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Your data matches 36 different statistics following compositions of up to 3 maps.
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Matching statistic: St000281
Values
([],2)
=> 0
([(0,1)],2)
=> 2
([],3)
=> 0
([(1,2)],3)
=> 0
([(0,1),(0,2)],3)
=> 0
([(0,2),(2,1)],3)
=> 4
([(0,2),(1,2)],3)
=> 1
([],4)
=> 0
([(2,3)],4)
=> 0
([(1,2),(1,3)],4)
=> 0
([(0,1),(0,2),(0,3)],4)
=> 0
([(0,2),(0,3),(3,1)],4)
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(1,2),(2,3)],4)
=> 0
([(0,3),(3,1),(3,2)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(3,2)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(1,3)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
([(0,3),(2,1),(3,2)],4)
=> 8
([(0,3),(1,2),(2,3)],4)
=> 4
([],5)
=> 0
([(3,4)],5)
=> 0
([(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(1,4)],5)
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 0
([(1,3),(1,4),(4,2)],5)
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 0
([(2,3),(3,4)],5)
=> 0
([(1,4),(4,2),(4,3)],5)
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(4,3)],5)
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,3)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0
Description
The size of the preimage of the map 'to poset' from Binary trees to Posets.
Matching statistic: St000205
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 91%●distinct values known / distinct values provided: 14%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 91%●distinct values known / distinct values provided: 14%
Values
([],2)
=> [1,1]
=> [1,1]
=> [1]
=> 0
([(0,1)],2)
=> [2]
=> [2]
=> []
=> ? = 2
([],3)
=> [1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,2)],3)
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,1,4}
([(0,1),(0,2)],3)
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,1,4}
([(0,2),(2,1)],3)
=> [3]
=> [2,1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,1,4}
([],4)
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [4]
=> []
=> ? ∊ {2,4,8}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [4]
=> []
=> ? ∊ {2,4,8}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [4]
=> []
=> ? ∊ {2,4,8}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [2,2]
=> [2]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [3,2]
=> [2]
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(2,3),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [3,2]
=> [2]
=> 0
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [3,2]
=> [2]
=> 0
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,4),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 91%●distinct values known / distinct values provided: 14%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 91%●distinct values known / distinct values provided: 14%
Values
([],2)
=> [1,1]
=> [1,1]
=> [1]
=> 0
([(0,1)],2)
=> [2]
=> [2]
=> []
=> ? = 2
([],3)
=> [1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,2)],3)
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,1,4}
([(0,1),(0,2)],3)
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,1,4}
([(0,2),(2,1)],3)
=> [3]
=> [2,1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,1,4}
([],4)
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [4]
=> []
=> ? ∊ {2,4,8}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [4]
=> []
=> ? ∊ {2,4,8}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [4]
=> []
=> ? ∊ {2,4,8}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [2,2]
=> [2]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [3,2]
=> [2]
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(2,3),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [3,2]
=> [2]
=> 0
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [4,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [3,2]
=> [2]
=> 0
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,4),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000791
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000791: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 90%●distinct values known / distinct values provided: 14%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000791: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 90%●distinct values known / distinct values provided: 14%
Values
([],2)
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,2}
([(0,1)],2)
=> [2]
=> []
=> []
=> ? ∊ {0,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,4}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,4}
([(0,2),(2,1)],3)
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,4}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,4}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,2,4,8}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,2,4,8}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,2,4,8}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,2,4,8}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,2,4,8}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,2,4,8}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,2,4,8}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(1,5),(3,4),(4,2),(5,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
Description
The number of pairs of left tunnels, one strictly containing the other, of a Dyck path.
The statistic counting all pairs of distinct tunnels is the area of a Dyck path [[St000012]].
Matching statistic: St000980
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 90%●distinct values known / distinct values provided: 14%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 90%●distinct values known / distinct values provided: 14%
Values
([],2)
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,2}
([(0,1)],2)
=> [2]
=> []
=> []
=> ? ∊ {0,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,4}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,4}
([(0,2),(2,1)],3)
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,4}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,4}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,2,4,8}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,2,4,8}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,2,4,8}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,2,4,8}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,2,4,8}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,2,4,8}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,2,4,8}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(1,5),(3,4),(4,2),(5,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
Description
The number of boxes weakly below the path and above the diagonal that lie below at least two peaks.
For example, the path $111011010000$ has three peaks in positions $03, 15, 26$. The boxes below $03$ are $01,02,\textbf{12}$, the boxes below $15$ are $\textbf{12},13,14,\textbf{23},\textbf{24},\textbf{34}$, and the boxes below $26$ are $\textbf{23},\textbf{24},25,\textbf{34},35,45$.
We thus obtain the four boxes in positions $12,23,24,34$ that are below at least two peaks.
Matching statistic: St001877
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,2}
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,2}
([],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,4}
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,4}
([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,4}
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,4}
([],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,2,4,8}
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,4,8}
([(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,4,8}
([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,4,8}
([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,4,8}
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,4,8}
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 0
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,2,4,8}
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 0
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
([],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(1,2),(1,3),(1,4),(1,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,4,4,8,8,8,16,16,16,16,32}
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001570
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,2}
([(0,1)],2)
=> ([],2)
=> ([],2)
=> ([],1)
=> ? ∊ {0,2}
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,4}
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,4}
([(0,2),(2,1)],3)
=> ([],3)
=> ([],3)
=> ([],1)
=> ? ∊ {0,0,1,4}
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,4}
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,2,4,8}
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,2,4,8}
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,2,4,8}
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,2,4,8}
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,2,4,8}
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,2,4,8}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,2,4,8}
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,2,4,8}
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,2,4,8}
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(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)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St000175
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 71%●distinct values known / distinct values provided: 14%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 71%●distinct values known / distinct values provided: 14%
Values
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,2}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {0,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,4}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
Description
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape.
Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial
$$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$
The statistic of the degree of this polynomial.
For example, the partition $(3, 2, 1, 1, 1)$ gives
$$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$
which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$.
This is the same as the number of unordered pairs of different parts, which follows from:
$$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Matching statistic: St000225
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 71%●distinct values known / distinct values provided: 14%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 71%●distinct values known / distinct values provided: 14%
Values
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,2}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {0,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,4}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000749
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 71%●distinct values known / distinct values provided: 14%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 71%●distinct values known / distinct values provided: 14%
Values
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,2}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {0,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,4}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,4}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,4,8}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4,4,8,8,16}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,4,8,8,8,16,16,16,16,32}
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].
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St000929The constant term of the character polynomial of an integer partition. St001651The Frankl number of a lattice. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. 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. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001498The normalised height of a Nakayama algebra with magnitude 1.
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