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Your data matches 61 different statistics following compositions of up to 3 maps.
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Matching statistic: St000970
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(load all 3 compositions to match this statistic)
St000970: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 0
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 1
Description
Number of peaks minus the dominant dimension of the corresponding LNakayama algebra.
Matching statistic: St000717
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000717: Posets ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 67%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000717: Posets ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [[1],[]]
=> ([],1)
=> ([],1)
=> ? = 0
[1,0,1,0]
=> [[1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0}
[1,1,0,0]
=> [[2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [[3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
Description
The number of ordinal summands of a poset.
The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$.
This statistic is the length of the longest ordinal decomposition of a poset.
Matching statistic: St000906
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 67%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [[1],[]]
=> ([],1)
=> ([],1)
=> ? = 0
[1,0,1,0]
=> [[1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0}
[1,1,0,0]
=> [[2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [[3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
Description
The length of the shortest maximal chain in a poset.
Matching statistic: St000460
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 67%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [2,1] => [2]
=> []
=> ? = 0
[1,0,1,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [2]
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [2]
=> 2
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000474
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000474: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000474: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [2,1] => [2]
=> []
=> ? = 0
[1,0,1,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [2]
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [2]
=> 2
Description
Dyson's crank of a partition.
Let $\lambda$ be a partition and let $o(\lambda)$ be the number of parts that are equal to 1 ([[St000475]]), and let $\mu(\lambda)$ be the number of parts that are strictly larger than $o(\lambda)$ ([[St000473]]). Dyson's crank is then defined as
$$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$$
Matching statistic: St000667
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [2,1] => [2]
=> []
=> ? = 0
[1,0,1,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [2]
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [2]
=> 2
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000668
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [2,1] => [2]
=> []
=> ? = 0
[1,0,1,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [2]
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [2]
=> 2
Description
The least common multiple of the parts of the partition.
Matching statistic: St000708
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [2,1] => [2]
=> []
=> ? = 0
[1,0,1,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [2]
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [2]
=> 2
Description
The product of the parts of an integer partition.
Matching statistic: St000770
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [2,1] => [2]
=> []
=> ? = 0
[1,0,1,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [2]
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [2]
=> 2
Description
The major index of an integer partition when read from bottom to top.
This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top.
For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
Matching statistic: St000870
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [2,1] => [2]
=> []
=> ? = 0
[1,0,1,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [2]
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [2]
=> 2
Description
The product of the hook lengths of the diagonal cells in an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
The following 51 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001279The sum of the parts of an integer partition that are at least two. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001432The order dimension of the partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000264The girth of a graph, which is not a tree. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001668The number of points of the poset minus the width of the poset. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000742The number of big ascents of a permutation after prepending zero. St001896The number of right descents of a signed permutations. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000871The number of very big ascents of a permutation. St001060The distinguishing index of a graph. St001964The interval resolution global dimension of a poset. St000872The number of very big descents of a permutation. St000451The length of the longest pattern of the form k 1 2. St001638The book thickness of a graph. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St001520The number of strict 3-descents. St001960The number of descents of a permutation minus one if its first entry is not one. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000682The Grundy value of Welter's game on a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001556The number of inversions of the third entry of a permutation. St001712The number of natural descents of a standard Young tableau. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001884The number of borders of a binary word. St000904The maximal number of repetitions of an integer composition. St000628The balance of a binary word. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000640The rank of the largest boolean interval in a poset.
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