{{1},{2,4},{3},{5},{6},{7}}
=> [1,2,1,1,1,1] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
{{1},{2,5},{3,4},{6,7}}
=> [1,2,2,2] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
{{1},{2,5},{3,4},{6},{7}}
=> [1,2,2,1,1] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
{{1},{2},{3,4,5},{6,7}}
=> [1,1,3,2] => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
{{1},{2},{3,4,5},{6},{7}}
=> [1,1,3,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
{{1},{2},{3,4},{5},{6,7}}
=> [1,1,2,1,2] => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
{{1},{2},{3,4},{5},{6},{7}}
=> [1,1,2,1,1,1] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
{{1},{2,5},{3},{4},{6,7}}
=> [1,2,1,1,2] => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
{{1},{2,5},{3},{4},{6},{7}}
=> [1,2,1,1,1,1] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
{{1},{2},{3,5},{4},{6,7}}
=> [1,1,2,1,2] => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
{{1},{2},{3,5},{4},{6},{7}}
=> [1,1,2,1,1,1] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
{{1},{2},{3},{4,5},{6,7}}
=> [1,1,1,2,2] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
{{1},{2},{3},{4,5},{6},{7}}
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
{{1},{2},{3},{4},{5},{6,7}}
=> [1,1,1,1,1,2] => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
{{1},{2},{3,5},{4,6},{7},{8}}
=> [1,1,2,2,1,1] => [1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
{{1},{2,4},{3,5,6},{7,8}}
=> [1,2,3,2] => [2,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St001879
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 57%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 57%
Values
{{1},{2},{3,4}}
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{1},{2,3,4},{5,6}}
=> [1,3,4,2,6,5] => [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 4
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 4
{{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 5
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 5
{{1},{2,4},{3},{5,6}}
=> [1,4,3,2,6,5] => [.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 1
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 1
{{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 5
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 5
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2,3,4,5},{6,7}}
=> [1,3,4,5,2,7,6] => [.,[[.,[.,[.,.]]],[[.,.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 4
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 4
{{1},{2,3,4},{5},{6,7}}
=> [1,3,4,2,5,7,6] => [.,[[.,[.,.]],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 5
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 5
{{1},{2,3,5},{4},{6,7}}
=> [1,3,5,4,2,7,6] => [.,[[.,[[.,.],.]],[[.,.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => [.,[[.,[[.,.],.]],[.,[.,.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
{{1},{2,3},{4},{5},{6,7}}
=> [1,3,2,4,5,7,6] => [.,[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 6
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 6
{{1},{2,4,5},{3},{6,7}}
=> [1,4,3,5,2,7,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ? = 2
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ? = 2
{{1},{2,4},{3,5},{6,7}}
=> [1,4,5,2,3,7,6] => [.,[[.,[.,.]],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 4
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 4
{{1},{2,4},{3},{5},{6,7}}
=> [1,4,3,2,5,7,6] => [.,[[[.,.],.],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => [.,[[[.,.],.],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
{{1},{2,5},{3,4},{6,7}}
=> [1,5,4,3,2,7,6] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 1
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 1
{{1},{2},{3,4,5},{6,7}}
=> [1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 5
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 5
{{1},{2},{3,4},{5},{6,7}}
=> [1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 6
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 6
{{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ? = 1
{{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ? = 1
{{1},{2},{3,5},{4},{6,7}}
=> [1,2,5,4,3,7,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
{{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
{{1},{2},{3},{4,5},{6,7}}
=> [1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 6
{{1},{2},{3},{4,5},{6},{7}}
=> [1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 6
{{1},{2},{3},{4},{5},{6,7}}
=> [1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 7
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,2,3,4,5,6,8,7] => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 7
{{1},{2},{3,5},{4,6},{7},{8}}
=> [1,2,5,6,3,4,7,8] => [.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,7),(4,7),(5,2),(6,3),(7,5)],8)
=> ? = 5
{{1},{2,4},{3,5,6},{7,8}}
=> [1,4,5,2,6,3,8,7] => [.,[[.,[.,.]],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,3),(2,5),(3,7),(5,6),(6,7),(7,4)],8)
=> ? = 6
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001514
(load all 44 compositions to match this statistic)
(load all 44 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001514: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 29%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001514: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 29%
Values
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
{{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 4
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 4
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 4
{{1},{2,3,4},{5,6}}
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 4
{{1},{2,3,4},{5},{6}}
=> [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 4
{{1},{2,3},{4},{5,6}}
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 5
{{1},{2,3},{4},{5},{6}}
=> [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 5
{{1},{2,4},{3},{5,6}}
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
{{1},{2,4},{3},{5},{6}}
=> [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
{{1},{2},{3,4},{5,6}}
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 5
{{1},{2},{3,4},{5},{6}}
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 5
{{1},{2},{3},{4},{5,6}}
=> [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 5
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
{{1},{2,3,4,5},{6,7}}
=> [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4
{{1},{2,3,4,5},{6},{7}}
=> [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4
{{1},{2,3,4},{5},{6,7}}
=> [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 5
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 5
{{1},{2,3,5},{4},{6,7}}
=> [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2
{{1},{2,3},{4},{5},{6,7}}
=> [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 6
{{1},{2,3},{4},{5},{6},{7}}
=> [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 6
{{1},{2,4,5},{3},{6,7}}
=> [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2
{{1},{2,4,5},{3},{6},{7}}
=> [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2
{{1},{2,4},{3,5},{6,7}}
=> [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
{{1},{2,4},{3,5},{6},{7}}
=> [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4
{{1},{2,4},{3},{5},{6,7}}
=> [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
{{1},{2,4},{3},{5},{6},{7}}
=> [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
{{1},{2,5},{3,4},{6,7}}
=> [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1},{2,5},{3,4},{6},{7}}
=> [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
{{1},{2},{3,4,5},{6,7}}
=> [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
{{1},{2},{3,4,5},{6},{7}}
=> [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5
{{1},{2},{3,4},{5},{6,7}}
=> [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 6
{{1},{2},{3,4},{5},{6},{7}}
=> [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
{{1},{2,5},{3},{4},{6,7}}
=> [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1},{2,5},{3},{4},{6},{7}}
=> [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1},{2},{3,5},{4},{6,7}}
=> [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
{{1},{2},{3,5},{4},{6},{7}}
=> [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
{{1},{2},{3},{4,5},{6,7}}
=> [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 6
{{1},{2},{3},{4,5},{6},{7}}
=> [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6
{{1},{2},{3},{4},{5},{6,7}}
=> [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7
{{1},{2},{3,5},{4,6},{7},{8}}
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 5
{{1},{2,4},{3,5,6},{7,8}}
=> [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6
Description
The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.
Matching statistic: St001948
(load all 46 compositions to match this statistic)
(load all 46 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 29%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 29%
Values
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 3
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,2,3,4,5] => 4
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,2,3,4,5] => 4
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,4,5] => 4
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => 4
{{1},{2,3,4},{5,6}}
=> [1,3,4,2,6,5] => [1,2,3,4,5,6] => ? = 4
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [1,2,3,4,5,6] => ? = 4
{{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => [1,2,3,4,5,6] => ? = 5
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => [1,2,3,4,5,6] => ? = 5
{{1},{2,4},{3},{5,6}}
=> [1,4,3,2,6,5] => [1,2,4,3,5,6] => ? = 1
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => [1,2,4,3,5,6] => ? = 1
{{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => [1,2,3,4,5,6] => ? = 5
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => [1,2,3,4,5,6] => ? = 5
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => [1,2,3,4,5,6] => ? = 5
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 5
{{1},{2,3,4,5},{6,7}}
=> [1,3,4,5,2,7,6] => [1,2,3,4,5,6,7] => ? = 4
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,2,3,4,5,6,7] => ? = 4
{{1},{2,3,4},{5},{6,7}}
=> [1,3,4,2,5,7,6] => [1,2,3,4,5,6,7] => ? = 5
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => [1,2,3,4,5,6,7] => ? = 5
{{1},{2,3,5},{4},{6,7}}
=> [1,3,5,4,2,7,6] => [1,2,3,5,4,6,7] => ? = 2
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => [1,2,3,5,4,6,7] => ? = 2
{{1},{2,3},{4},{5},{6,7}}
=> [1,3,2,4,5,7,6] => [1,2,3,4,5,6,7] => ? = 6
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6
{{1},{2,4,5},{3},{6,7}}
=> [1,4,3,5,2,7,6] => [1,2,4,5,3,6,7] => ? = 2
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => [1,2,4,5,3,6,7] => ? = 2
{{1},{2,4},{3,5},{6,7}}
=> [1,4,5,2,3,7,6] => [1,2,4,3,5,6,7] => ? = 4
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => [1,2,4,3,5,6,7] => ? = 4
{{1},{2,4},{3},{5},{6,7}}
=> [1,4,3,2,5,7,6] => [1,2,4,3,5,6,7] => ? = 2
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => [1,2,4,3,5,6,7] => ? = 2
{{1},{2,5},{3,4},{6,7}}
=> [1,5,4,3,2,7,6] => [1,2,5,3,4,6,7] => ? = 1
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => [1,2,5,3,4,6,7] => ? = 1
{{1},{2},{3,4,5},{6,7}}
=> [1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => ? = 5
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => ? = 5
{{1},{2},{3,4},{5},{6,7}}
=> [1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => ? = 6
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => ? = 6
{{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => [1,2,5,3,4,6,7] => ? = 1
{{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => [1,2,5,3,4,6,7] => ? = 1
{{1},{2},{3,5},{4},{6,7}}
=> [1,2,5,4,3,7,6] => [1,2,3,5,4,6,7] => ? = 2
{{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => [1,2,3,5,4,6,7] => ? = 2
{{1},{2},{3},{4,5},{6,7}}
=> [1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => ? = 6
{{1},{2},{3},{4,5},{6},{7}}
=> [1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => ? = 6
{{1},{2},{3},{4},{5},{6,7}}
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => ? = 6
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 6
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? = 7
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,7,8] => ? = 7
{{1},{2},{3,5},{4,6},{7},{8}}
=> [1,2,5,6,3,4,7,8] => [1,2,3,5,4,6,7,8] => ? = 5
{{1},{2,4},{3,5,6},{7,8}}
=> [1,4,5,2,6,3,8,7] => [1,2,4,3,5,6,7,8] => ? = 6
Description
The number of augmented double ascents of a permutation.
An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$.
A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.
The following 93 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000197The number of entries equal to positive one in the alternating sign matrix. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St000327The number of cover relations in a poset. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001430The number of positive entries in a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001684The reduced word complexity of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001925The minimal number of zeros in a row of an alternating sign matrix. St001927Sparre Andersen's number of positives of a signed permutation. St000029The depth of a permutation. St000134The size of the orbit of an alternating sign matrix under gyration. St000144The pyramid weight of the Dyck path. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000809The reduced reflection length of the permutation. St000890The number of nonzero entries in an alternating sign matrix. St000957The number of Bruhat lower covers of a permutation. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001117The game chromatic index of a graph. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001429The number of negative entries in a signed permutation. St001480The number of simple summands of the module J^2/J^3. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001817The number of flag weak exceedances of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001863The number of weak excedances of a signed permutation. St001869The maximum cut size of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001889The size of the connectivity set of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001896The number of right descents of a signed permutations. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001958The degree of the polynomial interpolating the values of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St000958The number of Bruhat factorizations of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001569The maximal modular displacement of a permutation. St001725The harmonious chromatic number of a graph. St001769The reflection length of a signed permutation. St001834The number of non-isomorphic minors of a graph. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001964The interval resolution global dimension of a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001468The smallest fixpoint of a permutation. St001856The number of edges in the reduced word graph of a permutation. St000528The height of a poset. St001330The hat guessing number of a graph. St000080The rank of the poset. St000906The length of the shortest maximal chain in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001782The order of rowmotion on the set of order ideals of a poset. St000454The largest eigenvalue of a graph if it is integral. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000264The girth of a graph, which is not a tree.
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