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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St001279
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 2
[1,1]
=> 0
[3]
=> 3
[2,1]
=> 2
[1,1,1]
=> 0
[4]
=> 4
[3,1]
=> 3
[2,2]
=> 4
[2,1,1]
=> 2
[1,1,1,1]
=> 0
[5]
=> 5
[4,1]
=> 4
[3,2]
=> 5
[3,1,1]
=> 3
[2,2,1]
=> 4
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 0
[6]
=> 6
[5,1]
=> 5
[4,2]
=> 6
[4,1,1]
=> 4
[3,3]
=> 6
[3,2,1]
=> 5
[3,1,1,1]
=> 3
[2,2,2]
=> 6
[2,2,1,1]
=> 4
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 0
[7]
=> 7
[6,1]
=> 6
[5,2]
=> 7
[5,1,1]
=> 5
[4,3]
=> 7
[4,2,1]
=> 6
[4,1,1,1]
=> 4
[3,3,1]
=> 6
[3,2,2]
=> 7
[3,2,1,1]
=> 5
[3,1,1,1,1]
=> 3
[2,2,2,1]
=> 6
[2,2,1,1,1]
=> 4
[2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> 0
[8]
=> 8
[7,1]
=> 7
[6,2]
=> 8
[6,1,1]
=> 6
[5,3]
=> 8
[5,2,1]
=> 7
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St000915
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000915: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 75%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000915: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 75%
Values
[1]
=> [1,0]
=> [1] => ([],1)
=> 0
[2]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> 0
[1,1]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 2
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 4
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 4
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 4
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ([(5,6)],7)
=> 2
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 4
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> 3
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ([(3,6),(4,6),(5,6)],7)
=> 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 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] => ([],8)
=> ? ∊ {0,2,3,4,5,6,7,8}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? ∊ {0,2,3,4,5,6,7,8}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> 4
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,5,6,7,8}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => ([(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,5,6,7,8}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5,3,4,6,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,5,6,7,8}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,5,6,7,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,5,6,7,8}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,5,6,7,8}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,8,9,9,9}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,8,9,9,9}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,8,9,9,9}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => ([(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,8,9,9,9}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,6,8,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,8,9,9,9}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,5,7,8,9,6] => ([(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,8,9,9,9}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,8,9,9,9}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,9,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,8,9,9,9}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,8,9,9,9}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => ([(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,8,9,9,9}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,8,9,9,9}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,9,3] => ([(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,8,9,9,9}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,8,9,9,9}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,8,9,9,9}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,8,9,9,9}
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => ([],10)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => ([(8,9)],10)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,9,8,7] => ([(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => ([(7,9),(8,9)],10)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,8,6,7,5] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,8,7,9,6] => ([(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,10,7] => ([(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,7,5,6,8,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,7,6,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,4,7,6,8,9,5] => ([(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,5,7,8,9,10,6] => ([(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,6,4,5,7,8,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,6,5,8,4] => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,9,4] => ([(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,9,10,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,5,3,4,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,6,5,4,7,8,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,9,3] => ([(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,10,4] => ([(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,5,4,3,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,8,9,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,9,10,3] => ([(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,10,2] => ([(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,9,9,9,9,10,10,10,10,10}
Description
The Ore degree of a graph.
This is the maximal Ore degree of an edge, which is the sum of the degrees of its two endpoints.
Matching statistic: St001879
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 31%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 31%
Values
[1]
=> [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? = 0
[2]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 0
[1,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0
[2,1]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,2,4}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,2,4}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,2,4}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {0,2,4,5}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,2,4,5}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,2,4,5}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {0,2,4,5}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[[[[[.,.],.],.],.],.],[.,.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? ∊ {0,2,4,5,6,6,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {0,2,4,5,6,6,6}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? ∊ {0,2,4,5,6,6,6}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,2,4,5,6,6,6}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,5,6,6,6}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {0,2,4,5,6,6,6}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,2,4,5,6,6,6}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? ∊ {0,2,3,5,6,6,6,7,7,7,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[[[[.,[.,.]],.],.],.],[.,.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? ∊ {0,2,3,5,6,6,6,7,7,7,7}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? ∊ {0,2,3,5,6,6,6,7,7,7,7}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[[.,[[.,.],.]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {0,2,3,5,6,6,6,7,7,7,7}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,2,3,5,6,6,6,7,7,7,7}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,2,3,5,6,6,6,7,7,7,7}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,2,3,5,6,6,6,7,7,7,7}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,2,3,5,6,6,6,7,7,7,7}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,2,3,5,6,6,6,7,7,7,7}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[[[[.,.],[.,.]],.],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? ∊ {0,2,3,5,6,6,6,7,7,7,7}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[[[[[.,[.,.]],.],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {0,2,3,5,6,6,6,7,7,7,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[[[[.,.],[.,.]],.],.],[.,.]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[[[.,[[.,.],.]],.],.],[.,.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[.,[[[.,.],.],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[[[.,.],.],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [.,[[[[[.,.],[.,.]],.],.],.]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [.,[[[[.,[[.,.],.]],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {0,2,3,4,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? ∊ {0,2,3,4,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? ∊ {0,2,3,4,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(5,4),(6,3)],8)
=> ? ∊ {0,2,3,4,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[[[[.,[[.,.],.]],.],.],.],[.,.]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? ∊ {0,2,3,4,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [.,[[[[.,[.,[.,.]]],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [.,[[[.,[[[.,.],.],.]],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [.,[[[.,[[.,[.,.]],.]],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [.,[[.,[[[[.,.],.],.],.]],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [.,[[[.,[.,[[.,.],.]]],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [.,[[.,[[[.,[.,.]],.],.]],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [.,[[[.,[.,[.,[.,.]]]],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [.,[[.,[[.,[[.,.],.]],.]],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[.,[[[[.,[.,.]],.],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[4,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [.,[[.,[[.,[.,[.,.]]],.]],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[3,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [.,[[.,[.,[[[.,.],.],.]]],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[[.,[[.,.],.]],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[4,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [.,[[.,[.,[[.,[.,.]],.]]],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[4,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[[[.,[.,[.,.]]],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[3,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [.,[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[4,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [.,[[.,[.,[.,[[.,.],.]]]],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 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: St001232
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 69%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 69%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {2,3}
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,3}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {0,4,4}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,4,4}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {0,4,4}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? ∊ {0,2,4,4,5,5}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {0,2,4,4,5,5}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {0,2,4,4,5,5}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,4,4,5,5}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,2,4,4,5,5}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {0,2,4,4,5,5}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,6}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,6}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {2,3,4,4,5,6,6,6,6}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,6}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,6}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,6}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,4,5,5,6,6,6,6,7,7,7,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? ∊ {0,4,5,5,6,6,6,6,7,7,7,7}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ? ∊ {0,4,5,5,6,6,6,6,7,7,7,7}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {0,4,5,5,6,6,6,6,7,7,7,7}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,4,5,5,6,6,6,6,7,7,7,7}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,4,5,5,6,6,6,6,7,7,7,7}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,4,5,5,6,6,6,6,7,7,7,7}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ? ∊ {0,4,5,5,6,6,6,6,7,7,7,7}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,4,5,5,6,6,6,6,7,7,7,7}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {0,4,5,5,6,6,6,6,7,7,7,7}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? ∊ {0,4,5,5,6,6,6,6,7,7,7,7}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,4,5,5,6,6,6,6,7,7,7,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? ∊ {0,2,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 5
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 8
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 6
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 7
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> 11
[4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[5,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> 5
[5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[4,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> 10
[4,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> 8
[5,5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> 6
[5,5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> 8
[5,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> 7
[5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[5,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> 10
[5,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> 7
[5,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> 8
[5,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 9
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[6,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 2
[5,5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> 7
[5,5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[5,4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> 4
[5,4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000718
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 44%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 44%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> 0
[2]
=> [[1,2]]
=> [1,2] => ([],2)
=> 0
[1,1]
=> [[1],[2]]
=> [2,1] => ([(0,1)],2)
=> 2
[3]
=> [[1,2,3]]
=> [1,2,3] => ([],3)
=> 0
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => ([(1,2)],3)
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => ([],4)
=> 0
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([],5)
=> 0
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 2
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 5
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([],6)
=> 0
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> 2
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 3
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,6}
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,6}
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([(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
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => ([],7)
=> 0
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ([(5,6)],7)
=> 2
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7)
=> 3
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 6
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? ∊ {5,6,7,7,7}
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {5,6,7,7,7}
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 6
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {5,6,7,7,7}
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {5,6,7,7,7}
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ([(0,1),(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {5,6,7,7,7}
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ([(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
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ([(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
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ([],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ([(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => ([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => ([(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => ([(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => ([(0,1),(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => ([(0,2),(0,3),(0,7),(1,2),(1,3),(1,7),(2,6),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => ([(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => ([(1,2),(1,7),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => ([(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)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => ([(0,1),(0,7),(1,6),(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)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => ([(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)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ([(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)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => ([(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => ([(5,7),(5,8),(6,7),(6,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => ([(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => ([(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => ([(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => ([(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => ([(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => ([(2,7),(2,8),(3,5),(3,6),(3,8),(4,5),(4,6),(4,8),(5,7),(6,7),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => ([(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => ([(3,4),(3,8),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => ([(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => ([(0,7),(0,8),(1,4),(1,5),(1,6),(1,8),(2,4),(2,5),(2,6),(2,8),(3,4),(3,5),(3,6),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => ([(1,2),(1,7),(1,8),(2,5),(2,6),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => ([(1,3),(1,4),(1,8),(2,3),(2,4),(2,8),(3,7),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => ([(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => ([(2,3),(2,8),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => ([(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => ([(0,1),(0,6),(0,8),(1,5),(1,7),(2,3),(2,5),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
Description
The largest Laplacian eigenvalue of a graph if it is integral.
This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral.
Various results are collected in Section 3.9 of [1]
Matching statistic: St000422
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 44%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 44%
Values
[1]
=> [1,0]
=> [1] => ([],1)
=> 0
[2]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> 0
[1,1]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 2
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 4
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {3,4}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {3,4}
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 4
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {3,5,5}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,5,5}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {3,5,5}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 4
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {3,5,5,6,6,6}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,5,5,6,6,6}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,5,5,6,6,6}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {3,5,5,6,6,6}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,5,5,6,6,6}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {3,5,5,6,6,6}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ([(5,6)],7)
=> 2
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 4
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ? ∊ {3,5,5,6,6,6,7,7,7,7}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,5,5,6,6,6,7,7,7,7}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,5,5,6,6,6,7,7,7,7}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ([(3,6),(4,6),(5,6)],7)
=> ? ∊ {3,5,5,6,6,6,7,7,7,7}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,5,5,6,6,6,7,7,7,7}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,5,5,6,6,6,7,7,7,7}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,5,5,6,6,6,7,7,7,7}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,5,5,6,6,6,7,7,7,7}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {3,5,5,6,6,6,7,7,7,7}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {3,5,5,6,6,6,7,7,7,7}
[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] => ([],8)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> 4
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => ([(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5,3,4,6,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => ([(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,6,8,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,5,7,8,9,6] => ([(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,6,4,5,7,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 8
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => ([(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)
=> 10
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ([(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)
=> 10
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ([(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)
=> 12
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph Kn equals 2n−2. For this reason, we do not define the energy of the empty graph.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Values
[1]
=> [1,0,1,0]
=> [2,1] => ([],2)
=> ? = 0
[2]
=> [1,1,0,0,1,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {0,2}
[1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {0,2}
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {0,2,3}
[2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,2,3}
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {0,2,3}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ? ∊ {0,2,3,4}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,2,3,4}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,2,3,4}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {0,2,3,4}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> ? ∊ {0,2,3,4,4,5,5}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {0,2,3,4,4,5,5}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,2,3,4,4,5,5}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,2,3,4,4,5,5}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,2,3,4,4,5,5}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ? ∊ {0,2,3,4,4,5,5}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? ∊ {0,2,3,4,4,5,5}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> ? ∊ {0,2,3,4,4,6,6,6,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> ? ∊ {0,2,3,4,4,6,6,6,6}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ? ∊ {0,2,3,4,4,6,6,6,6}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {0,2,3,4,4,6,6,6,6}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {0,2,3,4,4,6,6,6,6}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,2,3,4,4,6,6,6,6}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,3,4,4,6,6,6,6}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
=> ? ∊ {0,2,3,4,4,6,6,6,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? ∊ {0,2,3,4,4,6,6,6,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,8,7] => ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,4),(3,5),(4,3),(5,2),(5,6)],7)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => ([(0,6),(1,3),(1,6),(4,2),(5,4),(6,5)],7)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,9,8] => ([(0,7),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(8,2)],9)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => ([(0,7),(1,5),(3,4),(4,6),(5,3),(6,2),(6,7)],8)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,6,2,3,4,7,5] => ([(0,2),(0,5),(2,6),(3,4),(4,1),(4,6),(5,3)],7)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,5,1,2,6,3] => ([(0,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,2,3,4,5,5,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,5,2,3,6,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> 6
[4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7
[3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7
[3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,6,2,4,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> 6
[5,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,2,5,3,6,4,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 7
[4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,5,2,4,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 7
[5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 6
[4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,4,2,5,3,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 7
[4,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 7
[3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> 6
[5,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [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
[4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 6
[4,4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> 6
[5,5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,4,2,5,6,3,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> 6
[5,5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 7
[4,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> 6
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001468
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001468: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 38%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001468: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 38%
Values
[1]
=> [[1]]
=> [1] => [1] => 1 = 0 + 1
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 1 = 0 + 1
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 3 = 2 + 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 3 = 2 + 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 4 = 3 + 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 3 = 2 + 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => 5 = 4 + 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => 4 = 3 + 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [3,4,1,2] => 5 = 4 + 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 3 = 2 + 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => 5 = 4 + 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => 4 = 3 + 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,5,4,3,1] => 6 = 5 + 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [3,4,1,2,5] => 5 = 4 + 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [3,4,5,1,2] => 6 = 5 + 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1 = 0 + 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 3 = 2 + 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [4,3,2,1,5,6] => 5 = 4 + 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,3,1,4,5,6] => 4 = 3 + 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [6,5,4,3,2,1] => 7 = 6 + 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,5,4,3,1,6] => 6 = 5 + 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [3,4,1,2,5,6] => 5 = 4 + 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,3,5,6,2,1] => 7 = 6 + 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,6,1,5,4,2] => 7 = 6 + 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [3,4,5,1,2,6] => 6 = 5 + 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [4,5,6,1,2,3] => 7 = 6 + 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 1 = 0 + 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [4,3,2,1,5,6,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [2,3,1,4,5,6,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [6,5,4,3,2,1,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,5,4,3,1,6,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [3,4,1,2,5,6,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [2,7,6,5,4,3,1] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [4,3,5,6,2,1,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [3,6,1,5,4,2,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [3,4,5,1,2,6,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [5,6,4,2,7,3,1] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [3,4,7,1,6,5,2] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [4,5,6,1,2,3,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [4,5,6,7,1,2,3] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [4,3,2,1,5,6,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [2,3,1,4,5,6,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [6,5,4,3,2,1,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [2,5,4,3,1,6,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [3,4,1,2,5,6,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [8,7,6,5,4,3,2,1] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [2,7,6,5,4,3,1,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [4,3,5,6,2,1,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [3,6,1,5,4,2,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [3,4,5,1,2,6,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [4,3,7,8,6,5,2,1] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [3,8,1,7,6,5,4,2] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [5,6,4,2,7,3,1,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [3,4,7,1,6,5,2,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [4,5,6,1,2,3,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [6,5,8,7,2,1,4,3] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [3,6,7,5,2,8,4,1] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [4,5,8,1,2,7,6,3] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [4,5,6,7,1,2,3,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [2,3,1,4,5,6,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [6,5,4,3,2,1,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [2,5,4,3,1,6,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [3,4,1,2,5,6,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [8,7,6,5,4,3,2,1,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [2,7,6,5,4,3,1,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [4,3,5,6,2,1,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [3,6,1,5,4,2,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [3,4,5,1,2,6,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [2,9,8,7,6,5,4,3,1] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [4,3,7,8,6,5,2,1,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
Description
The smallest fixpoint of a permutation.
A fixpoint of a permutation of length n if an index i such that π(i)=i, and we set π(n+1)=n+1.
Matching statistic: St000673
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 38%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 38%
Values
[1]
=> [[1]]
=> [1] => [1] => ? = 0
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 2
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 3
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => 4
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => 3
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [3,4,1,2] => 4
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => 4
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => 3
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,5,4,3,1] => 5
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [3,4,1,2,5] => 4
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [3,4,5,1,2] => 5
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 2
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [4,3,2,1,5,6] => 4
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,3,1,4,5,6] => 3
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [6,5,4,3,2,1] => 6
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,5,4,3,1,6] => 5
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [3,4,1,2,5,6] => 4
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,3,5,6,2,1] => 6
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,6,1,5,4,2] => 6
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [3,4,5,1,2,6] => 5
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [4,5,6,1,2,3] => 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [4,3,2,1,5,6,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [2,3,1,4,5,6,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [6,5,4,3,2,1,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,5,4,3,1,6,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [3,4,1,2,5,6,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [2,7,6,5,4,3,1] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [4,3,5,6,2,1,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [3,6,1,5,4,2,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [3,4,5,1,2,6,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [5,6,4,2,7,3,1] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [3,4,7,1,6,5,2] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [4,5,6,1,2,3,7] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [4,5,6,7,1,2,3] => ? ∊ {2,3,4,4,5,5,6,6,6,6,7,7,7,7}
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [4,3,2,1,5,6,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [2,3,1,4,5,6,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [6,5,4,3,2,1,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [2,5,4,3,1,6,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [3,4,1,2,5,6,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [8,7,6,5,4,3,2,1] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [2,7,6,5,4,3,1,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [4,3,5,6,2,1,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [3,6,1,5,4,2,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [3,4,5,1,2,6,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [4,3,7,8,6,5,2,1] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [3,8,1,7,6,5,4,2] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [5,6,4,2,7,3,1,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [3,4,7,1,6,5,2,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [4,5,6,1,2,3,7,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [6,5,8,7,2,1,4,3] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [3,6,7,5,2,8,4,1] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [4,5,8,1,2,7,6,3] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [4,5,6,7,1,2,3,8] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8}
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [2,3,1,4,5,6,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [6,5,4,3,2,1,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [2,5,4,3,1,6,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [3,4,1,2,5,6,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [8,7,6,5,4,3,2,1,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [2,7,6,5,4,3,1,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [4,3,5,6,2,1,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [3,6,1,5,4,2,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [3,4,5,1,2,6,7,8,9] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [2,9,8,7,6,5,4,3,1] => ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9}
Description
The number of non-fixed points of a permutation.
In other words, this statistic is n minus the number of fixed points ([[St000022]]) of π.
Matching statistic: St001255
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001255: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 38%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001255: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 38%
Values
[1]
=> [[1]]
=> [1] => [1,0]
=> 1 = 0 + 1
[2]
=> [[1,2]]
=> [1,2] => [1,0,1,0]
=> 3 = 2 + 1
[1,1]
=> [[1],[2]]
=> [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 4 = 3 + 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 5 = 4 + 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 6 = 5 + 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[6]
=> [[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 = 6 + 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> 7 = 6 + 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> 7 = 6 + 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,1,1,1,0,0,1,0,0,0,1,0]
=> 6 = 5 + 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 7 = 6 + 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 5 = 4 + 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[7]
=> [[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]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7} + 1
[8]
=> [[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]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8} + 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
=> ? ∊ {0,2,3,4,4,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9} + 1
Description
The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.
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