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Your data matches 44 different statistics following compositions of up to 3 maps.
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Matching statistic: St001056
Values
([],1)
=> 0
([],2)
=> 0
([(0,1)],2)
=> 1
([],3)
=> 0
([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> 0
([],4)
=> 0
([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],5)
=> 0
([(3,4)],5)
=> 2
([(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
Description
The Grundy value for the game of deleting vertices of a graph until it has no edges.
Matching statistic: St000225
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {2,2}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {2,2}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [2,2,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,1,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [3]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [4]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,1,1]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000661
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1]
=> []
=> []
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,1)],2)
=> [2]
=> []
=> []
=> ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> []
=> ? ∊ {2,2}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> []
=> ? ∊ {2,2}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The number of rises of length 3 of a Dyck path.
Matching statistic: St000749
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {2,2}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {2,2}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [4,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,1,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [3]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [1,1,1,1]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,1,1]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The 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.
For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields
$$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3.
This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
Matching statistic: St001141
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001141: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001141: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1]
=> []
=> []
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,1)],2)
=> [2]
=> []
=> []
=> ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> []
=> ? ∊ {2,2}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> []
=> ? ∊ {2,2}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The number of occurrences of hills of size 3 in a Dyck path.
A hill of size three is a subpath beginning at height zero, consisting of three up steps followed by three down steps.
Matching statistic: St001214
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {2,2}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {2,2}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [4,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,1,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [3]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [1,1,1,1]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,1,1]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The aft of an integer partition.
The aft is the size of the partition minus the length of the first row or column, whichever is larger.
See also [[St000784]].
Matching statistic: St001586
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1]
=> []
=> []
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,1)],2)
=> [2]
=> []
=> []
=> ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> []
=> ? ∊ {2,2}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> []
=> ? ∊ {2,2}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [3]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [1,1,1]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [4]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The number of odd parts smaller than the largest even part in an integer partition.
Matching statistic: St001588
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001588: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001588: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {2,2}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {2,2}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [4,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,1,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [3]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [1,1,1,1]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,1,1]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The number of distinct odd parts smaller than the largest even part in an integer partition.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ? = 0
([],2)
=> ([],2)
=> ([],2)
=> ? ∊ {0,1}
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
([],3)
=> ([],3)
=> ([],3)
=> ? ∊ {0,2,2}
([(1,2)],3)
=> ([],2)
=> ([],2)
=> ? ∊ {0,2,2}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,2,2}
([],4)
=> ([],4)
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1}
([(2,3)],4)
=> ([],3)
=> ([],3)
=> ? ∊ {0,1,1,1,1,1,1,1}
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> ([],2)
=> ? ∊ {0,1,1,1,1,1,1,1}
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1}
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ? ∊ {0,1,1,1,1,1,1,1}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
([],5)
=> ([],5)
=> ([],5)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> ([],3)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],2)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([],6)
=> ([],6)
=> ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(4,5)],6)
=> ([],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(2,5),(3,4)],6)
=> ([],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([],3)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(0,2),(0,3),(0,4),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000621
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Values
([],1)
=> []
=> ?
=> ? = 0
([],2)
=> []
=> ?
=> ? ∊ {0,1}
([(0,1)],2)
=> [1]
=> []
=> ? ∊ {0,1}
([],3)
=> []
=> ?
=> ? ∊ {0,0,2,2}
([(1,2)],3)
=> [1]
=> []
=> ? ∊ {0,0,2,2}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> ? ∊ {0,0,2,2}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,2,2}
([],4)
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1}
([(2,3)],4)
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1}
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1}
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1}
([],5)
=> []
=> ?
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,4)],5)
=> [1]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([],6)
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(4,5)],6)
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3]
=> [3]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,1]
=> [3,1]
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,1]
=> 0
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,3]
=> [3]
=> 0
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is [[St000620]].
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000929The constant term of the character polynomial of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. 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. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001570The minimal number of edges to add to make a graph Hamiltonian. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition.
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