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Your data matches 21 different statistics following compositions of up to 3 maps.
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Matching statistic: St000668
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(load all 3 compositions to match this statistic)
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 2
[1,1]
=> 1
[3]
=> 3
[2,1]
=> 2
[1,1,1]
=> 1
[4]
=> 4
[3,1]
=> 3
[2,2]
=> 2
[2,1,1]
=> 2
[1,1,1,1]
=> 1
[5]
=> 5
[4,1]
=> 4
[3,2]
=> 6
[3,1,1]
=> 3
[2,2,1]
=> 2
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 1
[6]
=> 6
[5,1]
=> 5
[4,2]
=> 4
[4,1,1]
=> 4
[3,3]
=> 3
[3,2,1]
=> 6
[3,1,1,1]
=> 3
[2,2,2]
=> 2
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 1
[7]
=> 7
[6,1]
=> 6
[5,2]
=> 10
[5,1,1]
=> 5
[4,3]
=> 12
[4,2,1]
=> 4
[4,1,1,1]
=> 4
[3,3,1]
=> 3
[3,2,2]
=> 6
[3,2,1,1]
=> 6
[3,1,1,1,1]
=> 3
[2,2,2,1]
=> 2
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> 1
[8]
=> 8
[7,1]
=> 7
[6,2]
=> 6
[6,1,1]
=> 6
[5,3]
=> 15
[5,2,1]
=> 10
[5,1,1,1]
=> 5
Description
The least common multiple of the parts of the partition.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 22% ●values known / values provided: 25%●distinct values known / distinct values provided: 22%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 22% ●values known / values provided: 25%●distinct values known / distinct values provided: 22%
Values
[2]
=> []
=> [[],[]]
=> ([],0)
=> ? ∊ {1,2}
[1,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? ∊ {1,2}
[3]
=> []
=> [[],[]]
=> ([],0)
=> ? ∊ {1,2,3}
[2,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? ∊ {1,2,3}
[1,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {1,2,3}
[4]
=> []
=> [[],[]]
=> ([],0)
=> ? ∊ {1,2,2,4}
[3,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? ∊ {1,2,2,4}
[2,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,4}
[2,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,4}
[1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[5]
=> []
=> [[],[]]
=> ([],0)
=> ? ∊ {1,2,2,5,6}
[4,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? ∊ {1,2,2,5,6}
[3,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,5,6}
[3,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,5,6}
[2,2,1]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {1,2,2,5,6}
[2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[6]
=> []
=> [[],[]]
=> ([],0)
=> ? ∊ {1,2,2,2,6,6}
[5,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? ∊ {1,2,2,2,6,6}
[4,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,6,6}
[4,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,6,6}
[3,3]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[3,2,1]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {1,2,2,2,6,6}
[3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[2,2,2]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,6,6}
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[7]
=> []
=> [[],[]]
=> ([],0)
=> ? ∊ {1,2,2,2,6,6,7,10,12}
[6,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? ∊ {1,2,2,2,6,6,7,10,12}
[5,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,6,6,7,10,12}
[5,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,6,6,7,10,12}
[4,3]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[4,2,1]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {1,2,2,2,6,6,7,10,12}
[4,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[3,3,1]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,6,6,7,10,12}
[3,2,2]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[3,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,6,6,7,10,12}
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,2,2,1]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ? ∊ {1,2,2,2,6,6,7,10,12}
[2,2,1,1,1]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {1,2,2,2,6,6,7,10,12}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[8]
=> []
=> [[],[]]
=> ([],0)
=> ? ∊ {1,2,2,2,2,4,6,6,6,8,10,12,15}
[7,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? ∊ {1,2,2,2,2,4,6,6,6,8,10,12,15}
[6,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,4,6,6,6,8,10,12,15}
[6,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,4,6,6,6,8,10,12,15}
[5,3]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[5,2,1]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {1,2,2,2,2,4,6,6,6,8,10,12,15}
[5,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[4,4]
=> [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,1]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,2,4,6,6,6,8,10,12,15}
[4,2,2]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[4,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,2,4,6,6,6,8,10,12,15}
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,3,2]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ? ∊ {1,2,2,2,2,4,6,6,6,8,10,12,15}
[3,3,1,1]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {1,2,2,2,2,4,6,6,6,8,10,12,15}
[3,2,2,1]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ? ∊ {1,2,2,2,2,4,6,6,6,8,10,12,15}
[3,2,1,1,1]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,4,6,6,6,8,10,12,15}
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? ∊ {1,2,2,2,2,4,6,6,6,8,10,12,15}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? ∊ {1,2,2,2,2,4,6,6,6,8,10,12,15}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[9]
=> []
=> [[],[]]
=> ([],0)
=> ? ∊ {1,2,2,2,2,3,4,6,6,6,6,8,9,10,10,12,12,14,15,20}
[8,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,4,6,6,6,6,8,9,10,10,12,12,14,15,20}
[7,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,3,4,6,6,6,6,8,9,10,10,12,12,14,15,20}
[7,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,3,4,6,6,6,6,8,9,10,10,12,12,14,15,20}
[6,3]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[6,2,1]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {1,2,2,2,2,3,4,6,6,6,6,8,9,10,10,12,12,14,15,20}
[6,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[5,4]
=> [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[5,3,1]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,2,3,4,6,6,6,6,8,9,10,10,12,12,14,15,20}
[5,2,2]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[5,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,2,2,2,2,3,4,6,6,6,6,8,9,10,10,12,12,14,15,20}
[5,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,4,1]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {1,2,2,2,2,3,4,6,6,6,6,8,9,10,10,12,12,14,15,20}
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,3,3]
=> [3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[3,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[7,3]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[7,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[6,4]
=> [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[6,2,2]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[6,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[5,5]
=> [5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,3,3]
=> [3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[4,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[8,3]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[8,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[7,4]
=> [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[7,2,2]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[7,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[6,5]
=> [5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000771
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 22%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 22%
Values
[2]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> ? = 2
[1,1]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? ∊ {2,3}
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {2,3}
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,3,4}
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,3,4}
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,3,4}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ? ∊ {2,2,4,5,6}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {2,2,4,5,6}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,4,5,6}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,4,5,6}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[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)
=> ? ∊ {2,2,4,5,6}
[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)
=> 3
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> ? ∊ {1,2,3,4,5,6,6}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {1,2,3,4,5,6,6}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,3,4,5,6,6}
[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)
=> ? ∊ {1,2,3,4,5,6,6}
[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)
=> 2
[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)
=> ? ∊ {1,2,3,4,5,6,6}
[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)
=> ? ∊ {1,2,3,4,5,6,6}
[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)
=> 3
[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)
=> 2
[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)
=> ? ∊ {1,2,3,4,5,6,6}
[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)
=> 4
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> ? ∊ {2,2,3,4,4,6,6,6,7,10,12}
[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)
=> ? ∊ {2,2,3,4,4,6,6,6,7,10,12}
[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)
=> ? ∊ {2,2,3,4,4,6,6,6,7,10,12}
[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)
=> ? ∊ {2,2,3,4,4,6,6,6,7,10,12}
[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)
=> ? ∊ {2,2,3,4,4,6,6,6,7,10,12}
[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)
=> ? ∊ {2,2,3,4,4,6,6,6,7,10,12}
[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)
=> ? ∊ {2,2,3,4,4,6,6,6,7,10,12}
[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)
=> 1
[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)
=> ? ∊ {2,2,3,4,4,6,6,6,7,10,12}
[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)
=> ? ∊ {2,2,3,4,4,6,6,6,7,10,12}
[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)
=> ? ∊ {2,2,3,4,4,6,6,6,7,10,12}
[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)
=> 2
[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
[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)
=> ? ∊ {2,2,3,4,4,6,6,6,7,10,12}
[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)
=> 5
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> 2
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> 2
[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)
=> 2
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> 3
[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)
=> 2
[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)
=> 4
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> ? ∊ {1,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[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)
=> ? ∊ {2,2,3,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[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)
=> ? ∊ {2,2,3,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[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)
=> ? ∊ {2,2,3,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[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)
=> ? ∊ {2,2,3,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[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)
=> ? ∊ {2,2,3,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,2,3,4,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[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)
=> 4
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,2,3,4,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [5,2,4,3,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,3,4,2,6,7,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [6,2,3,5,4,7,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => ([(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)
=> 3
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,2,4,5,3,7,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [7,2,3,4,6,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => ([(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)
=> 4
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [6,2,5,4,3,7,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [7,2,3,5,6,4,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[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)
=> 5
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [7,2,4,5,6,3,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [7,3,4,5,6,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
(4−1−2−1−14−1−2−2−14−1−1−2−14).
Its eigenvalues are 0,4,4,6, so the statistic is 2.
The path on four vertices has eigenvalues 0,4.7…,6,9.2… and therefore statistic 1.
Matching statistic: St001645
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 18%●distinct values known / distinct values provided: 17%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 18%●distinct values known / distinct values provided: 17%
Values
[2]
=> 0 => [1] => ([],1)
=> 1
[1,1]
=> 11 => [2] => ([],2)
=> ? = 2
[3]
=> 1 => [1] => ([],1)
=> 1
[2,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 2
[1,1,1]
=> 111 => [3] => ([],3)
=> ? = 3
[4]
=> 0 => [1] => ([],1)
=> 1
[3,1]
=> 11 => [2] => ([],2)
=> ? ∊ {2,2,3,4}
[2,2]
=> 00 => [2] => ([],2)
=> ? ∊ {2,2,3,4}
[2,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {2,2,3,4}
[1,1,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {2,2,3,4}
[5]
=> 1 => [1] => ([],1)
=> 1
[4,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 2
[3,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 2
[3,1,1]
=> 111 => [3] => ([],3)
=> ? ∊ {3,5,6}
[2,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4
[2,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {3,5,6}
[1,1,1,1,1]
=> 11111 => [5] => ([],5)
=> ? ∊ {3,5,6}
[6]
=> 0 => [1] => ([],1)
=> 1
[5,1]
=> 11 => [2] => ([],2)
=> ? ∊ {2,2,2,3,4,4,5,6,6}
[4,2]
=> 00 => [2] => ([],2)
=> ? ∊ {2,2,2,3,4,4,5,6,6}
[4,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,5,6,6}
[3,3]
=> 11 => [2] => ([],2)
=> ? ∊ {2,2,2,3,4,4,5,6,6}
[3,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,1,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {2,2,2,3,4,4,5,6,6}
[2,2,2]
=> 000 => [3] => ([],3)
=> ? ∊ {2,2,2,3,4,4,5,6,6}
[2,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,4,5,6,6}
[2,1,1,1,1]
=> 01111 => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,5,6,6}
[1,1,1,1,1,1]
=> 111111 => [6] => ([],6)
=> ? ∊ {2,2,2,3,4,4,5,6,6}
[7]
=> 1 => [1] => ([],1)
=> 1
[6,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 2
[5,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 2
[5,1,1]
=> 111 => [3] => ([],3)
=> ? ∊ {3,3,4,5,6,6,6,7,10,12}
[4,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 2
[4,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4
[4,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,4,5,6,6,6,7,10,12}
[3,3,1]
=> 111 => [3] => ([],3)
=> ? ∊ {3,3,4,5,6,6,6,7,10,12}
[3,2,2]
=> 100 => [1,2] => ([(1,2)],3)
=> ? ∊ {3,3,4,5,6,6,6,7,10,12}
[3,2,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,4,5,6,6,6,7,10,12}
[3,1,1,1,1]
=> 11111 => [5] => ([],5)
=> ? ∊ {3,3,4,5,6,6,6,7,10,12}
[2,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {3,3,4,5,6,6,6,7,10,12}
[2,2,1,1,1]
=> 00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {3,3,4,5,6,6,6,7,10,12}
[2,1,1,1,1,1]
=> 011111 => [1,5] => ([(4,5)],6)
=> ? ∊ {3,3,4,5,6,6,6,7,10,12}
[1,1,1,1,1,1,1]
=> 1111111 => [7] => ([],7)
=> ? ∊ {3,3,4,5,6,6,6,7,10,12}
[8]
=> 0 => [1] => ([],1)
=> 1
[7,1]
=> 11 => [2] => ([],2)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[6,2]
=> 00 => [2] => ([],2)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[6,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[5,3]
=> 11 => [2] => ([],2)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[5,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[5,1,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[4,4]
=> 00 => [2] => ([],2)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[4,3,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[4,2,2]
=> 000 => [3] => ([],3)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[4,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[4,1,1,1,1]
=> 01111 => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[3,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 4
[3,3,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[3,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[3,2,1,1,1]
=> 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[3,1,1,1,1,1]
=> 111111 => [6] => ([],6)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[2,2,2,2]
=> 0000 => [4] => ([],4)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[2,2,2,1,1]
=> 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[2,2,1,1,1,1]
=> 001111 => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[2,1,1,1,1,1,1]
=> 0111111 => [1,6] => ([(5,6)],7)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[1,1,1,1,1,1,1,1]
=> 11111111 => [8] => ([],8)
=> ? ∊ {2,2,2,2,3,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[9]
=> 1 => [1] => ([],1)
=> 1
[8,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 2
[7,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 2
[7,1,1]
=> 111 => [3] => ([],3)
=> ? ∊ {3,3,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[6,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 2
[6,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4
[6,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[5,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 2
[5,3,1]
=> 111 => [3] => ([],3)
=> ? ∊ {3,3,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[4,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4
[4,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[10]
=> 0 => [1] => ([],1)
=> 1
[7,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[5,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[5,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 4
[4,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[11]
=> 1 => [1] => ([],1)
=> 1
[10,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 2
[9,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 2
[8,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 2
[8,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4
[7,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 2
[6,5]
=> 01 => [1,1] => ([(0,1)],2)
=> 2
[6,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4
[6,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[4,4,3]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4
[12]
=> 0 => [1] => ([],1)
=> 1
[9,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[7,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[7,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 4
[6,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[5,5,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 4
[5,4,3]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
Description
The pebbling number of a connected graph.
Matching statistic: St000306
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 39%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 39%
Values
[2]
=> 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1]
=> 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[3]
=> 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,1]
=> 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[4]
=> 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[3,1]
=> 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
[2,2]
=> 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[2,1,1]
=> 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[5]
=> 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[4,1]
=> 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[3,2]
=> 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,1]
=> 100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 6
[2,2,1]
=> 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[6]
=> 1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1
[5,1]
=> 1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 2
[4,2]
=> 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[4,1,1]
=> 1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {3,4,6}
[3,3]
=> 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[3,2,1]
=> 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {3,4,6}
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {3,4,6}
[2,2,2]
=> 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 4
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
[7]
=> 10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 1
[6,1]
=> 10000010 => [1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> 2
[5,2]
=> 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {2,3,4,6,6,10,12}
[5,1,1]
=> 10000110 => [1,5,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,6,6,10,12}
[4,3]
=> 101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[4,2,1]
=> 1001010 => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,3,4,6,6,10,12}
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,6,6,10,12}
[3,3,1]
=> 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> 3
[3,2,2]
=> 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,3,4,6,6,10,12}
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,6,6,10,12}
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,6,6,10,12}
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 5
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 7
[8]
=> 100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> 1
[7,1]
=> 100000010 => [1,7,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[6,2]
=> 10000100 => [1,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[6,1,1]
=> 100000110 => [1,6,1,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[5,3]
=> 1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 2
[5,2,1]
=> 10001010 => [1,4,2,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[5,1,1,1]
=> 100001110 => [1,5,1,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[4,4]
=> 110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[4,3,1]
=> 1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[4,2,2]
=> 1001100 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[4,2,1,1]
=> 10010110 => [1,3,2,1,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[4,1,1,1,1]
=> 100011110 => [1,4,1,1,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[3,3,2]
=> 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[3,3,1,1]
=> 1100110 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[3,2,2,1]
=> 1011010 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[3,2,1,1,1]
=> 10101110 => [1,2,2,1,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[3,1,1,1,1,1]
=> 100111110 => [1,3,1,1,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[2,2,2,2]
=> 111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[2,2,2,1,1]
=> 1110110 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 5
[2,2,1,1,1,1]
=> 11011110 => [1,1,2,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[2,1,1,1,1,1,1]
=> 101111110 => [1,2,1,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,6,7,10,12,15}
[1,1,1,1,1,1,1,1]
=> 111111110 => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 8
[9]
=> 1000000000 => [1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> 1
[8,1]
=> 1000000010 => [1,8,2] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[7,2]
=> 100000100 => [1,6,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[7,1,1]
=> 1000000110 => [1,7,1,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[6,3]
=> 10001000 => [1,4,4] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[6,2,1]
=> 100001010 => [1,5,2,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[6,1,1,1]
=> 1000001110 => [1,6,1,1,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[5,4]
=> 1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[5,3,1]
=> 10010010 => [1,3,3,2] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[5,2,2]
=> 10001100 => [1,4,1,3] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[5,2,1,1]
=> 100010110 => [1,4,2,1,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[5,1,1,1,1]
=> 1000011110 => [1,5,1,1,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[4,4,1]
=> 1100010 => [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[4,3,2]
=> 1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[4,3,1,1]
=> 10100110 => [1,2,3,1,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[4,2,2,1]
=> 10011010 => [1,3,1,2,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[4,2,1,1,1]
=> 100101110 => [1,3,2,1,1,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[4,1,1,1,1,1]
=> 1000111110 => [1,4,1,1,1,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[3,3,3]
=> 111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[3,3,2,1]
=> 1101010 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[3,3,1,1,1]
=> 11001110 => [1,1,3,1,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[3,2,2,2]
=> 1011100 => [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[3,2,2,1,1]
=> 10110110 => [1,2,1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[3,2,1,1,1,1]
=> 101011110 => [1,2,2,1,1,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,6,7,8,10,10,12,12,14,15,20}
[2,2,2,2,1]
=> 1111010 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 5
[1,1,1,1,1,1,1,1,1]
=> 1111111110 => [1,1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 9
[5,5]
=> 1100000 => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 2
[6,5]
=> 10100000 => [1,2,6] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 2
[2,2,2,2,2,1]
=> 11111010 => [1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 6
[6,6]
=> 11000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 2
Description
The bounce count of a Dyck path.
For a Dyck path D of length 2n, this is the number of points (i,i) for 1≤i<n that are touching points of the [[Mp00099|bounce path]] of D.
Matching statistic: St001232
(load all 44 compositions to match this statistic)
(load all 44 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 48%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 48%
Values
[2]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 3 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[4]
=> [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 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,4} - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[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]
=> ? ∊ {2,4} - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[5]
=> [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 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,5,6} - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,3,5,6} - 1
[3,1,1]
=> [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]
=> ? ∊ {2,3,5,6} - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[2,1,1,1]
=> [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]
=> ? ∊ {2,3,5,6} - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[6]
=> [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 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,4,6,6} - 1
[4,2]
=> [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]
=> ? ∊ {2,2,3,4,6,6} - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,4,6,6} - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,3,4,6,6} - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,4,6,6} - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {2,2,3,4,6,6} - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0 = 1 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,4,5,6,7,10,12} - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,3,3,4,5,6,7,10,12} - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,4,5,6,7,10,12} - 1
[4,3]
=> [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]
=> ? ∊ {2,2,3,3,4,5,6,7,10,12} - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ? ∊ {2,2,3,3,4,5,6,7,10,12} - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,4,5,6,7,10,12} - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[3,2,2]
=> [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]
=> ? ∊ {2,2,3,3,4,5,6,7,10,12} - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,3,3,4,5,6,7,10,12} - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,4,5,6,7,10,12} - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5 = 6 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [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 = 6 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {2,2,3,3,4,5,6,7,10,12} - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [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 = 5 - 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [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 = 8 - 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [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 = 7 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,2,2,2,2,3,4,4,4,6,6,6,6,6,10,12,15} - 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,6,6,6,6,6,7,8,9,10,12,12,14,15,20} - 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,6,6,6,6,6,7,8,9,10,12,12,14,15,20} - 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,6,6,6,6,6,7,8,9,10,12,12,14,15,20} - 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,6,6,6,6,6,7,8,9,10,12,12,14,15,20} - 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,6,6,6,6,6,7,8,9,10,12,12,14,15,20} - 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,6,6,6,6,6,7,8,9,10,12,12,14,15,20} - 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,6,6,6,6,6,7,8,9,10,12,12,14,15,20} - 1
[5,4]
=> [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]
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,6,6,6,6,6,7,8,9,10,12,12,14,15,20} - 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,6,6,6,6,6,7,8,9,10,12,12,14,15,20} - 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,6,6,6,6,6,7,8,9,10,12,12,14,15,20} - 1
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [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 = 4 - 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [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 = 6 - 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [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 = 6 - 1
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [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 = 10 - 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [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 - 1
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 7 = 8 - 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [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 = 8 - 1
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [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 = 4 - 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 7 = 8 - 1
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> 10 = 11 - 1
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [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 = 6 - 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [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 = 9 - 1
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 4 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000058
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 39%
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 39%
Values
[2]
=> [[1,2]]
=> {{1,2}}
=> [2,1] => 2
[1,1]
=> [[1],[2]]
=> {{1},{2}}
=> [1,2] => 1
[3]
=> [[1,2,3]]
=> {{1,2,3}}
=> [2,3,1] => 3
[2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> [3,2,1] => 2
[1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> [1,2,3] => 1
[4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> [2,3,4,1] => 4
[3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> [3,2,4,1] => 3
[2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> [2,1,4,3] => 2
[2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> [4,2,3,1] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => 1
[5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => 5
[4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => 4
[3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => 6
[3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 3
[2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 2
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 1
[6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 6
[5,1]
=> [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => 5
[4,2]
=> [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => 4
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> {{1,4,5,6},{2},{3}}
=> [4,2,3,5,6,1] => 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => 3
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => 6
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> {{1,5,6},{2},{3},{4}}
=> [5,2,3,4,6,1] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => 2
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => 1
[7]
=> [[1,2,3,4,5,6,7]]
=> {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => 7
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> {{1,3,4,5,6,7},{2}}
=> [3,2,4,5,6,7,1] => 6
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> {{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => 10
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> {{1,4,5,6,7},{2},{3}}
=> [4,2,3,5,6,7,1] => 5
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> {{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => 12
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> {{1,3,6,7},{2,5},{4}}
=> [3,5,6,4,2,7,1] => 4
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> {{1,5,6,7},{2},{3},{4}}
=> [5,2,3,4,6,7,1] => 4
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> {{1,3,4},{2,6,7},{5}}
=> [3,6,4,1,5,7,2] => ? = 3
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> {{1,2,7},{3,4},{5,6}}
=> [2,7,4,3,6,5,1] => 6
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> {{1,4,7},{2,6},{3},{5}}
=> [4,6,3,7,5,2,1] => 6
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> {{1,6,7},{2},{3},{4},{5}}
=> [6,2,3,4,5,7,1] => 3
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> {{1,3},{2,5},{4,7},{6}}
=> [3,5,1,7,2,6,4] => 2
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> {{1,5},{2,7},{3},{4},{6}}
=> [5,7,3,4,1,6,2] => 2
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> {{1,7},{2},{3},{4},{5},{6}}
=> [7,2,3,4,5,6,1] => 2
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> {{1,2,3,4,5,6,7,8}}
=> [2,3,4,5,6,7,8,1] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> {{1,3,4,5,6,7,8},{2}}
=> [3,2,4,5,6,7,8,1] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> {{1,2,5,6,7,8},{3,4}}
=> [2,5,4,3,6,7,8,1] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> {{1,4,5,6,7,8},{2},{3}}
=> [4,2,3,5,6,7,8,1] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> [2,3,7,5,6,4,8,1] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> [3,5,6,4,2,7,8,1] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> {{1,5,6,7,8},{2},{3},{4}}
=> [5,2,3,4,6,7,8,1] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> [2,3,4,1,6,7,8,5] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> [3,6,4,8,5,7,2,1] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> [2,7,4,3,6,5,8,1] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> [4,6,3,7,5,2,8,1] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> {{1,6,7,8},{2},{3},{4},{5}}
=> [6,2,3,4,5,7,8,1] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> {{1,2,5},{3,4,8},{6,7}}
=> [2,5,4,8,1,7,6,3] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> {{1,4,5},{2,7,8},{3},{6}}
=> [4,7,3,5,1,6,8,2] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> [3,5,8,7,2,6,4,1] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> {{1,5,8},{2,7},{3},{4},{6}}
=> [5,7,3,4,8,6,2,1] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> [7,2,3,4,5,6,8,1] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> [2,1,4,3,6,5,8,7] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> {{1,4},{2,6},{3,8},{5},{7}}
=> [4,6,8,1,5,2,7,3] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> {{1,6},{2,8},{3},{4},{5},{7}}
=> [6,8,3,4,5,1,7,2] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> [8,2,3,4,5,6,7,1] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,2,3,4,5,6,7,8] => ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> {{1,2,3,4,5,6,7,8,9}}
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> {{1,3,4,5,6,7,8,9},{2}}
=> [3,2,4,5,6,7,8,9,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> {{1,2,5,6,7,8,9},{3,4}}
=> [2,5,4,3,6,7,8,9,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> {{1,4,5,6,7,8,9},{2},{3}}
=> [4,2,3,5,6,7,8,9,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> {{1,2,3,7,8,9},{4,5,6}}
=> [2,3,7,5,6,4,8,9,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> {{1,3,6,7,8,9},{2,5},{4}}
=> [3,5,6,4,2,7,8,9,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> {{1,5,6,7,8,9},{2},{3},{4}}
=> [5,2,3,4,6,7,8,9,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> {{1,2,3,4,9},{5,6,7,8}}
=> [2,3,4,9,6,7,8,5,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> {{1,3,4,8,9},{2,6,7},{5}}
=> [3,6,4,8,5,7,2,9,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> {{1,2,7,8,9},{3,4},{5,6}}
=> [2,7,4,3,6,5,8,9,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> {{1,4,7,8,9},{2,6},{3},{5}}
=> [4,6,3,7,5,2,8,9,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> {{1,6,7,8,9},{2},{3},{4},{5}}
=> [6,2,3,4,5,7,8,9,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> {{1,3,4,5},{2,7,8,9},{6}}
=> [3,7,4,5,1,6,8,9,2] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> {{1,2,5,9},{3,4,8},{6,7}}
=> [2,5,4,8,9,7,6,3,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> {{1,4,5,9},{2,7,8},{3},{6}}
=> [4,7,3,5,9,6,8,2,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> {{1,3,8,9},{2,5},{4,7},{6}}
=> [3,5,8,7,2,6,4,9,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> {{1,5,8,9},{2,7},{3},{4},{6}}
=> [5,7,3,4,8,6,2,9,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> {{1,7,8,9},{2},{3},{4},{5},{6}}
=> [7,2,3,4,5,6,8,9,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> {{1,2,3},{4,5,6},{7,8,9}}
=> [2,3,1,5,6,4,8,9,7] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> {{1,3,6},{2,5,9},{4,8},{7}}
=> [3,5,6,8,9,1,7,4,2] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> {{1,5,6},{2,8,9},{3},{4},{7}}
=> [5,8,3,4,6,1,7,9,2] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> {{1,2,9},{3,4},{5,6},{7,8}}
=> [2,9,4,3,6,5,8,7,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> {{1,4,9},{2,6},{3,8},{5},{7}}
=> [4,6,8,9,5,2,7,3,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> {{1,6,9},{2,8},{3},{4},{5},{7}}
=> [6,8,3,4,5,9,7,2,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> [8,2,3,4,5,6,7,9,1] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> {{1,3},{2,5},{4,7},{6,9},{8}}
=> [3,5,1,7,2,9,4,8,6] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> {{1,5},{2,7},{3,9},{4},{6},{8}}
=> [5,7,9,4,1,6,2,8,3] => ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
Description
The order of a permutation.
ord(π) is given by the minimial k for which πk is the identity permutation.
Matching statistic: St001582
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 14%●distinct values known / distinct values provided: 13%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 14%●distinct values known / distinct values provided: 13%
Values
[2]
=> []
=> []
=> [1] => ? = 2
[1,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[3]
=> []
=> []
=> [1] => ? = 3
[2,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[4]
=> []
=> []
=> [1] => ? ∊ {3,4}
[3,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? ∊ {3,4}
[5]
=> []
=> []
=> [1] => ? ∊ {4,5,6}
[4,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? ∊ {4,5,6}
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? ∊ {4,5,6}
[6]
=> []
=> []
=> [1] => ? ∊ {2,3,4,4,5,6,6}
[5,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? ∊ {2,3,4,4,5,6,6}
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? ∊ {2,3,4,4,5,6,6}
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? ∊ {2,3,4,4,5,6,6}
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? ∊ {2,3,4,4,5,6,6}
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? ∊ {2,3,4,4,5,6,6}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {2,3,4,4,5,6,6}
[7]
=> []
=> []
=> [1] => ? ∊ {2,3,4,4,5,6,6,6,7,10,12}
[6,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? ∊ {2,3,4,4,5,6,6,6,7,10,12}
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? ∊ {2,3,4,4,5,6,6,6,7,10,12}
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? ∊ {2,3,4,4,5,6,6,6,7,10,12}
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? ∊ {2,3,4,4,5,6,6,6,7,10,12}
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? ∊ {2,3,4,4,5,6,6,6,7,10,12}
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? ∊ {2,3,4,4,5,6,6,6,7,10,12}
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {2,3,4,4,5,6,6,6,7,10,12}
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? ∊ {2,3,4,4,5,6,6,6,7,10,12}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {2,3,4,4,5,6,6,6,7,10,12}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {2,3,4,4,5,6,6,6,7,10,12}
[8]
=> []
=> []
=> [1] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[7,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? ∊ {2,2,3,4,4,4,4,5,6,6,6,6,6,7,8,10,12,15}
[9]
=> []
=> []
=> [1] => ? ∊ {2,2,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[8,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? ∊ {2,2,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? ∊ {2,2,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? ∊ {2,2,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? ∊ {2,2,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? ∊ {2,2,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? ∊ {2,2,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[9,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[8,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[8,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[10,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[9,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[9,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[11,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[10,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[10,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Matching statistic: St000454
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 30%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 30%
Values
[2]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1]
=> [1,1,0,0]
=> [2] => ([],2)
=> 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2} - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2} - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,3,6} - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,2,3,6} - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,3,6} - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,3,6} - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,3,4,6} - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,6} - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,6} - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,3,4,6} - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 0 = 1 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1] => ([(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)
=> ? ∊ {2,2,2,3,4,6} - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,3,4,6} - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 7 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,2] => ([(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)
=> 5 = 6 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,3,5,6,6,10,12} - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,5,6,6,10,12} - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,3,5,6,6,10,12} - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,3,5,6,6,10,12} - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,3,5,6,6,10,12} - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,3,5,6,6,10,12} - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,3,5,6,6,10,12} - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1] => ([(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)
=> ? ∊ {1,2,2,3,5,6,6,10,12} - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,3,5,6,6,10,12} - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => ([(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)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,2,1] => ([(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)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,2,1,1] => ([(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)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1,1] => ([(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)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1,1] => ([(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)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1,1] => ([(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)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,4,4,4,6,6,6,6,6,7,8,10,12,15} - 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(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)
=> ? ∊ {2,2,2,2,3,3,3,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20} - 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(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)
=> ? ∊ {2,2,2,2,3,3,3,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20} - 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => ([(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)
=> ? ∊ {2,2,2,2,3,3,3,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20} - 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,2,1] => ([(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(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)
=> ? ∊ {2,2,2,2,3,3,3,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20} - 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,3,3,3,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20} - 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,3,1] => ([(0,7),(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)
=> ? ∊ {2,2,2,2,3,3,3,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20} - 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,2,1,1] => ([(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(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)
=> ? ∊ {2,2,2,2,3,3,3,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20} - 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20} - 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,3,3,3,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20} - 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(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)
=> ? ∊ {2,2,2,2,3,3,3,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20} - 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> 0 = 1 - 1
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => ([(4,5)],6)
=> 1 = 2 - 1
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => ([],6)
=> 0 = 1 - 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is d-regular, then its largest eigenvalue equals d. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 35%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 35%
Values
[2]
=> [[1,2]]
=> [2] => ([],2)
=> 1
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 2
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> 2
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> 2
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> 2
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 2
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,4,6}
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,4,6}
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,4,6}
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(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]]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => ([],7)
=> ? ∊ {1,2,2,2,3,3,4,4,5,6,6,10,12}
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => ([(5,6)],7)
=> ? ∊ {1,2,2,2,3,3,4,4,5,6,6,10,12}
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [2,5] => ([(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,3,3,4,4,5,6,6,10,12}
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,3,3,4,4,5,6,6,10,12}
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,3,3,4,4,5,6,6,10,12}
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,3,3,4,4,5,6,6,10,12}
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,3,3,4,4,5,6,6,10,12}
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,3,3,4,4,5,6,6,10,12}
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,3,3,4,4,5,6,6,10,12}
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,3,3,4,4,5,6,6,10,12}
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,3,3,4,4,5,6,6,10,12}
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,3,3,4,4,5,6,6,10,12}
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,3,3,4,4,5,6,6,10,12}
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,2] => ([(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)
=> 6
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => ([],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [1,7] => ([(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [1,1,6] => ([(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [1,1,2,4] => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [1,1,1,2,3] => ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,3] => ([(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)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [1,1,2,2,2] => ([(1,7),(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)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [1,1,1,1,2,2] => ([(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)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,2] => ([(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)
=> ? ∊ {1,2,2,2,2,3,3,4,4,4,4,5,6,6,6,6,6,7,10,12,15}
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 8
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [9] => ([],9)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [1,8] => ([(7,8)],9)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [2,7] => ([(6,8),(7,8)],9)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [1,1,7] => ([(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [3,6] => ([(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [1,2,6] => ([(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [1,1,1,6] => ([(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [4,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [1,3,5] => ([(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [2,2,5] => ([(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [1,1,2,5] => ([(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [1,1,1,1,5] => ([(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,4,5,6,6,6,6,6,6,6,7,8,9,10,10,12,12,14,15,20}
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number HG(G) of a graph G is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000982The length of the longest constant subword. St001644The dimension of a graph. St001638The book thickness of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001488The number of corners of a skew partition. St001198The number of simple modules in the algebra eAe with projective dimension at most 1 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001206The maximal dimension of an indecomposable projective eAe-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module eA.
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