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Your data matches 74 different statistics following compositions of up to 3 maps.
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Matching statistic: St000276
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St000276: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000276: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1
[1,2] => ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> 0
[1,3,2] => ([(1,2)],3)
=> 0
[2,1,3] => ([(1,2)],3)
=> 0
[2,3,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> 0
[1,3,2,4] => ([(2,3)],4)
=> 0
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[2,1,3,4] => ([(2,3)],4)
=> 0
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 0
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> 0
[1,2,4,3,5] => ([(3,4)],5)
=> 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,3,2,4,5] => ([(3,4)],5)
=> 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
Description
The size of the preimage of the map 'to graph' from Ordered trees to Graphs.
Matching statistic: St001570
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 95%●distinct values known / distinct values provided: 12%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 95%●distinct values known / distinct values provided: 12%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 1
[1,2] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {0,1}
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1}
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,2,2}
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,2,2}
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,2,2}
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,2,2,3,3}
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,2,2,3,3}
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,2,2,3,3}
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,2,2,3,3}
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,2,2,3,3}
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,2,2,3,3}
[2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,1,3,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,2,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,2,2,4,4,8,8,8,8}
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,2,2,4,4,8,8,8,8}
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,2,2,4,4,8,8,8,8}
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,2,2,4,4,8,8,8,8}
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,2,2,4,4,8,8,8,8}
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,3,5,2,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,2,2,4,4,8,8,8,8}
[1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,2,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,3,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,5,2,3,4] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,2,4,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,3,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,5,4,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,2,2,4,4,8,8,8,8}
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,2,2,4,4,8,8,8,8}
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,2,2,4,4,8,8,8,8}
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,3,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,3,5,4,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[2,4,1,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,2,2,4,4,8,8,8,8}
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,3,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,6,3,5] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,5,2,4,6] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,4,6,3,5] => [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,4,1,3,5,6] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,4,1,3,6,5] => [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,4,1,6,3,5] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,4,6,1,3,5] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St000205
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Values
[1] => [1]
=> [1]
=> []
=> ? = 1
[1,2] => [2]
=> [1,1]
=> [1]
=> 0
[2,1] => [1,1]
=> [2]
=> []
=> ? = 1
[1,2,3] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,2,2}
[2,1,3] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,2,2}
[2,3,1] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,2,2}
[3,1,2] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,2,2}
[3,2,1] => [1,1,1]
=> [2,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[2,1,3,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? ∊ {2,2,3,3}
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,3,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,4,1,3] => [2,2]
=> [4]
=> []
=> ? ∊ {2,2,3,3}
[2,4,3,1] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,1,4,2] => [2,2]
=> [4]
=> []
=> ? ∊ {2,2,3,3}
[3,2,1,4] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[3,2,4,1] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[3,4,1,2] => [2,2]
=> [4]
=> []
=> ? ∊ {2,2,3,3}
[3,4,2,1] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[4,1,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,1,3,2] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[4,2,1,3] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[4,2,3,1] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[4,3,1,2] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [3,1]
=> [1]
=> 0
[1,2,3,4,5] => [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,5,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,3,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,4,3] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,5,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,4,2,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,5,2,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,5,4,2] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,2,5,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,4,3,2,5] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,4,5,3,2] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,5,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,4,3,5] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,4,5,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,5,3,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,5,4,3] => [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
[2,3,1,4,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,3,1,5,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,4,1,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,3,4,5,1] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,3,5,1,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,5,4,1] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,4,1,5,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,4,3,1,5] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,5,1,3,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,1,2,5,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,1,4,2,5] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,1,4,5,2] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,1,5,2,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,4,1,2,5] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,4,1,5,2] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,4,5,1,2] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,5,1,2,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[4,1,2,5,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[4,1,5,2,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[4,5,1,2,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,4,3,6,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,4,6,3,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,5,3,6,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,5,6,3,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,4,1,3,6,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,4,1,6,3,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,4,6,1,3,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,5,1,3,6,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,5,1,6,3,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,5,6,1,3,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[3,1,4,2,6,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[3,1,4,6,2,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[3,1,5,2,6,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[3,1,5,6,2,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[3,4,1,2,6,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Values
[1] => [1]
=> [1]
=> []
=> ? = 1
[1,2] => [2]
=> [1,1]
=> [1]
=> 0
[2,1] => [1,1]
=> [2]
=> []
=> ? = 1
[1,2,3] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,2,2}
[2,1,3] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,2,2}
[2,3,1] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,2,2}
[3,1,2] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,2,2}
[3,2,1] => [1,1,1]
=> [2,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[2,1,3,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? ∊ {2,2,3,3}
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,3,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,4,1,3] => [2,2]
=> [4]
=> []
=> ? ∊ {2,2,3,3}
[2,4,3,1] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,1,4,2] => [2,2]
=> [4]
=> []
=> ? ∊ {2,2,3,3}
[3,2,1,4] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[3,2,4,1] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[3,4,1,2] => [2,2]
=> [4]
=> []
=> ? ∊ {2,2,3,3}
[3,4,2,1] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[4,1,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,1,3,2] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[4,2,1,3] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[4,2,3,1] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[4,3,1,2] => [2,1,1]
=> [2,2]
=> [2]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [3,1]
=> [1]
=> 0
[1,2,3,4,5] => [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,5,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,3,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,4,3] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,5,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,4,2,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,5,2,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,5,4,2] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,2,5,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,4,3,2,5] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,4,5,3,2] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,5,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,4,3,5] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,4,5,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,5,3,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,5,4,3] => [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
[2,3,1,4,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,3,1,5,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,4,1,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,3,4,5,1] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,3,5,1,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,5,4,1] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,4,1,5,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,4,3,1,5] => [3,1,1]
=> [4,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,5,1,3,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,1,2,5,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,1,4,2,5] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,1,4,5,2] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,1,5,2,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,4,1,2,5] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,4,1,5,2] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,4,5,1,2] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,5,1,2,4] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[4,1,2,5,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[4,1,5,2,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[4,5,1,2,3] => [3,2]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,4,3,6,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,4,6,3,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,5,3,6,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,5,6,3,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,4,1,3,6,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,4,1,6,3,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,4,6,1,3,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,5,1,3,6,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,5,1,6,3,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,5,6,1,3,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[3,1,4,2,6,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[3,1,4,6,2,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[3,1,5,2,6,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[3,1,5,6,2,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[3,4,1,2,6,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000791
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000791: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000791: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Values
[1] => [1]
=> []
=> []
=> ? = 1
[1,2] => [1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1}
[2,1] => [2]
=> []
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,2] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,2,2}
[2,1,3] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,2,2}
[2,3,1] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,2,2}
[3,1,2] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,2,2}
[3,2,1] => [3]
=> []
=> []
=> ? ∊ {0,0,0,2,2}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,4,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,2,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,3,2] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,3,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,3,4,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,4,1,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,4,3,1] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[3,1,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,1,4,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,2,1,4] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[3,2,4,1] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[3,4,1,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,4,2,1] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[4,1,2,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,1,3,2] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[4,2,1,3] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[4,2,3,1] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[4,3,1,2] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[4,3,2,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,4,3,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,4,5,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,5,3,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,5,4,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,3,1,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,3,5,1,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,5,4,3,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,5,4,2,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[4,3,2,1,5] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[4,3,2,5,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[4,3,5,2,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[4,5,3,2,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,1,4,3,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,2,4,3,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,3,2,1,4] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,3,2,4,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,3,4,2,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,4,1,3,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,4,2,1,3] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,4,2,3,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,4,3,1,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,4,3,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,6,5,4,3,2] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,6,5,4,3,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[3,6,5,4,2,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[4,6,5,3,2,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[5,4,3,2,1,6] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[5,4,3,2,6,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[5,4,3,6,2,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[5,4,6,3,2,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[5,6,4,3,2,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[6,1,5,4,3,2] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[6,2,5,4,3,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[6,3,5,4,2,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[6,4,3,2,1,5] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[6,4,3,2,5,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[6,4,3,5,2,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
Description
The number of pairs of left tunnels, one strictly containing the other, of a Dyck path.
The statistic counting all pairs of distinct tunnels is the area of a Dyck path [[St000012]].
Matching statistic: St000980
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Values
[1] => [1]
=> []
=> []
=> ? = 1
[1,2] => [1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1}
[2,1] => [2]
=> []
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,2] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,2,2}
[2,1,3] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,2,2}
[2,3,1] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,2,2}
[3,1,2] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,2,2}
[3,2,1] => [3]
=> []
=> []
=> ? ∊ {0,0,0,2,2}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,4,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,2,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,3,2] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,3,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,3,4,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,4,1,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,4,3,1] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[3,1,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,1,4,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,2,1,4] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[3,2,4,1] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[3,4,1,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,4,2,1] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[4,1,2,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,1,3,2] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[4,2,1,3] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[4,2,3,1] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[4,3,1,2] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[4,3,2,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,3,3}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,4,3,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,4,5,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,5,3,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,5,4,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,3,1,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,3,5,1,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,5,4,3,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[3,5,4,2,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[4,3,2,1,5] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[4,3,2,5,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[4,3,5,2,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[4,5,3,2,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,1,4,3,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,2,4,3,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,3,2,1,4] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,3,2,4,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,3,4,2,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,4,1,3,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,4,2,1,3] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,4,2,3,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,4,3,1,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[5,4,3,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,6,5,4,3,2] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,6,5,4,3,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[3,6,5,4,2,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[4,6,5,3,2,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[5,4,3,2,1,6] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[5,4,3,2,6,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[5,4,3,6,2,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[5,4,6,3,2,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[5,6,4,3,2,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[6,1,5,4,3,2] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[6,2,5,4,3,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[6,3,5,4,2,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[6,4,3,2,1,5] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[6,4,3,2,5,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[6,4,3,5,2,1] => [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
Description
The number of boxes weakly below the path and above the diagonal that lie below at least two peaks.
For example, the path $111011010000$ has three peaks in positions $03, 15, 26$. The boxes below $03$ are $01,02,\textbf{12}$, the boxes below $15$ are $\textbf{12},13,14,\textbf{23},\textbf{24},\textbf{34}$, and the boxes below $26$ are $\textbf{23},\textbf{24},25,\textbf{34},35,45$.
We thus obtain the four boxes in positions $12,23,24,34$ that are below at least two peaks.
Matching statistic: St000379
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000379: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000379: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 1
[1,2] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {0,1}
[2,1] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {0,1}
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,2,2}
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,2,2}
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,2,2}
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,2,2}
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
Description
The number of Hamiltonian cycles in a graph.
A Hamiltonian cycle in a graph $G$ is a subgraph (this is, a subset of the edges) that is a cycle which contains every vertex of $G$.
Since it is unclear whether the graph on one vertex is Hamiltonian, the statistic is undefined for this graph.
Matching statistic: St000699
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000699: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000699: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 1
[1,2] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {0,1}
[2,1] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {0,1}
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,2,2}
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,2,2}
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,2,2}
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,2,2}
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
Description
The toughness times the least common multiple of 1,...,n-1 of a non-complete graph.
A graph $G$ is $t$-tough if $G$ cannot be split into $k$ different connected components by the removal of fewer than $tk$ vertices for all integers $k>1$.
The toughness of $G$ is the maximal number $t$ such that $G$ is $t$-tough. It is a rational number except for the complete graph, where it is infinity. The toughness of a disconnected graph is zero.
This statistic is the toughness multiplied by the least common multiple of $1,\dots,n-1$, where $n$ is the number of vertices of $G$.
Matching statistic: St001175
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Values
[1] => [1] => [1]
=> []
=> ? = 1
[1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,1}
[2,1] => [1,2] => [2]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,2,2}
[1,3,2] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,2,2}
[2,1,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,2,2}
[2,3,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,2,2}
[3,1,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[3,2,1] => [1,3,2] => [2,1]
=> [1]
=> 0
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,2,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,3,2,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,3,4,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,4,2,3] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[1,4,3,2] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[2,1,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,1,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,3,1,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,4,1,3] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[3,1,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[3,1,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[3,2,1,4] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[3,2,4,1] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[3,4,1,2] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[3,4,2,1] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[4,1,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,1,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[4,2,1,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,2,3,1] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[4,3,1,2] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[4,3,2,1] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,4,5,3] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,5,3,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> 0
[1,2,5,4,3] => [1,2,3,5,4] => [4,1]
=> [1]
=> 0
[1,3,2,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,2,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,4,2,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,4,5,2] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,5,2,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> 0
[1,3,5,4,2] => [1,2,3,5,4] => [4,1]
=> [1]
=> 0
[1,4,2,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0
[1,4,2,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> 0
[1,4,3,2,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0
[1,4,3,5,2] => [1,2,4,5,3] => [4,1]
=> [1]
=> 0
[1,4,5,2,3] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0
[1,4,5,3,2] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[1,5,2,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> 0
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[1,5,3,4,2] => [1,2,5,3,4] => [4,1]
=> [1]
=> 0
[1,5,4,2,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> 0
[1,5,4,3,2] => [1,2,5,3,4] => [4,1]
=> [1]
=> 0
[2,1,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,4,5,3] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,5,3,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> 0
[2,1,5,4,3] => [1,2,3,5,4] => [4,1]
=> [1]
=> 0
[2,3,1,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,1,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,4,1,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,4,5,1] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,5,1,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> 0
[2,3,5,4,1] => [1,2,3,5,4] => [4,1]
=> [1]
=> 0
[2,4,1,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0
[2,4,1,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> 0
[2,4,3,1,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0
[2,4,3,5,1] => [1,2,4,5,3] => [4,1]
=> [1]
=> 0
[2,4,5,1,3] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0
[2,4,5,3,1] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[2,5,1,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> 0
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[2,5,3,4,1] => [1,2,5,3,4] => [4,1]
=> [1]
=> 0
[2,5,4,1,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> 0
[2,5,4,3,1] => [1,2,5,3,4] => [4,1]
=> [1]
=> 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,5,6,2] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,3,4,5,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,3,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,3,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St001281
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001281: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001281: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 93%●distinct values known / distinct values provided: 12%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 1
[1,2] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {0,1}
[2,1] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {0,1}
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,2,2}
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,2,2}
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,2,2}
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,2,2}
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,2,2,3,3}
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,4,4,8,8,8,8}
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,5,5,5,5,10,10,10,10,10,10,10,10,10,10}
Description
The normalized isoperimetric number of a graph.
The isoperimetric number, or Cheeger constant, of a graph $G$ is
$$
i(G) = \min\left\{\frac{|\partial A|}{|A|}\ : \ A\subseteq V(G), 0 < |A|\leq |V(G)|/2\right\},
$$
where
$$
\partial A := \{(x, y)\in E(G)\ : \ x\in A, y\in V(G)\setminus A \}.
$$
This statistic is $i(G)\cdot\lfloor n/2\rfloor !$. We leave the statistic undefined for the graph without vertices and the graph with a single vertex.
The following 64 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001498The normalised height of a Nakayama algebra with magnitude 1. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000629The defect of a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001485The modular major index of a binary word. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000478Another weight of a partition according to Alladi. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000455The second largest eigenvalue of a graph if it is integral. St001651The Frankl number of a lattice. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000940The number of characters of the symmetric group whose value on the partition is zero. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001095The number of non-isomorphic posets with precisely one further covering relation.
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